Journal of Mechanical Engineering 2014 7-8

http://www.sv-jme.eu

60 (2014) 7-8

Since 1955

Strojniški vestnik Journal of Mechanical Engineering

Contents Papers

449

Frank Goldschmidtboeing, Alexander Doll, Ulrich Stoerkel, Sebastian Neiss, Peter Woias: Comparison of Vertical and Inclined Toothbrush Filaments: Impact on Shear Force and Penetration Depth

462

Matjaž Čebron, Franc Kosel: Stored Energy Predictions from Dislocation-Based Hardening Models and Hardness Measurements for Tensile-Deformed Commercial Purity Copper

475

Mihai Dupac, Siamak Noroozi: Dynamic Modeling and Simulation of a Rotating Single Link Flexible Robotic Manipulator Subject to Quick Stops

483

Benjamin Bizjan, Alen Orbanić, Brane Širok, Tom Bajcar, Lovrenc Novak, Boštjan Kovač: Flow Image Velocimetry Method Based on Advection-Diffusion Equation

495

Xiaoni Qi, Yongqi Liu, Hongqin Xu, Zeyan Liu, Ruixiang Liu: Modeling Thermal Oxidation of Coal Mine Methane in a Non-Catalytic Reverse-Flow Reactor

506

David Koblar, Jan Škofic, Miha Boltežar: Evaluation of the Young’s Modulus of Rubber-Like Materials Bonded to Rigid Surfaces with Respect to Poisson’s Ratio

512

Jelena R. Jovanovic, Dragan D. Milanovic, Radisav D. Djukic: Manufacturing Cycle Time Analysis and Scheduling to Optimize Its Duration

525

Feng Li, Yumo Qin, Zhao Pang, Lei Tian, Xiaohua Zeng: Design and Optimization of PSD Housing Using a MIGA-NLPQL Hybrid Strategy Based on a Surrogate Model

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Strojniški vestnik – Journal of Mechanical Engineering (SV-JME) Aim and Scope The international journal publishes original and (mini)review articles covering the concepts of materials science, mechanics, kinematics, thermodynamics, energy and environment, mechatronics and robotics, fluid mechanics, tribology, cybernetics, industrial engineering and structural analysis.

The journal follows new trends and progress proven practice in the mechanical engineering and also in the closely related sciences as are electrical, civil and process engineering, medicine, microbiology, ecology, agriculture, transport systems, aviation, and others, thus creating a unique forum for interdisciplinary or multidisciplinary dialogue.

The international conferences selected papers are welcome for publishing as a special issue of SV-JME with invited co-editor(s).

Editor in Chief Vincenc Butala University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Technical Editor Pika Škraba University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Founding Editor Bojan Kraut

University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Editorial Office University of Ljubljana, Faculty of Mechanical Engineering SV-JME, Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386 (0)1 4771 137 Fax: 386 (0)1 2518 567 info@sv-jme.eu, http://www.sv-jme.eu Print: Littera Picta, printed in 400 copies Founders and Publishers University of Ljubljana, Faculty of Mechanical Engineering, Slovenia University of Maribor, Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia Chamber of Commerce and Industry of Slovenia, Metal Processing Industry Association President of Publishing Council Branko Širok University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Vice-President of Publishing Council Jože Balič

University of Maribor, Faculty of Mechanical Engineering, Slovenia Cover: The picture shows the time history of quasi static filament shapes of an oscillatingrotating toothbrush with inclined filaments.

The blue filament shows the instantaneous filament shape while the green filaments denote the prior shapes. The filament typically slides along the tooth surface, but might also stick to the surface under certain circumstances. The filament exerts shear and normal forces on the tooth that depend on the filament shape. An analysis of these forces leads to the fact that filaments with an inclined angle of approximately 16° provide more efficient plaque removal than vertical bristles.

Courtesy: Procter & Gamble, Kronberg, Germany

International Editorial Board Koshi Adachi, Graduate School of Engineering,Tohoku University, Japan Bikramjit Basu, Indian Institute of Technology, Kanpur, India Anton Bergant, Litostroj Power, Slovenia Franci Čuš, UM, Faculty of Mechanical Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mechanical Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mechanical Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mechanical Engineering, Slovenia Imre Horvath, Delft University of Technology, Netherlands Julius Kaplunov, Brunel University, West London, UK Milan Kljajin, J.J. Strossmayer University of Osijek, Croatia Janez Kopač, UL, Faculty of Mechanical Engineering, Slovenia Franc Kosel, UL, Faculty of Mechanical Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mechanical Engineering, Slovenia Miroslav Plančak, University of Novi Sad, Serbia Brian Prasad, California Institute of Technology, Pasadena, USA Bernd Sauer, University of Kaiserlautern, Germany Brane Širok, UL, Faculty of Mechanical Engineering, Slovenia Leopold Škerget, UM, Faculty of Mechanical Engineering, Slovenia George E. Totten, Portland State University, USA Nikos C. Tsourveloudis, Technical University of Crete, Greece Toma Udiljak, University of Zagreb, Croatia Arkady Voloshin, Lehigh University, Bethlehem, USA General information Strojniški vestnik – Journal of Mechanical Engineering is published in 11 issues per year (July and August is a double issue).

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8 Contents

Contents Strojniški vestnik - Journal of Mechanical Engineering volume 60, (2014), number 7-8 Ljubljana, July-August 2014 ISSN 0039-2480 Published monthly

Papers Frank Goldschmidtboeing, Alexander Doll, Ulrich Stoerkel, Sebastian Neiss, Peter Woias: Comparison of Vertical and Inclined Toothbrush Filaments: Impact on Shear Force and Penetration Depth Matjaž Čebron, Franc Kosel: Stored Energy Predictions from Dislocation-Based Hardening Models and Hardness Measurements for Tensile-Deformed Commercial Purity Copper Mihai Dupac, Siamak Noroozi: Dynamic Modeling and Simulation of a Rotating Single Link Flexible Robotic Manipulator Subject to Quick Stops Benjamin Bizjan, Alen Orbanić, Brane Širok, Tom Bajcar, Lovrenc Novak, Boštjan Kovač: Flow Image Velocimetry Method Based on Advection-Diffusion Equation Xiaoni Qi, Yongqi Liu, Hongqin Xu, Zeyan Liu, Ruixiang Liu: Modeling Thermal Oxidation of Coal Mine Methane in a Non-Catalytic Reverse-Flow Reactor David Koblar, Jan Škofic, Miha Boltežar: Evaluation of the Young’s Modulus of Rubber-Like Materials Bonded to Rigid Surfaces with Respect to Poisson’s Ratio Jelena R. Jovanovic, Dragan D. Milanovic, Radisav D. Djukic: Manufacturing Cycle Time Analysis and Scheduling to Optimize Its Duration Feng Li, Yumo Qin, Zhao Pang, Lei Tian, Xiaohua Zeng: Design and Optimization of PSD Housing Using a MIGA-NLPQL Hybrid Strategy Based on a Surrogate Model

449 462 475 483 495 506 512 525

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 449-461 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1513 Original Scientific Paper

Received for review: 2013-10-18 Received revised form: 2014-01-29 Accepted for publication: 2014-04-14

Comparison of Vertical and Inclined Toothbrush Filaments: Impact on Shear Force and Penetration Depth Goldschmidtboeing, F. – Doll, A. – Stoerkel, U. – Neiss, S. – Woias, P. Frank Goldschmidtboeing1,* – Alexander Doll2 – Ulrich Stoerkel2 – Sebastian Neiss1 – Peter Woias1 1 University 2 Procter

of Freiburg, Department for Microsystems Technology, Germany & Gamble Manufacturing GmbH, Oral-B Laboratories, Germany

This paper presents a semi-analytical model to describe the action of vertical and 16°-inclined toothbrush filaments onto flat tooth surfaces and into approximal areas. The theory is based on the analytical solution of the beam equation about normal and transversal (shear) forces that deform the filaments. A 15% increase of penetration depth (approximal area model) and a 60% increase of transversal force (tooth surface model) for an inclination angle of 16° compared to a vertical filament were determined. Furthermore, that 16°-inclined filaments are able to transfer higher shear forces to the tooth compared to vertical ones was shown experimentally, which could account for the higher efficiency of plaque removal by inclined filaments compared to vertical filaments. The theoretical findings for manual and oscillating-rotating toothbrushes are discussed, and implications for their design are drawn.

Keywords: vertical and inclined toothbrush filaments, shear forces, penetration depth, power toothbrush, mechanical plaque removal, oscillating-rotating technology

0 INTRODUCTION Dental plaque is an oral biofilm that adheres to the teeth and consists of many species of bacterial cells, salivary polymers and microbial extracellular products.

Without proper control, dental plaque is known to be a causative and perpetuating factor of several common oral diseases, including gingivitis, periodontitis and caries. Therefore, the removal of plaque biofilm is the primary goal of oral hygiene.

Dental plaque formation occurs in several steps [1], starting with a layer of adsorbed molecules of bacterial and salivary origin. Subsequently, initial colonizing bacteria (generally streptococci) attach to this layer. Secondary colonizers attach via coadhesion and make the biofilm more diverse, resulting in a multispecies community. A biofilm matrix is formed by the attached bacteria via the synthesis of extracellular polymers. The combined metabolic activity of different bacteria can break down complex host macromolecules, which subsequently leads to the development of food chains. As a result, oral biofilms are highly organized, both structurally and functionally.

While numerous products are commonly recommended to achieve optimal plaque levels, including dental floss, fluoride dentifrice, and rinse, the toothbrush is a core component of any oral hygiene regimen. The main purpose of brushing teeth remains the effective disruption and removal of dental plaque. The plaque biofilm grows on teeth, resulting in a complex structure, depending on the composition, age and location, with different mechanical properties.

Independent of the particular structure and mechanical strength of the biofilm, it is common knowledge that mechanical energy and in particular shear forces are needed to remove the adhering plaque from the tooth surface [2]. Also well accepted is the fact that the tips of toothbrush filaments create higher shear forces than the respective flanks.

For effective plaque removal with the desired result of clean and healthy teeth and gums, several factors are important. To start with, the filament tips have to reach the plaque, which is easy to achieve for a large flat surface, but difficult for the approximal area around the connection sites of neighbouring teeth. Although access to those hard-to-reach areas is essential, access alone is not sufficient since the applied forces have to be greater than the adhesion and cohesion forces of the biofilm to destroy and remove it from the tooth surface.

Toothbrush innovations have to simultaneously address several consumer needs, with the improvement of the efficacy of plaque removal during the brushing routine of users being an important aspect for both manual- and power-driven toothbrushes. In 1999, Oral-B introduced an innovative manual toothbrush, called CrossAction®. Detailed investigation into the action of filaments during brushing was part of the development. As a result of the development process, the filament tufts were arranged in a unique CrissCross® design with filaments inclined at 16° (Fig. 1). Since its creation, the brush has proven to be a leading manual toothbrush that maximizes plaque removal, regardless of how the user brushes, which is supported by laboratory investigations, as well as clinical studies. Laboratory studies [3] have

*Corr. Author’s Address: University of Freiburg, Department for Microsystems Engineering, Laboratory for Design of Microsystems, Georges-Koehler-Allee 102, 79110 Freiburg, Germany, fgoldsch@imtek.de

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 449-461

demonstrated that this development significantly enhances approximal area penetration and cleaning effectiveness in comparison to an identical brush head with parallel, vertical (rather than inclined) filaments. Comparisons with more than 80 leading manual toothbrushes from around the world have demonstrated a consistent, significant advantage for the CrossAction® toothbrush in approximal area penetration and cleaning effectiveness [3]. According to a five-year literature review (2000 to 2005) [4] of clinical research, 14 single-brushing clinical comparisons and two longer-term clinical studies have demonstrated consistently greater plaque removal for the Oral-B CrossAction brush over the equivalent commercial standards, specifically for effective approximal cleaning. Several additional studies since then have confirmed those results [5] to [7]. Longerterm research also demonstrates significant benefits for CrossAction® in controlling gingivitis [8]. As with all Oral-B brushes and brush heads, CrossAction filaments are end-rounded to ensure gentle cleaning.

Fig. 1. Oral-B CrossAction® manual toothbrush

Power-driven toothbrushes have additional features for supporting the shear forces created by the filaments, and these features have been shown to enhance plaque removal performance even further and lead to increased performance in comparison to manual toothbrushes [9] and [10]. An example of such a feature is the 3D pulsation of the Oral-B Professional Care oscillating-rotating product line, which supports filament penetration into plaque. The pulsation provides an applied force in the direction along the filament axis, which leads to so-called poking events that provide additional shear forces (typically 340 pulsations per second). Other important features of power driven brushes are the short stroke length and the significantly higher frequency of motion (typically 78 oscillations per second) compared to manual brushes (typically 2 to 3 strokes per second).

The frequent changes of direction consequently lead 450

to a higher number of poking events. Shear forces are most effective at turning points of direction; thus, the combination of short movements with a high number of turning points at high frequency synergistically optimizes plaque removal efficacy. However, it is important that the stroke length remains long enough to allow the filaments to change their orientation and to apply sufficient shear forces.

In order to investigate and analyse the effect of combining the current advantages of power toothbrushes (3D motion of pulsations combined with oscillation/rotation) with the proven features of the CrossAction® manual toothbrush (i.e. the arrangement of most tufts at an inclined 16° angle), in-vitro experiments to explore this technology over current tuft geometries were designed. These in-vitro experiments may help to predict beneficial in-use relevance and thus demonstrate potential advances in brush design.

This paper is structured as follows: at the start, a first-principle model to describe the filament shapes and the forces onto the tooth surface by a semianalytical approach is presented. Thereafter, the resulting equations are solved, and the filament shapes are visualized in the solution section. After that, the results of a numerical simulation to the problem to judge the applicability of the simplified analytical model from the first section are presented. Then, an experimental validation of the hypothesis that 16°-inclined filaments transfer higher shear forces to the tooth in comparison to vertical filaments is presented. In the discussion section, some conclusions for optimizing toothbrush design, especially relevant for the design of oscillating-rotating power brushes, are drawn.

1 FIRST-PRINCIPLE MODELLING The aim here is to develop and analyse a simplified theoretical model to understand the basic effects of the inclination angle of toothbrush filaments on the shear force and on the penetration depth into approximal areas. Two types of toothbrushes are considered: a manual brush (Fig. 1) and the Oral-B oscillatingrotating power brush (Fig. 2), both with vertical and 16°-inclined end-rounded filaments.

For the sake of simplicity, both types of filament motion, the linear motion of a manual brush (Fig. 1) and the oscillating-rotating motion of a power brush (Fig. 2), are mapped onto a linear movement along the x axis.

The stroke length of a standard manual brush is assumed to be typically 10 mm while the typical

Goldschmidtboeing, F. – Doll, A. – Stoerkel, U. – Neiss, S. – Woias, P.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 449-461

stroke lengths of oscillating-rotating brush heads are in the range of 3.0 mm (inner ring) and 5.1 mm (outer ring) for the free moving head, respectively, and 2.3 mm (inner ring) and 4.0 mm (outer ring) for the oscillating-rotating brush head with a total recommended maximum brushing force of 2 N.

Fig. 2. Sketch of the oscillating-rotating brush with vertical and 16°-inclined filaments

The mapping of the oscillating-rotating motion to a linear one obviously introduces some error, as no radial forces and displacements are considered, but as these effects are understood to be small, a qualitative deviation from the real behaviour is not expected. Two geometric models are discussed (Fig. 3).

Fig. 3. Simplified tooth surface model (left) and approximal area model (right)

The first model describes the tooth surface as a perfect plane and this simplified tooth surface model is used to calculate the occurring forces. The second model describes the penetration depth of the filament into an approximal area. Both models were developed to allow for the analytical solution of the filament deformations, to show the principle effects of inclined filaments. More detailed tooth models could, for example, be generated by the digitization methods described by Budak et al. [11].

Nevertheless, the focus is on the demonstration of very basic effects that are meaningful for any tooth geometry and do not depend on details of the tooth

surface. Therefore, the simplest possible tooth model was chosen.

1.1 The Tooth Surface Model The model is schematically visualized in Fig. 4. The sketch is rotated so that the filament can be treated as a cantilever in the x-direction with the z-direction facing downwards. The length of the filament is symbolized

by L and the distance between the brush, and the perfectly planar tooth surface is given by d. The inclination angle α is defined as the angle between the perpendicular axis of the brush and the non-deformed filament. The position of the filament tip on the tooth surface is given by wtip with wtip = 0 at the incidence of the non-deformed filament. The filament’s projection onto the x-axis is denoted as Lx. The normal force Fx (defined in negative x-direction) and the transverse force Fz (defined in negative x-direction) deform the filament. Their projections in normal and tangential directions to the tooth surfaces are called the contact force Fn and shear force Ft, respectively.

Fig. 4. Idealized 2D-model of the Filament

The deformation of the filament w(x) is defined by the differential beam equation (Eq. (1)) [12], where E denotes the Young`s modulus of the filaments and I its second moment of area. Strictly speaking, this approach is only valid for small displacements and the filaments might show considerably high displacements. Therefore, its accuracy is checked in the numerical simulations section.

w( 4 ) ( x ) + λ 2 ⋅ w′′ ( x ) = 0, λ 2 =

Fx . (1) EI

This differential equation must be solved with the four boundary conditions given in Eq. (2) for the bearing (brush side) and for the tip (tooth side).

Comparison of Vertical and Inclined Toothbrush Filaments: Impact on Shear Force and Penetration Depth

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 449-461

w ( 0 ) = 0, w′ ( 0 ) = 0, w ( Lx ) = wtip , M ( Lx ) = − EI ⋅ w′′ ( x ) = 0.

(2)

The last part of Eq. (2) states that the tooth does not imply any torque on the filament. The solution of Eq. (1) with boundary conditions of Eq. (2) is given

by Eq. (3). wtip w( x) = ⋅ {sin ( λ ⋅ x ) − tan ( λ ⋅ Lx ) − λ ⋅ Lx

}

−λ ⋅ x + tan ( λ ⋅ Lx ) ⋅ (1 − cos ( λ ⋅ x ) ) . (3)

The only remaining unknown is the axial force parameter λ, as the projection length Lx and the tip displacement wtip are related by the geometrical condition Eq. (4). The axial force parameter λ is determined from the fact that the curved filament length equals the straight filament length L, Eq. (5).

Lx =

d + wtip ⋅ tan (α ) , (4) cos (α )

L=

Lx

∫

1 + w′ ( x ) dx. (5) 2

0

Eq. (5) with the bending line from Eq. (3) and the projected length Lx from Eq. (4) can be solved numerically for the parameter λ in dependence on α, d, L and wtip. This numerical solution yields an infinite number of positive solutions for the axial force parameter λ. The lowest value for λ is chosen because it is the only value that produces physically stable solutions. With this solution, the forces Fx and Fz can be calculated directly according to Eq. 6, while the projections Fn and Ft are calculated from Eq. (7).

(

)

Fx = EI ⋅ λ 2 , Fz = EI ⋅ λ 2 ⋅ w′ ( Lx ) + w′′′ ( Lx ) , (6)

The resulting tip displacement can then be calculated from the length condition Eq. (4), which yields two solutions wmin < 0 and wmax > 0 To judge whether the calculated bending lines w(x) represent stable states, a stability analysis for the steady state has to be performed. Stability is obtained as long as the second derivative of the elastically stored energy W with the tip displacement wtip is positive.

This requirement is equivalent to a positive slope of the transversal force Ft versus the tip displacement wtip (Eq. 9). This condition holds because the normal force Fn does not perform any work on the beam.

The model for an approximal area is shown in Fig. 5. The rounded edge of the tooth is modelled with a radius of R = 2 mm. The relative distance between the bearing of the filament and the beginning of the rounded shape is called z0. The position of the tip on the rounded shape is given by the angle ϕ. The modelling concept is the same as for the tooth surface model, but now two variables have to be considered.

While for the tooth surface model all forces merely depend on the tip displacement wtip, here the position z0 and the angle ϕ have to be considered. As we have only changed the geometry compared to the prior model, only the geometrical Eq. (3) and the Eq. (7) have to be replaced. Eq. (3) is replaced by Eq. (10) that relates the modelling variables Lx and wtip to the filament position z0 and the tip position ϕ.

Eq. (10) can be derived from Fig. 5 given that the opposite edges of the large rectangle are equal.

Lx = ( d + R ) ⋅ cos (α ) −

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− R ⋅ cos (α + φ ) − z0 ⋅ sin (α ) ,

Fn = − Fz ⋅ sin (α ) + Fx ⋅ cos (α ) . (7)

The limiting values for wtip are given by the solutions for λ = 0, i.e. solutions with zero axial force.

These solutions are found by taking the limit λ → 0 of Eq. (3), which is given in Eq. (8). This solution is obviously the pure bending solution that could be found directly by integrating the beam Eq. (1) with zero axial force. w 3 1  w ( x ) = tip ⋅  ⋅ Lx ⋅ x 2 − ⋅ x 3  . (8) 3 2 Lx  2 

∂Ft > 0. (9) ∂wtip

1.2 Model for Penetration into Approximal Areas

Ft = Fz ⋅ cos (α ) + Fx ⋅ sin (α ) ,

∂ 2W >0 ⇔ 2 ∂wtip

wtip = − z0 ⋅ cos (α ) −

− ( d + R ) ⋅ sin (α ) + R ⋅ sin (α + φ ) . (10)

Furthermore, the equations for the normal force FN and the tangential force FT have to be replaced by Eq. (11).

FN = Fz cos (α + φ ) + Fx ⋅ sin (α + φ ) ,

FT = Fz ⋅ sin (α + φ ) − Fx ⋅ cos (α + φ ) . (11)

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Fig. 5. Idealized 2D-model of the filament in an approximal area

2 SOLUTION 2.1 Tooth Surface Model and Brush Geometry Filaments with seven different angles α, from 0 to 24°, are compared. The lengths of the filaments L are adjusted in such a way that the thickness of the non-deformed brush d0 is constant (L = d0/cos(α)).

The round-shaped filaments have a diameter of 6 mil (0.152 mm) and a Young’s modulus of E = 2.7 GPa and, therefore, a resulting bending stiffness of EI = 7.15 108 Nm² (I =π/64·d4).

2.1.1 Solution: Bending Lines Each brush geometry is examined for four different constant distances d between brush and tooth: d = 7.46, 7.71, 7.96 and 8.21 mm, which means that the filaments are deformed in such a way that the distance between the brush head and tooth is reduced by 1, 0.75, 0.5 and 0.25 mm, respectively, compared to the no-load condition with a distance of d0 = 8.46 mm.

The deformation of 1 mm corresponds roughly to the recommend maximum normal force of the Oral-B Professional Care product line. First, the limiting bending lines with zero axial force are determined from Eq. (8) with Eq. (5). Then, the bending lines for 100 equidistant tip displacements wtip between wmin and wmax are calculated from Eq. (3) with the length condition, Eq. (5). The forces (Fx, Fz, Fn, Ft) are then easily determined from Eq. (6) and (7).

The filament movement is described via Figs.

6 and 7. It must be noted that the model is based on static forces and displacements, while the real process is inherently dynamic; nevertheless, some important issues may be derived from the static curves. The model yields two types of solutions: the “sticking” solution and the “sliding/sticking” solution. The sticking solutions (i.e. the solution violating the stability criterion in Eq. (9)) refer to the filament shapes that occur when the filament tip is stuck to

the tooth surface, while the sliding/sticking solutions describe shapes when the filament tip slides along the tooth surface with constant filament shape, and the friction force equals the bending force of the filament.

If the friction force exceeds the bending force, the tip again becomes stuck to the surface, and the filament shape evolves from sliding/sticking to a pure sticking solution. The friction force typically varies during a brushing event due to accumulated plaque or toothpaste particles. Therefore, a sliding filament typically becomes stuck after a while.

In Fig. 6, the bending lines for a simplified toothbrush filament are shown. Here x´ is the coordinate along the tooth surface, while z´ is the coordinate normal to the tooth surface. These coordinates are used to visualize the results in a clear manner. The filament shapes from the original x, z coordinate system are rotated to the new x´, z´ system. Different brush head positions are plotted to clarify the filament behaviour on a simplified surface when moving the brush head forward (from left to right in the diagrams as in Fig. 3). Fig. 6a shows the bending line just after touching the surface. With further movement, the tangential force increases. In Fig. 6b, the tangential force equals the friction force; therefore, the filament slides on the surface to Fig. 6c. In Fig. 6c, it is assumed that the filament sticks to the surface (i.e. it impacts a barrier, e.g. a biofilm agglomerate). The fixed point is indicated by a dot in the diagrams. In Fig. 6d, the filament undergoes a further bending until it reaches the last position before flipping (Fig. 6e). The filament will then flip; Fig. 6f shows the flipped shape. Moving further over the flip point will accordingly result in Fig. 6g and Fig. 6h. In the position shown in Fig. 6h, the normal force becomes negative; thus, the filament is drawn out of the sticking site and then slides along the tooth surface with a relatively low force until the brush changes direction in Fig 6i.

The described motion of the filament is one of many possible motions. These motions depend on many details, such as the local friction coefficient that may vary due to saliva, toothpaste particles and biofilms. We wish to describe these influences in further detail by discussing Fig. 6 in the context of the transversal forces shown in Fig. 7.

In Fig. 7, the transversal force in x´-direction is shown versus the relative tip to bearing distance wtip of the filament. We have chosen the negative sign in wtip to explain the filament shape from left to right, i.e. in the same direction as in Fig. 6. The red dots demonstrate the transversal forces of the filament shapes shown in Fig. 6.

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Fig. 6. Bending line of a simplified toothbrush filament with d0 – d = 0.75 mm for different brush head positions

The full curves show sliding/sticking solutions.

The filament may slide along the tooth surface with a constant shape as long as the friction force equals the transversal force of the filament. The dashed lines refer to sticking solutions.

Here, the slope of the transversal force versus tip position is negative and, according to Eq. (9), the filament shape is unstable. Physically speaking, this means that the filament tip sticks to the surface while the bearing moves; therefore, the shape varies permanently. The first filament shape (Fig. 6a) shows only a very low positive transversal force. This force increases (here in the negative direction, i.e. against the movement of the filament) until the transverse force and the friction force are equal at the second filament shape. This shape then remains constant, and the tip slides along the tooth surface until the friction force exceeds the maximum possible transverse force.

In Fig. 6, it is assumed that this sliding condition is valid for 1 mm; therefore, the filament shapes in Figs.

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6b and c are equal but separated by 1 mm of brush movement. We further assume that the filament suddenly becomes stuck, and the sticking force exceeds the maximum transverse force that is reached in Shape 3. If the brush moves further, the filament sticks to the tooth and flips between Shapes 4 and 5 in Figs. 6e and f. Next, the transversal force changes direction, i.e. it points against the direction of the brush movement. Depending on the friction force, in the backward direction it is possible that the filament’s tip moves backwards after that point, but for the sake of clear graphic representation it is assumed that the filament remains stuck even at Shape 6 with the highest backward force. The transversal force then decreases rapidly until the transversal force again becomes negative (Shape 7). Here, the transversal force is relatively small; nevertheless the friction force is overcome. This is because the normal force (shown in Section 2.1.2) becomes negative, i.e. the filament is pulled upwards out of the sticking site. Consequently,

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the filament slides over the surface until the direction of movement is changed.

the normal forces at the border between the sliding/ sticking and the sticking solutions, i.e. at the position of maximum transversal force. They are the maximum values in the sliding/sticking range, but even higher values are reached in the pure sticking range. Again, the maximum normal forces for the 16°-inclined filament are higher (only slightly for the forward direction) compared to the vertical filament.

Fig. 7. Bending line at different tip positions for a filament with α = 16° and d0 – d = 0.75 mm

2.1.2 Solution: Forces As seen in Section 2.1.1, the transversal force Ft and the normal force Fn play an important role for the modelling of the filament motion. These forces are shown in Figs. 8 and 9 for inclination angles of α = 16° and α = 0° at a distance of d0 – d = 0.75 mm.

Fig. 8. Transversal force FT versus tip position wtip (α = 16° and the α = 0° d0 – d = 0.75 mm)

The transversal force (Fig. 8) obviously takes its maximum value at the border between the sliding/ sticking and the sticking solutions. The maximum transversal force for the inclined filament is significantly higher than for the vertical one.

The two curves for the normal force in Fig. 9 show similar behaviour. The maximum normal forces FN for both α = 16° and α = 0° are defined as

Fig. 9. Normal force FN versus tip position wtip (α = 16° and α = 0° d0 – d = 0.75 mm)

These curves have been analysed for all geometries with the four defined brush-to-tooth distances.

Fig. 10. Maximum transversal force FT in the sliding/sticking branch (shape 3 in Fig. 7) versus inclination angle

Fig. 10 shows the maximum transversal force in the sliding/sticking region over the filament angle α for different filament compressions d0 – d. The results show that the transversal force increases with increasing inclination angle independent of the filament compression d0 – d. For a typical d0 – d of 0.5

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mm, about 60% more transversal force is obtained for a 16° angled filament compared to a vertical one.

In Fig. 11, the maximum normal forces to the surface are plotted over the filament angle for different d0 – d values. The results show that for d0 – d ≤ 0.75 mm the maximum normal forces increase slightly with the filament angle up to the angle of about 16°; for the very small value of d0 – d = 0.25 mm, the increase ends at an angle of α = 14.75°. Above this value, the maximum normal forces strongly decrease. For d0 – d = 1 mm the maximum normal forces slightly decrease (about 3%) with the angle but strongly decrease for angles >19°.

Fig. 12. Filament shapes depending on the brush position (α = 16°, d0 – d = 1 mm)

Fig. 11. Maximum normal force FN in the sliding/sticking branch (shape 3) versus α

This steep decrease of the normal force is the reason the angle of 16° is considered to be optimal for cleaning purposes. Details on this argumentation are given in the discussion section.

2.2 Penetration Model Fig. 12 shows the results for the penetration model for a 16° filament and a brush deformation of d0 – d = 1 mm.

All filaments are shown in their most backward position, i.e. equivalent to Shape 7 in the tooth surface model (Fig. 7).

The filament is moved from right to left in a forward direction and touches the tooth flank as a straight filament (Filament 1). When moving further, the filament penetrates deeper into the flank (Filament 2) until it reaches its deepest penetration (Filament 3); the related angle for our tooth model is φ16° = 69°.

After that turning point, the filament moves upwards again (Filaments 4 and 5) until it reaches the flat tooth surface (Filament 6).

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In comparison, the filament with a zero inclination angle would touch the tooth flank at the same position as the 16°-filament but would not be able to penetrate deeper into the flank. Therefore, the maximum reachable angle is φ0° = 60°. Consequently, the 16°-filament penetrates the gap deeper by 9° or 0.31 mm, which is equivalent to a 15% increase in the penetrated area.

A brush movement of approximately 8 mm is necessary to reach all positions from the red to the purple curve. Therefore, a filament from the oscillating brush would only reach a portion of these positions depending on its starting position. Nevertheless, only about a 2-mm travel range is necessary to reach the deepest penetration in Curve 3.

3 NUMERICAL SIMULATIONS The problem of filament shapes on the flat tooth surface model was numerically solved in order to judge whether the simplifications of our analytical model lead to valid results. The finite element method (FEM) with the COMSOL Multiphysics commercial software package (COMSOL, Inc. Burlington, USA) was used. A 2D model of the beam with equivalent second moment of inertia was implemented. The displacement was introduced on the neutral axis at the beam end while the other end was kept fixed. Since large deformations have to be included, geometric nonlinearities were considered in the model. The problem was solved on a triangular mesh with approximately 700 elements. In the FEM model, the same boundary conditions were used as in our analytical model, but include the effect of

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high displacements that is neglected in the analytical theory.

The resulting bending lines for the 16°-filament and d0 – d = 0.75 mm are shown in Fig. 13. It is clearly shown that both methods agree very well for negative tip displacements, i.e. for Shapes 1 to 3, but our analytical model loses accuracy in the range of high positive tip displacements, i.e. for Shapes 6 and 7.

Fig. 13. Filament bending lines (α = 16°, d – d0 = 0.75 mm) proposed model (full lines) and FEM simulations (dotted lines)

The same is true for the derived normal and transversal forces shown in Figs. 14 and 15. In the worst case, near the point of maximum force, the calculated (analytical) maximum normal and transversal forces are approximately a factor of two higher than the simulated (FEM) ones. This is supposed to be caused by the assumption of a spatially constant axial force in the analytical theory. In reality, the axial force is reduced due to a spatially varying bending away from the axial direction. Consequently, the analytical theory overestimates the normal force. The transversal force must therefore also be overestimated to compensate this axial force overestimation and to reach the same displacement as in the simulation. This argument also holds for the range of very high displacements (near Shape 7). As the normal force is reduced to zero, its effect on the filament shape vanishes and the deviation from the simulation is reduced. As a result, it must be stated that, for filaments with large strokes and large normal forces, i.e. for filaments in the range of Shape 6, the forces are significantly overestimated by the analytical theory. Nevertheless the analytical theory leads to a fairly good agreement for small displacements (Shapes 1 to 3) and for low normal forces (Shape 7).

A manual brush will reach all shapes, from 1 to 7, due to its high stroke. For these brushes, the analytical theory overestimates the cleaning effect, but the general trend derived from the analytical theory remains valid. Fortunately, the filaments of the oscillating-rotating brush will predominantly be bending in Shapes 1 to 3, where the analytical model coincides well with simulated curve. Therefore, the model quantitatively suits well the main goal of this paper, which is the characterization of an oscillatingrotating brush with 16°-inclined filaments.

Fig. 14. Normal forces versus tip position (α =16°, d0 – d = 0.75 mm)

Fig. 15. Transversal forces versus tip position (α =16° d0 – d = 0.75 mm)

4 EXPERIMENTAL VALIDATION A rheometer (Thermo Fisher Scientific, HAAKE Rheostress1) was used to qualitatively validate the increasing shear force with 16°-inclined filaments.

With the rheometer, the transferred torque from the brush head to a simplified tooth surface could be

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measured (according to our simplified surface model).

The transferred momentum is proportional to the transversal (shear) force, since the two brush heads have the same tufting configuration. The spread of the angled filament brush head that slightly increases the radius of the outside filament arrangement (outer ring) compared to the straight filament brush head can be ignored.

In the experiment, a brush head with the straight filaments and a brush head with the 16°-inclined filaments were compared (Fig. 2). The unidirectional rotational speed of the brush head and the normal force that is applied to the brush head were controlled, and the transferred torque to a sandblasted flat plastic plate was measured (typically used in in-vitro brushing tests). The rotation speed was controlled at 1 rpm, and its direction was changed (i.e. oscillated) after one full rotation. The normal force was controlled using weights on a balance (50 to 390 g) in such a way that the plastic plate was pressed with constant force against the oscillating-rotating brush head.

The transferred torque data were averaged and are given in Fig. 16.

Fig. 16. Transferred force versus normal force for two brushes with α = 0° and α = 16°

For small normal forces, both brush heads showed the same transferred torque. With increasing normal force, the forward movement (rotation) of the brush head with 16° angled filaments transferred significantly higher torques than the brush head with straight filaments. Oral-B’s advanced electric toothbrushes feature a pressure sensor that guides the user not to apply too much brushing pressure. This pressure sensor typically lights up at pressure/brushing forces above 2.5 N. At this recommended maximum brushing force, the brush head with 16° filaments was able to transfer about a factor of 1.8 more torque to the test plate. The experiment was repeated on other 458

materials (steel, ceramics) and similar qualitative behaviour as in Fig. 16 was found.

5 DISCUSSION An analytical model for the bending of 16°-inclined toothbrush filaments under the combined action of normal and transversal forces was developed and compared to numerical FEM solutions. It turned out that our analytical model was valid for small filament displacements, but became inaccurate with high filament displacements. As a consequence, for the given geometry of the oscillating-rotating brush, our model can be used for the defined Shapes 1 to 3 (Fig. 7), and to a lesser extent Shape 7, but overestimates the forces for the other sliding positions near Shape 6.

Our model was used to calculate the forces that act on a tooth depending on the brush to tooth distance d. In this section, how these observations for single filaments on idealized tooth geometries are related to the behaviour of a full brush on a real tooth are discussed. The basic idea is that a brush consists of many filaments that touch the tooth under different conditions. They may be bent differently due to different friction coefficients at different locations on the tooth and at different distances d due to the uneven tooth structure. Furthermore, the brush consists of different filaments made from different materials and cut to different lengths. A prediction of full brush behaviour from the collective action of the filaments requires significant simplification.

It is supposed that at any given time some filaments work in the most favourable way for cleaning while others are in a condition that does not contribute significantly to the cleaning process.

Optimizing the best possible conditions of a filament, therefore, optimizes the overall cleaning success. It is supposed that the transversal (shear) force is the most important parameter in teeth cleaning, but also that a certain amount of normal force is necessary.

Therefore, the transversal force should be maximized by choosing the optimum angle α such that the normal force is not reduced compared to the zero angle case.

One important factor for this discussion is the stroke length of the brush. The manual brush may be used with a stroke of 10 mm or more [3].

Consequently, all possible shapes from Fig. 6 may be accessed. The manual brush using 16°-angled filaments has already proved to have significantly greater plaque removal compared to straight brushes in experimental trials [3]. This result may be explained by the fact that the transverse forces for the 16°-angled filament are higher compared to the straight filament

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over the entire range of possible shapes (Fig. 7).

However, the results for high filament displacements are overestimated by the model, as shown earlier (Section 3: Numerical simulations).

When considering the oscillating-rotating brush, we must account for the reduced strokes compared to the manual brush. Here the stroke is reduced to 2.3 mm (inner ring) and to 4.0 mm (outer ring). Therefore, the shapes from Figs. 6a to e are relevant, while the other shapes play a lesser role. Fortunately, the model proved to be accurate for these shapes (Shapes 1, 2 and 3). As a result, we conclude that the model is appropriate to describe the cleaning process for the oscillating-rotating brush, despite its inaccuracy for high filament displacements. The calculated maximum normal and transversal forces in the range between Shapes 1 and 3 versus the inclination angle for the four calculated distances are shown in Figs. 10 and 11. It is clearly shown that the transversal force grows with the inclination angles for all distances, while the normal force shows a more complex behaviour.

Depending on the distance d, the normal force may rise or fall slightly with the inclination angles for low angles, but all curves start to decrease with a steep slope for angles higher than about 16°. Therefore, the optimum angle is supposed to be 16°. At this angle, the transversal force is increased by 60% compared to the common vertical configuration.

A second important parameter for the cleaning performance of a toothbrush is the penetration depth into approximal areas. Again, the full behaviour of the brush cannot be predicted, but we assume that some filaments will penetrate to the deepest amount possible into a given gap. Therefore, optimizing the maximum possible penetration depth is a good measure for the overall penetration performance. As the 16°-inclined filaments are longer than the vertical ones for the same brush thickness d0, it is likely that they may penetrate deeper into a gap, which was demonstrated for the 16°-filament. For the defined tooth geometry, the filament was able to penetrate about 0.3 mm deeper into the cavity. This corresponds to a 15% increase of the cleaned area; in experimental studies [3], the inclined filaments of the manual CrossAction® brush performed 10% better on penetration per stroke compared to commercially available brushes.

This study was conducted to improve the theoretical understanding of the filament-tooth interaction. Experimental studies typically consider the total normal brushing force and its impact on measures, such as efficacy and gingival abrasion [13] or dentin and enamel abrasion [14]. This study, however, shows how the physical process of global

brush deformation d0-d, which is closely related to the global normal force, leads to a variety of normal and transverse forces acting on the tooth surface or biofilm, respectively, depending on the individual filament position. Thus, it details the causal chain from global normal force to the mechanical impact on dental plaque as a prerequisite for the understanding of efficient removal. This approach is intended to help understand the physical impact of toothbrush filaments on tooth surfaces, which have been described as causative factors of cleaning efficacy or soft and hard tissue abrasion. Ultimately, such considerations will help to draw implications for a better toothbrush design. As with any consumer product of this kind, the final proof of efficacy benefit needs to be verified under in-use conditions, such as a consumer or clinical study.

5.1 Implication for Toothbrush Design Based on this study, some design conclusions for brush heads can be drawn: 1) Optimize filament action: To achieve the maximum transversal (shear) force during a brushing stroke on a flat surface, the filament has to reach Shape 3 in Fig. 7. This corresponds to a calculated stroke length of 2.32 mm (d0 – d <1 mm) for a straight filament and 0.53 mm (d0 – d <1 mm) for a 16°-inclined filament plus the distance of sliding prior to the stuck event (Shape 3 in Fig. 7).

2) Arrange filaments with angle: In order to increase the maximum absolute shear force and the penetration into approximal areas, the filament should be angled towards the surface.

3) Optimize the filament angle: With an increasing filament angle up to about 16°, the shear force increases while the normal force remains approximately constant. However, the filament angle should not be larger than about 16°, because the normal force decreases above this value and becomes smaller than the vertical filament version, which we wish to avoid.

4) Deliver maximum number of efficacy events: The brushing efficacy increases with the number of maximum shear force events per time interval.

The maximum frequency is the frequency in which filaments can still travel by at least a stroke length of 0.53 mm.

5) Ensuring travel of filaments at tooth surface: The configuration of the brush head (filament diameter/length, tuft arrangements, etc.) has to be designed in such a way that the movement of the brush head can be transferred across the filaments

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towards the tooth surface, i.e. it must be ensured that the movement is not reduced/suppressed with increasing brushing force.

6 CONCLUSIONS According to the derived model, the simulation data, and our experimental validation, we conclude that we can further increase the efficacy of a toothbrush in terms of biofilm removal. We have shown that oscillating-rotating electric toothbrushes deliver the necessary filament displacement (travel) to exert maximum shear force to the biofilm.

These high efficacy events occur during the deformation of the filament, just before filament flipping. In comparison to manual brushes, the electric toothbrush can increase the number of these high efficacy events during brushing (e.g. 78 events per second for a typical O/R power brush).

We have clearly demonstrated that filaments that are inclined result in higher shear forces, which is beneficial for plaque removal. A filament angle of about 16° was found to maximize shear force, while keeping the normal force almost constant compared to similar brush heads with straight filaments.

Furthermore, the penetration into approximal areas can be clearly increased using 16°-inclined filaments, in comparison to straight filaments.

A filament angle of 16° has proved to increase plaque removal in the design of manual brushes.

Our study shows that this angle also applies for oscillating-rotating power brushes. The new Oral-B CrossAction® replacement brush head features an outer and an inner ring with inclined filaments.

The outer ring has an angle of +16° for the forward direction of the oscillation; the inner ring has an angle of –16° for the backward direction of the oscillation.

Both the outer and the inner ring have sufficient deflection to exert maximum shear forces. The inner part of the brush head shows only minor movement.

This part was designed with straight filaments to provide the necessary stability (stiffness) of the filament design. The brush head in Fig. 2 was designed to meet the above-derived criteria for optimized shear force transmission and increased approximal area penetration (for the given oscillation frequency) and provides sufficient filament tip motion if used with the recommended brushing force (maximum 2 N). Based on our knowledge of the forces needed for biofilm removal, the Oral-B CrossAction® brush head with filament angles of 16° can deliver superior plaque removal efficacy in comparison to other tuft designs.

These benefits have been demonstrated in clinical 460

investigations, which are currently in the publication process.

7 ACKNOWLEDGEMENT The authors wish to thank Oral-B Laboratories for their financial support of this study.

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Comparative analysis of 3D digitization systems in the field of dental prosthetics. Technical Gazette, vol. 20, no. 2, p. 291-296.

[12] Popov, E.P. (1998). Engineering Mechanics of Solids, Second Edition. Prentice Hall, Upper Saddle River.

[13] Van der Weijden, G.A., Timmerman M.F., Versteeg, P., Piscaer, M., Van der Velden, U. (2004). High and low brushing force in relation to efficacy and gingival abrasion. Journal of Clinical Periodontolgy, vol. 31, no.

8, p. 620-624, DOI:10.1111/j.1600-051x.2004.00529.x [14] Wiegand, A., Burkhard, J.P.M., Eggmann, F., Attin, T. (2013). Brushing force of manual and sonic toothbrushes affects dental hard tissue abrasion, Clinical Oral Investigations, vol 17, no. 3, p. 815-822, DOI:10.1007/s00784-012-0788-z.

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 462-474 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1569 Original Scientific Paper

Received for review: 2013-11-23 Received revised form: 2014-02-12 Accepted for publication: 2014-05-07

Stored Energy Predictions from Dislocation-Based Hardening Models and Hardness Measurements for Tensile-Deformed Commercial Purity Copper Čebron, M. – Kosel, F. Matjaž Čebron1,* – Franc Kosel2 1 Hidria Rotomatika, Slovenia 2 University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Stored energy analyses by differential scanning calorimetry (DSC) and indentation hardness measurement were performed on crosssectional samples cut from the gauge length of tensile-deformed copper specimens. The stress-strain curve was described using dislocationbased hardening models integrated into a visco-plastic Taylor-type model of polycrystal deformation. Three approaches in reproducing the experimental stress-strain curve were used to evaluate the differences in dislocation density predictions resulting from different modelling options. A good description of hardening was achieved by all three approaches and constitutive models and only negligible differences were found in the predicted dislocation density between assumed homogeneous and heterogeneous dislocation distribution throughout the polycrystal. Measured values of stored energy are somewhat lower than those published in research studies in which one-step and slow annealing methods were used. A simple model predicting a nearly linear increase of stored energy with dislocation density was found to adequately describe retained energy evolution. Since different dislocation arrangements result in different yield stress and energy predictions, both results can be used to determine values of parameters in two-internal-variable hardening models. Even though both measured quantities were satisfyingly described, uncertainties regarding material parameters and the applied polycrystal and stored energy models prevent us from claiming that the evaluated dislocation density distributions represent the actual dislocation structure in the material. As expected for strongly hardening materials, the relationship between yield stress and hardness could not be adequately approximated by a linear function.

Instead, a linear combination of yield stress and hardening rate was used, finally providing a relation between hardness and stored energy through their mutual dependence on yield stress.

Keywords: dislocations, hardening models, crystal plasticity, stored energy, calorimetry, hardness

0 INTRODUCTION In the last decades considerable effort has been dedicated to the development of hardening models based on the underlying physical mechanisms involved in plastic deformation of crystalline materials. Due to the complexity of the processes involved, most physically based models to date combine the theoretical knowledge of deformational behaviour with a series of empirical rules obtained from experimental observations. The plastic response of metals is determined by the production and migration of defects in the crystal lattice called dislocations [1]. Despite suggestions that a complete description of work hardening in terms of dislocation theory may never be possible, in-depth research on pure metals and heterogeneous solids has clarified many aspects of the complex processes involved [2] and [3].

Dislocation-based models are expected to have a wider applicability and better predictability than phenomenological formulations over larger ranges of strain, strain rate and temperature. Since they are based on the description of microstructural evolution, they can be handily combined with models of other structure-dependent processes affecting physical 462

properties of the material like recristalization and recovery, e.g. in [4] to [6]. Earlier dislocation-based models of strain hardening, see [7] to [9], were mainly concentrating on explaining stage III of the hardening process and were able to describe a gradual decrease of the hardening rate, which in many cases is a nearly linear function of yield stress. This type of strain hardening behaviour can be explained in terms of constitutive models using a single internal variable related to the mean dislocation density [10]. Under deformation conditions in which plastic instabilities are avoided, the gradual decrease of the hardening coefficient is interrupted by a new hardening stage characterized by a nearly constant hardening rate which is eventually followed by a final drop-off leading to stress saturation [3].

Moreover, stored dislocations in crystalline materials are rarely distributed homogeneously throughout the structure, implying that a satisfactory description of the mechanical state in terms of the mean dislocation density is inadequate. To overcome this inconsistency and correctly describe the hardening behaviour following phase III, two-internal-variable models had to be invoked [11].

Estrin et al. [10] presented a dislocation-densitybased model aimed at describing the hardening

*Corr. Author’s Address: Hidria Rotomatika d.o.o., Spodnja Kanomlja 23, Slovenia, mat.cebron@gmail.com

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behaviour of cell-forming crystalline materials studying also the effect of texture variation on strain hardening. Argon and Haasen [3] proposed a mechanism of work hardening related to a cellular dislocation microstructure in which hardening of cell interiors occurs through the build-up of long range internal stresses associated with lattice misorientations between dislocation cells. In [12] Ma and Roters presented a constitutive model for fcc crystals which follows individual slip-system hardening and can therefore directly account for latent hardening. A model for heterogeneous alloys was developed in [13] with the density of mobile dislocations added as an additional internal variable. Recently, several dislocation models focused on understanding the dislocation mechanisms accompanying stress reversal have been proposed in an effort of explaining the material response following deformation path changes, e.g. in [14] and [15]. In the presence of a heterogeneous dislocation distribution, long range internal stresses are an unavoidable consequence of compatibility requirements during deformation in the stress-applied state [3]. Experiments by Mughrabi [11] and Mughrabi et al. [16] have shown that in the loaded state both cell walls and cell interiors are subjected to stresses in the same direction as the applied stress, however stresses in walls are far larger than the applied ones while those in cell interiors are smaller.

Essman [17] and [18] used fast neutron irradiation to pin dislocations of copper single crystals strained into hardening stages I and II. While he found no evidence of long range stresses in stage I the curvature of dislocations in the unstressed crystals strained into stage II allowed him to estimate long range internal stresses of the order of half the forward yield stress.

Dislocation-based hardening models provide evolutionary equations for dislocation densities and their relation to the shear strength of single crystals. In order to relate the latter to the macroscopic values of stress and strain measured on a polycrystal aggregate, the model has to be complemented by a description of polycrystal deformation. In this work a visco-plastic Taylor-type model is used. In recent years several variants of Taylor-type models have been introduced with the common quality of allowing for the active slip systems, slip rates, rates of lattice rotation and the deviatoric stress to be calculated for each grain separately. As a result, calculation time is much smaller compared to crystal plasticity FE methods or self-consistent models [19] and [20] in which grains are treated as inclusions within a homogenized medium having the average constitutive behaviour of the entire aggregate.

During plastic deformation of metals, most of the mechanical energy expended is dispersed as heat. The remaining part, often referred to as stored energy, is retained in the metal as the energy of the elastic field of the dislocation structure. The energy storage phenomenon in metals was discovered by Taylor and Quinney [21]. The retained energy can be measured by one or two-step methods. In onestep techniques the measurements take place directly during the deformation process. These methods are usually regarded as more accurate, however the need for special equipment and measuring procedures employed has favoured the use of two-step techniques with a shorter processing time and substantially lower costs [22]. In two-step methods the already deformed material is subjected to a protocol of heat treatment during which the heat flow caused by thermal processes in the specimen is measured.

It has been proposed in [23] and [24] that indentation hardness can provide a simple method for determining stored energy, since it is a local mechanical property which depends on yield stress and therefore on the local dislocation density. In this work the relationship between stored energy, yield stress and Vickers hardness for tensile-deformed copper has been studied and compared with previously published results and models.

1 CONSTITUTIVE MODELS 1.1 Bergström Hardening Model In 1970 Bergström [8] presented a dislocation-based constitutive model aimed at describing the hardening behaviour of polycrystalline α-Fe at room temperature.

The model distinguishes between three components of yield stress:

σ = σ 0 (T ) + σ * (ε p , T ) + σ d ( ρ , T ). (1)

In Eq. (1) σ0, σ* and σd are the lattice resistance, the strain-rate dependent temperature stress and the dislocation-density dependent hardening component, respectively. The contribution of σ* is usually small for fcc structured metals and is often neglected. Therefore the effect of temperature and strain rate on flow stress is usually introduced indirectly by describing their influence on the hardening rate affecting σd [25].

The hardening component σd is related to the mean dislocation density ρ by the Taylor relationship:

σ d = α ′Gb ρ , (2)

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where α´ is a constant describing the average interactions between dislocations, G is the shear modulus and b is the magnitude of the dislocation Burgers vector. Eq. (2) can be derived from a series of different interaction mechanisms (e.g. cutting of forest dislocations) and is widely used in dislocationdensity-based models as well as in more sophisticated single-crystal constitutive relations [26].

The density of mobile dislocations is considered to be constant while the variation of the total density is determined by four envisioned dislocation mechanisms: immobilization, re-mobilization, annihilation and creation, from which an evolutionary equation relating dislocation density to tensile strain ε p is deduced as:

d ρ / d ε p = U ′(ε p ) − A′ − Ω′ρ , (3)

where Uʹ is a measure of the rate at which mobile dislocations are immobilized or annihilated and is inversely proportional to the dislocations mean free path, Ωʹ expresses the probability of remobilization and annihilation through reactions between mobile and immobile dislocations and Aʹ gives the rate of density variation from strain-invariant sources (e.g. grain boundaries and free surfaces).

Since plastic deformation of metals occurs by irreversible shear in discrete slip systems, a more commonly used approach is to relate the dislocation density to the total shear strain of metal grains. In this case Eqs. (1) to (3) must be rewritten in terms of shear quantities. Neglecting σ* in Eq. (1), we obtain:

τ = τ 0 + τ d = τ 0 + α Gb ρ , (4)

d ρ / d γ = U ( γ ) − A − Ωρ , (5)

where τ and γ are the shear strength of active slip systems and the total shear deformation of the crystal, respectively. Since we only consider the total shear deformation, equal and concurrent hardening of all slip systems is implied. In the field of crystal plasticity this kind of hardening is often termed isotropic [10].

In order to integrate Eq. (5), the function U(γ) must be specified. In the original study [8] it was considered to be constant and the relation between strain and dislocation density was supposed to be nearly linear. This was justified by observing that a cellular dislocation structure consisting of hard dislocation walls and soft cell interiors forms in the very early stages of deformation and that the cell diameter d rapidly attains a constant value which does not appreciably change on further straining. As the mean 464

free path z is considered to be proportional to the cell diameter it follows that U has a constant value.

Under these assumptions an analytical expression linking strain and dislocation density can be deduced.

However, other investigations, e.g. in [26] and [27], have shown that typically in fcc structured metals the cell size d decreases with density according to:

d = K / ρ , (6)

where K is a dimensionless constant ranging approx. from 10 to 20 for Cu [26]. Assuming that the dislocation mean free path is proportional to the cell size, U can be written as:

U = U 0 ρ . (7)

where U0 is a constant. Since 1/ρ0.5 can be interpreted as the mean distance between dislocations, Eq. (6) is often seen as a manifestation of the so called principle of similitude, which states that dislocation structures refine themselves during straining in a self similar way thus retaining all ratios between different dimensions of the dislocation arrangement. The physical origin of what is called similitude is not yet understood and the related coefficient is not predictable from dislocation theory [26].

1.2 Estrin Hardening Model The majority of dislocation-based models developed for describing hardening after phase III consider the material as a two-phase composite consisting of cell walls with a high dislocation density and dislocation-poor cell interiors. Different evolutionary equations for dislocation densities in the two phases give rise to separate stages of hardening. In most of these models the shear strength of individual phases is determined by the local dislocation density. In [3] Argon and Hassen presented a comprehensive elastic analysis explaining the role of specific dislocation arrangements and their resulting long-range stresses.

Since these stresses are in self-equilibrium, they do not affect the yield strength of the material, however, they are fundamental for the two phases to deform in a coherent way. If such a deformation is supposed, the total shear strength is obtained by an expression resembling the so called rule of mixtures:

τ d = f τ w + (1 − f )τ c , (8)

where τw, τc and f are the shear strength of cell walls, the shear strength of cell interiors and the volume fraction of cell walls, respectively. In [10] Estrin et

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al. presented a 2D model for describing hardening of cell-forming metal crystals at large strains. The model was later generalized for the 3D case and arbitrary strain paths in [28]. One of the main assumptions of the model states that dislocation generation only occurs in walls, while annihilation takes place both in walls and cell interiors. The density in cell interiors ρc is affected by three processes, namely the creation of dislocations in walls, the loss of dislocations due to them becoming part of the wall structure and the mutual annihilation of dislocations, resulting in the following evolutionary equation:

* d ρc α ρ w 6β * = − − k0 ρc , (9) 1/ 3 dγ 3b bd (1 − f )

where α* is the fraction of dislocation sources operated by dislocations coming from walls, β* is the fraction of interior dislocations becoming part of the wall structure and k0 is a measure of the rate of annihilation of dislocations within the specific phase. Dislocation density in cell walls ρw is increased by dislocations coming from cell interiors as well as the activation of dislocation sources by interior dislocations and again reduced by mutual annihilation:

* d ρ w 6 β (1 − f ) = dγ bdf

2/3

+

3β * ρ w

fb (1 − f )

−1

d = K / ρa = K /

f = f ∞ + ( f 0 − f ∞ ) exp ( −γ / γ e ) , (12)

where f∞ and f0 are the saturated and initial values and γe is a constant describing the rate of decrease of f.

1.3 Polycrystal Model The hardening models presented above describe the relationship between the shear strength of slip systems and the accumulated shear deformation of individual crystals. In order to relate these to the macroscopic values of deformation and stress, a model of polycrystal deformation has to be used.

The following description of kinematic behaviour of crystals subjected to large deformations is based on the assumption of multiplicative decomposition of the total deformation gradient into an elastic and plastic part first proposed by Lee [29]. Three configurations of the material are introduced: initial, intermediate and current, see Fig. 1.

− k0 ρ w . (10)

The specific numeric values appearing in Eqs. (9) and (10) are a result of assuming a cubic shape of cells with side length d. Mechanisms leading to the development of cellular dislocation patterns are not considered in the model. Shear strengths of the two phases are related to the dislocation densities therein according to the same Taylor relationship, Eqs. (2) and (4), as is used in the Bergström model.

The constitutive description is completed by a scaling relation for the average cell size and a description of the evolution of the wall volume fraction. The average cell size is defined by Eq. (6) using the average dislocation density ρa as the scaling measure:

f ρ w + (1 − f ) ρc . (11)

The principle of similitude implies a constant volume fraction of cell walls. However, this assumption is not supported by experimental observations showing that after an initial increase f monotonically decreases to a saturation value at larger strains [10]. The variation of f is described by:

Fig. 1. Decomposition of deformation

The deformation is characterized by the velocity gradient tensor L and the total deformation gradient F describing the deformation of the material from the initial to the current configuration:

Fij = ∂xi / ∂X j , Lij = ∂xi / ∂x j , (13)

where x(X) and X are the current and initial coordinates of the material particle, respectively.

From Eq. (13) an expression for the time derivative of the deformation gradient can be deduced:

∂v  ∂v   ∂x  F = =    = LF. (14) ∂X  ∂x   ∂X 

Plastic deformation is a result of dislocation slip occurring on favourable crystallographic planes and in specified directions, together forming a slip system.

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A pair of orthogonal vectors {bs, ns}, where bs is a unit vector parallel to the slip direction and ns is a unit vector normal to the slip plane is used to describe slip system s.

The total deformation gradient F is decomposed into an elastic Fe and a plastic part Fp. It is supposed that during plastic deformation the material flows through the invariant lattice while in the following elastic part the lattice deforms attached to the material. The elastic deformation gradient comprises eventual elastic deformations and rigid body rotations.

Since our study is focused only on the rigid-plastic model of single crystal kinematics, no “real” elastic deformations are considered. The elastic part of the total deformation gradient thus reduces to a rigid rotation R:

e p = F F= F RF p . (15)

For the plastic deformation in the initial configuration we can write a relation similar to Eq. (14):

F p = Lp F p .

The plastic velocity gradient L p is given by a linear superposition of shear rates on all slip systems:

Lp = ∑ s γ s bis n sj ,

where γ s is the shear rate of system s. The velocity gradient L p can be decomposed into a symmetric strain rate Dp and a skew-symmetric rotation rate Wp, also known as spin:

Lp = D p + W p . (16)

Decomposing the dyadic product bsns into a symmetric part mijs and a skew-symmetric part qijs allows us to write the strain and rotation rate in terms of shear rates on active slip systems:

D p = ∑ s γ s mijs , (17)

W p = ∑ s γ s qijs . (18)

By inserting Eq. (15) into Eq. (14), the total velocity gradient can be written as:

 −1 = RR  T + RLp R T . (19) L = FF

Similarly to Eq. (16) the total velocity gradient can be additively decomposed into a strain and a rotation rate (L=D+W). Since RṘT is a skewsymmetric matrix, we obtain: 466

p T = D RD = R D p , R , (20)

 T + RW p R T = RR  T + W p , R . (21) W = RR

The strain rate D is a simple transformation of Dp from initial into current configuration, however the rotation rate W contains an additional contribution.

The polycrystal model must provide a value for the velocity gradient in crystals. In Taylor-type models L for each of the constituent grains is considered to be equal to the imposed macroscopic velocity gradient L*. Using Eqs. (17) and (20) the strain rate D* can be written as:

D* = D = ∑ s γ s mijs, R . (22)

In prescribing the macroscopic strain rate D* it is assumed that the volume does not change (Dii* = 0).

The sum of right-hand sides of Eq. (22) for ij = 11, 22, 33 is also zero because of the orthogonality between the rotated vectors b and n. As a result only five of the six Eqs. (22) are independent. The fcc crystal structure of copper has 12 potentially active slip systems of the {111}<110> type. Since there are more unknown slip rates than equations prescribing the deformation, the solution of Eq. (22) is not unique. In his original work Taylor assumed that out of the possible combinations of five active slip systems the one with the minimal rate of internally dissipated frictional work will be active. This assumption however still does not lead to a unique solution for strain rates [19]. To overcome this limitation of models based on a rate insensitive idealization of slip, Asaro and Needleman [30] proposed a simple rate-dependent model in which the shear rate γ s is uniquely defined by:

γ s = γ0 τ s / τ

1/ n

sign(τ s ), (23)

where τs is the shear stress in slip system s, τ is the shear strength of the system described by the hardening models, γ0 is a reference shear rate and the parameter n characterizes the material rate sensitivity.

If n approaches zero, a substantial amount of slip only occurs in systems where the shear stress reaches values near the shear strength. The rate-independent response of the material is therefore retrieved as n→0. In practice it is found that for values of 0.03 or lower, the stresses obtained from Eq. (23) differ only slightly from those obtained from the rate-insensitive Taylor-Bishop theory [30]. Using Eqs. (17) and (23), the deviatoric part of the prescribed strain rate can be written as:

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Dij* = γ0 ∑ mijs, R mkls, Rσ kl′ / τ s

1/ n

sign (mkls, Rσ kl′ ), (24)

where σ kl′ is the deviatoric part of the Cauchy stress tensor. Given a prescribed strain rate D* Eq. (24) represents a system of five independent nonlinear equations for five independent deviatoric components of stress. In our model the Newton-Raphson method was used in order to solve the system of stresses. It should be emphasized that in this approach all slip systems are considered to be active. Once the slip rates have been calculated, the spin of the crystal lattice is determined from Eqs. (18) and (21):

 T = W* − ∑ γ s q s , R . (25) RR ij s

The last expression is used to update the orientation of the crystal. A detailed description of evaluating the incremental rotation matrix using Eq. (25) and the Rodrigues formulation for finite rotations employed in our model can be found in [31].

The resulting rotation is used to calculate the new direction of vectors b and n from which follows the crystallographic texture evolution.

Using the polycrystal model, the increment of shear deformation of the ith crystal dγi and the shear strength of the crystal´s slip systems τi can be related to the imposed deformation increment dεp and the tensile stress acting on the crystal σi via the Taylor factor M: d γ i = ∑ s dγ is = M i (γ i )d ε p , σ i = M i (γ i )τ i . (26) Finally two approaches can be taken to express the macroscopic value of tensile stress. In most cases, due to smaller computational costs, all the constituent crystals are considered to harden equally in relation to the average deformation of the polycrystal aggregate. An average Taylor factor M̅ (γ) is used to relate the average shear deformation increment of the constituent grains dγ r to the macroscopic deformation increment:

d γ r = (1 / N ) ∑ i =1 M i d ε p = Md ε p , (27) N

where N is the number of grains included in the aggregate. Similarly the average Taylor factor is then used to link the macroscopic value of tensile stress to the average shear strength τr(γr) described by the hardening models, which in this case is supposed to be equal for all grains:

σ = M τ r . (28)

The second approach is to use the hardening models as constitutive equations directly describing the hardening of single crystals. The total stress is then computed as the average stress contribution from all grains in the polycrystal:

σ = (1 / N )∑ i =1 M iτ i . (29) N

In the first approach, a homogeneous distribution of dislocation density throughout all the grains of the polycrystal aggregate is supposed, with each grain hardening simultaneously to the same extent. In the second case, each grain is considered to harden according to the same hardening law but to a different extent due to the different total shear deformation imposed. Therefore a heterogeneous dislocation distribution is assumed with different dislocation densities in each of the constituent grains.

2 EXPERIMENTAL METHODS Commercial rods made of 99.98% purity copper supplied by AlCu d.o.o. were machined to a cylindrical shape of gauge length 35 mm and diameter 7 mm according to standard ISO 6892-1 to get proportional test pieces, see Fig. 2. After machining the specimens were annealed in a protective atmosphere at 500 °C for 30 minutes. The tensile tests were carried out at 24 °C using a Zwick/Roell Z050 testing machine at a constant deformation rate of 0.5 mm/min.

The specimens were subjected to various tensile forces in the plastic region up to fracture-point with two specimens loaded to each pre-set force value. As the differences between the force-displacement curves were found to be negligible, samples deformed to the same tensile force were assumed to be subjected to the same tensile stress.

Cross-sectional samples for DSC (2×) and hardness measurements were obtained from each deformed specimen by cutting circular discs with a thickness of 1 to 1.5 mm (calorimetry) and 5 to 6 mm (hardness) from the gauge length perpendicular to the tensile axis. Two cutting methods were used in order to estimate possible effects of the cutting process on stored energy and hardness results. One sample from each specimen was first cut using a wire ED machine.

The second group of samples was obtained using a Struers Secotom-15 cut-off machine at a cutting speed of approx. 42 m/s and a feed rate of 0.3 mm/min. Since ED machining causes a local increase of temperature and mechanical cutting induces further hardening in the surface layers, an opposite effect on hardness and stored energy was expected from the two cutting

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processes. However, no appreciable differences in either of the measured quantities for differently cut samples were found, from which it was concluded that neither of the cutting methods had a noticeable effect on the measured results.

Fig. 2. Dimensions of tensile test specimens (in mm)

Stored energy measurements were performed using a Mettler Toledo TGA/DSC1 machine incorporating a heat-flux type differential scanning calorimeter. Alumina crucibles were used and the DSC was purged by nitrogen gas with a flow of 50 ml/ min. Samples were heated from room temperature to 500 °C at a constant rate of 10 °C/min. The weight of the samples was between 250 to 450 mg.

Hardness measurements were carried out using an Emcotest Durascan 20 testing machine with a standard Vickers 136° pyramidal diamond indenter. All the measurements were made at room temperature; the load was set to 5 kg and applied for a standard dwell time of 15 s.

Fig. 3. Measured stress-strain curve and spec. load Table 1. Parameters of the Bergström model Approach

U0 [mm-1]

A [mm-2]

Ω [-]

A B C

1.059×105

-2.998×108

1.116×105 9.414×104

-2.779×108 -3.418×108

3.507 3.875 3.222

3 RESULTS AND DISCUSSION A random distribution of 1000 grain orientations was generated representing an initially isotropic polycrystal aggregate. Three approaches for reproducing the experimental stress-strain curve were used in order to evaluate the differences in dislocationdensity predictions resulting simply from different modelling options. In approach A Eqs. (27) and (28) were employed and the texture development was neglected, therefore the Taylor factors were constant and equal to their values at zero strain. The same equations with texture evolution taken into account were used in B, while in C Eq. (29) was employed with again considering texture changes.

The measured stress-strain curve of the material is presented in Fig. 3 along with the modelled curves and loading stresses of specimens. Material and model parameters used in the computations are given in Table 1 and Fig. 4. It is evident that a good description of hardening can be achieved by any of the employed modelling approaches and both hardening models.

The differences between the modelled curves are negligible and cannot be seen from Fig. 3 where they seemingly overlap each other and the experimental results.

468

Fig. 4. Predicted dislocation density (Bergström model)

Predicted dislocation densities from the Bergström model are presented in Fig. 4. Surprisingly, the mean densities from B and C are almost identical.

This results justifies using the simpler approach B even though the heterogeneous distribution of dislocation density related to the actual deformation of single crystals theorized in C is probably betterfounded. Approach C also allows us to predict the range of density present in the individual grains of the polycrystal. On the other hand, using approach A results in markedly different predictions from B and C (approx. 6% at final strain). Using constant values of Taylor factors is the most common approach for incorporating dislocation-based hardening models in continuum mechanics FE computations, e.g. in [32].

However if we are interested in dislocation density predictions, not accounting for texture evolution must lead to erroneous results [10], with approach A in our case overestimating the extent of hardening compared to B and C.

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Differential calorimeters measure differences in heat flow between the test sample and the reference as a function of time or temperature. With no appreciable thermodynamic processes taking place only a slight variation of this difference can be observed. However, at the onset of a thermal event in the material, an abrupt change in the difference of heat flows occurs which is visible as a peak in the DSC curve. The area bounded by the peak and a measured or approximated baseline is equal to the material`s change of enthalpy. In our study the non-isothermal heat treatment of samples produced exothermic peaks related to recrystallization.

Since the pressure work and the entropy contribution due to dislocation annihilation are both negligible [22], the measured enthalpy change can be equated to the free energy present in the form of the elastic energy of the dislocation structure. Contributions from other possible strain-induced defects (e.g. point defects and twins) along with the enthalpy change due to altering grain boundaries during recrystallization are neglected. In Fig. 5 the DSC curve of one of the calorimetry samples cut from tensile specimen 3 is presented. The measured stored energies were in the range of 20 to 160 mJ (0.07 to 0.36 J/g). Meaningful results could only be obtained from specimens 1 to 10 as the values from specimens 11 to 14 were too low to be accurately determined by the used equipment.

The measured values of stored energy are somewhat lower than those published in studies where one-step and slow annealing methods were used, see [22] and [33], however comparable to other results obtained by DSC, e.g. in [23].

Fig. 5. DSC curve of sample taken from specimen 3

Since two-phase dislocation models imply the existence of long-range internal stresses, the energy stored in the material after unloading is not only the self energy of dislocations, but also the energy of the long-range stress field. Different measuring techniques will produce different results, since they measure different portions of this energy. Long-range

stresses are often associated with the Bauschinger effect and other transient hardening phenomena observable during changes in the direction of straining [15] and [34]. It has been reported in [34] that a mild heat treatment below the recrystallization temperature tends to remove the permanent Bauschinger softening which is attributed to long-range compatibility stresses even though no appreciable changes in dislocation density can be observed at this stage [33]. It is reasonable to presume that during heating below the recrystallization point small changes in the dislocation structure occur causing long-range stresses to slowly anneal out. Although only one peak is present in the DSC curve some exothermic processes may still be proceeding during the flat phases of the curve in a slow, continuous manner therefore being undetectable by the DSC method. In one-step techniques this contribution is measured directly while in slow annealing methods it can usually only be estimated [22]. We can therefore assume that the energy measured by DSC is that of a relaxed dislocation structure with a negligible contribution from long-range stresses.

Dislocations densities resulting from approach C were used in stored energy evaluations. A relatively simple model for the energy of dislocations was employed in which predominantly edge-type dislocations are assumed to accumulate during the hardening process (ν = Poisson`s ratio, ρm = density): Es =

 ρi Gb 2   P  1 N ln  1/ 2  + Q . (30) ∑ m  i =1  4π (1 −ν ) ρ   ρi b  N 

Eq. (30) gives the self energy of edge-type dislocations with P/ρi0.5 taken as the upper and the magnitude of the Burgers vector b as the lower cut-off radius, while Q describes the energy of the dislocation core. Both P and Q are dimensionless parameters. By taking P = 1, an alternating dipolar arrangement of positive and negative dislocations is assumed with a shielding distance of the order of the mean dislocation spacing. A similar expression can be derived by considering the total energy of a dislocation dipole in a finite cylinder, see [35], yielding only negligibly different results from Eq. (30). For Q a value of 1 is used which is near the lower limit of energy usually attributed to dislocation cores [36].

Using P = 1 as the proportionality factor between the upper cut-off radius and the mean dislocation spacing generally leads to an underestimated value of stored energy [33] and [37], which is also the case in our study, see Fig. 6. The best fit with the measured results was obtained by taking P = 800. This value of

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P gives an unrealistically high upper cut-off radius for a random dislocation arrangement. However, it should also be acknowledged that there are two main reasons why the evaluated dislocation density is likely to be underestimated in our computations. The first is the applied polycrystal model. Taylor-type models represent an upper-bound regarding the stiffness of the polycrystal aggregate, therefore they predict the lower values of shear strength and dislocation density. The lower bound is represented by the so called Sachs-type models. The ratio of average Taylor factors between upper and lower bound models is approx. 1.37.

Fig. 6. Stored energy results

The second reason is the dependence of the strengthening coefficient α on dislocation density which is not described by the applied hardening models. It has been noted in [26] that dislocation densities evaluated by using constant values for α are probably underestimated since the line tension of dislocations declines with increasing density and the measured values of α usually pertain to lower concentrations in which experimental observations of dislocations are feasible. However, to the authors’s knowledge, this effect has not been included in any of the dislocation-density-based hardening models presented to date. A description of the drift of the hardening coefficient is provided in [26] by Eq. (A3). By using αref = 0.34 and ρref = 107 mm-2 from [38] in Eq. (A3) [26] and scaling the Taylor factors evaluated from our polycrystal model by 0.86, the computed dislocation density is increased enough to satisfactorily predict experimental stored energy results with P = 1. It should be noted that scaled values of Taylor factors and strengthening coefficient used are still well within reported experimental and computational values.

In one-internal-variable models, the relation between shear strength and dislocation density is unique. On the other hand, two-internal-variable models allow for different dislocation arrangements to 470

result in equal values of shear strength. Since different dislocation arrangements also result in different stored energy predictions, it follows that, given a known energy model, the measured values of stored energy can be used to determine the parameters of the hardening model. For evaluating the energy of the dislocation structure described by the Estrin model, Eq. (30) must be modified in order to account for the heterogeneous dislocation distribution:  1 N  ρ wi f wiGb 2   ln  ∑ N i =1  4π (1 −ν ) ρ m       ρ i f i Gb 2   1 N  ln  + ∑ c c N i =1  4π (1 −ν ) ρ m     

Es =

   + Q +  ρ wi b   P

 P   + Q  . (31)  ρci b  

Material parameters presented in Tables 1 and 2 were determined from the experimental data by the least square method in combination with a simplex optimization algorithm. For the Bergström model only the stress-strain curve was modelled with the three constants in Eqs. (5) and (7) used as fitting parameters.

Both, the stress-strain curve and the results of energy measurements (taking P = Q = 1 in Eq. (31)) were used to determine material parameters of the Estrin model in Eqs. (9) to (11), while those in Eq. (12) were only allowed to vary slightly within bounds estimated from experimental results presented by Müller et al. in [39].

Fig. 7. Predicted dislocation density (Estrin model) Table 2. Parameters of the Estrin model α* [-]

β* [/]

k0 [-]

K [-]

0.027738

0.018175

4.8876

42.726

f∞ [-] 0.21678

f0 [-] 0.29529

γe [/] 15.699

The mean dislocation density predicted by the Estrin model is, naturally, considerably higher than in the case of the Bergström model, see Fig. 7, since the

Čebron, M. – Kosel, F.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 462-474

calculated stored energy values were substantially too low using the same parameters in Eq. (30).

In [40] Zehetbauer and Seumer used TEM, calorimetry and electrical resistivity measurements to study the evolution of dislocation density in 99.95% purity copper deformed at room temperature. Despite the heterogeneous distribution of dislocations the shear strength was found to be proportional to the square root of the average dislocation density, perfectly in line with the one-internal-variable description of hardening. By using a constant Taylor factor of 3.06 they determined the strengthening coefficient α to be 0.27. Taking into account the evolution of the average Taylor factor given by our polycrystal model the best fit is obtained with α = 0.263. This result can be implemented into the Bergström model by simply setting the value of the strengthening coefficient to 0.263. On the other hand the parameters of the Estrin model have to be determined purposely to retrieve the same experimentally determined average density-strength relationship (and naturally reproduce the measured stress-strain curve). The coefficients used to describe the hardening of different phases in this instance need not to be equal to that determined for the average density. The predicted dislocation densities from both models (using approach C) in the aforementioned cases are presented in Fig. 8.

Fig. 8. Predicted dislocation densities

The best description of the measured stored energy values is achieved by taking P = 6 for the results from the Bergström model and P = 7.3 for the dislocation densities predicted by the Estrin model, see Fig. 9. Considering that dislocations are arranged in different slip systems these results present a very reasonable estimate of the upper cutoff radius of the order of the actual average spacing between dislocations. Naturally, using these results in the previously presented procedure of determining material parameters of the Estrin model, the same results as those already exhibited in Fig. 8 can be

retrieved. Interestingly, despite the vastly diverse density distribution predicted by the two models the average energy per line length differs by only ≈ 8%.

Fig. 9. Stored energy values

Fig. 10. Results of hardness measurements

In [23] and [24] an expression predicting a linear relationship between the stored energy and the squared value of yield stress was used. It is evident from results presented in Figs. 6 and 9 that a nearly linear model also results from our study and adequately describes stored energy evolution. In the same research a linear dependence was also supposed between yield stress and hardness, which in our case is not a satisfactory approximation. Fig. 10 shows that the measured hardness values cannot be described by a linear function but can be adequately approximated by a linear combination involving both yield stress and hardening rate (which in turn can also be described as a function of yield stress). This finally provides a relation between hardness and stored energy through their mutual dependence on yield stress, which for stored energy is given by Eqs. (4), (29), and (30) in the case of the one-internal-variable Bergström model.

4 CONCLUSIONS Standard specimens made from 99.98% purity copper were annealed and subjected to different tensile stresses in the plastic region of deformation. DSC and indentation hardness analyses were performed on

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cross-sectional samples cut from the gauge length of tensile specimens. The measured stress-strain curve was described using dislocation-based constitutive models. Two hardening models and their integration into a visco-plastic Taylor-type model of polycrystal deformation are presented. Three approaches for reproducing the experimental stress-strain curve were used to evaluate the differences in dislocationdensity predictions resulting from different modelling options. Unexpectedly, only negligible differences were found in the predicted mean dislocation densities between assumed homogeneous and heterogeneous density distribution through the polycrystal, while not accounting for texture evolution had a substantial influence on the results. A good description of the hardening curve was achieved by all the employed modelling approaches and both hardening models.

Values of stored energy were found to be lower than those measured in previous research studies by one-step and slow annealing methods. This probably results from the inability of the DSC method to measure energy released prior to recrystallization when during heating a slow relaxation of long range internal stresses occurs. For stored energy computations, dislocation densities from approach C were used. Since the part of Eq. (30) that is nonlinear in dislocation density is only logarithmic, stored energy predictions from B and C, similarly to the mean dislocation density, only barely differ. A simple model predicting a nearly linear increase of stored energy with dislocation density was found to adequately describe stored energy evolution. Supposing a dipolar dislocation distribution (P = 1) always resulted in an underestimation of the measured results. It should be mentioned that in many studies concerning the energy of lattice defects, the self energy of dislocations is still evaluated by using P = 1 and simply neglecting the core contribution Q, e.g. in [41], without any experimental validation. Besides from assuming an inadequately over-relaxed dislocation arrangement, the differences in measured and computed stored energy can also be attributed to an underestimated value of dislocation density. The scaled values of Taylor factors and strengthening coefficient used for illustrating the effect of parameter and model uncertainties on dislocation density and stored energy predictions are well within reported experimental and computational values.

Due to their good descriptive abilities dislocationbased models are becoming a popular option for describing hardening characteristics of fcc metals.

Frequently, these models are only calibrated to obtain a good description of hardening, while their 472

physical foundation in the theory of dislocations is disregarded. In these cases they can be viewed as purely phenomenological formulations. Recently a few approaches for ensuring dislocation densities and other dislocation structure parameters to stay within physically reasonable boundaries have been presented, e.g. in [42]. The use of two-internal variable models presents several difficulties regarding the determination of the exact values of material parameters. It is virtually impossible to determine the exact values of the hardening coefficients inside separate phases. Furthermore experimental studies regarding the exact density distribution are still quite scarce and hardly any general conclusions can be retrieved from them (for example the estimated values of the ratio of dislocation densities in cell interiors and cell walls vary from around 0.5 in [43] to 0 in [3]). Various researchers have questioned the use of composite type models especially since the vast majority of experimental studies have found the shear strength to be proportional to the square root of the average dislocation density regardless of the exact arrangement of dislocations, e.g. in [40]. This has also lead to the development of hybrid models in which the evolution of dislocation densities in separate phases of the cellular structure is still described by different evolutionary equations while the shear strength of the crystal is given by the Taylor relationship in terms of the average dislocation density only, e.g. in [42].

The dependence of yield stress and stored energy on dislocation structure in two-internalvariable models allows us to use the measured stress-strain curve and stored energy results to determine the parameters of the Estrin hardening model using pre-established yield stress and retained energy expressions. Even though both measured results are satisfyingly described by the model in combination with Eq. (31), it is evident from previous considerations that uncertainties regarding material parameters, the applied polycrystal model, the stored energy model and the accuracy of measurements of stored energy prevent us from claiming that the evaluated dislocation density and distribution are an accurate representation of the dislocation structure present in the material. Energy models and stored energy measurements are not accurate enough to capture the effects of smaller variations in the dislocation arrangement. Therefore they must not be regarded as an alternative to direct observations of the evolution of dislocations structures during hardening or other methods of measuring dislocation density usually used for validating dislocation-based hardening models. Their use in our research should

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instead be viewed as a quick and inexpensive way of assuring that the dislocation densities evaluated from hardening models remain within a physically acceptable range.

Finally very reasonable values of dislocation selfenergies which can be used in procedures of validation and determination of model parameters were obtained from experimentally determined results regarding dislocation density evolution presented by Zehetbauer and Seumer [40].

As expected for strongly hardening materials, the relationship between yield stress and Vickers hardness in our case could not be adequately approximated by a linear function. Instead a linear combination of yield stress and hardening rate was derived, giving a considerably better description of the measured hardness values. For slow and non-hardening materials our approximation closely resembles the relationship used in [23] and [24].

5 ACKNOWLEDGMENTS This research was partly financed by the European Union (European Social Fund, Programme for Human Resources Development for the Period 2007-2013).

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Čebron, M. – Kosel, F.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 475-482 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1544 Original Scientific Paper

Received for review: 2013-11-13 Received revised form: 2014-02-19 Accepted for publication: 2014-02-28

Dynamic Modeling and Simulation of a Rotating Single Link Flexible Robotic Manipulator Subject to Quick Stops Dupac, M., Noroozi, S. Mihai Dupac* – Siamak Noroozi

Bournemouth University, School of Design, Engineering and Computing, U.K. Single link robotic manipulators are extensively used in industry and research operations. The main design requirement of such manipulators is to minimize link dynamic deflection and its active end vibrations, and obtain high position accuracy during its high-speed motion. To achieve these requirements, accurate mathematical modeling and simulation of the initial design, to increase system stability and precision and to obtain very small amplitudes of vibration, should be considered.

In this paper the modeling of such a robotic arm with a rigid guide and a flexible extensible link subject to quick stops after each complete revolution is considered and its dynamical behavior analyzed. The extensible link that rotates with constant angular velocity has one end constrained to a predefined trajectory. The constrained trajectory allows trajectory control and obstacle avoidance for the active end of the robotic arm. The dynamic evolution of the system is investigated and the flexural response of the flexible link analyzed under the combined effect of clearance and flexibility.

Keywords: robotic arm, flexible manipulator, extensible mechanism, clearance, dynamics

0 INTRODUCTION The modeling and simulation of single link manipulators have received great consideration in current years in order to improve productivity and reduce production costs. Due to the field’s importance and applicability, such as to nuclear maintenance [1] and space applications [2], single link dynamics and control is considered to be a challenging research problem.

Since these manipulators contain interconnected rigid and flexible parts, their dynamic response is affected by the deformation and joint clearanceof the parts, which results in a high level of vibration and a low ability to perform accurate and safe operations.

To better understand and improve their mechanical response, accurate mathematical modeling and simulation of such systems discretized using finite elements or lumped mass methods have been considered by researchers worldwide.

The vibration control of an elastic link and some active methods to eliminate its vibration was discussed in [3]. Dynamic deflection of a flexible link and its active end vibrations have been approached

by Moulin and Bayo [4]. A nonlinear feedback controller for a single-link flexible manipulator derived using Lagrange approach was considered in [5]. A classical finite element approach Bricout et al.

[6] was considered to study flexible manipulators. The study of interconnected rigid and flexible links using finite elements has been discussed in [7]. Dynamic finite elements models of manipulators with a flexible links was analyzed by Tokhi et al. [8], and some

control strategies for damping vibration using shaping techniques have been approached in [9].

A dynamical analysis of a rotating single link manipulator was considered in [10]. Kinematic redundant manipulators with link flexibility and minimum deformation of the active end have been discussed in [11]. Lumped mass models to simulate trajectory tracking of the active end of a single-link flexible manipulator have been considered by Zhu et al. [12] and for two links manipulators in [13] and [14]. Dynamics and control of flexible-joint and dualarm robots and extensible members can be found in the papers given by [15] and [16]. The control of the vibrations of a flexible linkage mechanism and the impact effects have been discussed in [17]. The linear control a rotating flexible link of variable length undergoing periodic impacts was studied in [18].

The modeling of a translating flexible link with a prismatic joint and rotational motion and the study of its dynamical behavior have been considered in [19] and [20]. A dynamic finite element modeling of a translating and rotating flexible link was considered in [21]. The kinematic and dynamic simulation, impulsive/contact dynamics and stability of flexible mechanical systems have been presented and discussed in [22] to [24]. The dynamic analysis of some planar mechanisms with slider joints and clearance was considered in [25] and [26] and with lumped masses and impact in [27] and [28]. Mechanical systems with clearance and different types of impact/contact force models have been analyzed in [29] to [31]. A contact force model with hysteresis damping was considered in [32], and with a compliant contact in [33].

*Corr. Author’s Address: Bournemouth University, Talbot Campus, Fern Borrow, Poole, Dorset, BH12 5BB, mdupac@bournemouth.ac.uk

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In this paper the modeling and simulation of an extensible robotic arm with a rigid crank and a constrained flexible link subject to quick stops is considered. The flexible link, which rotates with constant angular velocity in a horizontal plane, has one end constrained to a predefined trajectory. The constrained trajectory allows trajectory control and obstacle avoidance of the end-effector of the robotic arm. The simulations performed on a circular/circularelliptic constrained trajectory, involves quick stops after each two complete revolutions, in order to explore manipulator behavior under real operational conditions. A clearance vs. a non-clearance model of the extensible flexible arm was considered in order to analyze the effect of clearance on the dynamic behavior of the system.

1 SINGLE LINK MANIPULATOR SYSTEM MODEL The flexible link PS of the single link manipulator in Fig. 1 is discretised as shown in Fig. 2 using n successive equal rods Ni Ni+1 (where N1 = P and Nn = S) connected with torsional springs. The flexible link, which can slide inside the rigid guide OD, is represented using a fixed reference frame O'xy with the origin at O'.

Each one of the successive equal rods has mass mi = m, length l = L/n, and moment of inertia J. Each one of the springs used to model link flexibility [34] and [35] has the stiffness k = EJ / dPS where E is the

The constrained trajectory that allows obstacle avoidance by the active end of the robotic arm is, in this case, an elliptic trajectory. The elliptical trajectory having its origin at O' has a transverse diameter of length dA'A'' and a conjugate diameter of length dB'B''.

The link PS is constrained to the circular/circularelliptic trajectory by a slot-joint.

The quick stop on the circular/circular-elliptic constrained trajectory takes place on the semielliptic trajectory at location A'' after every two complete revolutions, that is, a quarter-elliptic A''B' trajectory followed by a complete circular B'A'B''A'''B' trajectory, followed by a semi-circular B'A'B'' trajectory and finally followed by a quarter-elliptic B''A'' trajectory, and so on. The angle between two successive links Ni Ni+1 and Ni+1Ni+2 is denoted by θi+1 for any i = 1, n − 1 , while the angle between the link Ni Ni+1and the horizontal direction is denoted by Θi.

The angle between the guide and the horizontal direction denoted by θ, is always equal to the angle θ1 if there is no clearance between the guide and the flexible link. The motor torque, which acts on the end O of the rigid guide, performs quick stops after each complete revolution.

n

Young modulus and d PS = ∑ d Ni Ni+1 = L is the length i =1

of the flexible link, dOD is the length of the rigid crank and dOP is the distance between the end O of the guide and the constrained trajectory of the system.

Fig. 2. Lumped mass model of the flexible link with rigid support (guide)

2 DYNAMIC MODEL OF THE SINGLE LINK MANIPULATOR The dynamic model for the constrained flexible link can be expressed based on the position, velocity and acceleration vector of the centre of the mass Ci of each rigid rod Ni Ni+1, i = 1, n .

The position vector of the mass centre Ci of each rigid rod Ni Ni+1 is given by rCi = xCi i + yCi j , where i and j are the unit vectors of the fixed reference frame O'xy. The horizontal and the vertical coordinates xCi and yCi can be expressed as:

Fig. 1. Single link manipulator with a constrained flexible link and rigid support

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Dupac, M., Noroozi, S.

 xCi =  cos Θ j  cos θ  i + ∑ d N j N j +1    = dON1  −   sin θ  j =1  yCi =   sin Θ j  cos Θi  1 − d Ni Ni+1  (1) , 2  sin Θi 

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 475-482

where the distance dON1 (computed as in [36]) represents the variable distance between the centre of rotation of the guide denoted by O and the constrained trajectory, that is, the considered circular/circularelliptic trajectory. It can be observed that if first k connecting rods Nj Nj+1, j = 1, k are inside the driver OD, that is dON1 + ∑ j =1 d N j N j +1 ≤ dOD , then Θj = θ, for k

all j = 1, k .

The velocity vector of the mass centre Ci of each rigid rods Ni Ni+1 is the derivative with respect to time of the position vector rCi given by v Ci = rCi = xCi i + y Ci j where:  vx Ci   xCi   − sin θ  cos θ    =   = dON1  + dON1θ  +  v y Ci   y Ci   cosθ   sin θ 

i − sin Θ j  1     sin Θi  . (2) + ∑ d N j N j +1 Θ + d Ni Ni+1 Θ   j  i  j =1  − cos Θi   cos Θ j  2

and the subscript i represents the number of the generalized forces/coordinates.

The total kinetic energy can be expressed as in [27] by:

n

T = ∑ Ti = i =1

1 n ∑ mi vC2 i + ICi ωi2 , (5) 2 i =1

(

)

where the angular velocities ωi can be expressed as ωi = Θik for all i = 1, n . The model described in [27] was used to describe the generalized forces acting on each link.

2.2 Single Link Manipulator with Clearance The same clearance model considered in [34], where the flexible link of the manipulator model can translate and rotate about its support, was used for this study. Due to the clearance model, the flexible link may impact the guide as shown in Fig. 3.

The acceleration vector of the mass centre Ci of each rigid rods Ni Ni+1 is the double derivative with respect to time of the position vector rCi given by aCi = rCi = xCi i + yCi j where:  ax Ci  cos θ   − sin θ    = dON1  + 2dON1θ  +   a y Ci   sin θ   cosθ    − sin θ   2 cos θ   + dON1  θ   − θ  sin θ   −     cos θ  i  2 cos Θ j  − sin Θ j     +Θ −∑ d N j N j +1  Θ   j  j   sin Θ j  j =1  cos Θ j   

 +  

   sin Θi  1  2 cos Θi   . + d Ni Ni+1  Θ +Θ   i  i   2  sin Θi     − cos Θi 

(3)

2.1 Single Link Manipulator with No Clearance For the model with no clearance the flexible link translates parallel to its support and exhibits a continuous contact with the guide. One can write the Lagrange differential equation of motion with no impact as:

d  ∂T  dt  ∂qi

 ∂T = Qi , (4) −  ∂qi

where T is the total kinetic energy of the system, Ti is the kinetic energy of each ith link, qi = θi are the generalized coordinates, Qi are the generalized forces

Fig. 3. Constrained flexible extensible link with rigid support and clearance model; a) no impact, b) impact on one point, and c) impact on two points

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The impact model between the flexible link and the guide is shown in Fig. 3. Possible impact cases between the flexible link and the guide are: a) no impact, b) impact on a single point, and c) impact on two points. Since multiple impacts at the same time instant can be statistically excluded no such case was considered in this study. The discontinuous model considered here [27] and [18] assumes instantaneous impact and no change in the system configuration during contact, that is, the integration of equations of motion is halted at the time of impact, a momentum balance is performed to calculate the post impact velocities of the system components, and energy dissipation is quantified using the restitution coefficient. Other types of impact/contact force models have been discussed in [29], [30] and [31]. A hysteresis contact force model was discussed in [32], and a compliant contact in [33].

Using the discontinuous model [27] and [18] the impact differential equation of motion can be written as:

 ∂T   ∂u j

  ∂T  −  ts  ∂u j

the definition of the coefficient of restitution e and Eq. (9), one can write: –eva = vs . (10) 3 SIMULATION AND RESULTS In this section results from the computer simulations are presented for the following manipulator dimensions and material properties: guide length and clearance of 0.09 m and 3×10–3 m respectively, a constrained flexible link of 0.07 m, density of 7850 kgm−3, Young’s modulus of 2∙1011 Pa, and Poisson ratio of 0.3. The rotating end of the guide is located on the Ox axis at 0.02 m from the origin of the Cartesian reference frame. The constrained trajectory, that is, a semi-circular semi-elliptic trajectory is described as follows. The principal axes of the semi-elliptic trajectory, with the origin located on the Ox axis at

  = Pj , (6)  ta

∂T where ∂u are the generalized momenta and T is j the total kinetic energy of the system, Pj are the generalized impulses associated with the coordinate uj, and ta and ts represents the time of approach and separation. Considering:

ts

∫ Rdt = R i + R j, (7) x

y

ta

the force exerted during the impact of the flexible link with the guide. One can express the generalized impulses by:

Pj = ∑

t

∂v L s Rdt , (8) ∂q j t∫a

where vL = vG + ω × rN is the velocity of the impact point when the guide comes in contact with the flexible link. The velocity of approach and separation on the flexible link and guide can be expressed as:  v a = v L t − v G t a a

, (9)   v s = v L ts − v G ts where v L t and v G t , v L t and v G t are the flexible a s s a link velocity and guide velocity at time ta and ts before and after the impact. Using Newton’s formulation for 478

a)

b)

Fig. 4. Dynamic behaviour of the extensible flexible link with a quick stop and angular velocity ω1 ; a) Ox, and b) Oy, trajectories with (black/solid) and without (lighter/dashed) clearance for the active end of the single link manipulator

Dupac, M., Noroozi, S.

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0.01 m from the origin of the Cartesian reference frame, are described by a transverse diameter of 0.08 m, a conjugate diameter of 0.06 m. The semi-circle with the origin located on the Ox axis at 0.01 m from the origin of the Cartesian reference has a radius of 0.03 m.

The effect of friction has been neglected in this study, and the contacts/impacts of the flexible link with the crank are considered to be frictionless, that is, the friction coefficient between the parts was neglected.

The dynamic behaviour of the constrained single link manipulator rotating with the angular velocity ω1, with no clearance and quick stop is shown in Fig. 4a, that is, the Ox and Oy trajectories of the active end of the flexible link are plotted vs. the crank angle.

From Fig. 4a one can observe the perturbation the system due to the quick stop, as well as the time frame in which the system damped the perturbation.

Since a perturbation of the trajectory is an important factor affecting system dynamics, the behaviour of the extensible flexible link for a clearance vs. a nonclearance model have been considered.

The dynamic behaviour of the constrained single link manipulator, with clearance and quick stops is shown in Fig. 4b, that is, the Ox and Oy trajectories of the active end of the flexible link plotted vs. the crank angle. It was observed that the quick stop added more excitation to the end of the flexible link while impacts due to clearance act against the quick stop excitation and about the beam as a whole as shown in Figs. 4a and b.

Since an important design requirement of a rotating manipulator is the minimisation of the link dynamic deflection during high-speed motion, the effect of a higher angular velocity for the same clearance vs. a non-clearance model was also considered. The dynamic behaviour of the constrained single link manipulator with no clearance and angular velocity ω1 is shown in Fig. 5a.

A higher perturbation of the system trajectory can be observed for the angular velocity ω2 in Fig. 5a vs. the angular velocity ω1 in Fig. 4a. The dynamical behaviour of the single link manipulator with clearance and angular velocity ω2 is shown in Fig. 5b. A similar behaviour, that is, a higher trajectory perturbation, can be observed for the angular velocity ω2 in Fig. 5b vs. the angular velocity ω1 in Fig. 4b.

It was observed that more excitation is added to the system due to the quick stop against the impacts due to clearance acting as shown in Figs. 5a and 5b. It was also observed that the quick stops performed against the single link manipulator amplify the dynamic forces and flexible link vibration amplitude (against

the decrease in the flexible link natural frequency with length increase) as shown in Figs. 5 and 6.

a)

b) Fig. 5. Dynamic behaviour of the extensible flexible link with a quick stop and angular velocity ω2 ; a) Ox, and b) Oy, trajectories with(black/solid) and without (lighter/dashed) clearance for the active end of the single link manipulator

However, the decrease in the flexible link length due to the motion along the constrained circularelliptic trajectory increases the stiffness of the link, which was confirmed in a relatively similar study [37] regarding the behaviour of a flexible robot manipulator with a rotating-prismatic joint. Only the first mode of vibration have been observed for the vibrating flexible link with no clearance, while for the clearance model, the flexible link exhibits both the first and the second mode of vibration.

The displacement of the active end of the flexible manipulator with clearance along the active trajectory (Figs. 6a and b), immediately after the quick stop and for both angular velocities ω1 and ω2, that is, the interactions of each vibration mode, is shown in Figs.

7a and b, respectively.

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It can be seen that the amplitude of vibration suddenly increases immediately after the quick stops, due to the increase in the dynamic force. The two types of behaviour (Figs. 7a and b respectively) are nearly identical and the small differences are mainly due to the different angular velocities, which slightly change the dynamic force acting on the flexible link.

For the mechanism simulations presented here a nine rigid rods approximation has been used, that is, nine rigid links connected with torsional springs have been used to simulate the elastic rod.

a)

a)

b) Fig. 6. Trajectories of the active end flexible manipulator with clearance for; a) angular velocity ω1, and b) angular velocity ω2

b) Fig. 8. Dynamic behaviour of the active end flexible manipulator with clearance along the active trajectory immediately after the quick stop, Oy trajectories; a) without; and b) with clearance for 2 (gray/dotted), 4 (black) and 9 (white/dashed) links simulation

a)

b)

Fig. 7. Dynamic behaviour of the active end of the flexible manipulator with clearance along the active trajectory immediately after the quick stop, a) active vibrations for angular velocity ω1, and b) active vibrations for angular velocity ω2

480

To validate the obtained results, the effect of the number of elements on the dynamic response of the system was considered. Simulations involving 2 and 4 roads performed and compared to the nine rod simulation present visible differences between the obtained trajectories as shown in Fig 8. Simulations involving 8 and 11 rods have been performed too and no perceptible difference was found with respect to

Dupac, M., Noroozi, S.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 475-482

the dynamic behaviour of the 9 rod simulation. Thus n = 9 was considered adequate to accurately describe the mechanism dynamics.

One can conclude that the time evolution of the system is mainly affected by the quick stops and angular velocity, factors which accentuate trajectory divergence and affect system stability.

4 CONCLUSIONS In this paper the modelling and dynamic response of a constrained single link flexible rotating manipulator subject to quick stops was investigated. One end of the flexible manipulator was constrained to a predefined trajectory for possible trajectory control and obstacle avoidance. The effect of link flexibility, axial shortening and clearance has been considered in the analysis of the rotating flexible manipulator.

It was observed that the time evolution of the system was mainly affected by the quick stops and angular velocity. At the time of the quick stop (which represents the main impacts of the system) the flexible link makes successive shocks, which increase the amplitude of vibration. Further experimental tests will be performed in order to validate and generalize the simulations reported in this study.

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Dupac, M., Noroozi, S.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 483-494 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1614 Original Scientific Paper

Received for review: 2013-12-18 Received revised form: 2014-03-06 Accepted for publication: 2014-03-11

Flow Image Velocimetry Method Based on Advection-Diffusion Equation

Bizjan, B. – Orbanić, A. – Širok, B. – Bajcar, T. – Novak, L. – Kovač; B. Benjamin Bizjan1,2,* – Alen Orbanić2,3 – Brane Širok1 – Tom Bajcar1 – Lovrenc Novak1 – Boštjan Kovač2,3 1 University

of Ljubljana, Faculty of Mechanical Engineering, Slovenia d.o.o. – Research and Development, Slovenia 3 University of Ljubljana, Faculty of Mathematics and Physics, Slovenia 2 Abelium

This paper presents a non-contact method for velocity field calculation from a series of fluid flow images with illuminated planar layer of the flow and a mixed-in pollutant for flow visualization. The velocity field is calculated using a model similar to optical flow that is based on the advection-diffusion equation. The model is evaluated using a set of synthetic airfoil flow visualization images generated by a combination of computational fluid dynamics and inverse advection-diffusion equation approaches. The calculated velocity fields are in a good agreement with the true velocity fields.

Keywords: computer-aided visualization, image velocimetry, dense flow field calculation, optical flow, advection-diffusion equation, computational fluid dynamics

0 INTRODUCTION Flow visualization is a method for observation and monitoring of fluid flow structures in which we make flow structures visible by adding a pollutant such as a dye or a smoke to the fluid, unless the structures are already visible by themselves. Using high-speed cameras one is able to collect a large amount of spatial flow structure data in a very short time. Image data is further processed to determine velocity fields and other flow properties.

The main advantage of flow visualization methods is their very low impact on the flow dynamics as they are mostly non-contact methods. This is especially advantageous in processes with harsh measurement conditions (hot or corrosive fluids, etc.) where the choice of conventional measurement methods is very limited. Širok et al. [1] used the visualization principle to measure melt mass flow in mineral wool production. Another field of application includes large hydraulic structures such as irrigation and flood management systems where local flow velocities are of interest and would not be practical to be measured with conventional flow metering devices or simulated adequately enough due to potential complex flow phenomena [2]. Recently, Novak et al. [3] and [4] used flow visualization to study flow phenomena in side weirs. Furthermore, visualization methods are also preferred for multiphase flow analysis, especially in biological and chemical engineering, where other methods are too intrusive or do not allow measurements on the micro scale. For example, Bajcar et al. [5] and [6] used visualization techniques to study sedimentation efficiency in circular settling tanks.

There are many principles of flow visualization and a number of flow properties that can be assessed.

In this paper however we will focus on flow velocity measurement in incompressible flows. Local fluid velocities manifest themselves indirectly through movement of the brightness patterns in the image, known as the optical flow.

Numerous different approaches for optical flow computation have evolved so far. Horn and Schunck [7] developed an early optical flow calculation method based on image brightness analysis and an assumption of smooth velocity variations using a quadratic penalization scheme. Through the years, optical flow computation methods have been improved significantly in both accuracy and robustness. For example, Black and Anandan [8] improved the penalization term to make the algorithm less sensitive to noise and occlusions. Optical flow algorithms have been expanded further by the introduction of a temporal in addition to a spatial smoothing term, which allowed for use on image sequences longer than two frames. Flow-driven spatiotemporal smoothing term was used successfully by Brox et al. [9] and Bruhn and Weickert [10]. A detailed overview of optical flow analysis methods and their performance was provided by Barron et al. [11] and, more recently,

by Zimmer et al. [12].

While the majority of the earlier algorithms were primarily intended to be used for analysis of quasirigid motion with low divergence and practically no diffusion, in recent years significant research has been carried out on the use of optical flow for fluid flow visualization. Corpetti et al. [13] analyzed fluid flows using a minimization-based motion estimator based on a second order div-curl regularization and

*Corr. Author’s Address: Abelium d.o.o., Kajuhova 90, 1000 Ljubljana, Slovenia, benjamin.bizjan@abelium.eu

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on optical intensity continuity equation rather than the assumption of intensity conservation. Bajcar et al.

[14] presented a fluid flow visualization method based on the advection-diffusion equation. This method was used on experimental fluid flows with added passive tracer (smoke). All fluid flow analysis methods we have listed so far are two-dimensional. Recently however, three-dimensional approaches have become increasingly popular. For instance, Regart et al. [15] extended the 2D method by Corpetti et al. [13] to be used on 3D image sequences.

The majority of the models discussed lack the dedicated software solutions that could be used more generally. The purpose of our work was to develop a robust, software-supported visualization method that could be used on a wide variety of real lab- and industrial environment phenomena with a reasonable accuracy.

This paper is organized as follows. In Section 1 a theoretical formulation of our approach to velocity field calculation is presented. Section 2 describes the setup for generating a synthetic image sequence to be used for the evaluation of our method. Section 3 presents the method evaluation procedure on the generated image set while Section 4 provides calculation results and optimal calculation settings are determined. In Section 5 we list our conclusions and indicate the possibilities for further developments of our method.

fluids, hence div(v) = 0, we can further simplify Eq. (1):

Note that in this form with the right side set to 0, the equation is identical to the one of the optical flow as proposed by Horn and Schunck [6]. Derivatives are discretized using the central difference method:

f ( x + ∆xi ) − f ( x − ∆xi ) ∂f ( x) ≈ , (3) ∂xi 2∆xi

f ( x + ∆xi ) − 2 f ( x) + f ( x − ∆xi ) ∂2 f ( x) ≈ . (4) 2 ∂xi ∆xi 2

Let vx(t, i, j) and vy(t, i, j) denote velocity components at pixel (i, j) of the fluid at time t. Time t is advanced in discrete steps, t = {1, 2, 3,…, R}/f, where R is the number of images in the image sequence and f the number of frames per second. The integers ∆t and ∆xy represent the time and space displacements in the number of images in the series and the number of pixels in an image, respectively. The pixel size in meters will be denoted by s.

1 THEORETICAL BACKGROUND Our method is intended for approximate velocity field calculation from an image sequence of an incompressible fluid flow and has been implemented in our software package ADMflow (http://admflow. net). The method is based on the assumption that the concentration N of the pollutant at time t is proportional to the image brightness. The pixel at position (i, j) for image A at time t is characterized by its brightness (grayscale level) A(t, i, j) ranging from 0 (black) to 255 (white).

Our system of equations for velocity field calculation is derived from the advection-diffusion equation:  ∂2 N ∂2 N ∂N ∂ ( Nvx ) ∂ ( Nv y ) + + = D 2 + 2 ∂t ∂x ∂y ∂y  ∂x

  . (1) 

In Eq. (1) D is the diffusivity coefficient, and vx and vy are the components of the velocity vector v = (vx, vy). Since we are dealing with incompressible 484

 ∂2 N ∂2 N  ∂N ∂N ∂N + vx + v y = D  2 + 2  . (2) ∂t ∂x ∂y ∂y   ∂x

A(t + ∆t , i, j ) − A(t − ∆t , i, j ) + 2 D ⋅ ∆t / f A(t , i + ∆xy, j ) − A(t , i − ∆xy, j ) + vx (t , i, j ) + 2D D ⋅ ∆xy ⋅ s A(t , i, j + ∆xy ) − A(t , i, j − ∆xy ) + v y (t , i, j ) = 2 D ⋅ ∆xy ⋅ s A(t , i + ∆xy, j ) − 2 A(t , i, j ) + A(t , i − ∆xy, j ) = + (∆xy ⋅ s ) 2 A(t , i, j + ∆xy ) − 2 A(t , i, j ) + A(t , i, j − ∆xy ) . (5) + (∆xy ⋅ s ) 2

Using Eq. (5) we obtain one linear equation and two unknowns, namely vx(t, i, j) and vy(t, i, j), for each particular image t at the particular pixel (i, j). To differentiate between the actual velocities and the calculated ones, we shall use as the unknowns in the equations ux(t, i, j) and uy(t, i, j), respectively, and u will denote the calculated velocity. Equations can be obtained for all t < R – 1 and all the pixels in the images which are not within the ∆xy margin from the image borders. Clearly we get more unknowns than the equations, which is not surprising as our method basically solves Eq. (1), but without boundary

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conditions. To increase the number of equations in the model we use two approaches which can yield additional equations and the solution of the system is then obtained using the least squares method.

In the first approach called temporal smoothing, each triplet of images (t–∆t, t, t+∆t) introduces a set of equations with unknowns ux(t, i, j) and uy(t, i, j).

Taking the series of triplets on images t–∆t–|l/2|, t–∆t–|l/2| +1, …, t+∆t+|l/2|–1, we yield l sets of equations where all ux(t+r, i, j) and uy(t+r, i, j) for r ∈ {−  2l  , −  2l  + 1,...,  2l  − 1} are replaced by ux(t,i,j) and uy(t,i,j), respectively. The number of image triplets l used in temporal smoothing will be given by the parameter IIE (images in equations).

The second approach focuses on the penalization of rapid changes in the velocity field and has a similar effect as the regularization functional in the Horn and Schunck algorithm [7]. For each velocity vector u = (ux(t, i, j), uy(t, i, j)), we add the linear equations of the form β∙(grad ux) = 0 and β∙(grad uy) = 0 where we use Eq. (3) for the discretization with the displacement of 1 pixel (note: this may be different from the ∆xy used in discretization). The coefficient β ≥ 0 determines the amount of penalization of the spatial changes of the velocity field in the image - the larger the value, the more the change is penalized.

This approach is called spatial smoothing.

Similarly to the velocity gradient penalization, other flow field properties can be penalized as well. Flow divergence is regulated by equations γ∙(grad ux) = 0 and γ∙(grad uy) = 0 and flow curl by eqs. δ∙(curl ux) = 0, δ∙(curl uy) = 0, θ∙(grad(curl ux)) = 0 and θ∙(grad(curl uy)) = 0.

In an ideal case, two sequential images would be infinitesimally separated in the time domain and the spatial changes in the image would also be infinitesimal. To achieve as fine a discretization as possible, ∆t and ∆xy would therefore have to be 1.

However, when the spatial movements between two sequential images are large, this choice does not yield adequate results. Improved results are achieved by downsampling images by a coefficient k, which first smooths the image by filters and then takes every kth pixel in horizontal and vertical direction.

A direct downsampling would yield a much sparser velocity field, resulting in k2 times fewer vectors.

In the proposed model, we carry out downsampling implicitly, by first smoothing the image A by box filter and then, instead of the direct downsampling, we use ∆xy = k, thus still obtaining dense velocity fields and better results with respect to the ground truth, i.e. the real velocity fields. We use box filters with integer

parameter SWS (smoothing window size), where for each image pixel the average of the pixels in the square of size 2∙SWS+1 centered in the pixel is calculated.

The SWS parameter regulates image gray level smoothing and is not to be confused with β, which smoothes (penalizes) velocity gradients. Optimal ∆xy and SWS settings depend on the average flow feature displacement between consecutive images. If ζx and ζy are the horizontal and vertical displacement components, respectively, then we assume that the relevant displacement to be taken in consideration is the maximum value of both, ζ = max(ζx, ζy). We intend to find the optimal ratio for ∆xy/ζ and SWS/ζ.

2 SYNTHETIC IMAGE GENERATION For the purpose of our velocimetry method evaluation, we numerically generated a synthetic image sequence of smoke-traced air flow over a scaled-down NACA4421 airfoil in a low-speed wind tunnel. Due to low speeds, air flow can be treated as incompressible, allowing us to use our velocimetry method without modifications. We previously used experimentally obtained wind tunnel visualization images with hotwire-measured velocities to test the performance of our method. While such experimental images are a typical example of an industrial application, there is one significant drawback, namely the fact that the true velocity fields (the ground truth) are unknown as they can never be measured with total precision regardless of the choice of measurement method.

To overcome this limitation and thus ensure that the deviation of our method’s results from the ground truth is entirely a consequence of the method calculation error, a synthetic image sequence was produced. First, the airfoil was modeled in 3d modeling software, then meshed and imported into the Ansys Fluent computational fluid dynamics (CFD) software. A simulation of air flow over the airfoil was performed and the obtained velocity fields were exported. These velocity fields were then used in our software ADEsolver where smoke flow was visualized using an inverse advection-diffusion equation approach.

2.1 CFD Simulation The geometry of the numerical model was made to represent the actual wind tunnel where experiments had previously been conducted. Numerical simulations were performed for the airfoil NACA4421 with a chord of length 30 mm and oriented at a 3° angle of attack. The three-dimensional computational

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domain (Fig. 1) covered the entire height of the wind tunnel (100 mm) in the direction normal to the airfoil (y coordinate) and had a length of 200 mm in the streamwise direction (x coordinate) with the airfoil leading edge at x = 50 mm.

Fig. 1. Computational grid for the numerical simulation in Fluent

Top and bottom of the domain were defined as no slip walls since they represented the actual wind tunnel walls. Periodic boundary conditions were imposed on the spanwise boundaries. Inlet velocity profile was set accordingly to the experimentally measured data, obtained with a hot-wire anemometer at 18 equally spaced locations across the channel height (average inlet velocity was 4.945 m/s). Turbulence intensity of 1.5% was computed from the hotwire measurements and set as inlet turbulence, while the inlet turbulent length scale of 5 mm was estimated based on the upstream flow straightener geometry. Outlet was defined as a pressure outlet with constant relative pressure of 0 Pa. A time step of 4.11∙10–5 s was chosen taking in consideration both CFD calculation stability (namely, the CFL number which ranged 0.3 and 1.8 in the wake region, close to the recommended value of unity) and the flow feature displacement between images (ζ) which ranged between 1 and 5 pixels, assuring a good performance of our velocimetry method in all flow regions.

A combination of two turbulence models was used in the simulation. Initial conditions were provided by the steady state solution using the Shear stress transport (SST) model with the low Reynolds number correction enabled. Then, a transient simulation was run using the SST-SAS (SST-Scaleadaptive simulation) model. Both models were used with the default coefficients as set in Ansys Fluent 13 [16]. The SST model is a two-equation eddy viscosity turbulence model with improved prediction of separation compared to earlier models such as the k-ε model, and is one of the most frequently used 486

RANS (Reynolds-averaged Navier-Stokes) models for aerodynamics simulations [17].

While the SST model provides a good agreement with measured data for mean flow at reasonable computational expense, it fails to give sufficient information on turbulent structures. For this purpose, there are several different approaches that can yield more detailed results. A good compromise between calculation time and accuracy is to use one of the hybrid RANS-LES models such as the scale-adaptive simulation (SAS). The SAS model was developed by Menter and Egorov [18] and is in fact an improved unsteady-RANS (URANS) model with LES capability in unstable flow regions. The SAS model is based on introduction of the von Karman length scale into the turbulent length scale equation of a two-equation turbulence model, allowing for local detection of unsteadiness and automatic balancing between contributions of modeled and resolved turbulence stresses.

Air flow velocity fields were exported for a zoomed-in view defined by a planar cross-section of the computational domain in the streamwise direction (spanwise coordinate z = –0.05 m) Fig. 2.

A sequence of 100 consecutive grayscale contour plots sized 960×720 pixels were exported for both x and y velocity and the physical size of one pixel was s = 45.5 μm. These velocity contour plots were later imported into the ADEsolver software as the ground truth data (vx, vy) for flow tracer (smoke) visualization.

Fig. 2. An example of CFD-calculated x- (upper image) and y- (lower image) velocity contour plots around the airfoil; colormap range (black to white color) is [–3, 8] m/s for vx and [–6, 5] m/s for vy

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2.2 Inverse Advection-Diffusion Equation for Pollutant Simulation In this step, the x and y velocity fields calculated in Fluent were imported into our ADEsolver software along with the pollutant (smoke) sources defined as matrices of pollutant concentration (gray level) A with a range from 0 (zero source intensity) to 255 (maximum source intensity) in integer increments.

There was one source matrix for every time step with three sources (nonzero concentration areas) placed at different locations near the airfoil upper and lower surfaces. To ensure that the advection-diffusion equation based velocimetry method that was to be tested on generated synthetic images could function properly, gray level spatial and temporal gradients had to be sufficiently large. For this reason, the source intensity AS was not constant, but varied between AS,min = 100 and AS,max = 255, accordingly to Eq. (6) for all three sources.

AS (n) = AS ,min +

  2π 1 AS ,max − AS ,min ) ⋅ 1 + sin  (  2  t0 

 n   .(6)  

In Eq. (6), t0 is the sine wave period given in number of frames (images) between consequent concentration maxima (t0 = 20 was used). The independent variable n is the discrete time which is equal to the number of the current frame (n = 1, 2, 3, ..., R).

Once all the vx, vy and AS data were imported in ADEsolver, the pollutant concentration A was calculated for every pixel in the calculation area (960×720 pixels, the same as velocity calculation area in Fluent) using the simplified advection-diffusion equation given by Eq. (2). This differential equation was solved iteratively using the iterative equation (Eq. (7)), also known as the Jacobi iteration. With this method, the calculation using Eq. (7) is repeated several times using the results from the previous iteration until the solution has converged sufficiently.

In each iteration, the current time t is advanced by the time step dt and the new pollutant concentration is calculated for every pixel in the calculation domain.

The time step dt is typically much smaller than the time between two consecutive frames (δt = 1 / f ) in order to ensure stable convergence and the total number of iterations before the next frame is δt/dt.

A(t + dt , i, j ) = A(t , i, j ) + dt ⋅ D ⋅ [ A(t , i + 1, j ) + + A(t , i, j + 1) + A(t , i − 1, j ) + A(t , i, j − 1) −

−4 A(t , i, j )] / s 2 − adx(t , i, j ) − ady (t , i, j ).

(7)

Diffusivity D was estimated to 10–6 m2/s by visual comparison of simulated pollutant flow to the smoke flow from the actual experiment, meaning advection rather than diffusion was the predominant mechanism of pollutant spreading. The terms adx and ady represent the pollutant concentration change due to convection in x and y direction, respectively, and are defined using the upwind advection scheme [19] – Eqs. (8) and (9). adx(t , i, j; vx > 0) = vx (nδ t , i, j ) ⋅ dt ⋅ ( A(t , i, j ) − − A(t , i − 1, j )) / s, adx(t , i, j; vx < 0) = vx (nδ t , i, j ) ⋅ dt ⋅ ( A(t , i + 1, j ) − − A(t , i, j )) / s.

(8)

ady (t , i, j; v y > 0) = v y (nδ t , i, j ) ⋅ dt ⋅ ( A(t , i, j ) − − A(t , i, j + 1)) / s, ady (t , i, j; v y < 0) = v y (nδ t , i, j ) ⋅ dt ⋅ ( A(t , i, j ) − − A(t , i, j − 1)) / s.

(9)

Pollutant concentration field images were generated using the following procedure. First, we ran 200 time steps δt of simulation for constant ground truth vx and vy (the first from the 100 frame sequence acquired in Fluent) to obtain a steady state solution.

Then, a simulation of another 100 time steps in length was calculated using variable ground truth fields, with nth vx and vy fields being used for the calculation of nth time step pollutant concentration. For each time step, the pollutant concentration field was saved to an image of the same size as the Fluent velocity field images (960×720 pixels) to represent a synthetic flow visualization image that was to be used in ADMflow.

A section of a typical image is shown in Fig. 3.

3 METHOD EVALUATION PROCEDURE For evaluation of our velocimetry method, a sequence of 10 images was selected from the 100-image synthetic flow visualization sequence generated in ADEsolver, along with the corresponding ground truth velocity fields calculated in Fluent. The main selection criterion was to attain a good match in position of turbulent structures between Fluent and ADEsolver at the beginning of the chosen sequence and to find as many characteristic flow regions as possible. We identified three main regions of interest (Fig. 3). Region 1 was placed just behind the airfoil’s trailing edge where a small, slowly moving vortex was observed. In region 2, vortex shedding, namely the von Karmann vortex street occurred as a result of a low Reynolds number and the shape of the airfoil

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and manifested itself as an oscillating texture pattern.

Region 3 represents the laminar flow over the airfoil above the boundary layer zone where the highest velocities occurred. Windows shown by white squares were used as testing areas for our velocimetry method implemented in ADMflow. The sizing of the windows 1, 2 and 3 was 50×50, 50×50 and 40×40 pixels, respectively.

ea (t , i, j ) = ea , x (t , i, j ) 2 + ea , y (t , i, j ) 2 . (11)

To make a comparison between the results obtained in different flow regions easier, local relative error erl is introduced as a ratio between the absolute error magnitude and the average wind tunnel inlet velocity (V = 4.945 m/s): erl (t , i, j ) =

ea (t , i, j ) . (12) V

Spatial relative error average of er(t, i, j) over the calculation domain Ω and at time t is given as: Fig. 3. A section of a synthetic flow visualization image with the three distinctive flow regions and corresponding windows where our velocity calculation method was tested

Using the synthetic flow visualization sequence, velocity fields (ux and uy) were calculated in ADMflow in windows 1, 2 and 3 and compared to the actual velocity fields (ground truth) of the set, using the [m/s] unit for both. In order to evaluate our method’s accuracy, calculation errors are defined by equations (10) to (16) and an overview of the evaluation methodology is given in Fig. 4.

ers (t ) =

1 ∑ | erl (t , i, j ) |. (13) n (i , j )∈Ω

In Eq. (14), n is the number of points in the calculation domain Ω. Finally, we define the global relative error erg as a spatiotemporal average of the local relative error: erg =

1 n ⋅T

T =R/ f

∑ ∑

t =1/ f

( i , j )∈Ω

| erl (t , i, j ) | =

1 T =R/ f ∑ ers (t ). (14) T t =1/ f

Analogously to the velocity magnitude relative error erg, the x and y velocity relative errors ergx and ergy can be defined.

In addition to the velocity magnitude error, the velocity directional error is introduced as the angle φ between the calculated and true velocity vector (-π<φ<π), as given by Eq. (15) and (16). The local directional error at location (i, j) and time t is denoted by edl. edl (t , i, j;| a tan 2(u y , u x ) − a tan 2(v y , vx ) |< π ) = = a tan 2(u y , u x ) − a tan 2(v y , vx ), edl (t , i, j;| a tan 2(u y , u x ) − a tan 2(v y , vx ) |> π ) = = a tan 2(u y , u x ) − a tan 2(v y , vx ) − 2π , edl (t , i, j;| a tan 2(u y , u x ) − a tan 2(v y , vx ) |< −π ) =

Fig. 4. Flow chart of our velocimetry method evaluation procedure

The velocity field calculation error vector is introduced as the difference between calculated and true velocity fields, u and v, respectively. Local absolute error vector and magnitude at the pixel location (i, j) and time t are given by Eq. (10) and (11), respectively: e a (t , i, j ) = (ea , x (t , i, j ), ea , y (t , i, j )) =

488

= (u x (t , i, j ) − vx (t , i, j ), u y (t , i, j ) − v y (t , i, j )), (10)

= a tan 2(u y , u x ) − a tan 2(v y , vx ) + 2π .

(15)

In Eq. (15), atan2(y,x) is the four-quadrant inverse tangent defined by Eq. (16): arctan( y / x); x > 0 arctan( y / x) + π ; y ≥ 0, x < 0  arctan( y / x) − π ; y < 0, x < 0 . (16) a tan 2( y, x) = 

π / 2; y > 0, x = 0 −π / 2; y < 0, x = 0  undefined; y = 0, x = 0

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The spatial and global (spatiotemporal) averages of directional error (eds and edg, respectively) are defined analogously to the magnitude error definitions in Eq. (13) and (14).

4 RESULTS AND DISCUSSIONS In ADMflow, velocity fields in windows 1 to 3 were calculated for a number of different calculation parameter settings and compared to the ground truth fields using the magnitude and directional error definitions from the previous section. For every window, the optimal combination of calculation parameters was determined as the one where the calculation error was the lowest (Table 1).

Table 1. Optimal parameter settings for velocity calculation windows and the corresponding calculation errors win.

1 2 3

β

γ

θ

.002 .004 10-4

.01 .01 .03

10-6 10-6 10-7

Δxy 2 2 3

SWS 2 3 3

IIE 2 4 8

erg [%] 13.8 12.1 4.1

edg [°] 15.2 5.8 0.5

Other calculation parameters were identical for all the regions of interest presented in Table 1.

Diffusivity was set to the same value as in ADEsolver for pollutant simulation (D = 10–6). The calculation time step was set to ∆t = 1 as larger values resulted in excessive calculation errors. In Fig. 5, the velocity fields calculated by optimal settings given in Table 1 are compared to the true velocity fields of the image set.

Fig. 5. Comparison between calculated and true velocity fields for the first frame of the selected 10-image sequence

From Fig. 5, a good agreement between the calculated and true velocity fields can be observed

in terms of velocity magnitude as well as flow direction. In window 1, our calculation method was able to detect the vortex flow structure well, while at the bottom of window 3, a velocity drop near the boundary layer separation zone was also correctly identified. As we can see from Table 1, the largest velocity calculation error (about 14% in velocity magnitude and 13° in velocity direction) occurs in window 1. This can be largely attributed to a shift in the detected position of the vortex center (lowest velocity area) of the otherwise well reproduced vortex shape. The shift most likely occurs due to the fact that calculation parameters could only be optimized for window 1 as a whole and not separately for different parts of the vortex. Optimal calculation parameter values are largely dependent on the flow velocity magnitude, which is in fact much larger at the edge of the observed vortex than at its center as the vortex translatory motion is very slow and rotational motion is predominant.

In window 2, which is located further downstream in the von Karmann vortex street zone, the calculation error is already smaller (12% in velocity magnitude and 6° in velocity direction), especially the directional error. The error here largely depends on the flow visualization quality and may rise significantly if the pollutant concentration gradient becomes too low, especially if the concentration drops to near zero values.

The lowest calculation error was observed in window 3 both in terms of velocity magnitude (4%) and especially in the flow direction (only 0.5°). The flow in the window is almost steady, but periodic oscillations in pollutant concentration allow the advection-diffusion algorithm to perform properly.

This is true even for the area near the boundary layer separation zone where flow velocity is much lower but still accurately calculated.

Now, let us assess the effect of individual calculation parameters. In Fig. 6a and b, velocity magnitude and directional errors are shown as a function of parameters IIE and β. Other calculation parameters were set to the values given in Table 1.

From Fig. 6a and b it can be observed that the effect of temporal smoothing (parameter IIE) is predominant while for IIE ≥ 2, temporal smoothing (parameter β) has a relatively low impact on the calculation error. For windows 1 and 2, the lowest velocity magnitude and directional errors occur at IIE = 2 to 3 and β = 0.001 to 0.004 and for window 3 at IIE = 5 to 8 and β = 0 to 3∙10-4. Due to the higher error sensitivity to β at IIE = 1, only values of IIE ≥ 2 should be used. On the other hand, using too high IIE

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a)

b) Fig. 6. Effect of parameters IIE and β on a) velocity magnitude error; and b) velocity directional error

values for laminar flows such as the flow in window 3 is not advisable as the instantaneous velocity information is lost due to excessive time averaging and a good choice would therefore be IIE ≈ 5. While the calculation error at IIE = 8 is slightly lower than at IIE = 5, the difference is marginal (0.5% for velocity magnitude and 0.1° for direction, respectively).

Regarding parameter β, a nonzero setting is advisable as some flow image sets may produce excessive local errors for β = 0 but perform well above 10-3 to 10-2, with smooth, almost steady flows demanding lower β than the more unsteady flows (e.g. vortex shedding). The minimum required β also depends on the image signal-to-noise ratio higher the ratio, lower the needed β. Signal-to-noise ratio (SNR) in our synthetic image set is high due to the methodology used to visualize smoke - image gray level is defined by the calculated pollutant concentration, while for real experiments, images are 490

recorded with a camera and a higher level of noise may be present due to the non-ideal lighting, lenses and smoke generation. Regardless of the SNR, values of β >> 0.01 may cause the spatial smoothing effect to become too pronounced, filtering out not only the noise, but also the relevant flow structures.

In addition to IIE and β, another two very important parameters are ∆xy and the smoothing window size (SWS). The impact of these two parameters on velocity calculation error is shown in Figs. 7a and 7b.

From Figs. 7a and b we can see that the effect of smoothing window size (parameter SWS) is predominant while, for SWS ≥ 2, downsampling (parameter ∆xy) has a relatively low impact on the calculation error. For windows 1 and 2, the lowest velocity magnitude and directional errors occur at SWS and ∆xy in range of approximately 2 to 4. For window 3, the optimal parameter range is similar with

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a)

b) Fig. 7. Effect of parameters ∆xy and SWS on a) velocity magnitude error; and b) velocity directional error

the exception that the velocity directional error is large for SWS < 3 and becomes very low and practically independent of the smoothing window size for SWS ≥ 3. Due to the higher error sensitivity to SWS at IIE = 1 and vice versa, only values of IIE ≥ 2 and SWS ≥ 2 should be used (regardless of displacement ζ). On the other hand, if the value of these two parameters is too high, calculated velocity fields become scaled up in value, exceeding true values and causing the calculation error to increase again.

Similar to the coefficient β, SWS may improve the signal-to-noise noise ratio of the images, but with a different approach. β penalizes excessive local velocity gradients whereas SWS smoothes the gray level (and its gradient) by non-weighed arithmetic averaging, both methods effectively acting as low-pass filters [20]. The coefficient β should be set accordingly to the velocity gradient magnitude for proper penalization, while the primary criterion for choosing SWS

should be the flow feature displacement ζ to ensure downsampling (with factor ∆xy) is performed on a smooth enough pollutant concentration field.

To determine a more general rule for the selection of ∆xy and SWS, an optimal range of these two parameters can be analyzed as a function of ζ. In Table 2, the range of ∆xy and SWS that gives best calculation results is given for the existing image sequence as well as for the sequence where every second image is taken for calculation (subscript 2ζ), increasing ζ two-fold.

From Table 2 we can see that the optimal ∆xy and SWS range is proportional to the displacement ζ. An increase in ζ caused by increased flow velocity (e.g. window 3 compared to windows 1 and 2) or reduced frame rate (e.g. image set with one half of the original sampling rate, subscript 2 ζ ) results in the need for higher ∆xy and SWS settings. As seen in Table 1, frame rate reduction causes an increase in calculation errors, especially when ζ > 5. Calculations on image sets with

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ζ > 10 should generally be avoided to prevent issues due to the lack of temporal information.

∆xy =

ζ + 2, (17) 2

Table 2. Optimal ∆xy and SWS range depending on flow feature displacement ζ. Minimum velocity calculation errors are also listed

SWS =

ζ + 2. (18) 4

win.

ζ

∆xy

SWS

erg [%]

edg [°]

1

1.12

2 to 3

2 to 3

13.8

15.2

12ζ

2.24

3 to 5

2 to 6

13.1

19.3

2

3.30

2 to 4

2 to 5

12.1

5.8

22ζ

6.60

5 to 7

3 to 7

14.3

7.7

3

5.01

3 to 5

3 to 4

4.1

0.5

32ζ

10.0

3 to 8

3 to 5

6.8

0.5

In Fig. 8., the optimal ∆xy and SWS parameter range is shown bounded with solid lines according to the values from Table 2.

In addition to the calculation parameters studied so far, the effect of coefficients γ and θ was investigated as well. Coefficient γ (Fig. 9) regulates flow velocity divergence and reduces calculation error when set between 0.01 and 0.1. Velocity magnitude error is reduced by about 10%, in terms of the error’s own magnitude, for window 1 and as much as 40% for window 3 when compared to calculations with γ = 0. Directional error is only affected by γ in window 1 (2° reduction), while in windows 2 and 3 the error reduction is negligible. However, as γ exceeds 0.1, velocity magnitude and directional error start to grow very rapidly, making calculation less accurate than for γ = 0.

Fig. 9. Effect of γ on velocity magnitude error (upper panel) and directional error (lower panel)

Fig. 8. Parameters ∆xy (upper panel) and SWS (lower panel) as a function of displacement ζ

We propose a simple linear model for optimal ∆xy and SWS values (dashed line in Fig. 8) – Eqs. (17) and (18). The model predicts values within the optimal parameter range and fulfills the requirement IIE, SWS ≥ 2.

492

Coefficient θ (Fig. 10) regulates gradient of flow velocity curl and reduces calculation error when set between approximately 10–7 and 10–6. Velocity magnitude error is reduced by almost 15%, in terms of the error’s own magnitude, for window 1 and up to 30% for window 3 when compared to calculations with θ = 0. Directional error exhibits a similar dependence on θ, dropping to almost zero for window three in the optimal θ range. As with coefficient γ, calculation errors increase drastically if θ is raised above its optimal range.

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Fig. 10. Effect of θ on velocity magnitude error (above) and directional error (below)

In the end, we can summarize our findings regarding the calculation parameters of our velocimetry method into some recommendations: • ∆xy = ζ/2+2 (rounded to nearest integer), • SWS = ζ/4+2 (rounded to nearest integer), • ∆t = 1, • IIE = 2 to 5 (lower for more unsteady flows), • β = 10–4 to 10–2 (higher for more unsteady flows), • γ = 0.01 to 0.1, • θ = 10–7 to 10–6.

Of course, these recommendations are quite general and may not be optimal for all real-life flow problems. One is advised to consider the suggested parameter values as initial settings and then critically asses the results (velocity fields) by comparing them to the flow characteristics knowledgebase of the process determined by other methods (theoretical, experimental and/or numerical), if available.

5 CONCLUSIONS A non-contact, computer-aided velocimetry method was developed for the quantification of flow kinematic properties. The method takes recorded grayscale images of an observed process as input and calculates two-dimensional flow velocity fields. The calculation

algorithm is based on the advection diffusion equation, which couples the velocity field with the concentration field, and the image downsampling process, which compensates for the feature displacement between consecutive frames. The method was evaluated on a synthetic image set of airfoil flow visualization in a wind tunnel generated by a combination of CFD software and inverse advection-diffusion equation solver. This way, we simulated a quite complex flow with several characteristic flow types and a known ground truth.

Our velocimetry method was evaluated by comparison of the calculated velocity fields to their ground truth counterparts in three different regions. Multiple calculation parameters were varied, calculating the velocity field error for each set.

Calculation errors proved to be reasonably small for the optimal parameter settings – between 4% and 14% of mean inlet velocity for velocity magnitude and 0.5 to 15° for velocity direction. It is important to note that a lower error was attained in the higher flow velocity (thus more critical) regions above the airfoil and in the von Karmann vortex street behind it, compared to a higher error on the slowly moving vortex just behind the airfoil. Furthermore, visual comparison of the calculated and true velocity fields showed a good agreement between both, especially in terms of flow structures that were well preserved by our algorithm. In addition, our method proved to be quite robust in terms of the calculation parameter range that provides good results, although caution should be taken not to use excessively high parameter values, especially with penalization coefficients, as the error may surge. Based on the parameter variation results, some general conclusions regarding optimal parameter settings were provided, including a model for the downsampling coefficient and the smoothing window size.

While the conventional measurement methods may still be more accurate, they face other limitations that are overcome by our method. For example, the PIV method requires an expensive and rigid experimental setup and would perform poorly on smoke-type visualization. On the other hand, point velocimetry methods such as hotwire anemometry (HWA) and laser Doppler anemometry (LDV) are time consuming and only give average flow velocity fields. Therefore, the potential range of our method’s applications is much wider, including, but not limited to, the visualization and control of many industrial processes for which other velocimetry methods are not well suited. Further development of our velocimetry method should include numerical

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calculation algorithm refinements to improve the accuracy and introduce a certain level of automation in choosing the proper calculation parameters. The use of additional synthetic and real image sets for various flow types could also allow for a statistically verified knowledgebase of the method’s optimal employment.

At some point, an expansion of our method from 2D to 3D may be considered as an alternative to the other 3D velocimetry methods. Of course, development of the method must also be followed by improvements in flow visualization techniques, namely in a variable pollutant concentration to ensure optimal utilization of the method for areas of more stationary and laminar flows.

6 ACKNOWLEDGMENTS This work was supported in part by the Slovenian Research Agency (ARRS), grants P2-0167, L2-4270.

Operations were also financed in part by the European Union, European Social Fund and the Ministry of Economic Development and Technology, Republic of Slovenia, project KROP 2011 at Abelium d.o.o.

7 REFERENCES [1] Širok, B., Bajcar, T., Orbanić, A., Eberlinc, M. (2011). Melt mass flow measurement in mineral wool production. Glass Technology, vol. 52, no. 5, p. 161168.

[2] Tic, V., Lovrec D. (2012). Design of modern hydraulic tank using fluid flow simulation. International Journal of Simulation Modelling, vol. 11, no. 2, p. 77-88, DOI:10.2507/IJSIMM11(2)2.202.

[3] Novak, G., Kozelj, D., Steinman, F., Bajcar, T. (2013).

Study of flow at side weir in narrow flume using visualization techniques. Flow Measurement and Instrumentation, vol. 49, p. 45-51, DOI:10.1016/j. flowmeasinst.2012.10.008.

[4] Novak, G., Steinman, F., Müller, M., Bajcar, T. (2012). Study of velocity field at model sideweir using visualization method. Journal of Hydraulic Research, vol. 50, no. 1, p. 129-133, DOI:10.1080/00221686.20 11.648766.

[5] Bajcar, T., Gosar, L., Širok, B., Steinman, F., Rak, G. (2010). Influence of flow field on sedimentation efficiency in a circular settling tank with peripheral inflow and central effluent. Chemical Engineering and Processing: Process Intensification, vol. 49, no. 5, p.

514-522, DOI:10.1016/j.cep.2010.03.019.

[6] Bajcar, T., Steinman, F., Širok, B., Prešeren, T. (2011).

Sedimentation efficiency of two continuously operating circular settling tanks with different inlet- and outlet arrangements. Chemical Engineering Journal, vol. 178, p. 217-224, DOI:10.1016/j.cej.2011.10.054.

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[9] Brox, T., Bruhn, A., Papenberg, N., Weickert, J. (2004). High accuracy optical flow estimation based on a theory for warping. 8th European Conference on Computer Vision, Prague.

[10] Bruhn, A., Weickert, J. (2005). Towards ultimate motion estimation: combining highest accuracy with real-time performance. 10th International Conference on Computer Vision, Beijing.

[11] Barron, J.L., Fleet, D.J., Beauchemin, S. (1994).

Performance of optical flow techniques. International Journal of Computer Vision, vol. 12, no. 1, p. 43-77, DOI:10.1007/BF01420984.

[12] Zimmer, H., Bruhn, A., Joachim, W. (2011). Optic flow in harmony. International Journal of Computer Vision, vol. 93, no. 3, p. 368-388, DOI:10.1007/s11263-0110422-6.

[13] Corpetti, T., Heitz, D., Arroyo, G., Memin, E., SantaCruz, A. (2005). Fluid experimental flow estimation based on an optical-flow scheme. Experiments in Fluids, vol. 40, no. 1, p. 80-97, DOI:10.1007/s00348005-0048-y.

[14] Bajcar, T., Širok, B., Eberlinc, M. (2009). Quantification of flow kinematics using computer-aided visualization.

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[15] Regert, T., Tremblais, D., Laurent, D. (2010).

Parallelized 3D optical flow method for fluid mechanics applications. Paper presented at the 5th International Symposium on 3D Data Processing, Visualization and Transmission, Paris.

[16] ANSYS 13.0 Documentation (2010). Ansys, Pittsburgh.

[17] Sekavčnik, M., Mori, M., Novak, L., Smrekar, J., Tuma, M. (2008). Heat transfer evaluation method in complex rotating environments employing IR thermography and CFD. Experimental Heat Transfer, vol. 21, no. 2, p.

155-168, DOI:10.1080/08916150701815770.

[18] Menter, F.R., Egorov, Y. (2010). The Scale-Adaptive Simulation Method for Unsteady Turbulent Flow Predictions. Part 1: Theory and Model Description.

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113-138, DOI:10.1007/s10494-010-9264-5.

[19] Ngo, Q. A. (2010). Explicit one-step schemes for the advection equation: The upwind scheme. Retrieved on 19. 11. 2013 from http://anhngq.wordpress. com/2010/04/25/explicit-one-step-schemes-for-theadvection-equation-the-upwind-scheme/.

[20] Brüllmann, D. D., d’Hoedt, B. (2011). The modulation transfer function and signal-to-noise ratio of different digital filters: a technical approach. Dentomaxillofacial Radiology, vol. 40, p. 222-229, DOI:10.1259/ dmfr/33029984.

Bizjan, B. – Orbanić, A. – Širok, B. – Bajcar, T. – Novak, L. – Kovač; B.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 495-505 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1393 Original Scientific Paper

Received for review: 2013-08-20 Received revised form: 2013-12-23 Accepted for publication: 2014-01-24

Modeling Thermal Oxidation of Coal Mine Methane in a Non-Catalytic Reverse-Flow Reactor Qi, X. – Liu, Y. – Xu, H. – Liu, Z. – Liu, R. Xiaoni Qi1 – Yongqi Liu1,* – Hongqin Xu2 – Zeyan Liu1 – Ruixiang Liu1

1 Shandong

University of Technology, College of Traffic and Vehicle Engineering, China University of Technology, College of Mechanical Engineering, China

2 Shandong

Inspired by detailed designs of industrial porous burners, the combustion of methane–air mixtures in a non-catalytic reverse-flow reactor was studied numerically. The governing equations are the unsteady state equations of conservation of mass and chemical species, with separate energy equations for the solid and gas phases. These equations were solved using the commercial CFD code Fluent. In order to reveal the actual thermal oxidation in porous media, the user defined function (UDF) is used to extend the ability of FLUENT. The model has been used to investigate the effects of operating conditions such as the mixture inlet approach velocity (0.15 to 0.8 m/s) and methane concentration (0.3 to 0.7%) on the oxidation of methane within non-catalytic reactors packed with ceramic monolith blocks under adiabatic conditions. The calculated values of methane conversion showed good agreement with the corresponding available experimental data. Moreover temperature distribution characteristics in the oxidation bed were studied in order to maintain the autothermicity of TFRR with a high enough temperature in the hot zone.

Keywords: lean-methane, reverse flow reactor,thermal oxidation, modeling

0 INTRODUCTION Coal mine methane is not only a greenhouse gas but also a wasted energy resource if not utilised. As the air volume is large and the methane resource is dilute and variable in concentration and flow rate, ventilation air methane is the most difficult source of CH4 to use as an energy source. Because methane concentration in mine ventilation air is usually below 1 vol.% [1], the combustion of lean methane-air mixtures combined with the recovery of the heat of reaction is an important problem for the mining industry. Since the flowrates of this air from a single ventilation shaft usually exceed 500,000 m3/h, the methane gas should be somehow utilized rather than released into the atmosphere. One of the most reasonable options seems to be the combustion of CH4 in reverse-flow reactors with simultaneous heat recovery. A number of studies have been carried out to determine the best ways to utilize this lean methane via combustion and to recover the energy thus produced. Due to the very low CH4 concentrations the most promising solution seems to be autothermal combustion in reverse-flow reactors.

The combustion products of CH4 are CO2.

Although CO2 is also a greenhouse gas, its global warming potential is 21 times lower than that of CH4. So far, most studies have focused on catalytic combustion in catalytic flow-reversal reactors (CFRRs). The first attempt at ventilation air methane (VAM) combustion in CFRR was supposedly carried out at the Boreskov Institute of Catalysis [2]. At present CANMET in Canada is the most advanced in

terms of the development of the technology of lean air-methane mixtures combustion [3] and [4]. The principles of catalytic combustion and the general developments in the field have been described in a number of review articles [5] to [7]. Until now, many scholars [8] to [14] have studied the lean methane catalytic oxidation in a reverse flow reactor. Hayes [8] examined the combustion of methane on a palladium catalyst in a monolith reactor and determined the rate equation. Pablo M. [9] systematically compared the performance of particle beds and monolithic beds in catalytic reverse flow reactors. Shi [10] classified the existing technologies for coal mine methane mitigation and utilisation. Hayes [11] developed a comprehensive 2-D finite-element model for a single channel of a honeycomb type monolith catalytic reactor. Aubé [12] developed a mathematical model of CFRR that combines a transient twodimensional heterogeneous model with a numerical method allowing the fast formulation of new reactor configurations during the design phase. Tischer [13] developed transient two- and three-dimensional simulations of catalytic combustion monoliths. Veser [14] presented a one-dimensional two-phase reactor model for the oxidation of methane to synthesis gas over platinum in a monolith reactor. Shahamiri [15] developed a one-dimensional model to investigate the effects of operational conditions on CFRR. The authors in [16] studied the combustion of preheated lean mixtures of hydrogen with methane in a catalytic packed-bed reactor, it was demonstrated that the one-dimensional model could predict the effects of changes in operating conditions on the methane

*Corr. Author’s Address: Shandong University of Technology, College of Traffic and Vehicle Engineering, Zhangzhou Road 12, Zibo 255049, China, bmjw@163.com

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and hydrogen conversions, species concentrantions and gas temperature profiles along the bed. The simulations and experimental studies [17] performed show that CFRRs are characterised by high operating temperatures, leading to possible deactivation or even destruction of the relatively expensive catalyst.

Some investigations [18] and [19] carried out a comparative assessment of TFRR and CFRR. It has been shown that both solutions have advantages and drawbacks. Should the heat recovery be seriously taken into account, thermal flow-reversal reactors (TFRRs) are economically and technically the most advantageous solution. Therefore, non-catalytic oxidation in TFRRs is now frequently regarded as an attractive alternative [20]. Such reactors have long been used, e.g. for the homogeneous (thermal) combustion of volatile organic compounds (VOCs) [21]. However, obvious differences exist between the combustion of VOCs and the oxidation of ventilation air methane in reverse-flow reactors. In the former case, no heat is withdrawn from the system. Thermal combustion in a TFRR should be carried out under conditions that do not promote excessive formation of NOx , i.e. at temperatures below a maximum of 1300 °C in the reactor. The use of a TFRR for the combustion of lean methane mixtures therefore requires detailed studies to determine the reasonable operating conditions. The objective of the present study is to develop a model of reactive flow within a non-catalytic bed packed with ceramic monolith blocks, which include adequate heat and mass transfer models and consider both simultaneous gas phase and surface reactions. Such a model can be applied to investigate the effects of key operational parameters on thermal oxidation processes within the bed reactor, including the effect of fuel composition. It can also be used in combination with experimental data in deriving much needed data for thermal oxidation operations when employed in industry.

1 PHYSICAL AND MATHEMATICAL DESCRIPTION OF THE PROBLEM A schematic diagram of the honeycomb ceramic bed being considered in the simulation and experiment is shown in Fig. 1. The computational region, of which the length is 2100 mm, the section is 600×600 mm squared, which includes the porous ceramic block with a pore size is 2 to 3 mm. In a reverse flow reactor the feed is periodically switched between the two ends of the reactor using switching valves. When switching valves 1 and 4 are open, the feed flows to the reactor from left to right (forward flow), indicated by the 496

solid arrows. When switching valves 2 and 3 are open, the feed flows to the reactor from right to left (reverse flow), indicated by the dotted arrows. The total cycle consists of these two operations, and the term switch time denotes the time elapsed between two consecutive flow reversals. The sum of the times for forward and reverse flow is the cycle time. Thermal energy generated in an exothermic methane oxidation reaction can be captured with a solid heat storage medium. Then, by switching the flow direction, it is possible to keep the reactor core at high reaction temperatures. Therefore, autothermal operation is possible even when working with cold lean feeds. A one-dimensional physical model is presented here.

A premixed homogeneous fuel/air mixture enters a reactor packed with noncatalytic honeycomb ceramic monolith blocks. The bed is initially at a uniform temperature. air

methane

flowrate meter methane sensor burner

valve 1 valve 3

insulation blanket monolith 1 2 3 4

5 6

7 8 9 10 11 12

valve 2 valve 4

methane sensor outlet

Fig. 1. TFRR operating principle and experimental schema

The products formed leave the surface via a desorption process and travel from the surface to the gas mixture via mass diffusion. A portion of the heat released due to surface reactions increases the solid temperature, while the remainder is transferred to the gas phase. The heat received by the gas phase may be high enough to promote gas phase reactions.

The three modes of heat transfer (conduction, convection and radiation) contribute jointly to the transport of heat within the reactor. Conduction redirects heat from the downstream to the upstream regions of the bed, which contributes to an additional preheating of the fuel-air mixture. Convective heat transfer provides heat exchange between the solid and gas phases while the radiation mode becomes important mainly at sufficiently high temperature levels.

The transport processes taking place within the reactor are complex and the following assumptions

Qi, X. – Liu, Y. – Xu, H. – Liu, Z. – Liu, R.

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are employed in the developed model. The reactor operates adiabatically and at atmospheric pressure. The flow in the reactor is assumed to be one-dimensional.

The gas and solid are not in local thermal equilibrium.

Therefore, separate energy equations are considered for each phase. Radiation heat transfer in the gas phase is considered to be negligible in comparison to the solid ceramic radiation. The reactants and the products are treated as incompressible ideal gases.

The thermophysical properties of the gas species are functions of the local temperature and composition while those of the ceramic are assumed to be uniform and temperature independent. Based on these assumptions the governing conservation equations of mass, energy, and species are as follows: Conservative equation: ∂ρ g

∂t

+

∂( ρ g ug ) ∂x

= 0. (1)

Species balance equation: ∂Y ∂Y ∂ 2Yg ,i + ρ g g ,i + ρ g u g g ,i = − ρ g Di ,m ∂t ∂x ∂x 2 k + i ,m ac ρ g (Ys ,i − Yg ,i ) + M i R g ,i , (i = 1, , N g ), (2) ε

ki ,m ρ g (Yg ,i − Ys ,i ) =M i R g ,i , (i = 1, , N g ). (3)

Heat balance equation for the gas phase: ∂T ∂T ρ g cg g + ρ g cg u g g = ∂t ∂x 2 Ng ∂ Tg h + = −k g + a T − T M j R g , j H j . (4) ( ) ∑ g v s ∂x 2 ε j =1 Heat balance equation for the solid phase:

ρ s cs

∂Ts ∂ 2T ∂q h av (Tg − Ts ) . (5) = − k s 2s + rad + ∂t ∂x ∂x 1 − ε

Ideal gas law:

ρg =

RTg pM g

. (6)

The diffusion coefficients, Di,m for each species (Eq. (2)) are obtained using Taylor-Aris dispersion theory [22], convective mass transfer coefficient, ki,m, (Eqs. (2) and (3)) are calculated by analogy to the Taylor-Aris dispersion. The value of the thermal conductivity, ks, (Eq. (5)) is obtained from [23].

The diffusion approximation for radiation is used to consider the effect of radiation heat transfer with qrad (in Eq. (5)) calculated from the Rosseland model:

qrad =-

16σ T 4 , (7) 3β

where σ is the Stefan-Boltzmann constant, 5.67×10–8 W/(m2·K4), and β is the extinction coefficient calculated from [24]:

β =1.5ε r (1− ε )

Sr , (8) dH

where εr is the emissivity of the honeycomb ceramic, ε is the porosity of the bed and Sr is the scaling factor calculated as:

S r =1+1.84(1 − ε ) + 3.15(1 − ε ) 2 + 7.2(1 − ε )3 , (9)

ε > 0.3.

The flow in the monolith channel is laminar and entrance effects may be significant. The Nusselt number for heat transfer is calculated from the correlation of Groppi and Tronconi [25]. For square channels the Nusselt number is given by: Nu = hd H / k s = 2.977 (1 + 3.6Gz1/ 2 e( −50/ Gz ) ) . (10) The Graetz number is:

Gz = (d H ⋅ Re⋅ Pr) / x. (11)

The Reynolds number and Prandtl number are given by:

Re = ud H ρ g / µ g , (12)

Pr = c p , g µ g / k g . (13)

The local heat transfer coefficient is calculated from the local Nusselt number using the hydraulic diameter as the characteristic length. At the beginning of each new monolith section a new entry length is assumed to start.

The area to volume ratio for a monolith is calculated using the fractional open frontal area, or porosity, of the monolith structure and the hydraulic diameter of the channels.

a = 4ε / d H . (14)

The pressure drop is neglected and the velocity is corrected for temperature as follows:

us = us ,inletTg / Tg ,inlet . (15)

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The velocity, us, is defined in terms of the superficial velocity, which is based on the empty cross sectional area and the interstitial velocity, u, is the actual fluid velocity in the void space, thus: u = us / ε . (16)

The fluid is treated as an ideal gas, the density, heat capacity, thermal conductivity and viscosity of the fluid are described by a third-order polynomial function of temperature.

The boundary and initial conditions are: • Boundary conditions: x = 0 : Tg ( x, t ) = Tg ,inlet , h(Tg ( x, t ) − Ts ( x, t )) = − ks Yg ( x, t ) = Yg ,inlet

x = L: •

∂Tg ( x, t ) ∂x

= 0,

∂Ts ( x, t ) , ∂x

(17)

∂Yg ,i ( x, t ) ∂Ts ( x, t ) = 0, = 0. (18) ∂x ∂x

rate of depletion or generation of the gas phase species i due to the surface reactions over this time interval is: R s ,i =

t t X gt +∆ ,i g − X g ,i g

∆t

. (20)

This value was then substituted in Eqs. (3) and (4) for the numerical solution. Correspondingly, the amount of heat release due to surface reactions (Eq. (5)) was calculated. The same procedure of integration was performed for the gas phase reactions dX j R g , j = . dt The steady state solution for each condition was sought using the system of equations that are, nevertheless, in transient form. This was carried out by performing an adequate number of iterations until the solution reached a steady state operating point.

Initial condition:

= t 0= : Tg ( x) T= T= Yi ,inlet . (19) g ,inlet , Ts ( x ) g ,inlet , Yi ( x ) In this simulation, the initial temperature of the porous medium is set to the temperature distribution function in the oxidation bed after the startup process finished. The initialization of oxidation bed temperature field was realized by importing the UDF program in Fluent.

2 NUMERICAL METHOD Because the intermediate reaction products and the generation of free radicals are not involved, the singlestep oxidation of the methane reaction mechanism will not affect the accuracy of the reaction heat of the reaction and reaction residence time. The governing equations (Eqs. (1) to (6)) for a ceramic packedbed reactor could not be solved directly using the commercial software Fluent. However, some features of this code such as the QUICK scheme and second order method for discretization of temporal terms and easier post-processing of the numerical results contain useful code for solving Eqs.(1) to (6). The heat and mass transfer coefficients (e.g. Eqs (10)), which are not predefined in the software, were computed via modifying subroutines (UDFs). The values for these coefficients were updated at every flow time step. At each flow time step, t , the numerical integration of this system was performed while the flow variables were considered to remain unchanged. Therefore, the 498

Fig. 2. Sketch of part of the computational domain and the meshes

8000 computational cells for spatial discretization of the domain and a time step of 10-5 s were used in the simulations.

As the grids were refined, the results did not change significantly, which means that the current density of the grid sensitivity is low, the results meet the need. Fig. 2 shows the grids in the computational domain.

3 EXPERIMENTAL REACTOR SYSTEM Tests of methane thermal oxidation were carried out in a reverse flow reactor built at the Energy Research Institute of Shandong University of Technology. A general view of the experimental apparatus is shown in Fig. 3, while a simplified flowsheet of the apparatus is given in Fig. 4. The reactor is 600 mm wide, 600 mm high and 2100 mm long. Thermal energy generated in an exothermic methane oxidation reaction can be captured using the solid heat storage medium. Then, by switching the flow direction, it is possible to keep the reactor core at high reaction temperatures.

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Therefore, autothermal operation is possible even when working with cold lean feeds.

In the experiments, the ceramic honeycomb monoliths were used as the heat storage medium. The monoliths consist of a structure of parallel channels with porous walls. The properties of monolith are shown in Table 1.

Table 1. Properties of the honeycomb monolith Property Width of square hole [mm] Wall thickness [mm] Density [kg/m3] Specific surface area [m2/m3] Porosity [Void/%] Thermal conductivity [W/(m·K)] Heat capacity [J/(kg·K)] Bearing temperature [ºC]

Value 2.25 0.7 2400 1005 57 8 1200 1350

Fig. 3. General view of the research & demonstration TFRR apparatus

The oxidation bed is packed with ceramic monolith blocks with a large number of straight and parallel channels (2×2 to 3×3 mm), resulting in a low pressure drop. To prevent heat loss from the reactor to the surroundings, the reactor was surrounded with a layer of insulation ceramic fiber blanket 350 mm thick. The thermal conductivity of the ceramic fiber blanket is 0.144 W/(m·K). Therefore, heat loss from the outer surface can be ignored.

As shown in Fig. 4, the complete cycle consists of these two operations to ensure the symmetry of the temperature field of the TFRR. Thermal profiles from the reactor were obtained using twelve thermocouples (denoted from 1 to 12 in Fig. 4). All thermocouples were placed along the centerline of the reactor. The data acquisition system recorded all sensor values.

Fig. 4. General overview of the research & demonstration TFRR apparatus

The experimental system is divided into four parts: gas supply system, temperature acquisition system, startup system, gas composition analysis system and oxidation bed body. A schematic diagram of the experimental apparatus is shown in Fig. 4. Mine ventilation air used in the experiment was a mixture air and natural gas, where CH4 purity is 99.9%.

Compressed air from the air compressor through the gas regulating valve and pressure-relief valve mixes with the methane from Methane cylinder in the mixer, which is then fed into the oxidation bed. The time relay controls the periodic closing and opening of the electromagnetic valve to control the the cycle gas flow direction. To initiate the reaction, the reverse flow reactor was preheated from an ambient temperature to about 950 °C using an electrical heater in the central position of the reactor. Once the reactor was preheated, the burner was extinguished and the lean methane mixture was supplied to the reactor. The heating temperature and heating power in the startup process is controlled by the temperature control instrument.

The intake and exhaust composition is tested by gas chromatography. Electric heaters mounted in the middle of TFRR were used only for preheating the monoliths to enable the start of normal operations.

There are 12 thermocouple measuring points arranged along the axis of the oxidation bed, the output signal of the thermocouple connected to the computer can monitor the transient temperature field in the TFRR. The direction of the gas flow was controlled by four solenoid valves.

After the experimental system is well connected, and the experimental instrument, equipment and airtightness is checked, the experiment is started. The experimental procedures are as follows:

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(1) Turn on the power switch of the device, the system is then connected to the electricity supply and the air compressor is turned on; (2) Turn on the time relay, set the commutation switch time, and the reversing control system begins to work. Turn on the computer that is connected with the data acquisition card; open the data acquisition operating system on the computer, set the required acquisition channels and parameters, and the values of each parameter are displayed online; (3) open the methane cylinder and adjust the pressure-relief valve; (4) then open both the valves for air and for methane and set to the desirable flow rate values; (5) the device runs on a fixed commutation time, the system starts up successfully; (6) adjust the operating parameters, according to the experimental conditions, and collect and record the data. Repeat the above process for the next set of condition; (7) when the experiment is finished, first turn off the methane gas. Turn off the methane pressurerelief valve, stop data acquisition, and store the experimental data, then shut down the computer; (8) close the flowmeter and then turn off the air source, close the periodically reversing control system. Turn off the control power supply; (9) for the next experiment, repeat the above operations.

4 RESULTS AND DISCUSSION 4.1 Model Validation The experimental data we obtained were used to validate the developed model. Simulations were then conducted for the same operational conditions in the same reactor as those employed in the experimental investigation. Fig. 5 shows the centreline temperature profiles obtained from the experiments and the simulations at three times when the methane concentration is 0.5% volumetric concentration in air with a superficial velocity of 0.5 m/s and the switch time is 150 s. The lines represent the simulations and the points the experimental values. The results are in good agreement with the experimental value obtained for the same reactor. Therefore, this model is sufficient for use in the thermal design of objects on a larger industrial scale.

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4.2 Temperature Distribution Characteristics in the Oxidation Bed Fig. 5 shows the numerical results for a mixed gas flow velocity of 0.5 m/s, methane concentration of 0.5% (by volume), and a switching time of 150 s.

The abscissa and the ordinate of Fig. 5 represents the axial position in the oxidation bed and the temperature value, respectively. The curves indicate the regenerator temperature profile along the flow direction at 3, 63 and 123 s. The premixed low concentration methane-air mixture is induced into the preheated ceramic honeycomb bed. After a warmingup period caused by the chemical combustion reaction and the heat and mass transfer between the gas and solid, when the temperature of TFRR gets to about 900 °C, methane begins to react. The bed temperature begins to rise because of the heat generated by the oxidation reaction, which, in turn, maintains the oxidation reaction of methane. The bed temperature is higher in the middle and lower on both sides in the axial direction. After switching direction several times, a transient temperature profile that forms an M-shaped curve moves slowly along the axial direction. The curve is steep in the middle part and the curve’s moving velocity is much smaller than the feed velocity. It is the same as the wave propagation in physics, which is shown in Fig. 5. The curve parameters, i.e. peak temperature, the average temperature, the moving speed and the shape of the temperature distribution curve, determine the oxidation performance of the TFRR system, which suggests that the relationship between operating parameters and the curve parameters should be further studied.

Fig. 5. Moving curves of the temperature field in the oxidation bed

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Fig. 6. Axial temperature distribution as gas flows from left to right at same time in different cycles

Fig. 7. Axial CH4 component distribution as the gas flows from left to right and from right to left at different time in the same cycle

Fig. 9. Axial temperature distribution as gas flows from right to left within the same cycle

Fig. 10. Axial temperature distribution as gas flows from left to right within the same cycle

Fig. 8. Axial CH4 component distribution as the gas flows from left to right and from right to left at the same time in different cycles

4.3 Changing Principle of Temperature Field and Concentration Fields in the Oxidation Bed over One Cycle Fig. 6 shows the numerical results for a mixed gas flow velocity of 0.5 m/s, methane concentration of

Fig. 11. Axial temperature distribution as gas flows from right to left at same time in same cycle

0.5% (by volume), and a switching time of 150 s.

Fig. 6 shows the numerical results for a mixed gas flow velocity of 0.5 m/s, methane concentration

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of 0.5% (by volume), and a switching time of 150 s. The curves indicate the gas temperature profile along the flow direction for the first five cycles, i.e.

2, 302, 602, 902 and 1201 s, respectively, when the feed flows from left to right. As can be seen from Fig. 6, with the passage of time the width of the high temperature field obviously increases and the peak value increases slightly. The reason for this is that with the combustion reaction, a large amount of heat is released by burning methane accumulates in the oxidation bed and gradually spreads around so that the scope of the combustion reaction expands and a more involved reaction of methane is prompted, which in turn emits more heat, so that after a few cycles, the temperature distribution will gradually reach the edge of the oxidation bed. If the waveform overflows the oxidation bed, it will cause energy waste.

Fig. 7 shows the numerical CH4 component distribution results for a mixed gas flow velocity of 0.5 m/s, methane concentration of 0.5%, and a switching time of 150 s. Fig. 7 indicates that the gas temperature profile along the flow direction in the same cycle i.e.

2, 32, 62, 92 and 121 s, respectively, when the feed flows from left to right. In the same cycle, regardless of the gas flow direction, the curve trend is the same. At first, when the bed temperature has not yet reached the temperature of methane combustion, the curve is flat and the methane concentration is 1 (100%) (by volume), ,showing that the methane has not reacted. The more the curve moves to the middle of the bed oxidation, the higher the temperature. The methane then begins to react, so the curve of the methane concentration begins to decrease. Because the methane concentration is very low, the reaction finishes quickly, the methane gas mixture needs to be constantly ireintroduced into the bed, so the curve gradually moves to the middle from one side, then changes direction until the commutation time.

Fig. 8 shows the numerical CH4 component distribution results for a mixed gas flow velocity of 0.5 m/s, methane concentration of 0.5%, and a switching time of 150 s. There are two sets of curves in the figure, which indicates the methane concentration results of the feed flowing from left to right and from right to left, respectively. Fig. 8 indicates that, while at the same time of different cycles, the curves move to both side, because after one cycle the reaction of methane emits more heat to make the temperature field move to both sides, which promotes a faster reaction of methane and leads the concentration curves to move to the sides. By comparing the two sets of curves, it is shown that the interval of the two sets of curves are different, mainly because over time 502

the gradually increasing resistance leads to a decrease in the mixed gas flow rate, thus the spacing may be reduced.

4.4 Effect of Methane Concentration on Temperature Distribution Figs. 9 and 10 show the numerical gas temeprature results for a mixed gas flow velocity of 0.5 m/s, methane concentration of 0.3 to 0.7% (by volume), and a switching time of 150 s. Figs. 9 and 10 show that if the other conditions remain the same, when the methane concentration of the mixture varies from 0.3 to 0.7%, i.e. as the concentration increases, the peak temperature is higher, the temperature distribution curve is closer to the inlet side, the high temperature region of the temperature distribution increases, the concavity of the intermediate high temperature region is deeper, and the temperature gradient of the inlet and outlet increases.

The main reason is that when the oxidation bed is working under the high temperature conditions, the higher concentration of methane gas mixture should release a large amount of heat after a quick combustion reaction, but the flow rate does not increase, therfore the heat has little time to transfer to the exit end, so the heat accumulates in the middle to form a prominent peak temperature. However, it is important to note that if the methane concentration is too high, too much heat will accumulate in a local area bringing about a higher temperature which may cause damage to the ceramic regenerator; if the methane concentration is low, very little heat accumulated locally makes the peak temperature decrease, the high temperature zone narrows, the bed temperature began to drop and eventually drops below the critical methane oxidation temperature, resulting in an oxidation reaction that cannot proceed and an oxidation bed that cannot operate properly.

4.5 Effect of Velocity on Temperature Distribution Fig. 11 shows the numerical gas temeprature results for a mixed gas flow velocity of 0.15 to 0.8 m/s, methane concentration of 0.5% (by volume), and a switching time of 150 s. The effect of feed velocity on the gas temperature distribution is shown in Fig. 11. It indicates that with premixed methane gas flow increasing from 0.15 to 0.8 m/s, the maximum peak temperature and high-temperature zone change little.

How can this be?

The effect of the feed velocity on the temperature distribution has two trends. On one hand, when the

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concentration of methane is lower, the exothermic heat of the oxidation reaction increases with the flow rate, because for the same oxidation bed, if the methane concentration is fixed, a higher feed velocity means a higher flow rate and the combustion process in the oxidation bed will release more heat, so the temperature of the oxidation bed and the gas rises; on the other hand, the convective heat taken away by the air will increase as the feed velocity is higher. However, this depends on a specific condition: the convective heat being taken away caused by the increasing feed velocity must exceed the increasing thermal effects of oxidation also caused by the increasing feed velocity. Therefore, when operating TFRR, the maximum temperature value cannot be considered to simply change with the increasing or decreasing feed velocity. Therefore, conclusions should be made according to the actual situation in order to guide the actual operation correctly.

6 CONCLUSIONS The model developed simulates the thermal oxidation of lean homogeneous methane–air mixtures in a reverse flow reactor while employing the user defined function (UDF) to extend the ability of FLUENT. It was demonstrated that the model could predict the effect of changes in operating conditions such as inlet mixture composition, velocity of methane conversions, as well as the species concentrations and gas temperature profiles along the bed. The displayed trends were in good agreement with the corresponding experimentally observed ones.

It is doubtful, if any further complexity of the model, e.g. two-dimensional in space, could easily improve it. The important point is that the agreement of the temperature profiles shown in Fig. 6 seems to be sufficient to ensure good thermal design of objects on a larger industrial scale. The main goal of the design is to maintain autothermicity of TFRR via a high enough temperature in the hot zone, while realising that real CH4 conversion will be lower due to some other reasons that are hard to include in the model.

The main conclusions of the present study are: (1) In the TFRR the lean methane mixture ignition takes place at almost 900 °C. The heat release from the combustion of methane increases the gas and solid temperature, which, in turn, maintains the oxidation reaction of methane. The bed temperature is higher in the middle and lower on both sides along the axial direction. After switching direction several times, a transient temperature profile that forms an M-shaped curve moves slowly along the axial direction. The

curve is partially steep in the middle part, and the curve’s velocity is much lower than the feed velocity.

It is as same as the wave propagation in physics.

(2) After several cycles, the width of the high temperature field clearly increases and the peak value increases slightly. With the combustion reaction, a large amount of heat release by burning methane accumulates in the oxidation bed and gradually spreads around so that the scope of the combustion reaction broadens and more and more methane gets involved in the reaction, which, in turn, emits more heat, so that after some time, the temperature distribution will gradually reach the edge of the oxidation bed. If the waveform overflows from the oxidation bed, it will cause energy waste.

(3) Concentration curves move to both sides at the same time in different cycles. The interval of the two sets of curves are different, mainly because over time gradually increasing resistance leads to decreases in the mixed gas flow rate, the spacing may also be reduced.

(4) If the other conditions remain the same, when the methane concentration of the mixture varies from 0.3 to 0.7% (by volume), i.e. as the concentration increases, the peak temperature is higher, the temperature distribution curve is closer to the inlet side, the high temperature region of the temperature distribution increases, the concavity of the intermediate high temperature region is deeper, and the temperature gradient of the inlet and outlet is increased. However, it is important to note that if the methane concentration is too high, too much heat will accumulate in a local area leading to a high temperature that may cause damage to the ceramic regenerator; if the methane concentration is lower, very little heat accumulated locally makes the peak temperature decrease, the high temperature zone then narrows, the bed temperature begins to drop and eventually drops below the critical methane oxidation temperature, thus resulting in an oxidation reaction that cannot proceed and an oxidation bed that cannot operate properly.

(5) The effect of the feed velocity on the temperature distribution has two opposite trends.

Therefore, when operating TFRR, the maximum temperature value cannot be considered to simply change with increasing or decreasing feed velocity.

Conclusions should be made according to the actual situation in order to guide the actual operation correctly.

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6 NOMENCLATURE a monolith surface area to volume ratio [m2/m3] c constant pressure specific heat [J/(kg·K)] dH hydraulic diameter [m] di,m diffusivity of species i [m2/s] h convective heat transfer coefficient [W/(m2·K)] Hi enthalpy of species i [J] k thermal conductivity [W/(m·K)] ki,m convective mass transfer coefficient of species i [m/s] Mi molar mass of species i [mol] p pressure [Pa] qrad radiative heat flux [W/m2] R g species production rate in gas phase [mol/(m3·s)] R s species production rate on surface [mol/(m3·s)] t time [s] T temperature [K] u axial velocity [m/s] x Axial coordinate [m] Yi mass fraction of species i [%] Greek symbols: β extinction coefficient ε porosity εr emissivity of honeycomb ceramic μ viscosity [Pa·s] ρ density [kg/m3] Subscript: g gas phase s solid i component i at the reactor inlet inlet 7 ACKNOWLEDGMENTS This work was supported by the China National 863 High Technology Fund project (2009AA063202).

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[2] Gogin, L.L., Matros, L.L. Ivanov, A.G. (1990).

Catalytic Combustion in Catalytic Flow-Reversal Reactors. Nauka, Novosibirsk. (in Russian) [3] Salomons, S., Hayes, R.E., Poirier, M., Sapoundjiev, H. (2003). Flow reversal reactor for the catalytic combustion of lean methane mixtures. Catalysis Today, vol. 83, no. 1-4, p. 59-69, DOI:10.1016/S09205861(03)00216-5.

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[4] Litto,R., Hayes, R.E., Sapoundjiev, H., Fuxman, A., Forbes, F., Liu,B., Bertrand, F. (2006). Optimization of a flow reversal reactor for the catalytic combustion of lean methane mixtures. Catalysis Today, vol. 117, no.

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[16] Shahamiri, S.A., Wierzba, I. (2009). Simulation of catalytic oxidation of lean hydrogen-methane mixtures.

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14, p. 5785-5794, DOI:10.1016/j.ijhydene.2009.04.077.

[17] Recovery of Methane from Vent Gases of Coal Mines and Its Efficient Utilization as a High Temperature Heat Source (2003). European Union Project, Gliwice, p. 100-126.

[18] Gosiewski, K., Matros, Y.S., Warmuzinski, K., Jaschik, M., Tanczyk, M. (2008). Homogeneous vs. catalytic combustion of lean methane-air mixtures in reverseflow reactors. Chemical Engineering Science, vol. 63, no. 20, p. 5010-5019, DOI:10.1016/j.ces.2007.09.013.

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2-3, p. 287-294, DOI:10.1016/S1385-8947(02)001651.

[24] Singh, B.P., Kaviany, M. (1992). Modelling radiative heat transfer in packed beds. International Journal of Heat and Mass Transfer, vol. 35, no. 6, p. 1395-1397, DOI:10.1016/0017-9310(92)90031-M. [25] Groppi, G., Tronconi, E. (2000). Design of novel monolith catalyst supports for gas/solid reactions with heat exchange. Chemical Engineering Science, vol. 55, no. 12, p. 2161-2171, DOI:10.1016/S00092509(99)00440-6.

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DOI:10.5545/sv-jme.2013.1510 Original Scientific Paper

Received for review: 2013-10-17 Received revised form: 2014-01-31 Accepted for publication: 2014-03-05

Evaluation of the Young’s Modulus of Rubber-Like Materials Bonded to Rigid Surfaces with Respect to Poisson’s Ratio Koblar, D. – Škofic, J. – Boltežar, M. David Koblar1,* – Jan Škofic2 – Miha Boltežar3 1 Domel,

Slovenia Mehanizmi, Slovenia 3 University of Ljubljana, Faculty of Mechanical Engineering, Slovenia 2 Iskra

Axially loaded rubber blocks with a circular cross-section, whose ends are bonded to rigid plates, were studied. The frequency-response functions were obtained with a finite-element model on rubber specimens with respect to Poisson’s ratio and the shape factor. Then the apparent Young’s modulus was estimated and various equations that describe the relationship between the apparent Young’s modulus and the actual Young’s modulus of the rubber material were used to estimate the Young’s modulus of the rubber material. The subsequently estimated Young’s modulus of the rubber material was compared to the one defined in the finite-element model.

It is shown that there is a significant difference in the estimated Young’s modulus when different equations were used, especially when the Poisson’s ratio of the rubber material is smaller than the theoretical value of 0.5.

Keywords: Young’s modulus, finite-element model, rubber, rubber-like

0 INTRODUCTION Rubber materials are widely used for sound and vibration control and are usually bonded and compressed between rigid plates. In order to predict the vibration transferred through the rubber material in a finite-element model, knowledge of the material properties, such as the Young’s modulus, the damping factor and the Poisson’s ratio is essential.

An element of rubber material bonded to rigid plates, shown in Fig. 1, possesses an apparent Young’s modulus Ea, which is due to the restrained motion of the upper and lower bonded surfaces, greater in value than the Young’s modulus E of the rubber material [1] to [4].

For a block with a circular cross-section Gent and Lindley [5] derived a correlation in the form: Ea = E (1 + β S 2 ), (1)

where S is the ratio of one loaded surface to the forcefree surface and β is a numerical constant. For rubbers 506

1 1 1 = + , (2) Ea E (1 + β S 2 ) B

Fig. 1. Single-degree-of-freedom (SDOF) system, with ground excitation, measured displacement x1 of the mass M and the displacement x2 of excitation

where B is the Bulk modulus.

In deriving Eqs. (1) and (2) it was assumed that the material is virtually incompressible in terms of volume, that the cross-section of the block, normal to the direction of the applied load, remains plane and horizontal and also that the free vertical surfaces take up parabolic forms. Horton et al. [6] eliminated the assumption of a parabolic profile and derived a different expression:

that are square, circular or moderately rectangular in cross-section β = 2 should be used and for rubbers with the addition of carbon black, somewhat smaller values should be used [5].

For very large shape factors S some contribution to the total deflection may be anticipated from the bulk compression of the rubber. To account for this, Gent and Lindley [5] derived the equation:

1 1 2 1 = (1 + S tanh( Ea E 3 S

2 1 )) + , (3) 3 B

where B is again the Bulk modulus. Horton et al. [6] found that the results given with Eq. (3) are closely approximated by:

1 1 1 = + , (4) 2 Ea E (1.2 + 2 S ) B

and for a circular block of incompressible material, when B = ∞, can consequently be written as:

*Corr. Author’s Address: Domel, d.o.o., Otoki 21, 4228 Železniki, Slovenia, david.koblar@domel.si

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Ea = E (1.2 + 2 S 2 ). (5)

Williams and Gamonpilas [7] used the Timoshenko and Goodier [8] equilibrium equations and derived an expression in the form:

1 −ν 2 )S 1 + 3ν ( Ea 1 +ν = . (6) E 1 + 3ν (1 − 2ν ) S 2

Sim and Kim [9] developed a procedure to find a relationship between the ratio of the apparent Young’s modulus to the Young’s modulus, Poisson’s ratio and shape factor. The downside of this method is that the transmissibility needs to be measured on two different specimens (one thin and one thick).

To run FEM calculations it is essential to know the actual Young’s modulus. This is also important with other types of material models, see for instance [10].

In this paper a comparison of different equations for an estimation of the Young’s modulus from an apparent Young’s modulus derived from the transfer functions of a single-degree-of-freedom (SDOF) system will be presented. It will be shown that there is a significant difference in the estimated Young’s modulus when different equations are used, especially when the Poisson’s ratio of the rubber material is smaller than the theoretical value of 0.5. The point of view presented above will be verified in this paper by using transfer functions obtained from a finite-element model of specimens having various shape factors and Poisson’s ratios.

1 THEORETICAL BACKGROUND The rubber is utilized so that its behavior is governed by the complex Young’s modulus . Here it is assumed that the temperature remains constant with time, so that the complex Young’s modulus may be written as [11]:

Eω* = Eω (1 + iδ Eω ), (7)

where Eω is the real part and δ Eω is the ratio of the imaginary to the real part of the complex Young’s modulus Eω* , and is known as the damping factor and i is equal to −1 .

For a single-degree-of-freedom SDOF system, shown in Fig. 1, the transmissibility of the system, which is defined as the displacement ratio [11]:

T=

1 + iδ Eω X 2* = , (8) X 1 1 − ω 2 hM + iδ Eω AEaω

and with real and imaginary parts of the transmissibility, RE = Re(T) and IM = Im(T), known from the measurement, the frequency-dependent apparent Young’s modulus can be obtained [11]:

Eaω =

hM ω 2 [ IM 2 + ( RE − 1) RE ] . (9) A[ IM 2 + ( RE − 1) 2 ]

Now the apparent Young’s modulus needs to be converted to the Young’s modulus of the rubber material. In the literature, [5] to [7], several equations were used. By transforming the Gent and Lindley [5], Eq. (1), frequency-dependent Young’s modulus of the rubber material is derived:

Eω =

Eaω , (10) (1 + β S 2 )

or using Eq. (2) and taking the Bulk modulus:

B=

E , (11) 3(1 − 2ν )

where ν is Poisson’s ratio, into account indicates that the frequency-dependent Young’s modulus of the rubber material is:

Eω = Eaω (3 +

1 − 6ν ). (12) 1 + 2S 2

Similarly, by rearranging Horton et al. [6], Eqs. (3) and (4), and taking into account the Bulk modulus, Eq. (11), the frequency-dependent Young’s modulus of the rubber material is obtained:

1 1 Eω = Eaω (12 − 18ν − 6 S tanh( 3 S

2 )), (13) 3

and

Eω = Eaω (3 +

1 − 6ν ). (14) 1.2 + 2 S 2

Next, the frequency-dependent Young’s modulus is obtained by rearranging Horton et al. [6], Eq. (5), and is written in the form:

Eω =

Eaω . (15) 1.2 + 2 S 2

The last equation is obtained by rearranging Williams and Gamonpilas [7], Eq. (6), where the frequency-dependent Young’s modulus is expressed as:

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Eω = Eaω

(1 + ν )(−1 + 3S 2ν (−1 + 2ν )) . (16) −1 + (−1 + 3S 2 (−1 + ν ))ν

2 ESTIMATION OF THE YOUNG’S MODULUS FROM THE FREQUENCY-RESPONSE FUNCION The frequency-response functions were obtained with a finite-element model in Ansys Workbench v14.5. The finite-element model represents a cuboid aluminum mass with a cross-section of 20×20 mm and a height of 5 mm, a cylindrical rubber with a diameter of 20 mm and different heights (20, 10, 5, 2.5 and 1.25 mm), a cylindrical aluminum mass with a diameter of 50 mm and a height of 15 mm, and an

accelerometer on the top surface of the aluminum mass, which was modeled as a mass point of 4.6 g, shown in Fig. 2a. The mesh was generated with a higher-order 3-D 20-node solid element that exhibits quadratic displacement behavior, SOLID186 and element MASS21 for the mass point. The generated mesh is shown in Fig. 2b.

The material was assumed to behave as a linear elastic material with a constant Young’s modulus of 4 MPa, defined in Ansys Engineering data, as was shown in [11] that the numerical calculations are in good agreement with the experimental measurements.

Various Poisson’s ratios, from almost incompressible (ν = 0.4999) to compressible (ν = 0.45, 0.47 and 0.49) solids, were studied. Other material data are shown in Table 1.

Fig. 2. Model in a) ANSYS Workbench and b) mesh Table 1. Material parameters defined in ANSYS Workbench density [kg/m3] Young’s modulus [MPa] Poisson’s ratio [-] damping factor [-]

Aluminum 2850 7.1×104 0.33 -

Rubber 1200 4 0.4999 to 0.45 0.01

The harmonic analyses for combinations of five shape factors and four Poisson’s ratios, in total 20 cases, were made in the frequency band from 20 to 5000 Hz, with a frequency resolution of 1 Hz. Then Eq. (9) was used to calculate the frequency-dependent, apparent Young’s modulus and Eqs. (10) and (12) to (16) were used to calculate the values of the Young’s modulus from the apparent Young’s modulus. For the purpose of a clearer presentation the percentage error Eerror in the estimated values of the Young’s modulus Ew to the actual Young’s modulus defined in the 508

finite-element model Eactual was calculated with the following equation:

Eerror =

Eω − Eactual ⋅100%. (17) Eactual

The percentage errors for various shape factors and depending on the value of Poisson’s ratio are presented in Figs. 3 to 6.

3 DISCUSSION Percentage errors in the estimation of the Young’s modulus from the frequency-response functions calculated with the finite-element model, for ν = 0.4999, are presented in Fig. 3 and the percentage errors at 100 Hz are also shown in Table 2.

Koblar, D. – Škofic, J. – Boltežar, M.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 506-511

Fig. 3. Percentage errors of the estimated Young’s modulus to the actual Young’s modulus for different shape factors, (full) S = 0.25, (dashed) S = 0.5, (full with circle) S = 1, (dashed with circle) S = 2 and (full with x) S = 4, and equations (red) Eq. 10, (green) Eq. 12, (blue) Eq. 13, (magenta) Eq. 14, (cyan) Eq. 15, (yellow) Eq. 16, at ν = 0.4999

Fig. 5. Percentage errors of the estimated Young’s modulus to the actual Young’s modulus for different shape factors, (full) S = 0.25, (dashed) S = 0.5, (full with circle) S = 1, (dashed with circle) S = 2 and (full with x) S = 4, and equations (red) Eq. 10, (green) Eq. 12, (blue) Eq. 13, (magenta) Eq. 14, (cyan) Eq. 15, (yellow) Eq. 16, at ν = 0.47

Fig. 4. Percentage errors of the estimated Young’s modulus to the actual Young’s modulus for different shape factors, (full) S = 0.25, (dashed) S = 0.5, (full with circle) S = 1, (dashed with circle) S = 2 and (full with x) S = 4, and equations (red) Eq. 10, (green) Eq. 12, (blue) Eq. 13, (magenta) Eq. 14, (cyan) Eq. 15, (yellow) Eq. 16, at ν = 0.49

Fig. 6. Percentage errors of the estimated Young’s modulus to the actual Young’s modulus for different shape factors, (full) S = 0.25, (dashed) S = 0.5, (full with circle) S = 1, (dashed with circle) S = 2 and (full with x) S = 4, and equations (red) Eq. 10, (green) Eq. 12, (blue) Eq. 13, (magenta) Eq. 14, (cyan) Eq. 15, (yellow) Eq. 16, at ν = 0.45

From this it can be concluded that Eqs. (10) and (12) give the best result when the shape factor is small (large thickness of the rubber) and the error increases with higher shape factors (with decreasing thickness of the rubber). It can also be seen that with a

greater shape factor Eq. (12) gives better results than Eq. (10), since for very thin blocks some contribution to the total deflection may be anticipated from the bulk compression of the rubber, as discussed by Gent and Lindley [5].

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Table 2. Percentage error in the estimation of the Young’s modulus using different equations, at 100 Hz and ν = 0.4999 h [mm] S [-] Eq. (10) Eq. (12) Eq. (13) Eq. (14) Eq. (15) Eq. (16)

20 0.25 2.58 2.65 -8.08 -12.83 -12.9 11.91

10 0.5 0.90 0.99 -9.43 -10.88 -10.97 34.54

5 1 -3.66 -3.48 -9.28 -9.5 -9.68 92.74

2.5 2 -9.47 -8.98 -10.83 -10.95 -11.44 171.91

1.25 4 -29.72 -28.33 -28.75 -28.75 -30.14 158.93

Estimated values of the Young’s modulus with Horton et al.[6], Eq. (13), are underestimated by approximately 10%, compared to the actual value of the Young’s modulus defined in the finite-element model. Eqs. (14) and (15), which are approximations of Eq. (13), are very close together, where the maximum difference in the error compared to Eq. (13) is about 5% at S = 0.25 and smaller than 1.6% at S = 0.5. In the case of greater shape factors the difference is even smaller.

The difference in the estimation of the Young’s modulus from Eqs. (10) and (12) to (15) is due to the approximation that with compression free vertical surfaces take up a parabolic form, Gent and Lindley [5], Eqs. (10) and (12), and that the profile of the compressed block is not quite parabolic, Horton et al. [6], Eq. (13) and its approximations Eqs. (14) and (15). Apparently, the Gent and Lindley [5] approximation of the parabolic profile, Eqs. (10) and (12), is the best fit for the calculations in Ansys. The maximum difference in percentage errors between Eqs. (10) and (12) to (15) for a small shape factor is 15.5% at S = 0.25 and 12% at S = 0.5 and at higher shape factors it is only 1.8% at S = 4. From this it can be concluded that Eqs. (10) and (12) give the best results for all the addressed shape factors and Eq. (13), and its approximations Eqs. (14) and (15), give comparable results only for large shape factors (thin rubber).

The results given with Williams and Gamonpilas, Eq. (16), are not as expected. In the case of a small shape factor the percentage error is 11.9% at S = 0.25 and is comparable to Eqs. (13) to (15), but in the case of a larger shape factor the Young’s modulus is overestimated by 34.54% at S = 0.5 and even 158.93% at S = 4.

In the case that the rubber material is hardened by the addition of fillers, like carbon black, its Poisson’s ratio is smaller than the theoretical 0.5. Consider that the Poisson’s ratio is ν = 0.49. In this case the percentage errors in the estimation of the Young’s 510

modulus from the frequency-response functions calculated with finite-element models, are shown in Fig. 4 and the percentage errors at 100 Hz are also shown in Table 3.

It can be seen that Eqs. (10) and (15) do not take the Poisson’s ratio into account and compared to the first case, this time error is greater than with Eqs. (12) to (14) that include the Poisson’s ratio.

From this it can be concluded that Eqs. (10) and (15) are not appropriate for an estimation of the Young’s modulus when the shape factor S ≥ 1. In this case Eq. (16) provides better results than in the case of incompressible rubber (ν = 0.4999), but still does not come close to the results given with Eqs. (12) to (14).

For Poisson’s ratio ν = 0.49 Eq. (13) seems to offer the best performance in the estimation of the Young’s modulus in the whole range of shape factors, followed

by Eq. (14) and then Eq. (12).

Table 3. Percentage error in the estimation of the Young’s modulus using different equations, at 100 Hz and ν = 0.49 h [mm] S [-] Eq. (10) Eq. (12) Eq. (13) Eq. (14) Eq. (15) Eq. (16)

20 0.25 1.45 8.29 -2.32 -7.02 -13.87 10.85

10 0.5 -2.14 6.67 -3.44 -4.84 -13.65 31.35

5 1 -14.26 1.18 -3.98 -4.18 -19.62 76.16

2.5 2 -39.85 -7.37 -8.66 -8.68 -41.16 100.83

1.25 4 -72.19 -17.14 -17.30 -17.31 -72.36 49.08

In the case that the Poisson’s ratio is ν = 0.47 or ν = 0.45, the percentage errors in the estimation of the Young’s modulus from the frequency-response functions calculated with the finite-element model are shown in Fig. 5 and Fig. 6 respectively. It is clear that when the shape factor S ≥ 1, Eqs. (10) and (15) are again not appropriate for an estimation of the Young’s modulus. The estimation with Eq. (14) gives the best results and is followed by Eq. (13). Here, the results with Eq. (16) are improved compared to the rubber with the Poisson’s ratio ν = 0.4999 and ν = 0.49 and for S ≤ 0.5 are even better than the results given with Eqs. (12) to (14), but start to deviate for larger shape factors S ≥ 1.

It should be noted that for small shape factors the Young’s modulus can be estimated for a very narrow frequency band, e.g., for S = 0.25 about 100 Hz and in the case of higher shape factors the Young’s modulus can be estimated in a wider frequency band, e.g., for S = 0.5 about 1000 Hz at ν = 0.4999 as shown in Fig. 3.

The upper limit of the frequency band is limited with wave effects that may develop at high frequencies of the transmitted vibrations as intense peaks resulting in

Koblar, D. – Škofic, J. – Boltežar, M.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 506-511

a deterioration of the transmissibility, as reported by Rivin [12] and Snowdon [1], [13] and [14].

It is interesting that in the case S ≤ 0.5 for all variants of Poisson’s ratio Eq. (10) gives the best results and the percentage error is smaller than 10.8%, which appears at S = 0.5 and ν = 0.45 and even smaller in the case of S = 0.25 where it is 2.6% at ν = 0.4999.

From this a general conclusion can be made, that for an estimation of the Young’s modulus of rubber material to use with the finite-element analysis, it is possible to adapt the rubber dimensions to obtain a shape factor of around 0.5 and use the simplest Eq. (10) developed by Gent and Lindley [5]. With this approach it is possible to quite accurately estimate the Young’s modulus of a rubber material, only the Poisson’s ratio needs to be measured individually with a seperate procedure. Eq. (10) also gives the best results for all the shape factors if the rubber is almost incompressible, ν = 0.4999. Otherwise, if the shape factor S ≥ 1 Eq. (13) and its approximation Eq. (14) give the best result, but the Poisson’s ratio of the rubber needs to be known in advance.

4 CONCLUSIONS For an almost incompressible material (ν = 0.4999), Gent and Lindley [5], Eqs. (10) and (12) offer the best accuracy for the actual Young’s modulus, which was an input for the finite-element model, followed

by Horton et al. [6], Eq. (13), with its approximations Eqs. (14) and (15).

In the case that the Poisson’s ratio of the rubber material is not theoretical, ν = 0.5, for the shape factor S ≤ 0.5 Eq. (10) quite accurately estimates the Young’s modulus of the rubber material; however, as far as the shape factor S ≥ 1, Eq. (13) and its approximation Eq. (14) give the best results.

With the proper selection of rubber dimensions, to have a shape factor of around 0.5, it is possible to use the simplest equation for the estimation of the Young’s modulus from the apparent Young’s modulus, Eq. (10), and use the estimated Young’s modulus for the finite-element analysis.

The downside is that the Poisson’s ratio still needs to be measured individually, before it can be inputted into the appropriate equation.

5 ACKNOWLEDGEMENTS Operation part financed by the European Union, European Social Fund.

6 REFERENCES [1] Snowdon, J.C. (1968). Vibration and Shock in Damped Mechanical Systems, John Wiley and Sons, New York.

[2] Tsai, H.-C., Lee, C.-C. (1998). Compressive stiffness of elastic layers bonded between rigid plates. International Journal of Solids and Structures, vol. 35, no. 23, p.

3053-3069, DOI:10.1016/S0020-7683(97)00355-7.

[3] Koh, C.G., Lim, H.L. (2001). Analytical solution for compression stiffness of bonded rectangular layers.

International Journal of Solids and Structures, vol. 38, no. 1, p. 445-455, DOI:10.1016/S0020-7683(00)00057-3.

[4] Tsai, H.-C. (2005). Compression analysis of rectangular elastic layers bonded between rigid plates. International Journal of Solids and Structures, vol. 42, no. 11-12, p.

3395-3410, DOI:10.1016/j.ijsolstr.2004.10.015.

[5] Gent, A.N., Lindley, P.B. (1959). The compression of bonded rubber blocks. Proceedings of the Institution of Mechanical Engineers, vol. 173, no. 1, p. 111-122, DOI:10.1243/PIME_PROC_1959_173_022_02.

[6] Horton, J.M., Tupholme, G.E., Gover, M.J.C. (2002).

Axial loading of bonded rubber blocks. Journal of Applied Mechanics, vol. 69, no. 6, p. 836-843, DOI:10.1115/1.1507769.

[7] Williams, J.G., Gamonpilas, C. (2008). Using the simple compression test to determine young’s modulus, Poisson’s ratio and coulomb friction coefficient.

International Journal of Solids and Structures, vol. 45, no. 16, p. 4448-4459, DOI:10.1016/j. ijsolstr.2008.03.023.

[8] Timoshenko, S.P., Goodier, J.N. (1970). Theory of Elasticity, McGraw-Hill College, New York.

[9] Sim, S., Kim, K.-J. (1990). A method to determine the complex modulus and Poisson’s ratio of viscoelastic materials from fem applications. Journal of Sound and Vibration, vol. 141, no. 1, p. 71-82, DOI:10.1016/0022460X(90)90513-Y. [10] Mankovits, T., Szabó, T., Kocsis, I., Páczelt, I. (2014).

Optimization of the shape of axi-symmetric rubber bumpers. Strojniški vestnik - Journal of Mechanical Engineering, vol. 60, no. 1, p. 61-71, DOI:10.5545/svjme.2013.1315.

[11] Koblar, D., Boltežar, M. (2013). Evaluation of the frequency-dependent Young’s modulus and damping factor of rubber from experiment and their implementation in a FE analysis. Experimental Techniques, In Press, DOI:10.1111/ext.12066.

[12] Rivin, E. (2002). Vibration isolation theory. Braun, S.G. (ed.). Encyclopedia of Vibration. Academic Press, London, p. 1487-1506.

[13] Snowdon, J.C. (1979). Vibration isolation: Use and characterization. Journal of the Acoustical Society of America, vol. 66, no. 5, p. 1245-1274, DOI:10.1121/1.383546.

[14] Snowdon, J.C. (1958). The choice of resilient materials for anti-vibration mountings. British Journal of Applied Physics, vol. 9, no. 12, p. 461-469, DOI:10.1088/05083443/9/12/301.

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 512-524 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1523 Original Scientific Paper

Received for review: 2013-10-28 Received revised form: 2014-02-25 Accepted for publication: 2014-05-07

Manufacturing Cycle Time Analysis and Scheduling to Optimize Its Duration

Jovanovic, J.R. – Milanovic, D.D. – Djukic, R.D. Jelena R. Jovanovic 1,2,3,* – Dragan D. Milanovic2 – Radisav D. Djukic1,3 1 Technical

2 University

College of Applied Studies, Cacak, Serbia of Belgrade, Faculty of Mechanical Engineering, Serbia 3 ‘Sloboda’ Company, Serbia

This paper reports the results of investigations on manufacturing cycle times for special-purpose products. The company performs serial production characterized by complex and diverse technologies, alternative solutions and combined modes of workpiece movement in the manufacturing process. Because of various approaches to this problem, an analysis of previous investigations has been carried out, and a theoretical base is provided for the technological cycle and factors affecting the manufacturing cycle time. The technological and production documentation of the company has been analysed to establish the technological and real manufacturing cycle times, total losses and flow coefficients. This paper describes the original approach to production cycle scheduling on the grounds of investigations of manufacturing capacity utilization levels and causes of loss, in order to measure their effects and to reduce the flow coefficient to an optimum level. The results are a segment of complex studies on the production cycle management of complex products, accomplished in the company in the period from 2010 to 2012.

Keywords: manufacturing cycle, manufacturing capacity utilization level, production losses, material flow, serial production

0 INTRODUCTION AND PREVIOUS INVESTIGATIONS Inherent to the investigation of the manufacturing cycle time is a set of activities, from defining the optimum production lot, calculations of the quantity of required parts, production preparation and launching, cycle scheduling, management of production activities with current asset engagement, to the analysis and investigations of material flow.

Systems for the production of weaponry and military equipment (special-purpose products) have specific positions and roles. Production is regulated by special legal provisions, whereby business operations, among other things, are determined by the product choice and quality, manufacturing, financial and human resources, serial production, complex and diverse technologies, short-term deliveries (as a rule), demands for modification of standard product versions, possibilities to supply specific materials and parts, high-level security during the production, handling, storage and utilization of means.

The start-to-finish treatment cycle implies the choice of benchmark points within which time flows. In terms of production management, the manufacturing cycle determines the duration of business and production activities needed to carry out the overall manufacturing process of a certain quantity of product with minimum time flow, maximum utilization of manufacturing capacity and optimal engagement of financial resources.

Production planning and management is a complex set of activities, as confirmed by many 512

works dealing with this problem [1] to [2]. Eckert and Clarkson [3] describe current planning practice in the development processes for complex industrial products and the challenges associated with it, making suggestions for its improvement. Since they view planning as a project, they emphasize that in order to reduce the duration of project the overlap between tasks must be optimized. In contrast, Alfieri et al. [4] observe the manufacturing-to-order system producing complex items as a set of activities whereby a project scheduling approach should be applied to production planning. They have developed two mathematical models for the execution of activities, i.e. their overlap with a smaller number of activities (up to 30 and up to 60 respectively). Dossenbach [5] analyses the possibility of reducing manufacturing cycle times in the wood-processing industry.

In his work [6], Johnson provides a framework for reducing manufacturing throughput time.

It is based on identifying the factors that affect manufacturing throughput time, the actions that can be taken to diminish their impact, and their interactions. The framework is sufficiently detailed to provide guidance to the industry practitioner on how to reduce throughput time, but is sufficiently general to be applied in most manufacturing situations.

Lati and Gilad [7] have developed an algorithm for reducing losses in the semiconductor industry, called the MinBIT (minimizing bottleneck idle time) algorithm, which represents a new method for sequencing the handler’s moves; the authors also highlight its application in other industries to bring

*Corr. Author’s Address: Technical College of Applied Studies, Svetog Save 65, Cacak, Serbia, jelena.jovanovic@vstss.com

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 512-524

about cycle time reductions and throughput increases.

Hermann and Chincholkar [8] describe a decision support tool that can help a product development team to reduce manufacturing cycle time as early as in the product design phase. The design for production (DFP) tool determines how manufacturing a new product design affects the performance of the manufacturing system, taking into account the capacities available and estimating the manufacturing cycle time.

Bottleneck control in real time production [9], prioritizing machine fleet preventive maintenance [10], spare parts inventory for maintenance, optimization of initial buffer adjustment [11], reduction of machine setup time [12] and predicting order lead times [13], can lead to production effects improvement and manufacturing cycle time reduction.

Ko et al. [14] investigate the possibility of reducing cycle times in mass production by input and service rate smoothing.

Based on the analysis of cited works, the following can be concluded: • A generally accepted approach is to use the flow coefficient as a measure of the manufacturing process efficiency, which rests upon the comparison between the accomplished and technological (ideal) values of the manufacturing cycle.

• Investigations most commonly focus on the cycle within the framework of one-off and smallscale production, where technological values are determined in terms of the consecutive mode of workpiece movement and large-scale and mass production, and where technological values are determined in terms of parallel moves (analysis involves takt time and technological line productivity).

• The technological cycle duration, under conditions of serial production characterized by discontinuity, is calculated using the formulas for the consecutive or parallel mode of workpiece move, depending on the size of the production lot and the author’s assessments.

In terms of theoretical considerations and industrial practice, it is of crucial importance to master the key parameters that affect the manufacturing cycle duration under the conditions of complex-product serial production with dominating interruptible processes conditioned by complex and diverse technologies. The manufacturing process includes highly productive machines, having standard and specialized applications, with high concentrations of technological operations, but also the universaltype equipment with expressed differentiation of

operations. The technological procedure embraces both productive and non-productive operations with the involvement of technologies used to change the workpiece shape and features.

All the aforementioned manufacturing conditions require an integrated approach in investigating the manufacturing cycle that should enable its permanent reduction through dynamic and cyclic process oriented to: • generating an exact theoretical framework for calculating the technological cycle duration based on a combined mode of organizing the sequence of technological operations, • identifying the causes of losses, measuring their effects on manufacturing capacity utilization level and cycle duration, • manufacturing cycle design with scheduled losses that are lower than planned, taking into account the optimal overlap of activities, and • production launching, analysing and measuring the manufacturing process efficiency based on a comparison of real and designed values.

Based on all the above-mentioned factors, it can be inferred that cycle time duration is a stochastic quantity directly affected by: 1. factors related to product development and production program (e.g. total number, types, quantities and product complexity), 2. manufacturing capacity of the system and manufacturing process automotive level (e.g. human resources, equipment, space), 3. financial potentials (e.g. current assets, input inventories, size and structure of unfinished production), 4. technologies applied and manufacturing equipment layout (e.g. workplaces), 5. volume of production and modes of workpiece move in the manufacturing process (e.g. optimum production lot, type of production), 6. factors related to the adopted principles of manufacturing and informatics support in all material flow phases, 7. methods applied in production planning, monitoring and management, and 8. causes of cycle losses.

Total manufacturing cycle time (Fig. 1) is a highly complex quantity composed of a range of components, measurable and non-measurable, to be identified, whose conditionality and behaviour regularity has to be established. Cycle time duration consists of productive and non-productive time. Productive time is defined by technological operations related to changes in the workpiece’s shape and property, while

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non-productive time involves operations related to transport and control.

q, Eq. (2). Technological manufacturing cycle Tt , which is also an ideal cycle Tci, Eq. (3), includes the time needed for performing all n operations predicted by the technological procedure, on the products of a single lot. Production organization plays a critical role in determining the technological cycle, in which moves may be consecutive Tt(u) (Eq. (4)), parallel Tt(p) (Eq. (5)) and combined Tt(k) (Eqs. (6) or (7)), depending on the type of production, consist of a complex set of features.

P=

{( X , x ) j

}

j , i ∈ N , (1)

i

{

}

xi : A = {n,θ n , q} , θ n = ti i = 1, n , (2)

{

}

Tt = Tci = Tt ( p ) , Tt ( k ) , Tt (u ) , Tt ( p ) ≤ Tt ( k ) < Tt ( u ) , (3) n

Tt (u ) =

1 TECHNOLOGICAL (IDEAL) MANUFACTURING CYCLE The production program of the plant P, Eq. (1), consists of products Xj and parts xi. The process of parts manufacturing is defined by a set of data A composed of the number of technological operations n, ordered set θn of times per operation ti and lot size 514

(∀x i = 1, m ) ∈ P ∧ A = {n,θ , q}, (4)

,

i =1

H

i

n

n

Fig. 1. Several factors on manufacturing cycle time duration

The exposition of the investigation results will be organized into three sections treating: • Theoretical and technological bases for technological (ideal) manufacturing cycle time scheduling, depending on the mode of organizing the sequence of operations, with investigation results.

• Investigation of real manufacturing cycle time duration, manufacturing capacity utilization level (machines, human resources in manufacturing) and causes of losses.

• Manufacturing cycle scheduling of a chosen product, production launching according to the designed model and investigations of the flow coefficient K based on a comparison of realized and designed states in the production process (Kp), i.e. in terms of comparisons between real and technological (ideal) cycles (Kt) calculated using formulas for combined modes of workpiece movements.

q ⋅ ∑ ti

Tt ( p ) =

∑ t + ( q − 1) ⋅ t i =1

i

H

(∀x

max

)



n

∑ t + ( q − 1) ⋅  ∑ t − ∑ t i =1

Tt ( k ) = k −1

}

i = 1, m ∈ P ∧ A = {n,θ n , q} ,

i

(t

{

, tmax = max ti i = 1, n ,

i

) (

 H

k

k

j

j

(5)

  ,

)

< tk ≥ tk +1∀k = 1, n ∧ t j −1 ≥ t j < t j +1∀j = 2, n − 1 ,

(∀x

)

i = 1, m ∈ P ∧ A = {n,θ n , q} ,

i

n

n

Tt ( ) = k

∑ t + ( q − 1) ⋅ ∑ ( t i =1

i

i =1

i

− ti −1 ) ⋅ Fi

H

(6)

,

ti > ti −1 ⇒ Fi = 1 ∧ ti ≤ ti −1 ⇒ Fi = 0 ∧ t0 = 0,

(∀x

i

)

i = 1, m ∈ P ∧ A = {n,θ n , q}.

(7)

Depending on the time units in which the cycle time duration is expressed, the values of parameter H are determined by Eqs. (8) to (11).

H = 1 → Tt [norm hours/lot],

(8)

H = CS → Tt [shift/lot], (9)

H = CS · Sd → Tt [workdays/lot], (10)

H = Cs ⋅ S d ⋅

Jovanovic, J.R. – Milanovic, D.D. – Djukic, R.D.

D 1 , δ = k → Tt [ calendar days / lot ] ,(11) δ Dr

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 512-524

emerging at some workplaces (operations) of a parallel type due to the different durations of successive operations, Fig. 2.

where: ti total time per technological operation in norm hour/piece, Cs effective working hours in a shift, Sd number of shifts per day, Dk total number of calendar days in a corresponding period of time, Dr total number of workdays in a corresponding period of time, tmax run-time length of the longest technological operation, tk , tj technological operations that satisfy the condition from Eq. (6), Fi a constant that takes the value of 1 or 0, δ calendar and workdays ratio.

The combined type of work flow in a manufacturing process is most often encountered in serial production. Its goal is to eliminate downtimes

Fig. 2. Graphic representation of a combined mode of workpiece move in the manufacturing process

Order of operation

Machine code

Table 1. Sequence of technological operations with norms and technological cycle values for a job order lot of q = 30,000 pieces, in calendar days

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

2 M1 M2 M1 M3 M4 M5 M6 M7 M8 M9 Σ

Capacity in a shift qS

Time per operation

[piece/shift]

[cmh/piece] 4 6 40 8 134 8 40 8 172 8 40 8 172 172 300 8 50 92 40 92 30 92 2 1522

i

3 125000 18750 93750 5600 93750 18750 93750 4300 93750 18750 93750 4300 4300 2500 93750 15000 8200 18750 8200 25000 8200 375000

H = Cs ⋅ S d ⋅

ti

Parameters for technological cycle calculations

tmax

tk

tj

Fi

(ti – ti–1) · Fi

5 300 -

6 40 134 40 172 40 172 300 92 92 92 1174

7 8 8 8 8 8 172 8 40 30 290

8 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 -

9 6 34 126 32 164 32 164 128 42 42 52 62 884

1 Dk 365 = 7.5 ⋅1.3 ⋅ 0.704 = 6.866, δ = = = 1.42, δ Dr 257

Tt(u) = 66.50 ≈ 67 [cal. days/lot], Tt(p) =13.11 ≈ 14 [cal. days/lot], Tt(k) =38.62 ≈39 [cal. days/lot] Manufacturing Cycle Time Analysis and Scheduling to Optimize Its Duration

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1.1 Calculations of Technological Cycle for an Analysed Product The analysed product is manufactured in three displaced organizational units: 1120, 1170 and 1630.

The manufacturing process engages nine different machines (M1 to M9). Ten types of technology are applied, arranged into four groups: heat treatment (one type of technology), mechanical processing/ deformation, cutting (five types of technologies), chemical preparation (two types of technologies) and surface finish (two types of technologies). Of 22 technological operations, 17 are manufacturing (six operations are related to change in shape, 11 to change of characteristics) and five operations are nonmanufacturing (four operations refer to control, one to transport). The total time needed to produce a single part amounts to 0.017 norm hours. The norm structure is composed of 20% machine time (only a machine is operating), 59% combined time (both a handler and a machine are operating simultaneously) and 21% manual time (only handlers are engaged).

On the grounds of technological procedure and Eqs. (4) to (11) for a job order lot of 30,000 parts, Table 1 shows data needed for calculations of the technological cycle as well as the cycle values for consecutive, parallel and combined modes of workpiece movement, Eq. (12). Technological cycle Tt(k) is 2.95 times longer than Tt(p) and by 1.72 times shorter than Tt(u), respectively.

The obtained results confirm the correctness of the approach in which the flow coefficient Kt is defined as a real to technological Tt(k) ratio instead of Tt(u) or Tt(p) as has been the practice to date.

Tt ( p ) ≅ 14 < Tt ( k ) ≅ 39 < Tt (u ) ≅ 67. (12)

2 ANALYSIS OF CYCLE TIME AND CAUSES OF LOSSES IN MANUFACTURING CAPACITY Technological cycle Tt is an ideal manufacturing cycle Tci because the corresponding Eqs. (4) to (11) do not include losses in the cycle that are unavoidable in the manufacturing process. Unlike technological, cycle time, real duration Tcs includes all generated losses Gcs. Methods for data collecting on manufacturing cycle time duration Tcs can be arranged into three groups.

The first group of methods is based on the analysis of manufacturing, planning and other documentation for the system, when it is possible to establish the start and end dates for the manufacturing 516

process. The most commonly used manufacturing documentation items are job order documents (term cards, material requisition, worksheets, delivery notes, etc.), documents of technical control and various reports on the current state of the manufacturing process. For the processes that are rarely repeated, e.g. performed once a year or once in six months, planning documentation and other documents are used, relating to supply, reception, storage and sales for the approximate definition of benchmark points.

The second group of methods includes those based on the measurement of cycle time duration and its components. The chrono-metering method is applied for shorter cycle time durations, while the method of current observations is used for longer cycle time durations that are frequently repeated.

Cycle time duration is measured on a representative sample of parts.

The third group is based on estimating the total duration of cycle times. These methods are applied for cases in which the above methods are not applicable or require much effort.

2.1 Real Cycle, Losses in the Cycle and Flow Coefficient The analysis of manufacturing documentation (term cards and reports on the current state of the manufacturing process) was used to determine cycle time duration Tcs for a chosen part based on data about the realized start and end dates of production.

Total losses in the cycle Gcs are calculated with the help of Eq. (13) when the technological cycle duration Tt is subtracted from the real cycle time duration, paying attention to the type of production.

Total losses consist of intra-operational Guo and interoperational Gmo losses.

Since the company practices a serial type of production, the total losses in the cycle Gcs, average losses per operation ε and flow coefficient Kt, representing the correlation between real and technological cycle time duration, will be calculated using Eqs. (14) to (16). Various approaches to determining the correlation between real and theoretical cycle time duration can be also found in papers [15] and [16].

Gcs = Tcs − Tt = Guo + Gmo , (13)

Gcs = Tcs − Tt ( k ) , (14)

Jovanovic, J.R. – Milanovic, D.D. – Djukic, R.D.

ε=

Gcs , (15) n

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 512-524

Kt =

Tcs . (16) Tt ( k )

After the cycle time analysis has been completed, the results are presented in Table 2 and Figs. 3 and 4. A total of 13 job orders, seven in 2010 and six in 2011 for the quantity of 30,000 pieces each year were analysed.

production, it can be inferred that the production planning and management process is uncontrollable, experience-based, lacking identified and quantified causes of losses in the cycle.

Tt ( k ) = 38.62, 78 ≤ Tcs ≤ 200, 1.8 ≤ ε ≤ 7.3, 39.4 ≤ Gcs ≤ 161.4, 2.02 ≤ K t ≤ 5.18.

Fig. 4. Flow coefficient Kt per job order

Fig. 3. Technological and real cycle time per job order

Flow coefficient Kt (Fig. 4) takes values within 2.02 to 5.18 range. Total losses in the cycle (Gcs) measured against the technological value Tt(k) range from 39.4 to 161.4 calendar days, and losses are higher, on average, by 2.24 times than technological cycle duration. Taking into account the parameters from Eq. (17) and the fact that this is a serial repeating

(17)

2.2 Utilization of Machine Capacity and Structure of Losses per Cause of Downtime Taking into account the obtained results, Eq. (17), and the importance of the analysed product that is on a critical path of the manufacturing cycle of four complex products making up the framework of the 2010 and 2011 production programs, the causes and

Table 2. Real manufacturing cycle time duration Tcs, total losses in the cycle Gcs, losses in the cycle per operation ε and values of flow coefficient Kt, in calendar days

1 1 2 3 4 5 6 7 8 9 10 11 12 13

Lot 2 04/10 05/10 06/10 07/10 08/10 09/10 10/10 01/11 03/11 04/11 05/11 06/11 07/11

Job order (JO) Manufacturing date Quantity Start End 3 4 5 30000 30.08.2010 17.11.2010 30000 30.08.2010 17.01.2011 30000 30.08.2010 16.02.2011 30000 30.08.2010 17.03.2011 30000 18.09.2010 31.03.2011 30000 16.12.2010 08.04.2011 30000 16.12.2010 21.04.2011 30000 23.02.2011 06.06.2011 30000 09.04.2011 25.06.2011 30000 09.04.2011 06.07.2011 30000 07.07.2011 07.10.2011 30000 07.07.2011 23.10.2011 30000 08.09.2011 13.01.2012 Average value:

Cycle time duration

Losses

Tt(k)

Tcs

Gcs

ε

6

7 80 141 171 200 195 114 127 104 78 89 93 109 128 125.3

8 = 7–6 41.4 102.4 132.4 161.4 156.4 75.4 88.4 65.4 39.4 50.4 54.4 70.4 89.4 86.7

9 1.9 4.7 6.0 7.3 7.1 3.4 4.0 3.0 1.8 2.3 2.5 3.2 4.1 3.9

38.62

No

Manufacturing Cycle Time Analysis and Scheduling to Optimize Its Duration

Flow coefficient

Kt

10=7/6 2.07 3.65 4.43 5.18 5.05 2.95 3.29 2.69 2.02 2.30 2.41 2.82 3.31 3.24

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measurements of losses were investigated with regard to the manufacturing capacity. The causes of losses (i) were identified (Eq. (18)), and machine capacity utilization level ηm was established for the operations with machines and combined time cycles (Eq. (19)), with respect to cycle times in the structure of normhour per technological operation. Machine capacity utilization level and current losses, per downtime cause, can be measured with different techniques [17].

In this investigation, the method of current observations (MCO) was employed. The MCO is based on mathematical statistics and sampling theory. To apply the MCO, it is necessary to define the representative sample, choose the time period for screening (e.g. day, month, shift, number of observations required per shift), perform preparations for screening (e.g. train screeners, identify causes of losses, prepare screening sheets, define mode of data recording and processing, screener’s route and sequence of screening the machines).

Test screening identified eight causes of losses for machine capacity (i): machine breakdown (K), tool insufficiency and failure (A), waiting for a workpiece from the preceding operation (I), downtime caused by handler’s lack of discipline (C), material shortage (M), waiting for the workpiece from another organizational unit (V), lack of jobs (X) and other causes (e.g. downtimes due to power failure, fluid shortage, strikes) (O). In addition to the abovementioned eight characteristics related to current causes of losses, another two characteristics screened were associated with the operation of machines: machine is operating (T) and preparation-finish jobs (P).

8

g m = ∑ gi , i = {K , A, M , C , I ,V , O, X } , (18)

i =1

ηm =

n ( −) n( + ) , gi = i , (19) n n

where: total losses of machine capacity, gm gi machine capacity losses per downtime cause (i), n total number of observations, n(+) number of observations when the machine is running, ni(−) number of observations for machine

downtime per downtime cause (i).

Table 3. Experiment plan for measuring machine capacity utilization level and current losses, using the MCO method, accomplished in 2010 No

OU

No of machines

1 1 2 3

2 1120/I 1170 1120/II

3 2 4 3

Data for the experiment

n11 4 20 20 20

n12 5 20 20 20

n1

n2

6=4+5 40 40 40

n3

7 10 10 10

8 12 12 12

n*

m

No of features

9=6×7×8 4800 4800 4800

10=3×9 9600 19200 14400

11 10 10 10

Note: n11 no of observations per machine in shift 1, n12 no of observations per machine in shift 2, n1 no of observations per machine/day,

n2 no of screening days per month, n3 no of screening months per year, n* total number of observations per machine/year, m total number of observations for all machines per organizational unit

Table 4. Utilization level ηm, total gm and partial losses gi of machine capacity, per cause of downtime and machines, and scheduled parameters values μm No

Mi

ηm

1 1 2 3 4 5 6 7 8 9 Average

2 M1 M2 M3 M4 M5 M6 M7 M8 M9

3 0.4600 0.5600 0.5350 0.4500 0.4900 0.5150 0.5600 0.6950 0.3700 0.5150

518

gM

4 0.1250 0.0000 0.0500 0.0000 0.0850 0.1200 0.0000 0.0000 0.0000 0.0422

gA

Values of partial losses gi, per cause of downtime (i)

5 0.0200 0.0250 0.0650 0.0500 0.0500 0.0250 0.0450 0.0600 0.0050 0.0383

gK

6 0.1150 0.0000 0.1450 0.1050 0.0400 0.0150 0.0750 0.0200 0.1650 0.0756

gC

7 0.0450 0.0250 0.0300 0.0250 0.0550 0.0150 0.0500 0.0900 0.0200 0.0394

gX

8 0.1800 0.2150 0.1650 0.2150 0.2500 0.3000 0.0900 0.0000 0.3850 0.2000

gI

9 0.0000 0.1450 0.0000 0.1400 0.0000 0.0000 0.1350 0.0900 0.0550 0.0628

Jovanovic, J.R. – Milanovic, D.D. – Djukic, R.D.

gV

10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

gO

11 0.0550 0.0300 0.0100 0.0150 0.0300 0.0100 0.0450 0.0450 0.0000 0.0267

gm

μm

12 0.5400 0.4400 0.4650 0.5500 0.5100 0.4850 0.4400 0.3050 0.6300 0.4850

13=3+4+8 0.7650 0.7750 0.7500 0.6650 0.8250 0.9350 0.6500 0.6950 0.7550 0.7572

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 512-524

Table 3 shows the experiment plan, and Table 4 displays the results obtained by the program developed in paper [18]. Table 4 (column 13) shows scheduled values for machine capacity utilization levels μm, Eq. (20) to be used for scheduling manufacturing cycle time optimum values Tcp. The scheduled capacity utilization level (μm) is essentially the potential of each of nine engaged machines (Mi). Its value, Relation (20), includes losses caused by material shortages (gM) and lack of jobs (gX).

The average machine’s utilization engaged in the manufacturing process of an analysed product in 2010 amounts to 51.5%. Individually, per machine, it ranges from 37 to 69.5% (Table 4, column 3).

paid and unpaid leave (o), national holidays (dp), and engagement in other jobs (d). The total number of employed workers (zu) is the sum of productive (z) and clerical (za) workers. The average number of productive workers present at work every day (zr) is the difference between the total number of productive workers and the average number of productive workers absent from work for all the abovementioned reasons (zg), Eq. (23). The scheduled norm hours (nh) load per worker (Fnc), coefficient of productive worker overtime engagement (ξr) and scheduled productive worker utilization level (μr) can be calculated by employing Eqs. (24) to (26).

µm = η m + ( g M + g X ) . (20)

ηr =

8

2.3 Productive Human Resources Utilization Level and Current Losses per Cause of Working Hour Loss

zr , (21) z

}

j = b1 , b2 , i, g o , pr , o, d p , d , (22)

j =1

zu = z + za , zr = z − z g , (23)

The goal of each business-manufacturing system is to have the optimum number of employees (zu) in both administration (za) and production (z). Investigations of the causes for losses in working hours and productive human resources utilization level ηr are of significance for workplaces and technological operations with prevailing manual work [19].

In order to identify and reveal regularities in causes for working hour losses, corresponding data were collected and analysed in human resources department in the year 2010. The human resources utilization level ηr, Eq. (21), and current losses (zj), per cause (j), were established based on handlers’ work records. The analysis indicated eight causes of working hour losses (j), Eq. (22): sick leave up to 30 days (b1), sick leave over 30 days (b2), unexcused absence from work and tickets out (i), holiday (go), downtime (e.g. lack of jobs, strikes, vis major) (pr),

z

=

{

zg = ∑ z j ,

z − zg

Fnc = Dr ⋅ Cs ⋅ p n ,

ξr =

pn =

∑ NC , (24) ∑ EC

zr ⋅ Fnc + Pe , (25) zr ⋅ Fnc

µr = η r ⋅ ξ r . (26)

The scheduled norm hours load per worker (Fnc) is obtained as the product of the total number of working days (Dr), effective working hours in a shift (Cs) and the average norm hour [nh] execution (pn) for the observed organizational unit and a corresponding time period. The average norm-hour is obtained when the executed norm-hours (NC) are divided by the engaged effective working hours (EC) of productive

1 800

[workers/year]

zg

[%]

zb1

zb2

zi

zgo

zdp

zpr

zo

zd

[%]

[workers/ year]

2 6.15

3 1.8

4 1.35

5 11.4

6 2.85

7 1.8

8 1.35

9 0.3

10 27

11 216

zr

zj, j = {b1, b2, i, go, dp, pr, o, d}

z

[workers/year]

Table 5. Productive human resources utilization level ηr, total zg and partial zj losses of working hours, and scheduled parameter values μr

12=1-11 584

ηr

Pe

[%]

[nh/ year]

Fnc

ξr

13 73

14 163917

15 2217

16 1.13

Fnc = Dr ⋅ Cs ⋅ pn = 257 ⋅ 7.5 ⋅1.15 = 2217 [nh/worker annually],

ξr =

zr ⋅ Fnc + Pe 584 ⋅ 2217 + 163917 = = 1.13, µ r = η r ⋅ ξ r = 0.73 ⋅1.13 = 0.82. zr ⋅ Fnc 584 ⋅ 2217 Manufacturing Cycle Time Analysis and Scheduling to Optimize Its Duration

519

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workers per worksheet. The coefficient of overtime engagement depends on the specificity of workplaces and planned level of overtime hours (Pe).

Table 5 show parameters relevant to the utilization of productive human resources (PHR).

Due to an adverse age structure, working hours losses are high on account of sick leave (b1, b2) and holiday (go). Of the total of 800 workers, 27% or 216 workers are, on average, absent from work. Productive human resource (PHR) utilization level amounts to 73%.

Taking into account the coefficient of overtime engagement (ξr), the scheduled utilization level of PHR (μr) equals 82%.

3 MANUFACTURING CYCLES SCHEDULING

τi =

q ⋅ δ , (28) qsi ⋅ pi ⋅ S di ⋅ µi ⋅ ri k

(

)

Tcp = τ 1 + ( n − 1) ⋅ ∆τ + ∑ τ p − τ p −1 ,

p

(

)

p = i i = 2, n ∧ τ i > τ i −1 → ∑ p = k , (29) k

τ 1 + ( n − 1) ⋅ ∆τ + ∑ (τ p − τ p −1 ) = Tt + Gcp → p

k

∆τ =

(

Tt + Gcp − τ 1 − ∑ τ p − τ p −1 p

)

n −1

,

(30)

The optimization of manufacturing cycle times Tcp requires, first of all, investigations of losses causes, measurement of their values, minimization of their effects and scheduling of total losses Gcp to be lower than made Gcs, Fig. 5.

where: τi scheduled duration of technological operations, Δτ average partial loss between technological operations, p technological operation satisfying the condition from Eq. (29).

Fig. 5. Technological, scheduled and real manufacturing cycle time duration with corresponding losses

Fig. 6. Scheduled loss Δτi, τi > τi–1

When scheduling the manufacturing cycles, the scheduled cycle duration Tcp should tend to the optimum, Fig. 5. In other words, the goal of scheduling as a cyclic process is to permanently tend to the minimization of total losses, which means that scheduled losses (Gcp) should always be less than generated (Gcs) in all optimization steps, Eq. (27).

Tcp = Tt + Gcp , Tcs = Tt + Gcs = Tcp + G,

Tt < Tcp < Tcs → Gcp < Gcs .

(27)

The scheduled manufacturing cycle duration Tcp, Eq. (29), Figs. 6 and 7, aside from productive and non-productive cycle times, predicted by technological procedures, take into account scheduled manufacturing capacity utilization levels μi(μm, μr), Eqs. (20) and (26), real manufacturing conditions per operation, Eq. (28) and scheduled losses in a cycle (Gcp, ∆τ), Eq. (30).

520

Fig. 7. Scheduled loss Δτi, τi ≤ τi–1

3.1 Algorithm for Cycle Scheduling The first step in the scheduling process is calculating the manufacturing cycle time duration per operation (τi) using Eq. (28), with respect to real manufacturing conditions: the number of workplaces

Jovanovic, J.R. – Milanovic, D.D. – Djukic, R.D.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 512-524

per technological operation (ri), the number of shifts per day (Sdi), the average norm-hour execution (pi), manufacturing capacity utilization μi (μm, μr) and the norm-set capacity per shift (qSi), Table 6. Inventories in unfinished production and losses due to quality inadequacy are included in calculations via the formulas for planning the quantity (q) of the product to be produced.

3 1 2 1 2 1 2 1 2 1 2 1 2 2 2 1 2 1 2 2 1 1 1

6 0.820 0.765 0.820 0.775 0.820 0.765 0.820 0.750 0.820 0.665 0.820 0.825 0.935 0.650 0.820 0.820 0.820 0.695 0.820 0.755 0.820 0.820

7 0.38 0.73 0.51 1.57 0.49 0.73 0.50 1.50 0.51 0.81 0.51 1.43 1.29 2.08 0.49 0.75 1.89 0.74 1.34 2.11 1.87 0.12

τp – τp–1

5 1.10 1.02 1.08 1.04 1.13 1.02 1.10 1.10 1.09 1.06 1.09 1.05 1.03 1.05 1.13 1.15 1.12 1.10 1.18 1.07 1.13 1.18

[calendar days]

4 1 2 1 3 1 2 1 4 1 2 1 4 4 6 1 2 3 2 2 1 3 1

τi

operations

μi

p

Average execution of norm-hours pi

Sdi No of shifts

qSi 2 125000 18750 93750 5600 93750 18750 93750 4300 93750 18750 93750 4300 4300 2500 93750 15000 8200 18750 8200 25000 8200 375000

No of workplaces ri

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Capacity in a shift

Order of operation

Table 6. Parameters for scheduling the manufacturing cycle

8

9

√

0.35

√

1.06

√

0.24

√

1.00

√

0.30

The scheduled value of total losses Gcp can be also found using Eq. (32).

The expected cycle time duration Tc will be calculated by applying Eqs. (6) or (7) if the values of the technological times per operation ti (Table 1, column 4) are corrected by the corresponding coefficients μi (Table 6, column 6). Data required for calculating the expected values of cycle duration Tc are given in Table 7.

According to the investigations [20], total losses in the cycle in the company follow the beta distribution, where the value Gc˝= Mo = 15.97 has the highest probability (modal value).

Based on the above analysis, total loss Gcp of 16 calendar days was adopted, and partial loss Δτ was calculated in calendar days, Eq. (33): k

0.92

√

0.79

√ √

0.26 1.13

√ √

Σ

0.60 0.77

7.42

In the second step, it is necessary to adopt the total losses in the cycle Gcp, and, then using Eq. (30), to determine average partial losses between technological operations Δτ. Total scheduled losses in the cycle should be lower than those average (86.7 calendar days). They can be determined in a number of ways: by the help of Eq. (31) adopting the minimum value of losses in the cycle Gcs (Table 2, column 8).

Gcp ≤ min Gcs , Gcp ≤ 39.4, min Gcs = min {41.4 102.4 132.4 ... 89.4}. (31)

∆τ =

(

Tt + Gcp − τ 1 − ∑ τ p − τ p −1 p

n −1

)

= 2.2. (33)

The third step implies calculating scheduled values for the cycle Tcp, in calendar days, using data from Table 6 and Eq. (29):

√

Gcp = Tc − Tt ( k ) . (32)

k

(

)

Tcp = τ 1 + ( n − 1) ⋅ ∆τ + ∑ τ p − τ p −1 = 54. (34) p

Correlation between real Tcs and scheduled Tcp manufacturing cycle time duration is determined by the flow coefficient Kp, Eq. (35).

Kp =

Tcs . (35) Tcp

3.2 Production Launching According to the Scheduled Model and Analysis of Results The scheduled mode of manufacturing (Tcp) was realized in two production lots (job order lot 11/11 and 13/11) in 2011 and 2012. In both job order lots, the quantity of 30,000 pieces each was launched. Prior to production initiation, in the ‘Term card’ document, the scheduled start and end dates for the manufacturing process per operation were recorded (with respect to the results obtained in Section 3.1).

When determining and recording the dates in the ‘Term card’, it is necessary to calculate temporal reserves between the end and start dates of production

Manufacturing Cycle Time Analysis and Scheduling to Optimize Its Duration

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Table 7. Parameters for calculating expected cycle time duration Tc Order of operation

Machine code

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

2 M1 M2 M1 M3 M4 M5 M6 M7 M8 M9 -

Time per operation ti [cmh/piece] 3 6 40 8 134 8 40 8 172 8 40 8 172 172 300 8 50 92 40 92 30 92 2

Parameters for calculating technological cycle time

μi

Corrected time per operation [cmh/ piece]

tk

tj

Fi

4 0.820 0.765 0.820 0.775 0.820 0.765 0.820 0.750 0.820 0.665 0.820 0.825 0.935 0.650 0.820 0.820 0.820 0.695 0.820 0.755 0.820 0.820

5=3/4 7 52 10 173 10 52 10 229 10 60 10 208 184 462 10 61 112 58 112 40 112 2

6 52 173 52 229 60 208 462 112 112 112 -

7 10 10 10 10 10 184 10 58 40 -

8 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0

9 7 45 0 163 0 42 0 219 0 50 0 198 0 278 0 51 51 0 54 0 72 0

342

-

1230

G

Kp

8=7-6 17 7 12

9=7/6 1.31 1.13 1.22

1984 1572 Σ 1522 = (1984 + 29999·(1572–342)) / 100000 / 6.866 = 53.74 [calendar days/lot] Eq. (32) → Gcp = 53.74 – 38.62 = 15.12 ≈16 [calendar days/lot]

(ti

– ti–1)· Fi

Eq. (6) → Tc

Table 8. Parameters of the cycle established after production is finished according to the scheduled model Job order No 1 1 2

Lot

Quantity

2 11/11 13/11

3 30000

Manufacturing cycle Date

Start 4 24.10.2011 05.12.2011

End 5 02.01.2012 03.02.2012

Tcp

Tcs

6

7 71 61 66

54 Average value:

on a current operation. Their value depends on the scheduled partial loss Δτ and organizational mode of production (Figs. 6 and 7). During production activities per technological operation, the start and end dates of production were added to the column denoting scheduled dates.

After production is finished according to the designed model, Table 8 shows realized manufacturing cycle time duration (Tcs), losses in the cycle (G) and values of the flow coefficient Kp, Relation (35).

Table 9 displays parameters (Tcs, G, Kp, Kt) for 15 job orders of the analysed part, whose production 522

was realized in the period from 2010 to 2012. In the first 13 job orders (Nos 1 to 13), the manufacturing cycle was not scheduled; therefore production process management was performed based on experience and priorities defined by operational managers.

The achieved values of the cycle Tcs (Table 9, column 5, Nos 14 and 15) are considerably lower than the analysed values of the cycles per job order (Table 9, column 5, Nos 1 to 13). The flow coefficient Kp (1.13 to 1.31) takes considerably lower values than the coefficient Kt (2.02 to 5.18), which implies that

Jovanovic, J.R. – Milanovic, D.D. – Djukic, R.D.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 512-524

Table 9. Values of the flow coefficients (Kp, Kt) before and after production launching according to the designed model No 1 1 2 3 4 5

Lot 2 04/10 05/10 06/10 07/10 08/10

6

09/10

7 8 9 10 11

10/10 01/11 03/11 04/11 05/11

12 13

Job order Quantity 3

Manufacturing cycle and losses Tcp Tcs G 4 5 6= 5-4 80 26 141 87 171 117 200 146 195 141

Kt

Production

7=5/4 1.48 2.61 3.17 3.70 3.61

8 2.07 3.65 4.43 5.18 5.05

9

Experiencebased production

114

60

2.11

2.95

127 104 78 89 93

73 50 24 35 39

2.35 1.93 1.44 1.65 1.72

3.29 2.69 2.02 2.30 2.41

06/11

109

55

2.02

2.82

07/11

128 125.3 71 61 66

74 71.3 17 7 12

2.37 2.32 1.31 1.13 1.22

3.31 3.24 1.84 1.58 1.71

30000

54

Average value: 14 15

Kp

11/11 13/11

30000

54

Average value:

the coefficient Kp is more suitable for measuring the production process efficiency than the coefficient Kt. 4 CONCLUSION AND FURTHER RESEARCH The goal of the original approach demonstrated in this work is to reduce manufacturing cycle time to the maximum, taking into account serial production characterized by discontinuity and the considerable amount of current assets needed for financing the production process. This methodology is based on designed models that, on one hand, respect current technical-technological and manufacturing documentation and, on the other, real production constraints. The parameters for reducing manufacturing cycle are flow coefficients Kp and Kt (Fig. 8).

When measured by flow coefficients Kp and Kt, the average realized manufacturing cycle time duration, according to the designed model, is lower by 1.9 times (Table 9, columns 7 and 8), while the average losses in the cycle are smaller by 5.9 times (Table 9, column 6). Viewed from the angle of the manufacturing system, the flow coefficient Kp has a higher use value (Eq. (35)), because the accomplished values of the cycle are correlated with scheduled (planned) values. In this context, the model design becomes a cyclical process with the aim of minimizing total losses and reducing them to an optimal (i.e. acceptable) level. However, to compare the results with other business-manufacturing systems, from

Designed model-based production

the region and more distant areas, priority should be given to flow coefficient Kt, Eq. (16), because the values achieved for the cycle are compared to the technological (ideal) cycle, which is calculated for the case of serial production and combined workpiece move using Eqs. (6) or (7).

Fig. 8. Values of flow coefficients Kp and Kt per job order before and after scheduling

The results related to the identification of downtime causes and losses measurement are of importance not only for the cycle scheduling, but also for optimal production planning.

Further research should be directed to the analysis and scheduling of manufacturing cycle for complex products. It is necessary to define models for describing the structures of complex products

Manufacturing Cycle Time Analysis and Scheduling to Optimize Its Duration

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with respect to technological documentation and techniques that allow for the scheduling approach to realization of orders. The designed solutions, that are based on the principles of lean production, should make provisions for software application solutions from the domain of project management.

5 REFERENCES [1] Slak, A., Tavcar, J., Duhovnik, J. (2011). Application of genetic algorithm into multicriteria batch manufacturing scheduling. Strojniški vestnik - Journal of Mechanical Engineering, vol. 57, no. 2, p. 110-124, DOI:10.5545/sv-jme.2010.122.

[2] Suzic, N., Stevanov, B., Cosic, I., Anisic, Z., Sremcev, N. (2012). Customizing products through application of group technology: A case study of furniture manufacturing. Strojniški vestnik - Journal of Mechanical Engineering, vol. 58, no. 12, p. 724-731.

DOI:10.5545/sv-jme.2012.708.

[3] Eckert, C., Clarkson, P. (2010). Planning development processes for complex products. Research in Engineering Design, vol. 21, no. 3, p. 153-171, DOI:10.1007/s00163-009-0079-0.

[4] Alfieri, A., Tolio, T., Urgo, M. (2011). A project scheduling approach to production planning with feeding precedence relations. International Journal of Production Research, vol. 49, no. 4, p. 995-1020, DOI:10.1080/00207541003604844.

[5] Dossenbach, T. (2000). Manufacturing cycle time reduction - a must in capital project analysis. Wood & Wood Products, vol. 105, no. 11, p. 31-35.

[6] Johnson, J.D. (2003). A framework for reducing manufacturing through put time. Journal of Manufacturing Systems, vol. 22, no. 4, p. 283-298, DOI:10.1016/S0278-6125(03)80009-2.

[7] Lati, N., Gilad, I. (2010). Minimising idle times in cluster tools in the semiconductor industry. International Journal of Production Research, vol. 48, no. 21, p.

6443-6459, DOI:10.1080/00207540903280556.

[8] Herrmann, J.W., Chincholkar, M.M. (2000). Design for production: A tool for reducing manufacturing cycle time. Proceedings of the 2000 ASME Design Engineering Technical Conference, Baltimore, from: http://www.isr.umd.edu/Labs/CIM/projects/dfp/ dfm2000.pdf.

[9] Li, L., Chang, Q., Ni, J., Biller, S. (2009). Real time production improvement through bottleneck

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[10] Patti, A.L., Watson, K.J. (2010). Downtime variability: the impact of duration-frequency on the performance of serial production systems. International Journal of Production Research, vol. 48, no. 19, p. 5831-5841, DOI:10.1080/00207540903280572.

[11] Schultz, C.R. (2004). Spare parts inventory and cycle time reduction. International Journal of Production Research, vol. 42, no. 4, p. 759-776, DOI:10.1080/0020 7540310001626210.

[12] Kusar, J., Berlec, T., Zefran, F., Starbek, M. (2010).

Reduction of machine setup time. Strojniški vestnik Journal of Mechanical Engineering, vol. 56, no. 12, p.

833-845.

[13] Berlec, T., Govekar, E., Grum, J., Potocnik, P., Starbek, M. (2008). Predicting order lead times. Strojniški vestnik - Journal of Mechanical Engineering, vol. 54, no. 5, p. 308-321.

[14] Ko, S.S., Serfozo, R., Sivakumar, A.I. (2004). Reducing cycle times in manufacturing and supply chains by input and service rate smoothing. IIE Transactions, vol. 36, no. 2, p. 145-153, DOI:10.1080/07408170490245441.

[15] Puich, M. (2001). Are you up to the cycle-time challenge?. IIE Solutions, vol. 33, no. 4, p. 24-28.

[16] Hadas, L., Cyplik, P., Fertsch, M. (2009).

Method of buffering critical resources in maketo-order shop floor control in manufacturing complex products. International Journal of Production Research, vol. 47, no. 8, p. 2125-2139, DOI:10.1080/00207540802572582.

[17] Djukic, R., Jovanovic, R.J., Mutavdzic, M. (2009).

Investigations of machine capacity utilization level, downtime causes and structure of losses. Proceedings of Jupiter Conference, Belgrade, p. 4.11-4.16. (in Serbian) [18] Djukic, R., Milanovic, D., Jovanovic, R.J. (2010).

Program for establishing machine capacity utilization level. Proceedings of Quality Conference, Kragujevac, from: http://www.cqm.rs/2010/pdf/37/37.pdf. (in Serbian) [19] Djukic, R., Jovanovic, R.J. (2009). Effects of human resources on dynamic management of manufacturing systems. Proceedings of Jupiter Conference, Belgrade, p. 4.1-4.6. (in Serbian) [20] Djukic, R. (2010). Dynamic balance and managing of complex business-manufacturing systems. PhD Thesis.

University of Belgrade, Belgrade. (in Serbian)

Jovanovic, J.R. – Milanovic, D.D. – Djukic, R.D.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 525-535 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1492 Original Scientific Paper

Received for review: 2013-10-08 Received revised form: 2014-03-27 Accepted for publication: 2014-04-14

Design and Optimization of PSD Housing Using a MIGA-NLPQL Hybrid Strategy Based on a Surrogate Model Li, F. – Qin, Y. – Pang, Z. – Tian, L. – Zeng, X. Feng Li1 – Yumo Qin1 – Zhao Pang1 – Lei Tian2 – Xiaohua Zeng3* 1 Jilin

University, School of Mechanical Science and Engineering, China 2 BAIC MOTOR, China 3 Jilin University, State Key Laboratory of Automobile Dynamical Simulation, China A new power-split device (PSD) is designed as the core component of the multi-power coupling system in a hybrid electric vehicle. It is important to consider the influence of the factors of weight and stiffness on performance of PSD when designing PSD housing. In this paper, the overall arrangement of the power distribution system and PSD housing are redesigned. The finite element analysis is conducted to test PSD housing stiffness. According to the results of the analysis, this study adopts a hybrid optimization strategy based on surrogate models to obtain the least weight under the constraint of PSD housing stiffness. The surrogate models are established using a responsive surface method based on the data obtained by the optimal Latin hypercube design (OLHD). The hybrid optimization strategy combines a multi-island genetic algorithm (MIGA) with a nonlinear programming quadratic line search (NLPQL), which ensures obtaining optimal design parameters for PSD housing.

In comparison with optimization using a single MIGA, this hybrid optimization strategy is more efficient and feasible for optimizing housing.

Keywords: hybrid electric vehicle, power-split device, hybrid optimization, surrogate model, multi-island genetic algorithm, nonlinear programming quadratic line search

0 INTRODUCTION The hybrid electric vehicle (HEV), an eco-friendly and energy-saving vehicle, has been a very popular in vehicle design and manufacturing [1] to [4]. The power-split device (PSD) is a core component of a multi-power coupling system in a series-parallel HEV [5], because it can achieve energy coupling and conversion among the engine, motor, and generator under different working modes.

Based on the principle of differential, Yu et al.

[6] proposed a PSD by which the multiple power coupling and decoupling can be achieved. In order to avoid the difficulty in the arrangement and installation of transmission gears in the PSD housing, the design of a PSD is explored in this paper.

In a traditional drive vehicle, the transmission is very compact, and the space is insufficient. However, new drive source and drive components are added in the hybrid electric drive vehicle, so the minimum optimization of the volume and quality is more important. The design and optimization of this kind of PSD are challenging. In particular, lightweight design has attracted intensive effort due to its great contribution to cost, material volume and time savings in engineering design [7] and [8].

With the field of complex engineering, numerous single optimization algorithms, such as the KarushKuhn-Tucker method [9], non-gradient optimization [10], genetic algorithm [11], and particle swarm optimization [12] are proposed to optimize structures.

However, a single optimization algorithm does not

always achieve a global solution [13] and can be time-consuming. Hybrid optimization algorithms, such as the combination of the global search (genetic algorithm or evolution strategies) and the local search (descent method) [14], the SASP method (hybridization of simulated annealing and the descent method) [15], the simulated annealing and the local proximal bundle method [16], the combination of the particle swarm optimization and the genetic algorithm [17], are of significant interest for the rapid speed of convergence and robustness in seeking globally optimal solutions [18].

In order to obtain a globally optimal solution and reduce computing time, this study adopts a hybrid optimization strategy based on the combination of a multi-island genetic algorithm (MIGA; a global optimizer), and a nonlinear programming quadratic line search (NLPQL; a local optimizer). In the MIGA, each group is divided into several subgroups called “islands”. The selection, crossover, and mutation operations are performed in the subgroups, and the immigration operation is periodically performed among different targeted islands. The MIGA is a pseudo-parallel genetic algorithm, which can both avoid a local optimal solution as far as possible and accelerate convergence [19]. When solving constrained nonlinear mathematical problems, the NLPQL algorithm shows stability, rapid convergence, and the capacity to seek globally optimal solutions [20]. As the NLPQL can quickly determine the local optimal solution near the starting point, it can reduce the computing time during the optimization process.

*Corr. Author’s Address: Jilin University, 5988 Renmin Street, Changchun, China. zengxh@jlu.edu.cn

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This study proposes a new PSD and focuses on the sufficient level of stiffness of PSD housing, which ensures the minimum meshing misalignment between planet gear and half-axle gear. In such a case, a hybrid optimization strategy combining the MIGA and NLPQL based on a surrogate model is adopted. In this optimization, finite element analysis (FEA) is used to investigate the PSD housing stiffness. The surrogate models of weight and stiffness of the PSD housing are established. The hybrid strategy demonstrates higher efficiency and a stronger ability to find better solutions compared with a single MIGA.

1.2 Structure of PSD The PSD and the gear transmission train are shown in Fig. 2. The thrust bearing is applied at the shafts of each bevel planet gear to decrease the sliding friction between the bevel planet gears and the PSD housing; furthermore, the uneven distribution of the loading on the gear tooth is also improved.

1 DESIGN OF A NEW PSD 1.1 Transmission System with a PSD An integrated power distribution system including a PSD is shown in Fig. 1. The main parts of this system are the engine, the engine clutch, the gear reduction unit, the input clutch, the PSD, the motor, the gear acceleration unit for the generator, the generator clutch and the generator.

a) PSD housing

b) transmission train

c) mid-housing Fig. 1. Integrated power distribution system of parallel-series HEV with PSD

The torque generated by the engine is introduced by the input axle and is passed to the PSD through the gear reduction unit. Then, according to the principle of the differential mechanism, the torque drives three planet gears and two half-axle gears. The input clutch connects with the left-housing, so that the parts of PSD housing cannot rotate. In this way, the bevel planet gear in PSD only rotates on own axis. The left side of the motor connects with the half-axis gear, and the right side connects with the rear axle. The electricity, which is produced by the generator, is stored in the battery. When the motor is working, the battery supplies electricity to it. The generator is connected to the generator clutch, and its operation can be stopped when the generator clutch disengages.

526

Fig. 2. PSD; 1) left-housing; 2) mid-housing; 3) right-housing; 4) bevel planet gears; 5) left half-axle gear; 6) planet gear axle; 7) right half-axle gear; 8) outer ring 9) inner ring; 10) threaded hole; 11) axle holes; 12) radial plate; 13) ladder hole

The left and right-housing is connected to the mid-housing of the PSD, which is composed of an outer ring and an inner ring. The ladder hole, which is opened on the inner ring, accommodates the roller bearing as well as makes the left spline axle through it. Hence, the driving torque transmitted to left and right half-axle gears becomes an output from one side of PSD. 2 FEA OF PSD HOUSING 2.1 Forces and Boundary Conditions of PSD Housing Bevel planet gears and half-axle gears are the core transmission components of the PSD. The PSD

Li, F. – Qin, Y. – Pang, Z. – Tian, L. – Zeng, X.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 525-535

housing deflection, which is introduced by the insufficient stiffness, usually leads to the meshing misalignment of the gears. The meshing misalignment increases the transmission error and transmission noise, and decreases transmission efficiency and gear life. In this paper, FEA is adopted to evaluate the PSD housing stiffness.

Before the finite element models are created, the loads applied to the PSD housing are computed.

The engine torque M (100 Nm) is transmitted to the PSD housing by the left spline axle, and then M is allocated to the bevel planet gears through planet gear axles. The forces applied to PSD housing are shown in Fig 3.

Ftx = Fax =

4T , (2) dm

4T tan α cos δ + F , (3) dm

M is the torque applied on the bevel 2 nπ planet gears, F = m( ) 2 r is the centrifugal force 30 of the bevel planet gear, m is the weight of the bevel planet gear, n is the rotational speed of the revolution, r is the reference radius of the half-axle gear, α is the pressure angle of the bevel planet gear, δ is the reference cone angle of the bevel planet gear, dm is the reference diameter of the bevel planet gear at the tooth width midpoint.

The boundary conditions should be determined before a simulation is conducted. The engine transmits the input torque M (100 Nm) to the lefthousing, and the torque is distributed to the two halfaxle gears through the differential planetary gear train. In static load analysis, a hypothesis is proposed that the half-axle gears are fixed, so the torque M is applied to the left-housingof the PSD. Fax, Fal and Far can be converted into the pressure P acting on the corresponding plane, the loading results and the boundary conditions of the PSD housing are shown in Fig. 4.

where T =

Fig. 3. Forces applied to PSD housing

Fal and Far are the axial forces of the half-axle gears, which are applied to the left- and right-housing by half-axle gears. Ftx (Ft1, Ft2, Ft3) is the resultant force of the circumferential forces of the bevel planet gears, which is applied to the PSD housing. Fax (Fa1, Fa2, Fa3) is the resultant force of the axial force created by the bevel planet gear meshing transmission and the centrifugal force created by the bevel planet gear revolution, which is also applied to the PSD housing.

Fal, Far, Ftx and Fax are defined as:

Fal = Far =

6T tan α sin δ , (1) dm

Fig. 4. Loads and boundary conditions of PSD housing

Design and Optimization of PSD Housing Using a MIGA-NLPQL Hybrid Strategy Based on a Surrogate Model

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2.2 Results of FEA The loads and boundary conditions are applied to the finite element models, and FEA is conducted. The results of Von Mises equivalent stress are shown in Fig. 5. The maximum stress, approximately 92 MPa, occurs on the joint between the radial plate and inner ring of the mid-housing. Displacements are shown in Fig. 6 in the contour and vector forms. The maximum displacement is 0.025 mm.

stiffness for decreasing the meshing misalignment when optimizing the weight of the PSD housing.

Fig. 7. Deflections of PSD housing

3 OPTIMIZATION BASED ON SURROGATE MODEL

Fig. 5. Von Mises equivalent stress [MPa]

The surrogate model method uses a simple mathematic model to replace a complex structural optimization problem, which can reduce computing costs and simplify problems. This paper adopts it in the design optimization of PSD housing. First, the design variables, the objective functions and the constraint conditions are determined; then, the design of the experiment based on the OLHD is implemented, and the surrogate models are created with response surface method according to the experimental data.

3.1 Mathematical model of optimization 3.1.1 Design Variables Based on the FEA results, the structure sizes of the inner ring, the outer ring and the radial plate have significant effects on the stiffness and weight of the PSD housing. Therefore, their sizes are selected as the design variables, as shown in Fig. 8.

Fig. 6. Displacement [mm]

Deflections of the inner and outer rings are shown in Fig. 7. The inner ring turns a certain angle relative to the outer ring, which leads the planet gears to deflect β angle. The outer ring expands b along the radial direction, which leads the planet gear to deflect along the axial direction. All of these events lead to the meshing misalignment among the bevel planet gears and the half-axle gears. The insufficient stiffness of the PSD housing affects the accuracy of gear meshing, which is considered to be a key constraint condition in designing the structural parameters of the PSD. Thus, this study aims to secure the PSD housing 528

Fig. 8. Design variables

Design variables are expressed in matrix form, shown in Eq. (4):

Li, F. – Qin, Y. – Pang, Z. – Tian, L. – Zeng, X.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 525-535

X = [ X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 , X 8 ] . (4) T

3.1.2 Objective Function To meet the lightweight design requirement, minimizing the weight of the PSD housing is considered the objective function, as shown in Eq. (5): Min W = f (X), (5) where W is the weight of the PSD housing and X is the design variable matrix related to W.

The gear parameters are used in Eq. (6) to obtain Δg = 14.2 μm. An axial modification is adopted [22], i.e. the theoretical involute of the two end surfaces of the modification gear rotates from the central axis to the gear tooth. The rotation angle is determined by the size of the drum-shape. Finally, the endpoints of the two theoretical involutes are connected into an arc.

The arc radius is equal, which is used as the scanning trajectory of the tooth surface. With the variation of the skew angle β, the maximum contract stress of the modified and unmodified tooth profile is shown in Fig. 9.

3.1.3 Constraint Conditions (1) Constraints of Design Variables: The ranges of design variables are determined under the geometric non-interference of the PSD housing structure, as shown in Table 1.

Table 1. Ranges of design variables Design variables Upper bound [mm] Lower bound [mm]

X1 20 5

X2 20 5

X3 15 5

X4 12 5

X5 17 5

X6 20 5

X7 8 3

X8 8 3

(2) Stiffness Constraint of the PSD Housing: According to the results of the FEA, the insufficient stiffness of the PSD housing will make the bevel gears deflect along the axle, which will lead to the stress concentration. The tooth surface is shaped like a drum in order to alleviate the above problems. In this study, the stiffness index is determined by comparing the changes of the contact stress along with the skew angles, which is between standard and modified tooth profiles. The value of the drum-gear profile [21] is presented in Eq. (6):

∆g = 0.25b × 10−3 + 0.5 f g , (6)

where Δg [μm] is the drum size of the tooth profile, b [mm] is the gear width, fg = A(0.1b+10), A is the constant related to the accuracy magnitude of the gear as shown in Table 2. This paper adopts the accuracy magnitude 7, accordingly, A = 2.5.

Table 2. Constant A related to the accuracy Accuracy magnitude A Accuracy magnitude A

0 0.63 5 1.6

1 0.71 6 2.0

2 0.8 7 2.5

3 1.0 8 3.15

4 1.25

Fig. 9. Maximum contact stress of modified and unmodified tooth profile

As shown in Fig. 9, when the axial skew angle is smaller, the contact stress of the modified gear is heavier. However, with the increasing of the skew angle, the maximum contact stress of the modified gear increases dramatically. When the skew angle is more than 0.04°, the maximum contact stress of the unmodified gear exceeds that of the modified gear.

When the skew angle is 0.1°, the load distribution along the width of the gear tooth is very uneven, and the gear meshing line is shorter; the maximum contact stress increases by 85% compared with the normal meshing.

The skew angle β can be effectively reduced by improving the PSD housing stiffness. The uneven load distribution of the gear tooth can also be improved.

Therefore, in the optimization process, the PSD housing stiffness is selected as the constraint condition to ensure that the uneven load distribution of the gear tooth, which is mainly caused by the insufficient housing stiffness, is minimized.

According to the above analysis, in the PSD housing optimization, the skew angle of the bevel gear is restricted to less than about 0.04°. The relationship

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between the skew angle of the bevel gear axis and the deflection of axis endpoint is presented in Table 3.

Table 3. Relationship between the skew angle of the bevel gear axis and the deflection of axis endpoint Skew angle of the bevel gear axis [°] 0.01 0.03 0.05 0.07 0.1

Deflection of axis endpoint [mm] 0.0049 0.014 0.024 0.034 0.0489

By fitting calculation based on Table 3, the maximum deflection of the PSD housing is no more than about 0.02 mm. Thus, the constraint can be defined in the following form: U ≤ 0.02 mm.

(7)

3.2.2 Surrogate Model The surrogate model uses a simple mathematical model to replace a complex relation between the design variables and corresponding response during structural optimization process. At the same time, the calculation cost can be reduced effectively, and the results are very close to the true values [24] and [25].

RSM uses different order polynomials to express the relationship between the design variables and their responses. One of the most used RSM is the second order polynomial model, which has the advantages of low computational cost, better approximation effect, being easy to solve, etc. Therefore, in this paper, the second order polynomial RSM is adopted to establish the surrogate models. The mathematical model of the second-order polynomial RSM can be expressed in Eqs. (8) and (9): n

n

n

j −1

y = α 0 + ∑α i xi + ∑α ii xi + ∑ ∑α ij xi x j , 2

3.2 Establishment of the Surrogate Model

i =1

i =1

j = 2 i =1

(8) T

A = α 0 ,α1 ,…,α n ,α11 ,α 22 ,…,α nn ,α12 ,α13 ,…,α ( n−1) n  , (9)

3.2.1 Design of experiment based on OLHD The surrogate models are created based on the design matrix. In this study, the design matrix is generated using the design of experiment (DOE) and the real responses of the sample points are calculated by the FEA. The typical DOE includes the orthogonal design, uniform design, full factorial design, OLHD, etc. The OLHD improves the uniformity of the LHD by spacefilling and balance [23]. Thus, OLHD is selected as the DOE in this study.

where y is the output variable, xi is the design variable, n is the number of design variables, and A is the undetermined coefficient vector, αi, αii and αij are the regression coefficient.

The required sample points of RSM for establishing surrogate models are (n+1)×(n+2)/2, in which n is the number of design variables. In this optimization process, it has 8 design variables, so 50 sets of the sample points obtained using OLHD are used to establish the surrogate models of the PSD

Table 4. Coefficient of surrogate model of weight and maximum deflection Term constant x1 x2 x3 x4 x5 x6 x7 x8 x12 x22 x32 x42 x52 x62

530

Weight coefficient 0.72134406 0.00112139 –3.846E–04 6.6751E-04 –0.0058554 –6.502E-05 6.2272E-04 0.04281933 0.08717993 –5.226E–06 –2.143E–05 6.8029E–07 3.4382E–04 3.4432E–05 –8.587E–06

Deflection coefficient 0.1048929 –0.0016603 –0.0011450 –0.0017529 –0.0023501 –0.0028529 –0.0023899 –0.0050861 –0.0033694 2.66E–05 1.29E–05 1.86E–05 4.42E–05 3.25E–05 2.39E–05

Term x72 x82 x 1x 2 x 1x 3 x 1x 4 x 1x 5 x 1x 6 x 1x 7 x 1x 8 x 2x 3 x 2x 4 x 2x 5 x 2x 6 x 2x 7 x 2x 8

Weight coefficient 9.5887E–04 7.3471E–04 –3.775E–06 4.1338E–05 1.2887E–05 1.1903E–04 1.2612E–04 –2.012E–04 1.4735E–05 4.899E–06 2.8431E–04 1.3477E–04 2.0653E–04 –5.845E–05 –3.486E–05

Deflection coefficient 1.8691E–04 1.1818E–04 9.82E–06 5.65E–06 1.36E–05 1.21E–05 1.00E–05 2.65E–05 1.90E–05 6.81E–06 8.62E–06 1.46E–05 1.47E–05 3.83E–06 5.62E–06

Li, F. – Qin, Y. – Pang, Z. – Tian, L. – Zeng, X.

Term x 3x 4 x 3x 5 x 3x 6 x 3x 7 x 3x 8 x 4x 5 x 4x 6 x 4x 7 x 4x 8 x 5x 6 x 5x 7 x 5x 8 x 6x 7 x 6x 8 x 7x 8

Weight coefficient –1.008E–05 –4.582E–05 1.2846E–05 1.789E–05 –2.83E–05 1.9922E–04 –5.799E–05 –9.699E–05 –4.054E–05 –8.481E–06 –2.522E–04 –7.093E–05 –1.367E–05 5.5E–06 6.5998E–05

Deflection coefficient 2.02E–05 2.99E–05 1.22E–05 3.19E–05 3.83E–05 2.11E–05 3.47E–05 2.79E–05 1.82E–05 3.98E–05 6.35E–05 2.37E–05 3.04E–05 2.71E–05 4.39E–05

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housing weight and maximum deflection. In the meantime, the other 20 sets of sample points, which can be used to test the accuracy of the surrogate models, are also obtained in the same manner. Based on the 50 sets of sample points, the surrogate models are established using the second-order polynomial RSM, and the coefficients of weight and deflection are presented in Table 4.

According to the sample points obtained using OLHD, the influence degree of the design variables on output W and U are obtained based on the statistics principle, as shown in Fig. 10a and b, respectively.

n

(

S E = ∑ yi − y i i =1

)

2

. ST is the total deviation sum of

squares, ST = SE + SR. Using the error analysis, the R² values of the surrogate models of the housing weight and the maximum deflection equal 0.99995 and 0.96758, respectively, which indicates that the surrogate models can be used to predict performance.

3.3 Optimization Algorithm and Hybrid Optimization Strategy A hybrid optimization strategy that combines the MIGA for global optimization and the NLPQL for gradient optimization is used to obtain the optimal results. Brief explanations of the MIGA and NLPQL algorithms are given, and the hybrid strategy is then explained.

3.3.1 Optimization Algorithm

a)

b) Fig. 10. Influence degree of design variables on W and U; a) Influence degree of design variables on W, b) Influence degree of design variables on U

As the surrogate model is an approximate equation between design variables and response function, conducting an error analysis of a surrogate model is necessary before replacing the true model.

The accuracy of the surrogate model is usually measured via the multiple correlation coefficient R² in engineering application, which is defined as Eq. (10).

The value of R² is between 0 and 1. The accuracy of the surrogate model is higher when R² is close to 1, and the accuracy is acceptable when the value of R² exceeds 0.9 in engineering applications [26].

R2 = 1 −

SE SR = , (10) ST ST

where n SR is 2the regression sum of squares, S R = ∑ y i − y . SE is the residual sum of squares, i =1

(

)

A genetic algorithm (GA) is an effective method of global search optimization, which has been widely used in engineering optimization problems [27].

The MIGA is an enriched algorithm based on GA. However, compared with a traditional GA, the MIGA is more efficient in finding the global optimum and more suitable for problems that have difficulty in obtaining gradient information. The MIGA is an exploration optimization method that uses “selection”, “crossover” and “mutation” mechanisms to obtain the optimal design. The MIGA avoids obtaining a local optimal solution, thus can avoid the precocious phenomenon [28].

The NLPQL is used for gradient optimization to reduce the computing cost in the process of optimization. When the NLPQL is used to solve nonlinear mathematical problems, it has superior stability and fast convergence, and can easily obtain the global optimal solution [29].

3.3.2 Hybrid Optimization Strategy The MIGA-NLPQL hybrid optimization strategy is performed as follows: firstly, the MIGA is used to determine the target area of the extreme value in the design space. Next, the NLPQL is used for the accurate optimization in target area defined by the MIGA to obtain the optimal design results. The flowchart of the hybrid optimization based on surrogate models is shown in Fig. 11.

Design and Optimization of PSD Housing Using a MIGA-NLPQL Hybrid Strategy Based on a Surrogate Model

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Fig. 11. Flowchart of hybrid optimization based on surrogate models

First, the advantages of the MIGA are brought into full play in rapidly roughly locating the sensitive target area in the overall design space, so that the low efficiency of the global optimization algorithm in detail can be avoided. Then, the global optimal results, which are obtained from MIGA optimization, are used in the NLPQL optimization module. Thus, the advantages of the gradient optimization algorithm of the NLPQL are brought into full play on local optimization, and the optimal solution can be found accurately. Misguided results, which are caused by seeking optimal results in highly nonlinear or discrete design spaces using the gradient optimization algorithm, can be avoided.

3.4 Optimization Results In the optimization process, eight structural sizes are selected as design variables that belong to a multidimensional design, which is difficult to search in order to find the optimal value. To solve this problem, the MIGA-NLPQL hybrid optimization strategy is adopted in this study. In comparison with the optimization results of the MIGA-NLPQL and the single MIGA, the advantages of the hybrid optimization strategy are demonstrated. The iteration processes of the MIGA-NLPQL and MIGA are shown in Figs. 12 and 13, respectively.

The optimization process is time-consuming, so the number of optimization steps can determine 532

its efficiency. Fig. 12 shows that the total iteration of the MIGA is 2000 steps. The objective function begins convergence at about 400 steps, and the global optimal solution is obtained at 1662 steps.

Figs. 13a and b show the iteration process of hybrid optimization; the iteration steps total about 1360. In the hybrid optimization, the MIGA calculates about 1000 steps and the NLPQL calculates about 360 steps. In Fig. 13a, a preliminary optimal solution is found at about 770 steps by the MIGA. Although the convergence of the objective function is not ideal, it shows a convergent trend. In this hybrid optimization strategy, the MIGA is only used to find the preliminary global optimal solution, so the convergence does not affect the NLPQL for further optimization. Fig. 13b shows the optimization process of the NLPQL. The objective function begins to converge when iterating at about 200 steps and the global optimal solution is finally obtained at 352 steps around the preliminary optimal solution found by the MIGA. Therefore, the total iteration steps of the hybrid optimization strategy are 1122, which is less than the 1662 steps of a single MIGA. In comparison with the iteration process of the MIGA, the computing cost is obviously reduced; therefore, the MIGA-NLPQL is more efficient than the single MIGA algorithm.

Fig. 12. Iteration process of single MIGA

The results of the optimization are shown in Table 5. The second row, “Initial”, stands for the initial design without optimization; the third row, “MIGA”, stand for the results based on the MIGA technique; the fourth row, “MIGA-NLPQL”, stands for the results received from the hybrid strategy, “MIGA-NLPQL”.

“Rounded results” stands for the results after rounding according to the hybrid strategy of the MIGA-NLPQL. Comparing the optimization results of the single MIGA and MIGA-NLPQL with the original design, it is shown that the weight of PSD housing is reduced by 26% after optimization using the single MIGA and by 26.6% after optimization using the MIGA-NLPQL. Furthermore, the computing time of the MIGANLPQL 1360 steps is less than the computing time of

Li, F. – Qin, Y. – Pang, Z. – Tian, L. – Zeng, X.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 525-535

a)

b) Fig. 13. Iteration process of a) MIGA in MIGA-NLPQL, and b) NLPQL in MIGA-NLPQL

the MIGA 2000 steps. Therefore, the MIGA-NLPQL strategy can perform more efficiently to reduce calculation time and has a stronger ability to obtain optimal results than a single MIGA. The rounded results of the MIGA-NLPQL are selected as the final design parameters. Table 6 shows the geometric dimensioning and the performances of the original and the final designs of PSD housing.

The values of the objective functions U and W obtained by the surrogate models are 0.0095 and 1.1686, respectively. The optimal design is verified

by FEA as shown in Fig. 14. The error between the surrogate model and the FEA of U is 13.6%, and that of W is 10.4%; which the error values of U and W are within the acceptable range, it can be verified that the surrogate model can be used to replace the performance of PSD. As shown in Fig. 14, the maximum displacement of FEA results is 0.011 mm, which satisfies the constrict condition of deflection. In

Fig. 14. FEA of optimal PSD; a) Displacement [mm], b) Von Mises equivalent stress [MPa]

Table 5. Optimization results of MIGA-NLPQ and MIGA

MIGA MIGA-NLPQL Rounded results

X1 5.203 5.0 5.0

X2 5.130 5.0 5.0

X3 14.656 15.0 15.0

Design variables [mm] X4 X5 X6 5.525 13.248 19.843 5.371 12.359 17.603 5.4 12.4 17.6

X7 3.004 3.0 3.0

X8 3.0 3.0 3.0

Response value U [mm] W [kg] 0.0088 1.1774 0.0095 1.1686 / /

Computing steps 1360 2000 /

Table 6. Comparison of the original and the final designs X1 Original designs

15

X2 15

Final designs Optimization rates

5.0 /

5.0 /

X3 7 15.0 /

Design variables [mm] X4 X5 7 15 5.4 /

12.4 /

X6 15

X7 6

X8 5

Response value U [mm] W [kg] 0.025 1.592

17.6 /

3.0 /

3.0 /

0.0095 62%

Design and Optimization of PSD Housing Using a MIGA-NLPQL Hybrid Strategy Based on a Surrogate Model

1.1686 26.6%

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comparison with previous optimization, the maximum displacement is reduced by 56%. The maximum stress of the optimal PSD is 93.2 MPa, which occurs at the holes of end covers, and the stress of the joint between the radial plate and inner ring of the midhousing is improved. The above analysis shows that the optimization of PSD is reasonable, and the performances of PSD are improved.

4 CONCLUSIONS This paper proposes a new PSD design scheme used in HEVs to improve the support form of bevel planet gears. To decrease the meshing misalignment of the internal gears caused by the lack of PSD housing stiffness, improving the stiffness is imperative.

However, increasing stiffness and reducing weight are often contradictory. Therefore, the optimization design of PSD housing is performed, where the minimum weight is selected as the optimization objective and the stiffness is selected as the constraint. To enhance the optimization efficiency and obtain global optimal results simultaneously, the hybrid optimization strategy of the MIGA-NLPQL, based on a surrogate model, is adopted in this paper. In this way, the weight of PSD housing is reduced by 26.6% under the stiffness constraint. The design optimization shows that the MIGA-NLPQL hybrid optimization strategy has strong optimization ability and rapid convergence speed, as proven by the comparison between the single MIGA and MIGA-NLPQL results.

5 ACKNOWLEDGEMENTS The authors acknowledge the financial support of the National Natural Science Foundation of China (no. 51075179 and no. 51375202) and the Scientific Frontier and Interdisciplinary Merit Aid Projects of Jilin University, China (no. 2013ZY08).

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Vsebina Strojniški vestnik - Journal of Mechanical Engineering letnik 60, (2014), številka 7-8 Ljubljana, julij-avgust 2014 ISSN 0039-2480 Izhaja mesečno

Razširjeni povzetki Frank Goldschmidtboeing, Alexander Doll, Ulrich Stoerkel, Sebastian Neiss, Peter Woias: Primerjava strižnih sil in globine penetracije pri navpičnih in poševnih filamentih zobne ščetke Matjaž Čebron, Franc Kosel: Vrednotenje shranjene energije v natezno trajno deformiranih bakrenih vzorcih komercialne čistote na osnovi dislokacijskih modelov deformacijskega utrjevanja in merjenja trdote Mihai Dupac, Siamak Noroozi: Dinamično modeliranje in simulacija fleksibilnega rotacijskega robotskega manipulatorja z enim segmentom, ki deluje v režimu hitrih zaustavitev Benjamin Bizjan, Alen Orbanić, Brane Širok, Tom Bajcar, Lovrenc Novak, Boštjan Kovač: Vizualizacijska metoda za izračun hitrostnih polj toka na osnovi advekcijsko-difuzijske enačbe Xiaoni Qi, Yongqi Liu, Hongqin Xu, Zeyan Liu, Ruixiang Liu: Modeliranje toplotne oksidacije premogovniškega metana David Koblar, Jan Škofic, Miha Boltežar: Ocena modula elastičnosti gumi podobnih materialov vezanih na toge površine z ozirom na Poissonovo število Jelena R. Jovanovic, Dragan D. Milanovic, Radisav D. Djukic: Analiza in terminiranje za optimizacijo časa proizvodnega cikla Feng Li, Yumo Qin, Zhao Pang, Lei Tian, Xiaohua Zeng: Snovanje in optimizacija ohišja PSD s hibridno strategijo MIGANLPQL na osnovi modela surogata Osebne vesti Doktorske disertacije, magistrska dela, diplomske naloge

SI 87 SI 88 SI 89 SI 90 SI 91 SI 92 SI 93 SI 94 SI 95

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 87 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-10-18 Prejeto popravljeno: 2014-01-29 Odobreno za objavo: 2014-04-14

Primerjava strižnih sil in globine penetracije pri navpičnih in poševnih filamentih zobne ščetke

Goldschmidtboeing, F. – Doll, A. – Stoerkel, U. – Neiss, S. – Woias, P. Frank Goldschmidtboeing1,* – Alexander Doll2 – Ulrich Stoerkel2 – Sebastian Neiss1 – Peter Woias1 1 Univerza

v Freiburgu, Oddelek za mikrosisteme, Nemčija & Gamble, Oral-B Laboratorij, Nemčija

2 Procter

Namen raziskave je preučitev in analiza učinka kombiniranja trenutnih prednosti električnih zobnih ščetk s 3D-gibanjem (pulzno gibanje, združeno z oscilacijami/rotacijo) in preizkušenih lastnosti naprednih ročnih zobnih ščetk, kjer so šopi ščetin nagnjeni pod kotom 16°.

Za opis delovanja navpičnih in za 16° nagnjenih filamentov zobne ščetke na ravne zobne površine in bližnje predele je bil uporabljen polanalitični model. Teorija bazira na analitični rešitvi enačbe nosilcev ob upoštevanju normalnih in prečnih (strižnih) sil, ki deformirajo filamente.

Začeli smo z načelnim modelom, ki popisuje obliko filamentov in sile na zobne površine s polanalitičnim pristopom. Sledi reševanje tako pridobljenih enačb in vizualizacija oblike filamentov v poglavju z rešitvami.

Nastale sile in globine penetracije so ovrednotene za različne kombinacije kotov nagiba in deformacij ščetke.

Nato so predstavljene rešitve numeričnih simulacij za oceno uporabnosti poenostavljenega analitičnega modela iz prvega dela. Sledi eksperimentalna validacija hipoteze, da filamenti, nagnjeni pod kotom 16°, prenašajo na zob večje strižne sile kot vertikalni filamenti. Članek se konča z razdelkom o primernosti predlaganega modela za snovanje zobnih ščetk.

Ugotovljeno je bilo 15-odstotno povečanje globine penetracije (model sosednjih območij) in 60-odstotno povečanje prečne sile (model zobne površine) za kot nagiba 16° v primerjavi z navpičnim filamentom. Kot 16° je optimalen, ker se pri njem poveča prečna sila, ne da bi se zmanjšala normalna sila, ki deluje na zobno površino.

Eksperimentalno je dokazano, da lahko filamenti pod kotom 16° prenašajo na zob večje strižne sile kot vertikalni filamenti, to pa lahko pomeni večjo učinkovitost poševnih filamentov pri odstranjevanju zobnih oblog.

Razvita teorija ne podaja kvantitativnega razmerja med kotom nagiba in učinkovitostjo odstranjevanja zobnih oblog, podaja pa način za optimizacijo sil na zobno površino ter povečanje globine penetracije v sosednje vrzeli.

Klinični pomen poševnih filamentov je že dokazan za ročne zobne ščetke: filament pod kotom 16° pri ročnih ščetkah izboljša odstranjevanje zobnih oblog. Naša študija je pokazala, da je isti kot uporaben tudi za električne ščetke, ki oscilirajo in se vrtijo. Prednosti pri odstranjevanju zobnih oblog so bile nedavno potrjene tudi v klinični primerjavi ščetk z oscilacijsko-rotacijskim gibanjem v primerjavi s kontrolnimi ščetkami z glavami različnih oblik.

Avtorji menijo, da je kombinacija teoretičnega modeliranja in kliničnih študij učinkovita metoda za izboljšanje učinkovitosti ščetk za umivanje zob.

Ti eksperimenti in-vitro lahko napovejo prednosti pri uporabi ter ponazorijo potencialne prednosti pri oblikovanju ščetk.

Ključne besede: vertikalni in poševni filamenti zobne ščetke, strižne sile, globina penetracije, električna zobna ščetka, mehansko odstranjevanje zobnih oblog, tehnologija oscilacije-rotacije

*Naslov avtorja za dopisovanje: Univerza v Freiburgu, Oddelek za mikrosisteme, Georges-Koehler-Allee 102, 79110 Freiburg, Nemčija, fgoldsch@imtek.de

SI 87

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 88 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-11-23 Prejeto popravljeno: 2014-02-12 Odobreno za objavo: 2014-05-07

Vrednotenje shranjene energije v natezno trajno deformiranih bakrenih vzorcih komercialne čistote na osnovi dislokacijskih modelov deformacijskega utrjevanja in merjenja trdote Matjaž Čebron1,* – Franc Kosel2 1 Hidria Rotomatika, Slovenija, 2 Univerza v Ljubljani, Fakulteta za strojništvo, Slovenia Od fizikalno osnovanih reoloških modelov v osnovi pričakujemo boljši opis mehanskega odziva materiala kot pri fenomenološko tvorjenih zvezah v širšem področju deformacij, hitrosti deformiranja in temperature. Zgodnji dislokacijski modeli so bili osredotočeni na opis tretje faze utrjevalnega procesa, za katero je značilen enakomeren padec stopnje utrjevanja. Zveza med napetostmi in deformacijami je v tem področju opisljiva z enoparametričnimi modeli, v katerih je strižna trdnost kristala odvisna od ene same notranje spremenljivke. Potreba po opisu dodatnih faz utrjevanja zaradi njihovega pomena v tehnoloških procesih izrazitega preoblikovanja kovin in odkritje nastanka celične dislokacijske strukture že kmalu po začetku trajnega deformiranja sta botrovala razvoju dvoparametričnih modelov, v katerih je material obravnavan kot dvofazni kompozit, sestavljen iz dislokacijskih celic z nizko gostoto dislokacij v notranjostih in visoko gostoto dislokacij v celičnih mejah. Modeli deformacijskega utrjevanja podajajo evolucijske enačbe gostote dislokacij v odvisnosti od strižne deformacije ter zvezo med gostoto in strižno trdnostjo, povezava med mehanskim odzivom kristalov in vrednostmi napetosti in deformacij na velikostnem nivoju polikristaliničnega sestava pa je tvorjena z uporabo modelov deformiranja polikristalov. V prispevku sta dva dislokacijska modela utrjevanja (Bergström-ov in Estrin-ov) uporabljena v povezavi s Taylor-jevim modelom polikristaliničnega deformiranja za opis napetostno-deformacijske krivulje standardno oblikovanih bakrenih vzorcev pri nateznem preizkusu.

Večina dela, opravljenega pri plastičnem deformiranju kovin, se pretvori v toploto in izgubi ob stiku materiala z okolico, manjši del pa ostane shranjen v materialu v obliki deformacijske energije napetostno-deformacijskega polja dislokacij. Pri enoparametričnih modelih je zveza med strižno trdnostjo kristalov in povprečno gostoto dislokacij enolična, v dvoparametričnih modelih pa lahko različna razporeditev dislokacij podaja isto vrednost strižne trdnosti.

Ker imajo različne razporeditve dislokacij tudi različno shranjeno energijo, lahko ob podanem energijskem modelu dislokacijske strukture izmerjene vrednosti energije uporabimo za določanje parametrov ali vsaj za preverjanje dislokacijskih modelov deformacijskega utrjevanja.

Diferenčna dinamična kalorimetrija sodi med dvostopenjske metode termične analize in omogoča spremljanje termodinamičnih sprememb snovi ter merjenje prenosa toplote med procesi v odvisnosti od temperature in časa. Za merjenje shranjene energije v trajno deformiranih bakrenih vzorcih je uporabljen toplotno-tokovni kalorimeter, ki določa razliko v toplotnem toku med vzorcem in referenco z merjenjem temperaturne razlike med njima. Za ocenjevanje vrednosti shranjene energije je predlagana uporaba merjenja trdote, saj je le-ta odvisna od utrditve materiala in posledično od lokalne gostote dislokacij. Ugotovljeno je, da je mogoče zvezo med shranjeno energijo in kvadratom vrednosti meje tečenja pri enoosnem obremenjevanju dobro aproksimirati z uporabo linearne enačbe, medtem ko izmerjena odvisnost trdote od meje tečenja s premosorazmerno zvezo ni dovolj dobro opisana.

Dislokacijski modeli deformacijskega utrjevanja postajajo predvsem zaradi dobrih opisnih lastnosti vse bolj razširjeni. V večini primerov uporabe modelov za opis mehanskega odziva konstrukcijskih elementov so materialni parametri prilagojeni le dobremu opisu eksperimentalnega napetostno-deformacijskega odziva, njihova fizikalna osnova pa je pri tem večkrat zanemarjena. Fizikalno osnovani modeli so tako uporabljeni za čisto fenomenološki opis utrjevanja.

Določitev natančne vrednosti parametrov tako, da ti ohranjajo zvezo z dejanskimi procesi v materialu, predstavlja enega večjih izzivov na področju fizikalno osnovanih materialnih modelov. Eksperimentalna opredelitev vrednosti nekaterih parametrov je praktično neizvedljiva, študije natančne razporeditve dislokacij v heterogenih strukturah pa so še vedno zelo redke in le pogojno uporabne. Energijski modeli dislokacijske strukture in same meritve shranjene energije so v splošnem preveč nenatančni, da bi lahko preko njih opisali ali zaznali manjše spremembe v dislokacijski strukturi.

Prikazana metoda preverjana dislokacijskih modelov zato ne more predstavljati zamenjave neposrednemu opazovanju razvoja dislokacijske strukture med utrjevanjem ali drugim metodam natančnejšega določevanja gostote dislokacij.

Uporabo energijskih modelov tako predlagamo le kot hiter in enostaven način zagotavljanja fizikalno še sprejemljivih vrednosti izračunanih veličin (npr. gostote dislokacij) in ne kot metodo preučevanja podrobnosti dislokacijske strukture.

Ključne besede: dislokacije, modeli deformacijskega utrjevanja, plastičnost kristalnih struktur, shranjena energija, kalorimetrija, trdota

SI 88

*Naslov avtorja za dopisovanje: Hidria Rotomatika d.o.o., Spodnja Kanomlja 23, Slovenija, mat.cebron@gmail.com

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 89 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-11-13 Prejeto popravljeno: 2014-02-19 Odobreno za objavo: 2014-02-28

Dinamično modeliranje in simulacija fleksibilnega rotacijskega robotskega manipulatorja z enim segmentom, ki deluje v režimu hitrih zaustavitev Dupac, M., Noroozi, S. Mihai Dupac* – Siamak Noroozi

Univerza v Bournemouthu, Znanstveno-tehniška fakulteta, Združeno kraljestvo

Modeliranje in simulacija manipulatorjev z enim segmentom sta v zadnjih letih deležna velike pozornosti, cilj pa je izboljšanje produktivnosti in zmanjšanje proizvodnih stroškov. Dinamika in upravljanje manipulatorjev z enim segmentom zaradi pomena in uporabnosti tega področja (npr. pri vzdrževanju jedrskih naprav in pri aplikacijah v vesolju) štejeta za zanimiv raziskovalni problem, s katerim se ukvarja na tisoče raziskovalcev. Na dinamični odziv manipulatorjev, sestavljenih iz povezanih fleksibilnih in togih delov, vplivajo predvsem deformacije komponent in mehanski udarci, ki povzročajo visoko stopnjo vibracij ter ogrožajo natančnost in varnost delovanja. Zvezne deformacije segmentov, redni udarci zaradi zračnosti in materialne lastnosti, ki niso enake za natezni in za tlačni režim delovanja, povzročajo oblikovanje in rast razpok. Različni materiali se različno odzivajo na udarce, porazdelitev kinetične energije pa je pomembna za ugotavljanje odziva takšnih sistemov. Članek obravnava natančno matematično modeliranje in simulacijo takšnih sistemov za boljše razumevanje in izboljšanje njihovega mehanskega odziva. Članek obravnava modeliranje in simulacijo iztegljive rotacijske robotske roke s togo ročico in omejenim fleksibilnim segmentom, ki je izpostavljena režimu hitrih zaustavitev. Gibanje enega konca fleksibilnega segmenta, ki se vrti v vodoravni ravnini s konstantno kotno hitrostjo, je omejeno z vnaprej določeno trajektorijo. Omejena trajektorija omogoča nadzor nad trajektorijo, orodje na koncu robotske roke pa se izogne oviram. Opravljene simulacije na krožni/ krožno-eliptični omejeni trajektoriji vključujejo hitro zaustavitev po dveh kompletnih obratih za določitev vedenja manipulatorja v realnih delovnih pogojih. Vpliv zračnosti na dinamično vedenje sistema, torej na stik/udarec iztegljivega segmenta ob togo ročico ob nenadnih zaustavitvah in vrtenju segmenta, je bil opredeljen s primerjavo modela z upoštevano zračnostjo in takšnega brez zračnosti.

Model stika/udarcev v sistemu več teles je običajno opisan z zveznimi ali nezveznimi modeli. Zvezni pristop uporablja naslednja razširjena modela: (i) model zvezne kontaktne sile, ki privzema, da je sila pri trku zvezna funkcija deformacije in je lahko linearna (Kelvin-Voigtov model) ali nelinearna (po Hertzovem zakonu), in (ii) model/ metodologijo enostranske omejitve, ki privzema tlačne sile v kinematičnih omejitvah enačb gibanja; te sile izginejo, ko se stik ob udarcu teles/segmentov prekine. Model nezvezne kontaktne sile, ki je bil uporabljen za vrednotenje udarcev v tem članku, privzema hipen udarec in ohranitev konfiguracije sistema med trkom. Pri matematičnem pristopu se integracija enačb gibanja konča v trenutku trka, ko se izračuna bilanca navora za določitev hitrosti komponent sistema po trku. Med trkom, ki je običajno razdeljen na fazo kompresije in na fazo ekspanzije, se zanemarijo vse sile v mehanskem sistemu, ki niso povezane s sunkom. Newtonov zakon trka poveže obe fazi tako, da poda razmerje med relativno hitrostjo po trku ter koeficientom trka (ki določa energijske izgube) in relativno hitrostjo pred trkom. Pri modeliranju sunka sile in hitrosti pri trku so bile uporabljene naslednje kinematične predpostavke: i) telesa so toga in se ob trku ne deformirajo; ii) do trka pride med ogliščem in robom; iii) sila udarca in impulz sile delujeta na oglišča; iv) trk je modeliran s pomočjo koeficienta trka; v) položaj teles se od začetka do konca trka ne spreminja; vi) hitrost teles se ne spreminja med trkom; vii) sprememba hitrosti teles ob koncu trka je nezvezna. Vpliv trenja lahko zanemarimo in stik/udarec fleksibilnega segmenta ob ročico je zato brez trenja. Možne vrste udarcev med fleksibilnim segmentom in vodilom so: i) brez udarca, ii) udarec v eni točki in iii) udarec v dveh točkah.

Pri snovanju rotacijskega manipulatorja je pomembno zmanjšanje dinamičnega odklona segmenta med visokohitrostnim gibanjem, zato je bil upoštevan vpliv višje kotne hitrosti in zračnosti v primerjavi z modelom brez zračnosti. Ugotovljeno je bilo, da hitre zaustavitve povečujejo vzbujanje sistema. Opaženo je bilo tudi, da se amplituda vibracij hipno poveča (ob zmanjšanju lastne frekvence fleksibilnega segmenta, ko se poveča dolžina) takoj po nenadni zaustavitvi, in sicer zaradi povečanja dinamičnih sil. Ugotovljeno je bilo tudi to, da nenadne zaustavitve manipulatorja z enim segmentom ojačijo dinamične sile, s čimer je pojasnjeno povečanje amplitude vibracij. Zmanjšanje dolžine fleksibilnega segmenta zaradi gibanja po omejeni krožno-eliptični trajektoriji pa poveča togost segmenta, kot potrjuje študija vedenja fleksibilnega robotskega manipulatorja z rotacijskim prizmatičnim sklepom. Pri vibrirajočem fleksibilnem zglobu brez zračnosti je bil opažen samo prvi način vibracij, medtem ko se pri modelu zračnosti s fleksibilnim segmentom pojavita prvi in drugi način vibracij. Za validacijo in posplošitev simulacij, o katerih poroča študija, bodo opravljeni dodatni eksperimentalni preizkusi.

Ključne besede: robotska roka, manipulator, zračnost, dinamika, analiza udarca, kontaktne sile *Naslov avtorja za dopisovanje: Univerza v Bournemouthu, Znanstveno-tehniška fakulteta, Oddelek za konstruiranje in inženiring Talbot Campus, Fern Borrow, Poole, Dorset, BH12 5BB, Združeno kraljestvo, mdupac@bournemouth.ac.uk

SI 89

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 90 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-12-18 Prejeto popravljeno: 2014-03-06 Odobreno za objavo: 2014-03-11

Vizualizacijska metoda za izračun hitrostnih polj toka na osnovi advekcijsko-difuzijske enačbe Bizjan, B. – Orbanić, A. – Širok, B. – Bajcar, T. – Novak, L. – Kovač; B. Benjamin Bizjan1,2,* – Alen Orbanić2,3 – Brane Širok1 – Tom Bajcar1 – Lovrenc Novak1 – Boštjan Kovač2,3 1 Univerza v Ljubljani, Fakulteta za strojništvo, Slovenija 2 Abelium d.o.o., Slovenija 3 Univerza v Ljubljani, Fakulteta za matematiko in fiziko, Slovenija

V današnjem času obstajajo številne vizualizacijske metode, ki omogočajo izračun hitrostnih polj opazovanega toka tekočine, pri tem pa je njihova glavna prednost, da so brezdotične in zato ne vplivajo na tok. Najpogosteje se uporabljajo metode, ki delujejo na osnovi križne korelacije (npr. PIV), znane pa so tudi metode na osnovi optičnega toka. PIV metode so dokaj uveljavljene za meritve hitrostnih polj, vendar pa je njihova slabost kompleksnost in razmeroma visoka cena merilnega sistema. Metode na osnovi optičnega toka lahko po drugi strani delujejo na tokovih z manj zahtevno izvedbo vizualizacije (npr. dim, delci…), vendar so v primerjavi s korelacijskimi še dokaj neuveljavljene in slabo računalniško podprte.

Naš namen je bil razviti robustno računalniško podprto metodo na osnovi optičnega toka, ki bo omogočala analizo različnih tokovnih pojavov v laboratorijskem in industrijskem okolju. Metoda, implementirana v programskem okolju ADMflow, temelji na reševanju advekcijsko-difuzijske (AD) enačbe in predpostavki gostega tokovnega polja, pri čemer v AD enačbi koncentracijo polutanta v toku nadomestimo kar z lokalno sivinsko stopnjo na sliki. Z vidika natančnosti metode je ključen del računskega algoritma podvzorčenje slik (ang. downsampling, parameter ∆xy) in uvedba penalizacijskih členov (β, γ, θ), ki zgladijo nenadne spremembe hitrostnih gradientov, divergence in gradienta rotorja. Pred samim izračunom se slika zgladi z neuteženim filtrom (velikost okna za glajenje določa parameter SWS), glajenje po času pa posredno določimo s številom zaporednih slik (parameter IIE), ki se uporabijo za izračun hitrostnega polja v danem trenutku. Poleg predstavitve metode je glavni namen prispevka ovrednotiti njeno natančnost pri izračunu hitrostnih polj na različnih tipih tokovnih struktur.

Našo vizualizacijsko metodo smo ovrednotili na sekvenci sintetičnih slik zračnega toka čez krilo (profil NACA4421) v vetrovniku, ki smo jo ustvarili v dveh korakih. Prvi korak je bila numerična simulacija zračnega toka čez krilo v okolju za računalniško dinamiko tekočin Fluent, pri čemer smo uporabili model SST-SAS. S tem smo dobili izhodiščna hitrostna polja (ang. ground truth), ki smo jih nato uporabili še za pripravo sintetičnih slik - simulacijo dima z uporabo AD enačbe v obratni smeri. Koncentracijo dima na izviru smo sinusno pulzirali, da smo zagotovili zadostne časovne in prostorske gradiente sivin na slikah za pravilno delovanje naše metode. Iz dobljenih sintetičnih slik smo nato v programu ADMflow izračunali hitrostna polja in jih primerjali z izhodiščnimi vrednostmi. Izvedli smo variacijo parametrov izračuna v ADMflow-u, pri čemer smo za vsak nabor parametrov (β, γ, θ, ∆xy, SWS, IIE) izračunali napako hitrostnih polj po iznosu (absolutni hitrosti) in smeri. Z variacijo smo ugotovili, pri katerem naboru parametrov je dosežena najmanjša računska napaka, na podlagi česar smo lahko sklepali o optimalnih nastavitvah teh parametrov.

Rezultati kažejo, da je metoda najbolj natančna na slojevitih, brezvrtinčnih tokovih, nekoliko manj pa na vrtinčnih tokovih. Napaka po iznosu hitrosti se je za različne tipe toka gibala med 4 in 14%, napaka po kotu pa med 0.5 in 15°. Na natančnost izračuna najizraziteje vpliva nastavitev parametra za podvzorčenje (∆xy), ki ga je potrebno prilagoditi hitrosti pomikanja tokovnih struktur med zaporednima slikama, le-ta pa naj bo 1 do 10 pikslov na sliko. Pomemben vpliv imajo tudi penalizacijski koeficienti za hitrostne gradiente (β), divergenco (γ) in gradiente rotorja (θ), ki znatno znižajo računsko napako glede na primer, ko so nastavljeni na ničelne vrednosti (pri prevelikih vrednostih pa povzročijo hiter porast napake). Optimalno število zaporednih slik v enačbah (IIE) je tem nižje, čim bolj je tok nestacionaren.

Glede na to, da se vizualizacija v vetrovniku z dimom običajno uporablja le za kvalitativno analizo toka, so dobljeni rezultati (razpon napak) zelo zadovoljivi, saj predstavljena vizualizacijska metoda odpira nove možnosti za hitro, robustno in z vidika merilne opreme nezahtevno oceno hitrostnih polj toka. Glede nadaljnjega razvoja metode se kaže možnost uporabe strojnega učenja z namenom vsaj delne avtomatizacije pri izbiri parametrov izračuna, smiseln je tudi razvoj 3D metode za opazovanje kompleksnejših tokov.

Ključne besede: računalniško podprta vizualizacija, hitrostno polje, izračun gostega tokovnega polja, optični tok, advekcijsko-difuzijska enačba, računalniška dinamika tekočin SI 90

*Naslov avtorja za dopisovanje: Abelium d.o.o., Kajuhova 90, 1000 Ljubljana, Slovenija, benjamin.bizjan@abelium.eu

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 91 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-08-20 Prejeto popravljeno: 2013-12-23 Odobreno za objavo: 2014-01-24

Modeliranje toplotne oksidacije premogovniškega metana v nekataličnem reaktorju z izmenjavanjem smeri toka Qi, X. – Liu, Y. – Xu, H. – Liu, Z. – Liu, R. Xiaoni Qi1 – Yongqi Liu1,* – Hongqin Xu2 – Zeyan Liu1 – Ruixiang Liu1

1 Tehniška

univerza v Shandongu, Kolidž za promet in avtomobilsko tehniko, Kitajska univerza v Shandongu, Kolidž za strojništvo, Kitajska

2 Tehniška

Metan iz premogovnikov ni le toplogredni plin, neizkoriščen je tudi izgubljen vir energije. Zaradi zelo nizkih koncentracij CH4 se kot najobetavnejša rešitev kaže avtotermično zgorevanje v reaktorjih z izmenjevanjem smeri toka. Večina dosedanjih študij je bila posvečena katalitičnemu zgorevanju v katalitičnih reaktorjih z izmenjavanjem smeri toka (CFRR). Ekonomsko in tehnično najprimernejša rešitev za izkoriščanje toplotne energije pa so termični reaktorji z izmenjevanjem smeri toka (TFRR). Nekatalitična oksidacija v TFRR je zato pogosto privlačna alternativa. Tovrstni reaktorji so že dolgo v uporabi, npr. za homogeno kurjenje hlapnih organskih spojin (VOC).

Med zgorevanjem VOC in oksidacijo metana iz prezračevalnih sistemov v reaktorjih z izmenjevanjem smeri toka obstajajo očitne razlike. Termično zgorevanje v TFRR mora potekati pri pogojih, ki ne sprožajo čezmernega nastajanja NOx, t.j. maksimalna temperatura v reaktorju je lahko največ 1300 °C. Uporaba TFRR za kurjenje revnih zmesi metana zato zahteva podrobno preučitev delovnih pogojev.

Cilj prispevka je razvoj modela reaktivnega toka v nekatalitičnem reaktorju, napolnjenem s keramičnimi monolitnimi bloki, ki vključuje primerne modele prenosa toplote in snovi ter upošteva plinsko fazo in površinske reakcije. Računsko polje je dolžine 2100 mm in prereza 600×600 mm, vsebuje pa porozno keramiko s porami velikosti 2 do 3 mm. V tem reaktorju se smer dovoda izmenjuje med eno in drugo stranjo reaktorja s pomočjo preklopnih ventilov. Vodilne enačbe popisujejo ohranitev mase, energije in elementov. Enostopenjski mehanizem oksidacije metana ne vpliva na natančnost reakcijske toplote in časa. Vodilnih enačb za reaktor s keramično polnitvijo ni bilo mogoče rešiti neposredno s komercialno programsko opremo Fluent. Koeficienti prenosa toplote in snovi, ki niso vnaprej opredeljeni v programski opremi, so bili izračunani z modifikacijskimi podrutinami (UDF).

Razviti model je bil validiran s pomočjo eksperimentalnih podatkov. Nato je bila opravljena simulacija z enakimi delovnimi pogoji v reaktorju, kot so bili uporabljeni pri eksperimentih. Preskusi termične oksidacije metana so bili izvedeni v reaktorju z izmenjevanjem smeri toka, ki so ga zgradili na inštitutu za energetske raziskave pri Tehniški univerzi v Shandongu. Rezultati se dobro ujemajo z eksperimentalnimi vrednostmi, ki so bile pridobljene za isti reaktor. Model je torej primeren za termično snovanje objektov na industrijski ravni.

Glavni zaključki študije so: (1) Revna zmes metana v TFRR se vžge pri skoraj 900 °C. Temperatura je višja v sredini in se niža v aksialni smeri proti koncem reaktorja. Po več spremembah smeri toka se prehodni temperaturni profil v obliki črke M počasi premakne v aksialni smeri.

(2) Visokotemperaturno polje se po nekaj ciklih razširi, vršna vrednost pa se nekoliko poveča.

(3) Krivulje koncentracij se premaknejo v obe smeri. Intervali dveh setov krivulj so različni, predvsem ker se sčasoma poveča upor in se zmanjša pretok zmesi plina.

(4) Pri nespremenjenih ostalih pogojih se s povečevanjem koncentracije metana v zmesi od 0,3 do 0,7 % (vol.) poveča vršna temperatura, krivulja temperaturne porazdelitve je bližje vstopni strani, visokotemperaturno območje temperaturne porazdelitve se poveča, vmesno visokotemperaturno območje je bolj konkavno, poveča pa se tudi temperaturni gradient na vhodu in izhodu.

(5) Vpliv hitrosti dovoda na temperaturno porazdelitev ima dva nasprotna trenda. Vrednost maksimalne temperature v TFRR torej ne kaže enostavne odvisnosti od naraščajoče ali pojemajoče hitrosti dovoda.

Ključne besede: revna zmes metana in zraka, reaktor z izmenjevanjem smeri toka, termična oksidacija, modeliranje, nekatalitični, homogeno

*Naslov avtorja za dopisovanje: Tehniška univerza v Shandongu, Kolidž za promet in avtomobilsko tehniko, Zhangzhou 12, Zibo 255049, Kitajska, bmjw@163.com

SI 91

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI-92 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-10-17 Prejeto popravljeno: 2014-01-31 Odobreno za objavo: 2014-03-05

Ocena modula elastičnosti gumi podobnih materialov vezanih na toge površine z ozirom na Poissonovo število Koblar, D. – Škofic, J. – Boltežar, M. David Koblar1,* – Jan Škofic2 – Miha Boltežar3 1 Domel,

3 Univerza

2 Iskra

Slovenija Mehanizmi, Slovenija

v Ljubljani, Fakulteta za strojništvo, Slovenija

Gumeni materiali se pogosto uporabljajo za zmanjševanje prenosa vibracij in posledično tudi zmanjševanje hrupa.

Pri vgradnji v aplikacije pa so gume ali gumi podobni materiali pogosto vezani na ali med toge plošče, zaradi česar pride do spremembe obnašanje gume, saj se pri obremenitvi vezana površina materiala ne more deformirati tako kot ostali del nevezani material, kar daje občutek večje togosti od dejanske. V kolikor želimo napovedati prenos vibracij preko gumenega materiala z modelom iz končnih elementov potrebujemo materialne podatke kot sta modul elastičnosti in Poissonovo število.

Zasledimo lahko več eksperimentalnih člankov, ki podajajo povezave med modulom elastičnosti in navideznim modulom elastičnosti za gume ali gumi podobne materiale vezane med toge površine, ni pa mogoče zaslediti medsebojne primerjave pri izračunu z modelom iz končnih elementov.

Namen raziskave je narediti kvalitativno primerjavo med različnimi enačbami, ki popisujejo razmerje med navideznim modulom elastičnosti in dejanskim modulom elastičnosti, katerega potrebujemo pri analizi prenosnosti vibracij v modelu s končnimi elementi.

V prispevku so obravnavani aksialno obremenjeni gumijasti bloki okroglega prereza, ki so vezani na toge plošče. Za gumijaste vzorce je bil narejen model iz končnih elementov s katerim so izračunane frekvenčne prenosne funkcije, pri katerih smo spreminjali oblikovni faktor vzorcev (višino vzorca) in Poissonovo število.

Skupaj je bilo narejeno 20 primerov v frekvenčnem območju od 20 do 5000 Hz, pri frekvenčni resoluciji 1 Hz. Iz izračunanih frekvenčnih prenosnih funkcij je bila izračunana frekvenčna odvisnost navideznega modula elastičnosti pri posameznem oblikovnem faktorju in Poissonovem številu, nato pa je bil z uporabo različnih enačb, ki popisujejo odvisnost med navideznim modulom elastičnosti in modulom elastičnosti, ocenjen dejanski modul elastičnosti gumijastega materiala. V naslednjem koraku je bila narejena primerjava med tako ocenjenim modulom elastičnosti in tistim podanim v modelu iz končnih elementov. Rezultati analize so pokazali, da je pri uporabi različnih enačb bistvena razlika v ocenjenih vrednostih ocenjenega modula elastičnosti, še posebej, ko je Poissonovo število manjše od teoretične vrednosti 0,5.

Keywords: modul elastičnosti, navidezni modul elastičnosti, model iz končnih elementov, guma, gumi podoben

SI 92

*Naslov avtorja za dopisovanje: Domel, d.o.o., Otoki 21, 4228 Železniki, Slovenija, david.koblar@domel.si

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 93 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-10-28 Prejeto popravljeno: 2014-02-25 Odobreno za objavo: 2014-05-07

Analiza in terminiranje za optimizacijo časa proizvodnega cikla Jovanovic, J.R. – Milanovic, D.D. – Djukic, R.D. Jelena R. Jovanovic 1,2,3,* – Dragan D. Milanovic2 – Radisav D. Djukic1,3 1 Visoka

tehnično-strokovna šola, Čačak, Srbija v Beogradu, Fakulteta za strojništvo, Srbija 3 »Sloboda« Co. Čačak, Oddelek za proizvodnjo in inženirsko upravljanje, Srbija 2 Univerza

Članek podaja rezultate raziskave časovne dimenzije časa proizvodnega cikla sestavnih delov kompleksnih izdelkov. Rezultati predstavljajo del zahtevne raziskave upravljanja s proizvodnim ciklom kompleksnih izdelkov, ki se je začela 2010 ter je bila zaradi svojega pomena, obsega in zahtevnosti problematike razdeljena na več faz.

Raziskave so z ozirom na teoretične predpostavke in industrijsko prakso osredotočene na čase ciklov v pogojih ponavljajoče se serijske proizvodnje, kjer lahko zaradi kompleksnih in raznovrstnih tehnologij prihaja do zastojev.

Tehnološke proizvodne operacije, s katerimi se spreminjajo oblika in lastnosti obdelovancev, se tukaj prekrivajo z neproizvodnimi operacijami. Tehnološke operacije so jasno diferencirane: v proizvodnem ciklu se uporabljajo tako visokoproduktivni stroji za standardne in posebne namene, kjer so zgoščene tehnološke operacije, kakor tudi univerzalna oprema. Za proizvodnjo je značilno kombinirano gibanje obdelovancev po korakih procesa. Članek podaja preplet teorije in prakse za integriran pristop k preučevanju proizvodnega cikla, identifikaciji vzrokov izgub in merjenju njihovega vpliva, snovanju modelov za izboljševanje kakovosti načrtovanja in upravljanje proizvodnje, kakor tudi k eksperimentalni verifikaciji modelov v izbranem podjetju.

Izvirne formule za računanje tehnološkega proizvodnega cikla na podlagi gibanja obdelovanca skozi proizvodni proces, ter časovnih enot, s katerimi se izraža trajanje cikla, so bile pridobljene s pomočjo teorije grafov.

Za določitev trajanja proizvodnega cikla, preučitev vzrokov in merjenje izgub proizvodnih zmogljivosti so bile uporabljene metode statistične obravnave in analize. Članek za optimizacijo trajanja cikla opredeljuje izviren algoritem za terminiranje, ki upošteva realne pogoje in omejitve pri izvajanju proizvodnega procesa. Učinkovitost skrajšanja proizvodnega cikla je merjena s koeficientom pretočnosti. Znanstveni in strokovni javnosti je namenjen tudi kazalnik skrajšanja časa proizvodnega cikla, ki se določi po koncu proizvodnje na podlagi oblikovanih modelov. Koeficient pretočnosti je v tem kontekstu opredeljen kot razmerje dejanskega in terminiranega časa cikla, v proizvodnem sistemu pa ima večjo uporabnost, saj stimulira zmanjšanje celotnih izgub na sprejemljivo (optimalno) raven. Doseženo povprečno trajanje proizvodnega cikla v skladu z oblikovanimi modeli je v primerjavi z rezultati obstoječih metod načrtovanja in upravljanja proizvodnje krajše za 47,3 % oz. za 59 dni.

Prihodnje raziskave bodo usmerjene v oblikovanje modela za popis kompleksne strukture izdelkov ter v uporabo metod in tehnik za realizacijo naročil s terminiranjem, ob uporabi ustreznih programskih rešitev.

Ključne besede: proizvodni cikel, stopnja izkoriščenosti proizvodnih kapacitet, izgube pri proizvodnji, tok materiala, serijska proizvodnja

*Naslov avtorja za dopisovanje: Visoka tehnično-strokovna šola, Svetog Save 65, Čačak, Srbija, jelena.jovanovic@vstss.com

SI 93

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 94 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-10-08 Prejeto popravljeno: 2014-03-27 Odobreno za objavo: 2014-04-14

Snovanje in optimizacija ohišja PSD s hibridno strategijo MIGANLPQL na osnovi modela surogata Li, F. – Qin, Y. – Pang, Z. – Tian, L. – Zeng, X. Feng Li1 – Yumo Qin1 – Zhao Pang1 – Lei Tian2 – Xiaohua Zeng3* 1 Univerza

Jilin, Fakulteta za tehniške vede in strojništvo, Kitajska 2 BAIC MOTOR, Kitajska 3 Univerza Jilin, Državni laboratorij za simulacijo in sistemsko dinamiko, Kitajska

Članek podaja novo shemo za konstruiranje razdelilnih gonil PSD (Power-Split Device) pri HEV (hibridna vozila) z izboljšano obliko stožčastih planetnih zobnikov. Predstavlja tudi hibridno strategijo optimizacije ohišja PSD za zmanjšanje teže, izboljšanje togosti, optimalno konstrukcijo in skrajšanje časa snovanja. Članek podaja novo shemo za snovanje PSD. Za razliko od tradicionalnih ohišij diferencialov sta levi in desni del povezana s srednjim delom ohišja novega gonila PSD, ki je sestavljeno iz zunanjega in notranjega obroča. Takšen konstrukcijski pristop omogoča izboljšanje podporne oblike stožčastih planetnih zobnikov. Članek predstavlja hibridno strategijo optimizacije na osnovi modela surogata za snovanje ohišja PSD, ki bazira na modelu polinomskih odzivnih površin drugega reda in na hibridni strategiji optimizacije (genetski algoritem z več otoki (MIGA) in nelinearno kvadratično programiranje (NLPQL)). Strategija uporablja zasnovo optimalne latinske hiperkocke za določitev vzorčnih točk, polinomske odzivne površine drugega reda za določanje globalnih aproksimacij in hibridno strategijo optimizacije MIGA-NLPQL za doseganje globalnega optimuma. Članek obravnava optimizacijo zasnove ohišja PSD za zmanjšanje teže ohišja PSD pri omejeni togosti. Za vrednotenje togosti ohišja PSD je bila uporabljena simulacija po metodi končnih elementov. Vzorčne točke so bile določene z zasnovo optimalne latinske hiperkocke. Indeks togosti je bil določen s primerjavo sprememb kontaktne napetosti skupaj s kotom nagiba med standardnim in modificiranim profilom zob. Model polinomske odzivne površine drugega reda je bil vzpostavljen za zmanjšanje računskih stroškov, algoritem MIGA-NLPQL pa je bil uporabljen za doseganje globalnega optimuma.

Ukrepi za povečanje togosti in zmanjšanje teže si pogosto nasprotujejo. Zato je bila opravljena optimizacija konstrukcije ohišja PSD, kjer je bila za cilj optimizacije postavljena minimalna teža, za omejitev pa togost. Z optimizacijo se zmanjša teža ohišja PSD in se izboljša togost. Metoda modela surogata poenostavlja probleme in porabi manj računskega časa, hibridna strategija optimizacije pa je učinkovitejša, prihrani čas in privede do optimalnih rezultatov hitreje kot metoda enojne optimizacije.

Nov pristop k snovanju PSD lahko izboljša obliko stožčastih planetnih zobnikov, ki je drugačen kot pri tradicionalnih diferencialih. V članku je predstavljena hibridna strategija optimizacije na osnovi modela surogata za optimizacijo tipičnega ohišja. Optimalni rezultati so pridobljeni s hibridno strategijo optimizacije, ki združuje algoritma MIGA za globalno optimizacijo in NLPQL za gradientno optimizacijo. Hibridna strategija optimizacije prinaša več prednosti v primerjavi z metodo enojne optimizacije.

Ključne besede: hibridna vozila, razdelilno gonilo (PSD), optimizacija ohišja, hibridna optimizacija, model surogata, MIGA, NLPQL

SI 94

*Naslov avtorja za dopisovanje: Univerza Jilin, 5988 Renmin, Changchun, Kitajska. zengxh@jlu.edu.cn

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 95-98 Osebne objave

Doktorske disertacije, magistrska dela, diplomske naloge

DOKTORSKE DISERTACIJE

MAGISTRSKO DELO

Na Fakulteti za strojništvo Univerze v Ljubljani sta obranila svojo doktorsko disertacijo: ● dne 5. junija 2014 Jernej MELE z naslovom: »Katalitično uplinjanje biomase v lebdeči plasti z visoko-temperaturno vodno paro« (mentor: izr. prof. dr. Andrej Senegačnik, somentor: doc. dr. Dušan Klinar); V delu je obravnavano optimiranje katalitičnega uplinjanja biomase v lebdeči plasti z minimiziranjem dotoka visokotemperaturne vodne pare. Uplinjanje se izvaja v uplinjevalniku z ločenim uplinjevalnim in zgorevalnim reaktorjem. Razžarjeni sipki zrnasti katalizator v sistemu kroži in transportira toploto med reaktorjema. Za zaznavo minimalne hitrosti fluidizacije in stanja popolne fluidizacije je predlagana izvirna modalna zaznavno-nadzorno metoda. Dinamika krožeče lebdeče plasti se je najprej preizkušala v hladni laboratorijski napravi. S podobnostno teorijo po Glicksmanovem kriteriju je bil načrtovan pilotni uplinjevalnik toplotne moči 750 kW. Kemijske analize pridobljenega sinteznega plina so potrdile, da je obratovanje zgrajenega pilotnega uplinjevalnika ustrezno. Primerjava eksperimentalnih in analitičnih rezultatov potrjuje, da se v uplinjevalnem reaktorju izvaja razpad vodne pare in spremljajoče reakcije uplinjanja, ter reformiranje produktov; ● dne 30. junija 2014 Georgije BOSIGER z naslovom: »Ablacija bioloških tkiv z adaptivno vodenim laserskim snopom« (mentor: prof. dr. Janez Diaci); Delo zajema proučevanje praktičnih metod spremljanja laserske obdelave bioloških tkiv z računalniško vodenim Er:YAG laserskim snopom za učinkovitejšo izvedbo posegov v medicini in kirurgiji s poudarkom na ciljni aplikaciji - lasersko podprti implantologiji. Razvite metode temeljijo na osnovi piezoelektrične detekcije udarnih valov, ki se ob lasersko povzročenih mikroeksplozijah širijo po zraku nad tkivom. In vitro raziskave z razvitim prototipnim adaptivnim sistemom kažejo, da je z razvitimi metodami mogoče spremljati lasersko pripravo mest za zobne vsadke.

Na Fakulteti za strojništvo Univerze v Mariboru je z uspehom zagovarjal svoje magistrsko delo: dne 20. junija 2014: Zdenko KOLARIČ z naslovom: »Vpliv katalitične oksidacije SO2 na moker kalcitni postopek« (mentor: prof. dr. Aleksandra Lobnik).

DIPLOMSKE NALOGE Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv univerzitetni diplomirani inženir strojništva: dne 18. junija 2014: Jurij BREZAVŠČEK z naslovom: »Vpliv geometrije obtočnih kanalov na temperaturno polje v peči za sušenje kolektorjev elektromotorjev« (mentor: prof. dr. Branko Širok); Dejan MARJETIČ z naslovom: »Sistem za administracijo mehatronskega učnega laboratorij« (mentor: prof. dr. Janez Diaci); Gaia OŽBOT z naslovom: »Dinamika strukture filma kapljevine - glicerola na rotirajočem disku« (mentor: prof. dr. Branko Širok, somentor: izr. prof. dr. Marko Hočevar); dne 19. junija 2014: Dejan MODIC z naslovom: »Razvoj prijemala za prijemanje plošč kamene volne« (mentor: izr. prof. dr.

Niko Herakovič); dne 20. junija 2014: Bojan FAGANELI z naslovom: »Razvoj in raziskave protipovratnega ventila za vodno hidravliko« (mentor: doc. dr. Franc Majdič); Dejan KLEMENČIČ z naslovom: »Naprava za vtiskovanje kovinskih puš v plastična ohišja« (mentor: prof. dr. Marko Nagode); Ambrož KNE z naslovom: »Raziskava pretočnih rež znotraj proporcionalnega potnega ventila« (mentor: doc. dr. Franc Majdič); dne 24. junija 2014: Nada BREGAR z naslovom: »Neparametrično statistično modeliranje topografije površin« (mentor: prof. dr. Edvard Govekar); Klemen FRANCETIČ z naslovom: »Razvoj prototipa temperaturno obstojnega 2K-polimernega tečaja« (mentor: izr. prof. dr. Tomaž Pepelnjak).

SI 95

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 95-98

* Na Fakulteti za strojništvo Univerze v Mariboru sta pridobila naziv univerzitetni diplomirani inženir strojništva: dne 23. junija 2014: Uroš KOLAR z naslovom: »Sodobne tehnologije ravnanja z odpadnimi olji« (mentor: izr. prof. dr.

Darko Lovrec, somentor: prof. dr. Niko Samec); dne 26. junija 2014: Peter ŠMIGOC z naslovom: »Konstrukcija avtomobilske tovorne prikolice« (mentor: doc. dr.

Aleš Belšak, somentor: izr. prof. dr. Miran Ulbin). * Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv magister inženir strojništva: dne 18. junija 2014: Damir DEBOGOVIĆ z naslovom: »Vpliv zamenjave visokotlačne turbine na energijske in masne tokove v sekundarnem delu jedrske elektrarne« (mentor: izr. prof. dr. Mihael Sekavčnik); Anton PRESKAR z naslovom: »Hladilni sistem rečno hlajenega kondenzatorja v jedrski elektrarni« (mentor: izr. prof. dr. Mihael Sekavčnik); dne 19. junija 2014: Miha BLATNIK z naslovom: »Stroj za krivljenje konusov« (mentor: prof. dr. Jožef Duhovnik); Jurij KRANJC z naslovom: »Razvoj fleksibilnega prijemala za plazemsko lotanje strehe osebnega vozila« (mentor: izr. prof. dr. Jernej Klemenc); Mitja PETACI z naslovom: »Meritev hitrosti gibanja zraka povzročenega s tlačnim udarnim valom« (mentor: doc. dr. Viktor Šajn, somentor: prof. dr. Franc Kosel); dne 20. junija 2014: Damjan LUKANČIČ z naslovom: »Razvoj manipulatorja za surovce poltovornih avtoplaščev« (mentor: prof. dr. Marko Nagode, somentor: izr.prof. dr. Niko Herakovič); Dejan PLOS z naslovom: »Razvoj postopka konstruiranja za portalne vozne enote« (mentor: izr. prof. dr. Jože Tavčar, somentor: prof. dr. Jožef Duhovnik). * Na Fakulteti za strojništvo Univerze v Ljubljani sta zagovarjala svoje magistrsko delo (II. stopnja Erasmus): dne 24. junija 2014: Alejandra PEREZ RODRIGUEZ z naslovom: »Meritve lastnosti šobe sušilnika za lase z vizualizacijo toka / Measurements of flow properties SI 96

of an innovative nozzle of a hairdryer by using flow visualization « (mentor: izr. prof. dr. Marko Hočevar, somentor: prof. dr. Branko Širok); Teresa PEREZ RODRIGUEZ z naslovom: »Meritve lastnosti šobe sušilnika za lase z anemometrijo na vročo žičko / Measurements of flow properties of an innovative nozzle of a hairdryer using hot wire anemometry« (mentor: izr. prof. dr. Marko Hočevar, somentor: prof. dr. Branko Širok). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv univerzitetni diplomirani gospodarski inženir: dne 20. junija 2014: Markus KRISTOF z naslovom: »Analiza učinkovitosti toplotne črpalke« (mentor: doc. dr.

Matjaž Ramšak, doc. dr. Barbara Bradač Hojnik). * Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv magister inženir strojništva: dne 20. junija 2014: Tomaž KASTELIC z naslovom: »Analiza konstrukcije sedeža Sitty z uporabo računalniških simulacij in eksperimentov« (mentor: izr. prof. dr.

Matej Vesenjak, somentor: prof. dr. Srečko Glodež); Nejc ŠTROVS z naslovom: »Analiza in optimiranje zapenjalnega mehanizma zaščitnega stikala« (mentor: izr. prof. dr. Matej Vesenjak, somentor: red. prof. dr. Srečko Glodež); dne 24. junija 2014: Marta PLIBERŠEK z naslovom: »Vpliv dveh metod za sproščanje zaostalih napetosti na varjencu za transport tekoče žlindre« (mentor: doc. dr. Tomaž Vuherer, somentor: izr. prof. dr. Karl Gotlih). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv magister gospodarski inženir: dne 26. junija 2014: Krešimir GORIŠEK z naslovom: »Tržna analiza in razvoj aluminijastih ohišij LED svetil« (mentorja: izr. prof. dr. Bojan Dolšak, doc. dr. Matjaž Iršič). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv magister inženir oblikovanja izdelkov: dne 24. junija 2014:

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 95-98

David ROŽMARIN z naslovom: »Inženirsko oblikovanje igral in njihova umestitev« (mentor: izr. prof. Vojmir Pogačar). * Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva (UN): dne 10. junija 2014: Klemen LAVKA; dne 11. junija 2014: Vid LUŽNIK, dne 12. junija 2014: Peter HORVAT, in dne 16. junija 2014: Matic POGORELEC. * Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv diplomirani inženir strojništva (UN): dne 23. junija 2014: Rok POTOČNIK z naslovom »Vpliv zraka na delovanje hidravlične naprave« (mentor: izr. prof. dr.

Darko Lovrec, somentor: dr. Vito Tič); dne 26. junija 2014: Andrej PETEK z naslovom »Ergonomski vidiki razvoja vrtnega traktorja« (mentorja: izr. prof. dr.

Bojan Dolšak, somentor: asist. dr. Jasmin Kaljun); Jožef ZADRAVEC z naslovom »Nadgradnja visokozmogljivega konvencionalnega športnega avtomobila v visoko zmogljiv hibridni športni avtomobil« (mentor: izr. prof. dr. Jožef Predan, somentor: prof. dr. Nenad Gubeljak). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv diplomirani inženir mehatronike (UN): dne 26. junija 2014: Denis KOKOL z naslovom »Strega CNCobdelovalnega stroja z robotom KUKA KR30« (mentorja: izr. prof. dr. Karl Gotlih, izr. prof. dr. Aleš Hace). * Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva: dne 4. junija 2014: Marko ANDREJAŠ z naslovom: »Pisarna projektnega vodenja podjetja« (mentor: izr. prof. dr.

Janez Kušar, somentor: prof. dr. Marko Starbek); Blaž HOLC z naslovom: »Proces izdelave podvozij organiziran po sistemu ISO« (mentor: prof. dr. Janez Kopač); Zdenko KUMER z naslovom: »Analiza časovnih struktur dejavnosti delavca« (mentor: izr. prof. dr.

Janez Kušar, somentor: prof. dr. Marko Starbek);

Aleš PREVODNIK z naslovom: »Energetska učinkovitost procesa za proizvodnjo asfalta« (mentor: izr. prof. dr. Ivan Bajsić); dne 19. junija 2014: Aleš ERPE z naslovom: »Priprava in distribucija vode za injekcije« (mentor: prof. dr. Iztok Golobič); Gorazd FURLAN z naslovom: »Sanitarna armatura z vgrajenim filtrirnim sistemom« (mentor: prof. dr. Iztok Golobič); Igor ROŽMANC z naslovom: »Procesni sistemi za pripravo in dobavo prečiščene vode v farmaciji« (mentor: izr. prof. dr. Ivan Bajsić, somentor: prof. dr.

Iztok Golobič); Miha STAVBER z naslovom: »Energetski pregled industrijskega proizvodnega obrata« (mentor: izr. prof. dr. Andrej Senegačnik); Marko FERLAN z naslovom: »Dvig energijske učinkovitosti sistema za vračanje toplote z lamelnimi prenosniki toplote« (mentor: prof. dr. Vincenc Butala); Vitomir KUGLER z naslovom: »Optimizacija montažnega procesa varilnih priprav in prijemal« (mentor: izr. prof. dr. Niko Herakovič); Rok LAZNIK z naslovom: »Oblikovanje ohišja podkrilne kamere« (mentor: izr. prof. dr. Tadej Kosel); Bojan VERCE z naslovom: »Zasnova brezpilotnega letalskega sistema za oddaljen nadzor in dostavo tovora« (mentor: izr. prof. dr. Tadej Kosel); dne 20. junija 2014: Matic COTIČ z naslovom: »Raziskava osnovnih karakteristik mešanja v modelnem fermentorju z večstopenjskimi mešali« (mentor: doc. dr. Andrej Bombač); Mitja FORŠTNER z naslovom: »Optimizacija preoblikovanja pločevinskega izdelka« (mentor: izr. prof. dr. Tomaž Pepelnjak); dne 24. junija 2014: Bojan JUG z naslovom: »Obdelava vlaknocementnih fasadnih plošč s postopki odrezavanja« (mentor: prof. dr. Janez Kopač); Tomaj TAVŽELJ z naslovom: »Zasnova, izdelava in določitev zmogljivosti prototipnega motocikla na električni pogon« (mentor: izr. prof. dr. Tomaž Katrašnik). * Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv diplomirani inženir strojništva: dne 17. junija 2014: Teodor TOT z naslovom: »Optimiranje orodja za brizganje plastike v podjetju Elektromaterial Lendava« (mentor: izr. prof. dr. Igor Drstvenšek, somentor: dr. Tomaž Brajlih);

SI 97

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, SI 95-98

dne 26. junija 2014: Alfred PFIFER z naslovom: »Vzdrževanje ogrevalnih in hladilnih sistemov v poslovnem objektu Elektro Maribor« (mentor: prof. dr. Boris Aberšek); Andrej SUVALJ z naslovom: »Dimenzioniranje jeklene nadstrešnice« (mentor: doc. dr. Janez Kramberger). * Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva (VS): dne 4. junija 2014: Ambrož SIRK z naslovom: »Konstruiranje naprave za montažo žice v betonske distančnike« (mentor: izr. prof. dr. Jože Tavčar, somentor: prof. dr.

Jožef Duhovnik);

SI 98

dne 19. junija 2014: Simon TROHA z naslovom: »Kurilnost ukapljenega naftnega plina« (mentor: izr. prof. dr.

Andrej Senegačnik); dne 24. junija 2014: Aljaž KOTNIK z naslovom: »Programska oprema za analizo turbulentnega toka pasivnega polutanta« (mentor: izr. prof. dr. Marko Hočevar, somentor: prof. dr. Branko Širok). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv diplomirani inženir strojništva (VS): dne 26. junija 2014: Mitja ŠIJANEC z naslovom: »Vzdrževanje stroja za razrez in obrez aluminija kot integralni del zagotavljanja kakovosti« (mentor: doc. dr. Marjan Leber).

Strojniški vestnik – Journal of Mechanical Engineering (SV-JME) Aim and Scope The international journal publishes original and (mini)review articles covering the concepts of materials science, mechanics, kinematics, thermodynamics, energy and environment, mechatronics and robotics, fluid mechanics, tribology, cybernetics, industrial engineering and structural analysis.

The journal follows new trends and progress proven practice in the mechanical engineering and also in the closely related sciences as are electrical, civil and process engineering, medicine, microbiology, ecology, agriculture, transport systems, aviation, and others, thus creating a unique forum for interdisciplinary or multidisciplinary dialogue.

The international conferences selected papers are welcome for publishing as a special issue of SV-JME with invited co-editor(s).

Editor in Chief Vincenc Butala University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Technical Editor Pika Škraba University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Founding Editor Bojan Kraut

University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Editorial Office University of Ljubljana, Faculty of Mechanical Engineering SV-JME, Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386 (0)1 4771 137 Fax: 386 (0)1 2518 567 info@sv-jme.eu, http://www.sv-jme.eu Print: Littera Picta, printed in 400 copies Founders and Publishers University of Ljubljana, Faculty of Mechanical Engineering, Slovenia University of Maribor, Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia Chamber of Commerce and Industry of Slovenia, Metal Processing Industry Association President of Publishing Council Branko Širok University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Vice-President of Publishing Council Jože Balič

University of Maribor, Faculty of Mechanical Engineering, Slovenia Cover: The picture shows the time history of quasi static filament shapes of an oscillatingrotating toothbrush with inclined filaments.

The blue filament shows the instantaneous filament shape while the green filaments denote the prior shapes. The filament typically slides along the tooth surface, but might also stick to the surface under certain circumstances. The filament exerts shear and normal forces on the tooth that depend on the filament shape. An analysis of these forces leads to the fact that filaments with an inclined angle of approximately 16° provide more efficient plaque removal than vertical bristles.

Courtesy: Procter & Gamble, Kronberg, Germany

International Editorial Board Koshi Adachi, Graduate School of Engineering,Tohoku University, Japan Bikramjit Basu, Indian Institute of Technology, Kanpur, India Anton Bergant, Litostroj Power, Slovenia Franci Čuš, UM, Faculty of Mechanical Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mechanical Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mechanical Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mechanical Engineering, Slovenia Imre Horvath, Delft University of Technology, Netherlands Julius Kaplunov, Brunel University, West London, UK Milan Kljajin, J.J. Strossmayer University of Osijek, Croatia Janez Kopač, UL, Faculty of Mechanical Engineering, Slovenia Franc Kosel, UL, Faculty of Mechanical Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mechanical Engineering, Slovenia Miroslav Plančak, University of Novi Sad, Serbia Brian Prasad, California Institute of Technology, Pasadena, USA Bernd Sauer, University of Kaiserlautern, Germany Brane Širok, UL, Faculty of Mechanical Engineering, Slovenia Leopold Škerget, UM, Faculty of Mechanical Engineering, Slovenia George E. Totten, Portland State University, USA Nikos C. Tsourveloudis, Technical University of Crete, Greece Toma Udiljak, University of Zagreb, Croatia Arkady Voloshin, Lehigh University, Bethlehem, USA General information Strojniški vestnik – Journal of Mechanical Engineering is published in 11 issues per year (July and August is a double issue).

Institutional prices include print & online access: institutional subscription price and foreign subscription €100,00 (the price of a single issue is €10,00); general public subscription and student subscription €50,00 (the price of a single issue is €5,00). Prices are exclusive of tax.

Delivery is included in the price. The recipient is responsible for paying any import duties or taxes. Legal title passes to the customer on dispatch by our distributor.

Single issues from current and recent volumes are available at the current single-issue price. To order the journal, please complete the form on our website. For submissions, subscriptions and all other information please visit: http://en.sv-jme.eu/.

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We would like to thank the reviewers who have taken part in the peerreview process.

ISSN 0039-2480 © 2014 Strojniški vestnik - Journal of Mechanical Engineering.

All rights reserved. SV-JME is indexed / abstracted in: SCI-Expanded, Compendex, Inspec, ProQuest-CSA, SCOPUS, TEMA. The list of the remaining bases, in which SV-JME is indexed, is available on the website.

The journal is subsidized by Slovenian Research Agency.

Strojniški vestnik - Journal of Mechanical Engineering is also available on http://www.sv-jme.eu, where you access also to papers’ supplements, such as simulations, etc.

Instructions for Authors All manuscripts must be in English. Pages should be numbered sequentially. The maximum length of contributions is 10 pages. Longer contributions will only be accepted if authors provide justification in a cover letter. Short manuscripts should be less than 4 pages. For full instructions see the Authors Guideline section on the journal’s website: http://en.sv-jme.eu/.

Please note that file size limit at the journal’s website is 8Mb. Announcement: The authors are kindly invited to submitt the paper through our web site: http://ojs.sv-jme.eu. Please note that file size limit at the journal’s website is 8Mb. The Author is also able to accompany the paper with Supplementary Files in the form of Cover Letter, data sets, research instruments, source texts, etc. The Author is able to track the submission through the editorial process - as well as participate in the copyediting and proofreading of submissions accepted for publication - by logging in, and using the username and password provided.

Please provide a cover letter stating the following information about the submitted paper: 1. Paper title, list of authors and affiliations.

2. The type of your paper: original scientific paper (1.01), review scientific paper (1.02) or short scientific paper (1.03).

3. A declaration that your paper is unpublished work, not considered elsewhere for publication.

4. State the value of the paper or its practical, theoretical and scientific implications. What is new in the paper with respect to the state-of-the-art in the published papers?

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Every manuscript submitted to the SV-JME undergoes the course of the peer-review process.

THE FORMAT OF THE MANUSCRIPT The manuscript should be written in the following format: - A Title, which adequately describes the content of the manuscript.

- An Abstract should not exceed 250 words. The Abstract should state the principal objectives and the scope of the investigation, as well as the methodology employed. It should summarize the results and state the principal conclusions.

- 6 significant key words should follow the abstract to aid indexing.

- An Introduction, which should provide a review of recent literature and sufficient background information to allow the results of the article to be understood and evaluated.

- A Theory or experimental methods used.

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- A Results section, which should clearly and concisely present the data using figures and tables where appropriate.

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- Conclusions, which should present one or more conclusions that have been drawn from the results and subsequent discussion and do not duplicate the Abstract.

- References, which must be cited consecutively in the text using square brackets [1] and collected together in a reference list at the end of the manuscript.

Units - standard SI symbols and abbreviations should be used. Symbols for physical quantities in the text should be written in italics (e.g. v, T, n, etc.).

Symbols for units that consist of letters should be in plain text (e.g. ms-1, K, min, mm, etc.)

Abbreviations should be spelt out in full on first appearance, e.g., variable time geometry (VTG).

Meaning of symbols and units belonging to symbols should be explained in each case or quoted in a special table at the end of the manuscript before References.

Figures must be cited in a consecutive numerical order in the text and referred to in both the text and the caption as Fig. 1, Fig. 2, etc. Figures should be prepared without borders and on white grounding and should be sent separately in their original formats.

Pictures may be saved in resolution good enough for printing in any common format, e.g. BMP, GIF or JPG. However, graphs and line drawings should be prepared as vector images, e.g. CDR, AI. When labeling axes, physical quantities, e.g. t, v, m, etc. should be used whenever possible to minimize the need to label the axes in two languages.

Multi-curve graphs should have individual curves marked with a symbol. The meaning of the symbol should be explained in the figure caption.

Tables should carry separate titles and must be numbered in consecutive numerical order in the text and referred to in both the text and the caption as

Table 1, Table 2, etc. In addition to the physical quantity, e.g. t (in italics), units (normal text), should be added in square brackets. The tables should each have a heading. Tables should not duplicate data found elsewhere in the manuscript.

Acknowledgement of collaboration or preparation assistance may be included before References. Please note the source of funding for the research.

REFERENCES A reference list must be included using the following information as a guide. Only cited text references are included. Each reference is referred to in the text by a number enclosed in a square bracket (i.e., [3] or [2] to [6] for more references). No reference to the author is necessary.

References must be numbered and ordered according to where they are first mentioned in the paper, not alphabetically. All references must be complete and accurate. All non-English or. non-German titles must be translated into English with the added note (in language) at the end of reference. Examples follow.

Journal Papers: Surname 1, Initials, Surname 2, Initials (year). Title. Journal, volume, number, pages, DOI code.

[1] Hackenschmidt, R., Alber-Laukant, B., Rieg, F. (2010). Simulating nonlinear materials under centrifugal forces by using intelligent crosslinked simulations. Strojniški vestnik - Journal of Mechanical Engineering, vol. 57, no. 7-8, p. 531-538, DOI:10.5545/sv-jme.2011.013.

Journal titles should not be abbreviated. Note that journal title is set in italics.

Please add DOI code when available and link it to the web site.

Books: Surname 1, Initials, Surname 2, Initials (year). Title. Publisher, place of publication.

[2] Groover, M.P. (2007). Fundamentals of Modern Manufacturing. John Wiley & Sons, Hoboken.

Note that the title of the book is italicized.

Chapters in Books: Surname 1, Initials, Surname 2, Initials (year). Chapter title. Editor(s) of book, book title. Publisher, place of publication, pages.

[3] Carbone, G., Ceccarelli, M. (2005). Legged robotic systems. Kordić, V., Lazinica, A., Merdan, M. (Eds.), Cutting Edge Robotics. Pro literatur Verlag, Mammendorf, p. 553-576.

Proceedings Papers: Surname 1, Initials, Surname 2, Initials (year). Paper title. Proceedings title, pages.

[4] Štefanić, N., Martinčević-Mikić, S., Tošanović, N. (2009). Applied Lean System in Process Industry. MOTSP 2009 Conference Proceedings, p.

422-427.

Standards: Standard-Code (year). Title. Organisation. Place.

[5] ISO/DIS 16000-6.2:2002. Indoor Air – Part 6: Determination of Volatile Organic Compounds in Indoor and Chamber Air by Active Sampling on TENAX TA Sorbent, Thermal Desorption and Gas Chromatography using MSD/FID. International Organization for Standardization. Geneva. www pages: Surname, Initials or Company name. Title, from http://address, date of access.

[6] Rockwell Automation. Arena, from http://www.arenasimulation.com, accessed on 2009-09-07.

EXTENDED ABSTRACT By the time the paper is accepted for publishing, the authors are requested to send the extended abstract (approx. one A4 page or 3.500 to 4.000 characters). The instructions for writing the extended abstract are published on the web page http://www.sv-jme.eu/ information-for-authors/.

COPYRIGHT Authors submitting a manuscript do so on the understanding that the work has not been published before, is not being considered for publication elsewhere and has been read and approved by all authors. The submission of the manuscript by the authors means that the authors automatically agree to transfer copyright to SV-JME and when the manuscript is accepted for publication. All accepted manuscripts must be accompanied by a Copyright Transfer Agreement, which should be sent to the editor. The work should be original by the authors and not be published elsewhere in any language without the written consent of the publisher.

The proof will be sent to the author showing the final layout of the article. Proof correction must be minimal and fast. Thus it is essential that manuscripts are accurate when submitted.

Authors can track the status of their accepted articles on http://en.svjme.eu/.

PUBLICATION FEE For all articles authors will be asked to pay a publication fee prior to the article appearing in the journal. However, this fee only needs to be paid after the article has been accepted for publishing. The fee is 300.00 EUR (for articles with maximum of 10 pages), 20.00 EUR for each addition page.

Additional costs for a color page is 90.00 EUR.

http://www.sv-jme.eu

60 (2014) 7-8

Since 1955

Strojniški vestnik Journal of Mechanical Engineering

Contents Papers

449

Frank Goldschmidtboeing, Alexander Doll, Ulrich Stoerkel, Sebastian Neiss, Peter Woias: Comparison of Vertical and Inclined Toothbrush Filaments: Impact on Shear Force and Penetration Depth

462

Matjaž Čebron, Franc Kosel: Stored Energy Predictions from Dislocation-Based Hardening Models and Hardness Measurements for Tensile-Deformed Commercial Purity Copper

475

Mihai Dupac, Siamak Noroozi: Dynamic Modeling and Simulation of a Rotating Single Link Flexible Robotic Manipulator Subject to Quick Stops

483

Benjamin Bizjan, Alen Orbanić, Brane Širok, Tom Bajcar, Lovrenc Novak, Boštjan Kovač: Flow Image Velocimetry Method Based on Advection-Diffusion Equation

495

Xiaoni Qi, Yongqi Liu, Hongqin Xu, Zeyan Liu, Ruixiang Liu: Modeling Thermal Oxidation of Coal Mine Methane in a Non-Catalytic Reverse-Flow Reactor

506

David Koblar, Jan Škofic, Miha Boltežar: Evaluation of the Young’s Modulus of Rubber-Like Materials Bonded to Rigid Surfaces with Respect to Poisson’s Ratio

512

Jelena R. Jovanovic, Dragan D. Milanovic, Radisav D. Djukic: Manufacturing Cycle Time Analysis and Scheduling to Optimize Its Duration

525

Feng Li, Yumo Qin, Zhao Pang, Lei Tian, Xiaohua Zeng: Design and Optimization of PSD Housing Using a MIGA-NLPQL Hybrid Strategy Based on a Surrogate Model

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