Journal of Mechanical Engineering 2011 6

57 (2011) 1 6

Platnica SV-JME 57(2011)6_kor4.pdf 1 20.6.2011 14:45:31

Since 1955

C

M

Y

CM

MY

CY

CMY

Journal of Mechanical Engineering - Strojniški vestnik

K

6 year 2011 volume 57 no.

Platnica SV-JME 57(2011)6_kor4.pdf 2 20.6.2011 14:45:31

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 Co-Editor Borut Buchmeister University of Maribor Faculty of Mechanical Engineering, Slovenia Technical Editor Pika Škraba University of Ljubljana Faculty of Mechanical Engineering, Slovenia

C

M

Y

CM

MY

CY

CMY

K

Editorial Office University of Ljubljana (UL) Faculty of Mechanical Engineering SV-JME Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386-(0)1-4771 137 Fax: 386-(0)1-2518 567 E-mail: info@sv-jme.eu http://www.sv-jme.eu Founders and Publishers University of Ljubljana (UL) Faculty of Mechanical Engineering, Slovenia University of Maribor (UM) Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia Chamber of Commerce and Industry of Slovenia Metal Processing Industry Association Cover: Top: Test rig for measurement of energetic, cavitational and dynamic characteristics of Francis turbines. For better flow observation the conical part of the draft tube was made of plexiglass. At part load cavitating vortex rope can be observed and pressure pulsations caused by its rotation were measured. Bottom: The same shape of the rotating vortex rope was obtained by numerical simulation. Also pressure distribution on the runner and streamlines in the inlet part of the draft tube are presented.

Image courtesy: Turboinštitut d.d., Slovenia

ISSN 0039-2480 © 2011 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 Book Agency.

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 Mech. Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mech. Engineering, Slovenia Imre Felde, Bay Zoltan Inst. for Mater. Sci. and Techn., Hungary Jože Flašker, UM, Faculty of Mech. Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mech. 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 Mech. Engineering, Slovenia Franc Kosel, UL, Faculty of Mech. Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mech. 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 Mech. Engineering, Slovenia Leopold Škerget, UM, Faculty of Mech. 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 President of Publishing Council Jože Duhovnik UL, Faculty of Mechanical Engineering, Slovenia Print Tiskarna Present d.o.o., Ižanska cesta 383, Ljubljana, Slovenia General information Strojniški vestnik – The 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/.

You can advertise on the inner and outer side of the back cover of the magazine. The authors of the published papers are invited to send photos or pictures with short explanation for cover content.

We would like to thank the reviewers who have taken part in the peer-review process.

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.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6 Contents

Contents Strojniški vestnik - Journal of Mechanical Engineering volume 57, (2011), number 6 Ljubljana, June 2011 ISSN 0039-2480 Published monthly

Papers Dragica Jošt, Andrej Lipej: Numerical Prediction of Non-Cavitating and Cavitating Vortex Rope in a Francis Turbine Draft Tube 445 Paulo Flores: A Methodology for Quantifying the Kinematic Position Errors due to Manufacturing and Assembly Tolerances 457 Jurij Prezelj, Mirko Čudina: A Secondary Source Configuration for Control of a Ventilation Fan Noise in Ducts 468 Dejan Dragan: Fault Detection of an Industrial Heat-Exchanger: A Model-Based Approach 477 Henrik Zaletelj, Gorazd Fajdiga, Marko Nagode: Numerical Methods for TMF Cycle Modeling 485 Dobrivoje Ćatić, Branislav Jeremić, Zorica Djordjević, Nenad Miloradović: Criticality Analysis of the Elements of the Light Commercial Vehicle Steering Tie-Rod Joint 495 Aleksander Preglej, Rihard Karba, Igor Steiner, Igor Škrjanc: Mathematical Model of an Autoclave 503 Wai Chi Wong, Ishak Abdul Azid, Burhanuddin Yeop Majlis: Theoretical Analysis of Stiffness Constant and Effective Mass for a Round-Folded Beam in MEMS Accelerometer 517 Instructions for Authors 526

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456 DOI:10.5545/sv-jme.2010.068

Paper received: 25.03.2010 Paper accepted: 23.03.2011

Numerical Prediction of Non-Cavitating and Cavitating Vortex Rope in a Francis Turbine Draft Tube Jošt, D. – Lipej, A. Dragica Jošt* – Andrej Lipej Turboinštitut, Ljubljana, Slovenia

The paper presents a prediction of vortex rope in a draft tube obtained by the numerical flow analysis. The main goal of the research was to numerically predict pressure pulsation amplitude versus different guide vanes openings and compare the results with experimental ones. Three turbulent models (SAS-SST, ω-RSM and LES) were used. Also the effect of different domain configurations, grid density and time step size on results was examined. At first analysis was done without cavitation, while later at one operating point the cavitation model was included. ©2011 Journal of Mechanical Engineering. All rights reserved.

Keywords: Francis turbine, vortex rope, cavitation, pressure pulsation, turbulent models

0 INTRODUCTION Pressure fluctuations are a serious problem in hydraulic machinery and are usually the result of a strong vortex created in a centre of a flow at the outlet of a runner. The draft tube vortex appears at partial load operating regimes usually in radial turbines and also at single regulated axial turbines.

The consequences of the vortex developed in the draft tube are very unpleasant pressure pulsation, axial and radial forces and torque fluctuation as well as turbine structure vibration.

The intensive investigations of draft tube pressure pulsation on various specific speed turbines were performed in 1990’s and the results are published in [1] to [3]. Tests were performed in the wide turbine operating range from partial to full load and from runaway to head above optimum. Pressure pulsations were measured on models and prototypes. The influence of cavitation and various kinds of air admission on the pressure pulsation as well as on the efficiency characteristics were investigated. The conclusion was that the intensity of the vortex depends on specific speed of the turbine, operating regime and especially on a shape of runner blades and channel.

Significantly different dynamic characteristics can be obtained with different runner shapes at the same specific speed and approximately the same energetic characteristics. Unfortunately, the results from the model test are not always valid for the prototype and an acceptable model can be an unpleasant surprise as a prototype. Therefore, *Corr. Author’s Address: Turboinštitut, Rovšnikova 7, 1210 Ljubljana, Slovenia, dragica.jost@turboinstitut.si

an accurate numerical prediction of the vortex existence and intensity in a design stage is an important task.

The first attempt to numerically simulate the unsteady flow pattern accompanied with the helical vortex was made by Skotak [4] and [5].

Large Eddy Simulation (LES) model was used and in spite of for this model very coarse grid he managed to get a low pressure zone, which agrees well with the rotating rope observed in experiment. In the following years several papers about this topic were published. Usually unsteady analysis was performed only in the draft tube and the results of the previous steady state analysis of the runner were used as inlet boundary conditions.

In some cases the runner and the draft tube were analyzed simultaneously. Most of the authors reported that by standard k-ε model no rotating rope was obtained, while by the extended k-ε model of Kim and Chen [6] and realizable k-ε model [7] the rotating rope was obtained, but it was overly damped. Better results were obtained

by Reynolds Stress Models (RSM) [7] and [8] and Large Eddy Simulation (LES) [9] and [10].

Generally, where the experimental results were available, the frequency of pressure pulsation matched the measured values quite well, but the prediction of amplitudes was less accurate. When the cavitation was included [9], [11] and [12], it was reported that numerically obtained cavitating rope was smaller than in the experimental observation while the frequencies and amplitudes of pressure pulsation were not compared with 445

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

the experimental values. The frequencies were the same whether cavitation was modeled or not, but amplitudes were smaller in case of cavitation modeling. Due to long computational time, in most papers mentioned above, unsteady numerical analysis was limited to one or two operating points, computational grids were rather coarse and in some cases more time steps would be needed to get reliable results. In addition to the numerical flow analysis, an interesting approach to better understand these phenomena is simultaneous flow visualization and measurements of structural fluctuations [13].

This paper is a continuation of the work presented in [14] and [15]. In [14] the results obtained by the SAS-SST model at four operating points were presented. In [15] a numerical simulation at one operating point was performed by three turbulent models (SAS-SST, RSM and LES), with and without cavitation. Due to very long computational time work was not completed and only preliminary results were presented.

In this paper the results of [14] and [15] are summarized and completed.

1 FLOW DISTRIBUTION AT THE RUNNER OUTLET AT PART LOAD For part load conditions, the vortex (also called rope) spirals outward and processes in the direction of the turbine's rotation typically between one fourth and one third of the turbine's rotational frequency. These phenomena can cause large pressure fluctuations, low frequency vibrations and undesirable variations in the turbine output.

At part load a turbine works with a relative guide vanes opening, which is smaller than at optimal operating regime. The runner channel is not uniformly fulfilled with the flow because the main flow is near the shroud. A great secondary backflow zone is formed near the hub (Fig. 1).

On the border between backflow and mainstream there is a strong tangential shear, which is the main reason for the vortices.

Due to the the direction of circumferential component of absolute velocity the vortex has the same direction as the runner and goes downstream towards the draft tube. The vortex has a spiral shape and depending on the outlet velocity conditions it can have one, two or even three 446

branches. Inside the strong vortex the pressure is very low and if it reaches the value of vapour pressure the cavitation is also present.

The shape of the runner channel and the distribution of circumferential and meridional velocity components, which depends on the runner blade shape, have an important influence on the distribution of the vorticity. At some runners the vortex is very weak also at extreme part load regime, but in some cases the vortex can be very strong with high amplitudes of the pressure pulsations. Unfortunately, it is not possible to predict the direct connection between the runner shape and dynamic characteristics of the turbine.

It is for this reason necessary to complete the research work in this area, especially numerical prediction of the vortex formation and pressure pulsations characteristics, before any part of the turbine is constructed.

2 NUMERICAL MODELING Flow in water turbines is turbulent and unsteady. While the efficiency and cavitation in Francis, Kaplan and bulb turbines can be predicted by a steady state flow analysis and the results are usually accurate enough, unsteady flow analysis has to be performed when unsteady phenomena such as rotating vortex rope are the objects of interest. In these cases also more advanced turbulent models as Reynolds Stress Models (RSM), Large Eddy Simulation (LES), Detached Eddy Simulation (DES) or Scale-Adaptive Simulation (SAS) models have to be used. In this paper three turbulent models are used: SAS-SST, RSM and LES. The Scale-Adaptive Simulation (SAS) is an improved URANS (Unsteady Reynolds Averaged Navier-Stokes) formulation, which allows the resolution of the turbulent spectrum in unstable flow conditions. The SAS concept is based on the introduction of the von Karman length-scale into the turbulence scale equation. The information provided by the von Karman length-scale allows SAS models to dynamically adjust to resolved structures in a URANS simulation, which results in a LES-like behaviour in unsteady regions of the flow-field. At the same time, the model provides standard RANS capabilities in stable flow regions.

SAS-SST model is the combination of SAS and SST (Shear Stress Transport) model.

Jošt, D. – Lipej, A.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

Reynolds Stress Turbulence Models (RSM) based on transport equations for all components of the Reynolds stress tensor and the dissipation rate. The exact production term and the inherent modelling of stress anisotropies make RSM more suited to complex flows, for example a flow with a rotating vortex rope in the draft tube, where standard two-equation models fail. However, due to six additional transport equations the computational time increases significantly.

a)

b)

Fig. 1. Velocity distribution at the outlet of the runner for different operating regimes; a) BEP, b) part load Details on the structure of turbulent flows, such as pressure fluctuations, can be obtained

by LES. LES is an approach which solves for large-scale fluctuating motions and uses »subgrid« scale turbulence models for the smallscale motion. In ANSYS three LES models are available: the Smagorinsky model, the walladapted local eddy-viscosity model (LES WALE) and the Dynamic Smagorinsky-Lilly model. The first two models are algebraic. The Smagorinsky model is available together with two different formulations of the wall damping function. The LES WALE model needs no wall damping, while the Dynamic Smagorinsky-Lilly model uses the information contained in the resolved turbulent velocity field to evaluate the model coefficient.

The method needs explicit (secondary) filtering and it is therefore more time consuming than the algebraic models. LES requires fine grids and small time steps, particularly for wall bounded flows, as well as a large number of time steps to generate statistically meaningful correlations for the fluctuation velocity components [16].

The cavitating vortex rope has to be modeled by one of the multiphase models. The

homogeneous and inhomogeneous multiphase models are available. In the inhomogeneous model each fluid possesses its own field and the fluids interact via interphase transfer terms. There is one solution field for each separate phase.

Sub-models differ in the way they model the interfacial area density and the interphase terms.

The homogeneous model assumes that transported quantities (with the exception of volume fraction) for the process are the same for all phases.

Therefore, it is sufficient to solve bulk transport equations for shared fields rather than solving individual transport equations. Density and viscosity are calculated from density and viscosity of all phases in the fluid. The cavitation is usually modeled by homogeneous model.

Cavitation refers to the process by which vapour forms in low pressure regions of a liquid flow. In the ANSYS CFX-12, The RayleighPlesset model is implemented in the multiphase framework as an interphase mass transfer model.

The growth of bubbles is given by RayleighPlesset equation. The equation for volume fraction is fully coupled with flow equations, because the term, which represents the sources, depends on the pressure. The bubble grows if the pressure is low.

On the other hand, the vaporization causes change of density in flow equations.

A detailed description of turbulent models and multiphase models can be found in ANSYS CFX-12 Solver Theory Documentation [16].

3 MODEL TEST Pressure fluctuations on Francis model turbine at Turboinštitut are observed and measured in accordance with IEC 90193. For this purpose KISTLER piezoresistive absolute pressure transducers are located on different locations at the spiral casing and at the inlet part of the draft tube. The main purpose of these measurements was to obtain enough information about the magnitude and the nature of pressure fluctuations, their dominant frequency and additionally the dampening effect of air admission.

Signals from the pressure transducers are wired to the multi channel data acquisition system, based on National Instruments multifunction card with additional SCXI signal conditioning modules. Signals are acquired continuously with

Numerical Prediction of Non-Cavitating and Cavitating Vortex Rope in a Francis Turbine Draft Tube

447

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

5 kHz sampling frequency and 16 bit resolution.

Binary data samples stored on the computer hard disk are at least 30 seconds long. LabVIEW software is used to record and analyze pressure signals.

4 NUMERICAL PREDICTION OF VORTEX ROPE Accuracy of numerical prediction of vortex rope was verified on two Francis turbines.

For the first one (case 1) numerical analysis was performed at four operating points by SAS-SST turbulent model on different computational grids and domain configurations and with different time steps. Analysis was done without cavitation.

The second turbine (case 2) was analyzed at one operating point by RSM, SAS-SST and LES turbulent models. To obtain the cavitating vortex rope homogenious two-phase flow model was used.

4.1 Test Case 1: Prediction of Pressure Pulsation Amplitudes and Frequencies for Different Operating Regimes For the first turbine different computational grids and domain configurations were used (Table 1). At first, the computational grid of the complete turbine including spiral casing with stay vanes, guide vane cascade, runner and draft tube with 3,300,000 elements was used. The second computational grid of the complete turbine consists of the total 17,000,000 elements. The third configuration consists of the runner and the draft tube with 25,000,000 elements. The grid of the spiral casing is made using tetrahedral elements with prism layers near the walls, all other turbine parts are meshed by hexahedral elements. The length of the time step in unsteady calculation was equal to one, three or six degrees of the runner revolution. Convergence criteria was prescribed to 5×10-5 for RMS and in the average a converged solution was obtained after three to ten iterations in each time step, depending on the flow rate operating regime. Shape of the vortex rope at OP1, OP2 and OP3 is presented as isosurface of constant pressure (Fig. 2). Its exact value is in this case insignificant as cavitation was not modeled and the value of reference pressure 448

value has no influence on the results. At OP4 no vortex rope was observed and pressure pulsations were negligible.

The numerically obtained frequency of the vortex rope is about 20 to 26% of the runner rotational speed depending on the operating regime. The frequency spectrum obtained from the measurements on the model test rig is very similar to the numerically obtained values. The difference between numerical and experimental results is about 1% of the runner rotational speed. In Table 3 the values of pressure pulsation frequency obtained by measurement and numerical analysis are presented.

The effect of grid density and time step can be seen in Fig. 3 where pressure pulsations at OP2 for two grid configurations are presented.

For the coarse grid the time step was equal to 3 deg. of runner revolution. For the fine grid with 17 million elements the first part of the graph shows the pressure pulsation for time step equal to 6 deg. and the last part for time step equal to 1 deg. of runner revolution. In this case, the differences in frequency and amplitude due to the time step are negligible, but the effect of grid density can be clearly seen. The amplitudes of pressure pulsation calculated on the fine grid are significantly higher than those obtained on the coarse grid.

The calculation took a considerable amount of time since during the calculation the shape of the vortex rope was forming and only after several vortex revolutions the correct frequency was obtained. In Fig. 4 the history of the frequency during the calculation is shown for operating points OP1 and OP2. At the beginning of the calculation the vortex rope frequency was lower than the frequency obtained by the measurements. After five complete vortex revolutions the numerical and experimental values of the frequency differ for about one percent.

In addition to the vortex rope frequency the main challenge of the numerical analysis was the prediction of the pressure pulsation amplitudes for different operating regimes. The comparison between the measurements and the numerical results shows that the numerical prediction of the position of maximal pressure pulsation amplitude is quite accurate, but the predicted values of the amplitudes are lower than the experimental ones, except for the operating regime with the

Jošt, D. – Lipej, A.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

Table 1. Number of elements in particular geometry configuration, case 1

Configuration 1 Configuration 2 Configuration 3

Spiral casing, stay and guide vanes 1,400,000 1,400,000 -

Runner

Draft tube

Total

1,000,000 12,600,000 12,600,000

900,000 3,000,000 12,400,000

3,300,000 17,000,000 25,000,000

a)

b)

c)

d)

e)

f)

Fig. 2. Vortex rope at different operating points; a) experiment OP1, b) experiment OP2 , c) experiment OP3, d) numerical simulation OP1, e) numerical simulation OP2 , f) numerical simulation OP3; case 1

smallest flow rate. A significant difference is obtained in pressure pulsation amplitudes for two different computational grids. The detailed results and comparison with measurements are presented in Fig. 5. It can be seen that with coarse computational grid (configuration 1) the difference between the numerical and experimental results Table 2. Operating points for numerical analysis, case 1 Operating point OP1 OP2 OP3 OP4

A0 /A0 BEP

φ / φ BEP

ψ / ψ BEP

0.656 0.800 0.840 1.000

0.66 0.81 0.85 1.00

0.97 0.97 0.97 0.97

at OP2 is about 44%, but with grid refinement (configuration 2) the accuracy improved and the difference is less than 14%.

4.2 Test Case 2: The Effect of Turbulent Models In case 2 we focused on the effect of different turbulent models on the results. Analysis was done by SAS-SST, RSM and LES with and without cavitation. Cavitation was modeled by homogenious two-phase model. Mass transfer was done by the Rayleigh-Plesset model. In all the cases steady state solution without cavitation was used as an initial for steady state calculation with cavitation and these results were used as an initial for unsteady analysis with cavitation.

Numerical Prediction of Non-Cavitating and Cavitating Vortex Rope in a Francis Turbine Draft Tube

449

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

Table 3. Pressure pulsation frequency – experimental and numerical results, case 1 Operating point OP1 OP2 OP3 OP4

Experimental values [Hz] 3.20 4.00 4.10 4.45

configuration 1 3.09 4.01 3.80 -

0.0

0.5

1.0 1.5 t [s]

configuration 3 3.80 -

1.10E+05 1.05E+05 1.00E+05 9.50E+04 9.00E+04 8.50E+04

p [Pa]

p [Pa]

1.10E+05 1.05E+05 1.00E+05 9.50E+04 9.00E+04 8.50E+04

Numerical values [Hz] configuration 2 3.22 3.95 3.82 -

2.0

0.0

0.5

1.0 1.5 t [s]

2.0

4

Rotation 1

3

Rotation 2

2 1 0

a)

Frequency [Hz]

Frequency [Hz]

time step 3° time step 6° time step 1° a) b) Fig. 3. Pressure pulsations at OP2 for two computational grids; a) 3.3 million elements , b) 17 million elements, case 1

Rotation 3 Rotation 4 Rotation 5 Measurements

Rotation 1

3

Rotation 2

2

Rotation 3 Rotation 4

1

Rotation 5

0

Measurements

b) Fig. 4. Vortex rope frequency variations during the calculation at; a) OP1, b) OP2 compared to experimental values, case 1

Analysis was done for an operating point at part load (A0/AoBEP = 0.8572, φ/φBEP = 0.7716, ψ/ψBEP = 0.7850), for which pressure was measured at two positions on the conical part of the draft tube (Fig. 6) and quite strong pressure pulsations were detected. The value of cavitation coefficient σ was equal to 0.1497. For this value of σ the cavitation was present and due to the water vapour in the rope its long thick shape was seen (Fig. 7).

When cavitation is modeled the value of cavitation coefficient in numerical analysis has to be the same as in reality. Therefore, the value of 450

4

static pressure prescribed at the outlet of the draft tube was obtained from the measurements.

The domain of calculation, when SAS-SST and RSM were used, was the complete turbine.

Computational grid consists of 5.6 million nodes and is rather coarse, only the grid in the draft tube was a bit refined and it consists of 3.4 million nodes. Unsteady analysis by ω-RSM model started from the steady state solution obtained by the same turbulent model, while the analysis with SAS-SST model started from the steady state solution obtained by the SST model.

Time step was equal to 2 degrees of runner revolution. When the cavitation model was

Jošt, D. – Lipej, A.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

Fig. 5. Pressure pulsation, 98% peak–to-peak amplitudes for different operating regimes; comparison between experimental and numerical values, case 1

Fig. 6. The positions of pressure pulsation measurement, case 2

included the ω-RSM model was replaced by BSLRSM and the time step was reduced to 1 degree of runner revolution.

LES requires fine grid, therefore the domain of the calculation was reduced to the draft tube with prolongation at the outlet. The grid consists of 23.5 million nodes. At the inlet of the draft tube velocity components obtained by an analysis of the whole turbine by SST turbulent model were prescribed. The results of the steady state solution obtained by the SST model were used as an initial condition. The runner rotation was simulated by unsteady inlet condition. Time step corresponded to 0.5 deg. of runner revolution.

Fig. 7. Cavitating vortex rope on test rig, case 2 The average Courant number was around 0.2.

For each time step 4 to 6 iterations were needed.

Calculation without cavitation was done by the Smagorinsky model. When analysis was repeated with cavitation LES WALE model was used. Due to the two-phase model for each time step more iterations (ten) were needed.

When cavitation was modeled the homogenious two-phase model was used and fluid density was calculated from the values of water density ρ1 and water vapour density ρ2 by the term ρ = r1ρ1 + r2ρ2, where r1 and r2 are volume fractions of water and water vapour respectively and r1 + r2 = 1. In regions without cavitation

Numerical Prediction of Non-Cavitating and Cavitating Vortex Rope in a Francis Turbine Draft Tube

451

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

Fig. 8. Density at five horizontal cross-sections and a detail at one section, results of SAS-SST model with cavitation, case 2

a)

b)

c)

d) e) f) Fig. 9. Numerically obtained shape of the vortex rope, case 2; a) SAS- SST, 20 runner revolutions, no cavitation, b) ω-RSM, 20 runner revolutions, no cavitation, c) LES, 17 runner revolutions, no cavitation, d) SAS-SST, 20 runner revolutions, cavitation, e) BSL-RSM,20 runner revolutions, cavitation, f) LES, 20 runner revolutions, cavitation density is equal to water density. The cavitating vortex rope consists of water and water vapour, therefore the value of density inside the rope is in range between ρ1 and ρ2. Density at five horizontal cross-sections is presented in Fig. 8. It can be seen that the core of the vortex rope mostly consists of water vapour and density there is close to ρ2. The results in Fig. 8 were obtained by the SAS-SST model, but the pictures obtained by the other two 452

turbulent models (RSM and LES) would be quite similar.

The numerically obtained vortex rope can be presented as an iso-surface of the appropriate value of absolute pressure and in case of cavitation also as an iso-surface of the appropriate value of water vapour volume fraction or fluid density.

Furthermore, the images are almost equal. In this case the shape of the vortex rope is presented as

Jošt, D. – Lipej, A.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

an iso-surface of vaporization pressure. When cavitation was not included all three turbulent models predict a shape of the vortex rope similar to the one observed on the test rig (Figs. 9a, b and c). When cavitation was included the formation of the cavitating vortex rope was slower. In case of BSL-RSM model about 5 runner revolutions were needed to get the final length of the rope while in case of SAS-SST only after 15 runner revolutions the final length of the rope was obtained. The formation of cavitating rope by LES model was even slower. The vortex ropes obtained by the cavitation model and different turbulent models can be seen in Figs. 9d, e and f. They have slightly different shapes. The difference is significant just behind the runner where the ropes obtained by SAS-SST and BSL-RSM go nearly straight down.

The rope obtained by LES WALE model is after 20 runner revolutions still not fully developed, but its shape is in better agreement with the experiment.

Numerically obtained pressure pulsations at two positions at the cone of the draft tube are presented in Fig. 10. In all the cases it took a long time before the frequency and amplitudes stabilized. Looking at first the curves obtained without cavitation, it can be seen that at the beginning the results of different turbulent models were quite different, but with time they were becoming more similar. The situation was worse when cavitation was modeled. By SAS-SST model the pressure pulsation frequency stabilized after 25 runner revolutions and it was nearly the same as without cavitation, but the amplitude at position 2 was much too small. By BSL-RSM only 32 runner revolutions were done. It seems that numerical errors were accumulating during

the calculation and pressure oscillation was increasingly irregular. LES with the cavitation needed more than 20 runner revolutions before the oscillations stabilized. The frequency and amplitudes are slightly larger than those obtained without cavitation and are therefore, closer to experimental values. By LES besides pressure oscillation of high amplitudes also small oscillations of high frequency and small amplitudes were obtained.

In Table 4 experimental and numerical values of pressure pulsation frequency and amplitudes are presented. Frequencies of pressure pulsation were obtained by Fast Fourier Transform, Hanning’s window was used. The calculated values of frequency are smaller than the experimental ones for all turbulent models with and without cavitation, except for LES WALE model with cavitation, where the calculated value is 1.4% larger than the measured one and where the best agreement between numerical and experimental values was obtained. The calculated values of amplitudes are smaller than the measured one, especially for position 2, where the discrepancies obtained without cavitation are 31.26, 30.83 and 34.75% for SAS-SST, RSM and LES respectively. When cavitation was modeled only LES WALE model gave useful results, but the amplitudes are still much smaller than the measured ones. The grid in the draft tube for LES is more refined than for the other two turbulent models, therefore it is difficult to say, whether the results are better because of the more advanced turbulent model or because of grid refinement. The results by BSL-RSM and SAS-SST model with cavitation may be improved by grid refinement and double precision calculation.

Table 4. Numerical and experimental results for case 2 Experiment SAS-SST, cavitation not modeled ω-RSM, cavitation not modeled LES Smagorinsky, cavitation not modeled SAS-SST, cavitation modeled BSL-RSM, cavitation modeled LES WALE , cavitation modeled

Frequency [Hz] 3.50 3.26 3.21 3.12 3.32 Results too irregular 3.55

Position 1 App [%] Position 2 App [%] 2.77 4.67 2.25 3.21 2.57 3.23 2.32 2.96 2.67 2.00 Results too Results too irregular irregular 2.43 3.20

Numerical Prediction of Non-Cavitating and Cavitating Vortex Rope in a Francis Turbine Draft Tube

453

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

cavitation not modeled, cavitation modeled Fig. 10. Pressure pulsation obtained by different turbulent models with and without cavitation modeling, case 2

5 COMPUTATIONAL EFFORT AND HARDWARE FOR VORTEX ROPE PREDICTION The numerical simulation of vortex rope is considerably time consuming. It takes a long 454

time before the vortex acquires its final shape and the frequency of its rotation stabilizes. Usually more than 40 runner revolutions are needed to get stable values of pressure pulsation frequency and amplitudes. In case of RSM there are additional equations for Reynolds stresses, while for LES the

Jošt, D. – Lipej, A.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

grid has to be refined and time step reduced. The situation is even worse when cavitation is also modeled.

When the project Numerical simulation of vortex rope in a Francis turbine started, our computer capacities were limited to the cluster with 32 dual core processors. In May 2008 a supercomputer cluster with 2048 processor cores – 512 Quad-Core Intel Xeon processors L5335 was installed. In total the cluster has more than 2 TB RAM and for high performance computing communications InfiniBand is used. In case 1 the first mesh configuration (3.3 millions elements) was analyzed by the old computer and the CPU time for one complete runner revolution was about 24 hours on eight processors. For the second and third configurations of case 1 and for all analysis of case 2 the new supercomputer cluster was used. For the grid with 17 millions elements 32 quad-core processors (128 cores) were used and the total computational time was about ten hours for one runner revolution. For LES, where computational grid consists of about 23 millions elements, 64 quad-core processors (256 cores) were used and for one-phase flow two runner revolutions were performed per day. For the two phase flow with cavitation model more than one day was needed for one runner revolution. The usage of the higher number of processors for problems of about 20 million elements is not sensible because the reduction of the calculation time is too insignificant.

6 CONCLUSION Nowadays powerful supercomputers enable the prediction of the rotating vortex rope in a Francis turbine. The results of case 1 showed that the experimentally and numerically obtained frequencies are very close, while the prediction of amplitudes is less accurate, but with grid refinement it approaches the experimental values.

In case 2 it was shown that SAS-SST, RSM and LES models are suitable for vortex rope prediction and when cavitation was not modeled there was no significant difference in the accuracy of the results obtained by these three turbulent models.

When cavitation was modeled the results obtained

by SAS-SST and RSM were less accurate.

However, we expect that they can be improved by

grid refinement and double precision calculation.

When LES model was used modeling of cavitation improved the results, amplitudes and especially frequency came closer to experimental values.

7 ACKNOWLEDGEMENTS The research is partially financed by the Slovenian Research Agency ARRS – Contract no L2-1067-0263-08.

8 NOMENCLATURE A0 [-] App [%]

relative guide vane opening peak to peak pressure pulsation amplitude, App =

∆p .100% ρ gH

H [m] head Hat [m] head corresponding to

atmospheric pressure Hp [m] vaporization head Hs [m] suction head g [m/s2] gravity p [Pa] pressure paver [Pa] average pressure prel [-] relative pressure,

v [m/s] η [-] ηBEP [-] σ [-]

prel =

p − paver ρ gH

velocity efficiency efficiency at BEP cavitation coefficient,

σ=

H at − H s − H p H

φ [-] discharge coefficient ρ [kg/m3] density ρ1 [kg/m3] water density 3 ρ2 [kg/m ] water vapour density ψ [-] pressure coefficient BEP Best Efficiency Point 9 REFERENCES [1] Kercan, V., Bajd, M., Djelić, V., Lipej, A., Jošt, D. (1995). Model and prototype draft tube pressure pulsations. Proceedings of the 7th International Meeting of IAHR Work Group on the Behavior of Hydraulic

Numerical Prediction of Non-Cavitating and Cavitating Vortex Rope in a Francis Turbine Draft Tube

455

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 445-456

Machinery under Steady Oscillatory Conditions, A6.

[2] Kercan, V., Bajd, M., Djelić, V., Lipej, A., Jošt, D. (1996). Model and prototype draft tube pressure pulsations. Proceedings of the XVIII. IAHR Symposyum on Hydraulic Machinery and Cavitation, Volume II, p.

994-1003.

[3] Kercan, V. (2001). The influence of Francis runner shape on phenomena of vortex rope in a draft tube. Ph.D thesis, University in Rijeka, Technical faculty. 94 p. (in Croatian) [4] Skotak, A. (1999). Draft tube swirl modeling.

Proceedings of the 9th International Meeting of IAHR Work Group on the Behavior of Hydraulic Machinery under Steady Oscillatory Conditions, D4.

[5] Skotak, A. (2000). Of the helical vortex in the draft tube turbine modeling. XXth IAHR Symposium on Hydraulic Machinery and Systems.

[6] Ruprecht, A., Helmrich, T., Aschenbrenner, T., Scherer, T. (2002). Simulation of vortex rope in a turbine draft tube. Proceedings of the XXIst IAHR Symposium on Hydraulic Machinery and Systems.

[7] Skotak, A., Mikulašek, J., Lhotakova, L. (2002). Effect of the inflow conditions on the unsteady draft tube flow. Proceedings of the XXIst IAHR Symposium on Hydraulic Machinery and Systems.

[8] Sick, M., Doerfler, P., Sallaberger, M., Lohmberg, A., Casey, M. (2002). CFD simulation of the draft tube vortex.

Proceedings of the XXIst IAHR Symposium on Hydraulic Machinery and Systems.

[9] Guo, Y., Kato, C., Miyagawa, K. (2006).

Large-eddy simulation of non-cavitating and

456

cavitating flows in an elbow draft tube. 23rd IAHR Symposium on Hydraulic Machinery and Systems.

[10] Kurosawa, S., Satou, S. (2006). Turbulent flow simulation for the draft tube of a Kaplan turbine. 23rd IAHR Symposium on Hydraulic Machinery and Systems.

[11] Miyagawa, K., Tsuji, K., Yahara, J., Nomura, Y. (2002). Flow instability in an elbow draft tube for a Francis pump-turbine.

Proceedings of the XXIst IAHR Symposium on Hydraulic Machinery and Systems.

[12] Zhou, L, Wang, Z, Tian, Y. (2006).

Numerical simulation of vortex cavitation in draft tube. 23rd IAHR Symposium on Hydraulic Machinery and Systems.

[13] Širok, B., Blagojević, B., Bajcar, T., Trenc, F. (2003). Simultaneous study of pressure pulsation and structural fluctuations of a cavitated vortex core in the draft tube of a Francis turbine. J. Hydraul. Res., vol. 41, no.

5, p. 541-548.

[14] Lipej, A., Jošt, D., Mežnar, V., Djelić, V. (2008). Numerical prediction of pressure pulsation amplitude for different operating regimes of Francis turbine draft tube. 24th IAHR Symposium on Hydraulic Machinery and Systems.

[15] Jošt, D., Lipej, A. (2009). Numerical prediction of the vortex rope in the draft tube. 3rd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, p. 75-85.

[16] ANSYS CFX, Release 12, Solver Theory Documentation (2009).

Jošt, D. – Lipej, A.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467 DOI:10.5545/sv-jme.2009.159

Paper received: 16.11.2009 Paper accepted: 06.03.2011

A Methodology for Quantifying the Kinematic Position Errors due to Manufacturing and Assembly Tolerances Flores, P. Paulo Flores* CT2M/ Department of Mechanical Engineering, University of Minho, Portugal

A systematic and general methodology for kinematic position errors analysis of multibody systems is investigated throughout this work, taking into account the influence of the manufacturing and assemble tolerances on the performance of planar mechanisms. The generalized Cartesian coordinates are used to mathematically formulate kinematic constraints and equations of motion of the multibody systems. Thus, the systems are defined by a set of generalized coordinates, which represents the instantaneous positions of all bodies, together with a set of generalized dimensional parameters that defines the functional dimensions of the system. These generalized dimensional parameters take into account the tolerances associated with the lengths, fixed angles, diameters and distance between centers, among others. This paper highlights the relation among kinematic constraints, dimensional parameters and output kinematic parameters. Based on the theory of dimensional tolerances, the variation of the geometrical dimensions is regarded as a tolerance grade with an interval associated with each dimension and, consequently, a kinematic amplitude variation for the bodies’ position. The methodology presented is implemented in a computational code developed for kinematic analysis of multibody systems, capable of automatically generating and solving the equations of motion for general multibody systems. Finally, a slider-crank mechanism is used as a numerical example to demonstrate the accuracy of the presented methodology, as well as to discuss the main assumptions and procedures adopted in this work. ©2011 Journal of Mechanical Engineering. All rights reserved.

Keywords: positional error, manufacturing tolerances, assembly systems, planar mechanisms 0 INTRODUCTION It is well known that the tolerances play a key role in the modern design process by introducing quality improvements and limiting manufacturing costs. According to standard ANSI Y14.5M-1994, tolerances are used to define the allowable limits of geometric variation that are inherent in manufacturing and assemble processes [1]. Thus, the assignment of geometric tolerances is always a trade-off between two distinct situations: (i) a part with tight tolerances is good for assembly, but the cost to manufacture the part is increased; (ii) alternatively, loose tolerance in one part may make the whole assemble infeasible.

The tolerance analysis deals with the study of the aggregate behaviors of given individual tolerances [2] and [3].

Over the last few decades a number of research papers have been published on the influence of the tolerance and clearance on the kinematic performance of multibody systems [4] to [14]. However, most of these research works

lack generality, that is, they are developed for specific mechanisms and situations. Garrett and Hall [4] defined the concept of mobility bands to study the effect of tolerance and clearance in the design of linkages. Dhande and Chakraborty [5] presented a stochastic model for the analysis and synthesis of the four-bar mechanism considering tolerances and clearances. Hummel and Chassapis [6] described an approach to the design and optimization of Cardan joints with manufacturing tolerances. Based on the reliability concept, Shi [7] presented and developed a probabilistic model of mechanical error in spatial mechanisms. Choi et al. [8] presented an analytical approach to tolerance optimization for planar mechanisms with lubricated joints based on mechanical error analysis. Wittwer et al. [9] established the direct linearization method applied to position error in kinematic linkages due to the link-length and angle variation. Fogarasy and Smith [10] presented a complete investigation on the study of the influence of manufacturing tolerances on the kinematic response of mechanisms. However, this

*Corr. Author’s Address: University of Minho, Department of Mechanical Engineering, Campus de Azurem, 4800-058 Guimaraes, Portugal, pflores@dem.uminho.pt

457

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467

approach required that constraint equations to be known and independent. Flores et al. [11] and [12] studied the influence of clearance in joints on the kinematic and dynamic performance of multibody systems. These works are valid for both planar and spatial systems and for dry and lubricated joints; however, they do not account for tolerance effects. Dong and Ye [13] modeled and studied a reheat-stop mechanism considering the effects of tolerance, misalignment and thermal action.

A good survey on the research work developed in the field of tolerance analysis of kinematic mechanism is provided by Chase and Parkinson [14].

By and large, there are two main approaches to study the effect of the manufacturing tolerances on the kinematic position errors, namely, the deterministic and probabilistic methods. The deterministic method involves fixed values or constraints that are used to find an exact solution.

These methods are used mostly when tolerances are known and the worst position error is to be determined. In contrast, the probabilistic or statistical methods deal with random variables that result in a probabilistic response. The statistical approaches are utilized when dimensions have some type of random distribution and the probability of being within a given tolerance band is to be evaluated.

Chase and Greenwood [15] introduced a statistical model to sum tolerances of characteristics affected by variability with mean shift. This model weights the values of tolerances sum between two extremes cases: (i) the worst case and (ii) the root sum square case. The worst case model assumes that all the component dimensions occur, in each assembly, at their extreme and worst limit simultaneously. The root sum square method considers that the component dimensions occur statistically having a Gaussian distribution. From statistical point of view the worst model is the most pessimistic evaluation of the sum variability, meanwhile the root sum square is the more optimistic evaluation of the sum variability.

In general, the manufacturing cost increases geometrically for uniform incremental tightening of tolerances [16]. The cost is also related to the characteristics of manufacturing processes used, and the degree of maturity of the workers. The 458

allocated tolerances should be as large as possible for the sake of economy and ease of manufacture.

However, large tolerances usually increase mechanical errors. Thus, designers should allocate tolerances to minimize the manufacturing cost while keeping mechanical errors below a certain specific limit [17].

The main purpose of this research work is to present a general and systematic approach to quantify the kinematic position errors due to manufacturing and assemble tolerances. Based on the worst case the deterministic method is utilized.

The kinematic constraints and equations of motion of the multibody systems are formulated under the framework of multibody systems methodologies.

The system is defined by a set of generalized coordinates, which represents the instantaneous positions of all bodies, together with a set of generalized dimensional parameters that defines the functional dimensions of the system. The generalized dimensional parameters take into account the tolerances associated with the lengths.

The relation between the kinematic constraints, dimensional parameters and output kinematic parameters is demonstrated. Finally, the proposed methodology is applied to an elementary planar multibody system in order to demonstrate its features.

1 KINEMATIC ANALYSIS The kinematic analysis is the study of the motion of a system, independently of the forces that produce it. Since in the kinematic analysis the forces are not considered, the motion of the system is specified by driving elements that govern the system motion during the analysis, while the position, velocity and acceleration of the remaining elements are defined by kinematic constraint equations that describe the system topology. It is clear that in the kinematic analysis, the number of driver constraints must be equal to the number of degrees of freedom of the multibody mechanical system. In short, the kinematic analysis is performed by solving a set of equations that result from the kinematic and driver constraints.

When the configuration of a multibody system is described by n Cartesian coordinates, then a set of algebraic kinematic independent

Flores, P.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467

holonomic constraints Φ can be written in a compact form as [18] to [20],

Φ(q,t) = 0,

(1)

where q is the vector of generalized coordinates and t is the time variable, usually associated with the driving elements.

The velocities and accelerations of the system elements are evaluated through the velocity and acceleration constraint equations. Thus, the first time derivative with respect to time of Eq. (1) provides the velocity constraint equations:

Φq q = −Φt ≡ υ, (2)

where Φq is the Jacobian matrix of the constraint equations, that is, the matrix of the partial derivates, ∂Φ/∂q, q is the vector of generalized velocities and υ is the right hand side of velocity equations, which contains the partial derivates of Φ with respect to time, ∂Φ/∂t. It should be noticed that only rheonomic constraints, associated with driver equations, contribute with non-zero entries to the vector υ. Furthermore, it is assumed that this vector does not present any dependency on the vector of coordinates.

A second differentiation of Eq. (1) with respect to time leads to the acceleration constraint equations, obtained as:

 = −(Φq q )q q − 2Φqt q − Φtt ≡ γ , (3) Φq q

 is the acceleration vector and γ is the right where q hand side of acceleration equations, i.e., the vector of quadratic velocity terms, which contains the terms that are only function of velocity, position and time. In the case of scleronomic constraints, that is, when Φ is not explicitly dependent on the time, the term Φt in Eq. (2) and the Φqt and Φtt terms in Eq. (3) vanish.

The constraint equations represented by Eq. (1) are, in general, non‑linear in terms of q and are usually solved by the Newton-Raphson method. Eqs. (2) and (3) are linear in terms of  , respectively, and can be solved by any q and q usual method for linear equations’ systems. Thus, the kinematic analysis of a multibody system can be carried out by solving the set of Eqs. (1) to (3).

The necessary steps to perform this analysis are sketched in Fig. 1, and described as, (i) Specify the initial conditions for positions q0 and initialize the time counter t0. (ii) Evaluate the position constraint Eq. (1) and solve it for positions, q. (ii) Evaluate the velocity constraint Eq. (2) and solve it for velocities, q . (iv) Evaluate the acceleration constraint Eq. (3)  . and solve it for accelerations, q

Fig. 1. Flowchart of computational procedure for kinematic analysis of a multibody system A Methodology for Quantifying the Kinematic Position Errors due to Manufacturing and Assembly Tolerances

459

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467

(v) Increment the time. If the time is smaller than final time, go to step ii), otherwise stop the analysis.

2 MATHEMATICAL FORMULATION OF THE KINEMATIC POSITION ERRORS DUE TO TOLERANCES In order to evaluate, in a systemic and general way, the influence of the manufacturing and assemble tolerances on the kinematic position errors, special attention needs to be given to the mathematical formulation of the description of the systems’ configuration. Thus, according to the previous section, the equations of constraints can be written as:

Φ(q1, q2, ..., qn, d1, d2, ..., dm)=0 , (4)

where q1, q2, ..., qn represent the generalized vector of coordinates that define the kinematic system’s configuration at any instant, and d1, d2, ..., dm are the generalized vectors of the dimensional parameters defining the functional dimensions of the system. It should be noted that Eq. (4) represents the kinematic system’s constraints, which can easily be written using, for instance, Cartesian coordinates. Furthermore, the number of generalized coordinates, n, and the number of generalized dimensional parameters, m, must be adequately selected bearing in mind the correct system’s description and system’s degrees of freedom.

The kinematic analysis of any multibody system implies the resolution of Eq. (4) for q1, q2, ..., qn and their derivatives, according to what was presented in the previous section. In this process, it is assumed that vectors d1, d2, ..., dm are constants, meaning that there is no variation of the dimensional parameters and, consequently, that they do not affect the global system’s performance. However, it is well known that this is not the case in practical engineering design and manufacturing processes.

Since, in general, multibody systems are conducted by driving elements, excluding these elements and considering that the kinematic constraints are independent, the Jacobian matrix can be written as follows:

460

Φq =

∂Φk , (k = 1, …, n-dr; l = 1, …, n), (5) ∂ql

where indices k and l represent, respectively, the n-dr kinematic constraints and n Cartesian coordinates. The number of driving elements is represented by variable dr.

Considering all coordinates and dimensional parameters as global system’s variables, the variation of the constraint equation is expressed as: ∂Φk δ q1 + ... + ∂Φk δ q n + ∂Φk δ d1 + ... + ∂Φk δ d m = 0., (6) ∂q1

∂q n

∂d1

∂d m

In a compact form, Eq. (6) is be written as: n − dr m ∑ ∂Φk δ qi + ∑ ∂Φk δ d j = 0 , (k = 1, …, n-dr) (7) i =1 ∂q i j =1 ∂d j

in which, δqi is the variation of the generalized system’s coordinates and δdj is the variation of the dimensional parameters. This last term represents the manufacturing tolerances of the corresponding functional dimensions, such as the lengths of the multibody system parts.

In a matrix form, Eq. (7) is expressed as: Φqi δ qi = −Φd j δ d j , (i = 1, …, n-dr, j = 1, …, m), (8) where Φqi is the Jacobian matrix and Φdi represents the derivative of the constraint equations in relation to the dimensional parameters.

Flores, P.

Fig. 2. Double pendulum

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467

Since in the kinematic analysis of multibody systems the Jacobian matrix is known, as illustrated in the previous section, specifying the manufacturing tolerances, δdj , only the matrix Φdi needs to be evaluated in order to obtain the kinematic position errors of all system’s bodies, by solving Eq. (8). It should be noted, that with this approach very low computational effort is added to the standard kinematic analysis procedure. Moreover, it should be highlighted that Eq. (8) represents a linear system of equations that can easily be solved by employing any numerical method, such as the LU factorization procedure, available in the thematic literature [21] and [22].

With the intent to illustrate the application of this approach, a double pendulum system demonstrative example follows. Fig. 2 shows the double pendulum, where the body numbers, local and global coordinate systems are illustrated.

Since this simple multibody system has two ideal revolute joints, the corresponding constraint equations expressed in Cartesian coordinates are written as:

Φ1 ≡ − x2 −

r2 cos θ 2 = 0, (9) 2

Φ 2 ≡ − y2 −

r2 sin θ 2 = 0, (10) 2

Φ 3 ≡ x2 +

r r2 cos θ 2 − x3 − 3 cos θ3 = 0, (11) 2 2

Φ 4 ≡ y2 +

r r2 sin θ 2 − y3 − 3 sin θ3 = 0, (12) 2 2

where x2, y2, θ2, x3, y3 and θ3 are the global system coordinates and r2 and r3 are the selected dimensional tolerance parameters. The input parameters corresponding to the driving elements are the angles θ2 and θ3, being x2, y2, x3, and y3 the output parameters. The differentiation of the constraints’ Eqs. (9) to (12) yields the variation of the constraints as follows:

cos θ 2 δ r2 = 0, (13) 2 sin θ 2

δΦ 2 ≡ −δ y2 − δ r2 = 0, (14) 2 cos θ3 cos θ 2 δΦ 3 ≡ δ x2 + δ r2 − δ x3 − δ r3 = 0, (15)

δΦ1 ≡ −δ x2 −

2

2

δΦ 4 ≡ δ y2 +

sin θ3 sin θ 2 δ r2 − δ y3 − δ r3 = 0. (16) 2 2

Rearranging Eqs. (13) to (16) results: -1 0  1  0

cosθ 2 0  δ x2   2  sin θ 0  δ y2   2 2 =   -1 0  δ x3  - cos2θ2   0 -1 δ y3   - sin θ2  2

0 0 -1 0 0 1

0   0  δ r2   , (17) cosθ3   δ r3  2  sin θ3  2 

which represents a linear system of equations that can be solved for the unknowns δx2, δy2, δx3 and δy3, since the values of the variables θ2 and θ3 are specified as inputs and δr2 and δr3 represent the tolerance amplitude for the double pendulum arm lengths r2 and r3.

In a compact form Eq. (17) is be written as:

Φqδq = –Φdδd (18)

Again, from Eq. (18) great simplicity and generality of the proposed methodology for the study of kinematic position errors due to manufacturing tolerances is evident. In Eq. (18) the term Φq, which corresponds to the partial derivatives of the constraint equations with respect to the dimensional parameters, represents the quantitative influence of the individual variation of the selected dimensional tolerance parameters on the kinematic accuracy of the output parameters. Fig. 3 shows the flowchart of the computational procedure to perform kinematic analysis of a multibody system, in which the kinematic position errors owing to tolerances are evaluated.

The basic and fundamental idea of this study is to present a simple and general approach to kinematic position error analysis of mechanisms.

This analysis is simple and general because it is valid for any planar or spatial mechanism.

Furthermore, the proposed approach is systematic as it is developed under the framework of multibody system methodologies being included in a standard procedure for kinematic analysis. The main reason for that is based on the fact that the additional computational effort is not significant because almost all the necessary parameters to solve Eq. (18) are already known from the standard kinematic analysis. However, this is not necessary to embed this kinematic position error analysis in the general kinematic analysis

A Methodology for Quantifying the Kinematic Position Errors due to Manufacturing and Assembly Tolerances

461

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467

procedure. Obviously, it is possible to complete the standard kinematic analysis, and then go back at each step of time simulated and evaluate the position error analysis be solving Eq. (18).

However, this way requires to store the history response, being easier and computationally more efficient to perform the position error analysis at each step of the standard kinematic procedure.

The problem of sensitivity analysis of mechanical systems has grasped the attention of several authors such as Arora and Haug [23], Lee et al.

[24], Schulz and Brauchli [25], only to mention a few. Furthermore, the problem of optimization and parameter identification of multibody system models using gradient based optimization methods has been developed by Bock and his co-workers [26] and [27].

In order to quantify the kinematic accurate position of a multibody system link, it is first necessary to define the amount of tolerance allowed for each of the dimensional parameters considered. Thus, according to standard ISO 2861, for common mechanical operating conditions the IT grades are usually in the range IT8 to IT11

[17]. For the manufacturing tolerances of the dimensional parameters of a multibody system, the bilateral tolerances specified in ISO 286-1 are commonly used. Hence, in Eq. (18), the variation of the functional parameters δd can be regarded as such tolerance fields. Therefore, it is possible to write the following relation:

δd = ±½T , (19)

where T represents the total manufacturing tolerance range corresponding to the dimensional parameters.

3 MANUFACTURING AND ASSEMBLY TOLERANCE ANALYSIS It is known that tolerances are used to define the allowable limits of geometric variations that are inherent in the manufacturing and assembly processes [1]. Broadly, there are two approaches to solve this problem; worst case assemblability and statistical assemblability. In the first case, it is assumed that all the component dimensions

Fig. 3. Flowchart of the computational procedure to perform kinematic analysis of a multibody system including the evaluation of the kinematic position errors due to manufacturing tolerances 462

Flores, P.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467

occur, in each assembly, at their extreme and worst limit simultaneously. When this approach is employed, the designer desires to ensure that the components can always be assembled. This means that the probability of having a kinematic error exceeding the specific limits in a particular system is null. On the other hand, the statistical assemblability can be used to take advantage of statistical averaging over of components, allowing for the use of less restrictive tolerances in exchange for admitting the small probability of non-assembly. In general, the standard process is defined at the confidence level corresponding to ±3σ interval [28] and [29]. From the statistical point of view the worst case model is the most pessimistic, while the statistical assemblability is the most optimistic case. In practical situations it is expected that they fall between the worst and statistical models. These two approaches are discussed in detail in the following paragraphs.

Using Eq. (18) as reference, the variation of the generalized system’s coordinates can be rewritten as:

δq = sδd , (20) where s represents the sensitive coefficients given by: s = −Φq−1Φd . (21)

Considering a system with m generalized dimensional parameters and that the maximum tolerance or error is specified, based on the worst scenario assemblability, it is possible to evaluate the maximum error of a general output coordinate as: m 1 1 2 Tmaximum ≥ δ q = ∑ s j 2 T j = 1 = j

(22) = 12 ( s1T1 + s2T2 + ... + smTm ) .

The average tolerance can be determined by the following expression: m

Taverage =

∑s j =1

1 j 2

m

Tj =

Tmaximum . m

(23)

In a similar way, the tolerance associated with any system component can be given by:

Ti =

Tmaximum . (24) si m

It is worth noting that Tmaximum is a given quantity, allowing the evaluation of average

tolerance values, which ultimately can be used as a guiding reference to specify the manufacturing precision requirements of the system components by selecting the IT grades.

Tolerance is statistical in nature since the output of a random variable is, in general, normally or Gaussian distributed, with the level of confidence three-sigma considered. This means that only two or three cases in a thousand have the probability to be outside the ±3σ range.

This confidence level of tolerance becomes an important design parameter to be evaluated and optimized. Thus, using the statistical approach, the root mean square considers that the component dimensions occur statistically having a Gaussian distribution and can be expressed by:

1 2

Tmaximum ≥ δ q =

m

∑s ( j =1

2

j

1 2

T j ) . (25) 2

In the statistical approach, the average tolerance can be determined by the following expression:

Taverage =

Tmaximum . m

(26)

In a similar way, the tolerance associated with any system component can be given by:

Ti =

Tmaximum . si m

(27)

In order to convert the tolerance value to the standard process-tolerance, the γ factor is introduced [3]: T

(28) γ = maximum .

2σ Solving Eq. (28) for σ yields:

σ=

Tmaximum .

2γ

(29)

Consequently, the admissible tolerance of any component can be expressed by: 3T

Tadmissible = 6σ = maximum . (30) γ In Table 1, the γ factor is listed as a function of confidence p value of the tolerance.

This table was constructed by integrating the standard normal distribution function f(γ), which can be defined by the error integral in the form:

A Methodology for Quantifying the Kinematic Position Errors due to Manufacturing and Assembly Tolerances

463

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467

p 1 = f (γ ) = 100 2π

∫

γ

−γ

et / 2 d γ . (31) 2

Table 1. Values of γ factor for different confidence levels p = 99.7% p = 99.0% p = 97.0% p = 95.0%

f(γ) = 0.997 f(γ) = 0.990 f(γ) = 0.970 f(γ) = 0.950

γ = 2.96 γ = 2.58 γ = 2.17 γ = 1.96

4 EXAMPLE OF APPLICATION: SLIDER-CRANK MECHANISM An elementary slider-crank mechanism is used here to show the influence of manufacturing tolerances on kinematic performance. Fig. 4 shows the configuration of the mechanism, comprising four rigid bodies that represent the crank, connecting rod, slider and ground, three revolute joints and one ideal translational joint.

The inertia properties and length characteristics of each body, as well as the associated tolerance according to ISO 286-1 standard are given in Table 2. In the present example the tolerance grade IT 10 was selected establishing the number of generalized dimensional parameters equal to two (m = 2), which are related to the lengths of the crank (r2) and connecting rod (r3).

velocity of 500 rpm clockwise. The initial system configuration corresponds to the top dead point. In the numerical simulation, r2 and r3 were selected as dimensional tolerance parameters, being δx4 and δθ3 the output parameters. Thus, applying the methodologies presented in the previous sections, Figs. 5 and 6 show, respectively, the maximum absolute errors on the linear slider position and the angular position of the connecting rod, when the worst case approach is considered. These maximum position errors were evaluated at 25 crank angular positions. By observing the obtained global results it can be concluded that the maximum position errors vary during the computational simulation of the slider crank mechanism.

Fig. 5. Maximum linear position error of the slider evaluated over a complete crank cycle

Fig. 4. Schematic representation of the slidercrank mechanism Table 2. Geometric properties of the slider-crank mechanism Body Nr. 1 2 3 4

Description Ground Crank Connecting rod Slider

Length [m] 0.050 0.120 -

Tolerance range [mm] ± 50 ± 70 -

In the kinematic simulation, the crank is the driving element and rotates at a constant angular 464

Fig. 6. Maximum angular position error of the connecting rod evaluated over a full crank cycle In order to clearly show the differences of the worst case and the statistical model, let us consider the mathematical equation that allows the evaluation of the position error of the slider, which can be written as [30]:

Flores, P.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467

´ x4 = ± ( cos¸ 2 + sin¸ 2tan¸ 3 ) ´ r2 + + ( cos¸ 3 + sin¸ 3tan¸ 3 ) ´ r3  ,

(32)

Tmaximum = ± (1× 31) + (1× 44 ) = ±54 µm .

2

or in another form:

δ x4 = ± ( s1 12 T1 + s2 12 T2 ) , (33)

where the manufacturing tolerances on dimensions r2 and r3 are represented by T1 and T2, and the sensitive coefficients s1 and s2 can be written as follows: (34)

s1 = cosθ2 + sinθ2 tanθ3 ,

s2 = cosθ3 + sinθ3 tanθ3 ,

(35)

From the analysis of Fig. 5, it can be observed that the most critical situation occurs when the crank and the connecting rod are collinear, that is, when θ2 = 0 or 180º and θ3 = 180º. In these circumstances, the sensitive coefficients s1 and s2 are equal to 1. When the worst case is considered, the maximum tolerance can be calculated using Eq. (22) yielding:

Tmaximum = ± ( s1 12 T1 + s2 12 T2 ) =

= ± (1× 50 + 1× 70 ) = ±120 µm.

It is obvious that this value is relatively high from a practical engineers view point. On the other hand, when the statistical model is considered, the maximum tolerance is determined using Eq. (25), that is:

Tmaximum = ±

( s1 12 T1 ) + ( s2 12 T2 ) 2

2

=

= ± (1× 50 ) + (1× 70 ) = ±86 µm.

2

2

This value is still high for practical purposes. Therefore, considering, for example that the maximum admissible tolerance is equal to ±55 μm, then the γ factor can be evaluated using Eq. (30):

Tadmissible =

for crank and connecting rod lengths are ±31 μm and ±44 μm. Thus, the maximum tolerance is now:

3Tmaximum 3 × 55 ⇒γ = = 1.92 . γ 86

This value corresponds to a level of confidence less than 95%, which is far too high.

With the intent to increase the confidence level, the sensitive coefficients associated with the tolerance dimensions should have larger values.

This desideratum can be achieved by reducing the tolerance grade IT, for instance from 10 to 9 [17].

Consequently, according to ISO 286-1 standard the corresponding tolerance ranges of grade IT 9

2

Hence, the γ factor is evaluated as: 3 × 55 γ= = 3.06 .

54

The corresponding confidence level is over 99.7%, which can be considered to be clearly satisfactory.

5 CONCLUDING REMARKS In this paper, a general and systematic methodology for kinematic positional error analysis of multibody systems was investigated, taking into account the influence of the manufacturing and assembly tolerances on the performance of planar mechanisms. In the process, the main aspects for kinematic analysis of multibody systems were revised. Based on the theory of dimensional tolerances, the variation of the geometrical dimensions is regarded as a tolerance grade with an interval associated with each dimension and, consequently, a kinematic amplitude variation for the positions. The presented deterministic method evaluates the relation between variations in the dimensional parameters and variation in the generalized coordinates. The statistical approach based on the confidence level three-sigma was also studied. The methodologies proposed have been exemplified through the application of kinematics to a slidercrank mechanism. The simplicity and generality of the proposed methodology for the study of kinematic position errors due to manufacturing tolerances was thus demonstrated. It should be highlighted that in this paper, only dimensional parameters such as length of links have been addressed. However, other parameters can be integrated in the general methodology presented throughout this work, namely those related to roundness of circular surfaces and clearances in joints.

6 ACKNOWLEDGMENTS This work is supported by the Portuguese Foundation for the Science and Technology under the project (PTDC/EME-PME/099764/2008).

A Methodology for Quantifying the Kinematic Position Errors due to Manufacturing and Assembly Tolerances

465

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467

7 REFERENCES [1] ANSY, Y14.5M-1994. (1994). Dimensional and Tolerancing, ASME, New York.

[2] Lee, W.J., Woo, T.C. (1990). Tolerances: Their Analysis and Synthesis. Journal of Engineering for Industry, vol. 112, p. 113121.

[3] Di Stefano, P. (2006). Tolerances analysis and cost evaluation for product life cycle.

International Journal of Production Research, vol. 44, no. 10, p. 1943-1961.

[4] Garrett, R.E., Hall, A.S. (1969). Effect of Tolerance and Clearance in Linkage Design.

Journal of Engineering for Industry, vol. 91, p. 198-202.

[5] Dhande, S.G., Chakraborty, J. (1973).

Analysis and Synthesis of Mechanical Error in Linkages – A Stochastic Approach.

Journal of Engineering for Industry, vol. 95, p. 672-676.

[6] Hummel, S.R., Chassapis, C. (2000).

Configuration design and optimization of universal joints with manufacturing tolerances. Mechanism and Machine Theory, vol. 35, p. 463-476.

[7] Shi, Z. (1997). Synthesis of mechanical error in spatial linkages based on reliability concept. Mechanism and Machine Theory, vol. 32, no. 2, p. 255-259.

[8] Choi, J.H., Lee, S.J., Choi, D.H. (1998).

Tolerance Optimization for Mechanisms with Lubricated Joints. Multibody System Dynamics, vol. 2, p. 145-168.

[9] Wittwer, J.W., Chase, K.W., Howell, L.L. (2004). The direct linearization method applied to position error in kinematic linkages. Mechanism and Machine Theory, vol. 39, p. 681-693.

[10] Fogarasy, A.A., Smith, M.R. (1998). The influence of manufacturing tolerances on the kinematic performance of mechanisms.

Journal of Mechanical Engineering Science, vol. 212, p. 35-47.

[11] Tian, Q., Zhang, Y., Chen, L., Flores, P. (2009). Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints. Computers and Structures, vol. 87, p. 913-929.

466

[12] Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M. (2007). Dynamic behaviour of planar rigid multibody systems including revolute joints with clearance. Journal of Multi-body Dynamics, vol. 221, p. 161-174.

[13] Dong, X., Ye, J. (2009). Performance Analysis of the Reheat-Stop-Valve Mechanism under Dimensional Tolerance, Misalignment and Thermal Impact. Strojniški vestnik - Journal of Mechanical Engineering, vol. 55, no. 9, p.

507-520.

[14] Chase, K.W., Parkinson, A.R. (1991). A survey of research in the application of tolerance analysis to the design of mechanical assemblies. Research in Engineering Design, vol. 3, p. 23-37.

[15] Chase, K.W., Greenwood, W.H. (1988).

Design issues in mechanical tolerance analysis. Manuf. Rev., vol. 1, no. 1, p. 50-59.

[16] Drozda, T.J., Wick, C. (1983). Tool and Manufacturing Engineers Handbook, Vol. I, Machining. 4th ed., SME, New York.

[17] ISO 286-1. (1988). ISO system of limits and fits – Parts 1: Base of tolerances, deviations and fits.

[18] Nikravesh, P.E. (1988). Computer Aided Analysis of Mechanical Systems. Prentice Hall, Englewood Cliffs, New Jersey.

[19] Shabana, A.A. (1989). Dynamics of Multibody Systems. John Wiley and Sons, New York.

[20] Haug, E.J. (1989).

Computer-Aided Kinematics and Dynamics of Mechanical Systems - Volume I: Basic Methods, Allyn and Bacon, Boston, Massachusetts.

[21] Atkinson, K.E. (1989). An Introduction to Numerical Analysis, 2nd ed. John Wiley & Sons, New York.

[22] Amirouche, F.M.L. (1992). Computational Methods for Multibody Dynamics. Prentice Hall, Englewood Cliffs, New Jersey.

[23] Arora, J.S., Haug, E.J. (1979). Methods of design sensitivity analysis in structural optimization. AIAA Journal, vol. 17, no. 9, p.

970-974.

[24] Lee, S.J., Gilmore, B.J., Ogot, M.M. (1993). Dimensional tolerance allocation of stochastic dynamic mechanical systems through performance and sensitivity analysis.

Journal of Mechanical Design, vol. 115, p.

392-402.

Flores, P.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 457-467

[25] Schulz, M., Brauchli, H. (2000). Two methods of sensitivity analysis for multibody systems with collisions. Mechanism and Machine Theory, vol. 35, no. 10, p. 1345-1365.

[26] Körkel, S., Kostina, E., Bock, H., Schlöder, J. (2007). Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes.

Optimization Methods and Software, vol. 19, p. 327-338.

[27] Diehl, M., Bock, H.G., Kostina, E. (2006). An approximation technique for robust nonlinear optimization. Mathematical Programming:

Series A and B Archive, vol. 107, no. 1, p.

213-230.

[28] Ang, A.H.S., Tang, W.H. (1984). Probability concepts in engineering planning and design.

Vol. I - Basic principles. John Wiley & Sons, New York.

[29] Ang, A.H.S., Tang, W.H. (1984). Probability concepts in engineering planning and design.

Vol. II - Decision, risk, and reliability. John Wiley & Sons, New York.

[30] Flores, P., Claro, J.C.P. (2007). Kinematics of Mechanisms. Edições Almedina, Coimbra.

(In Portuguese)

A Methodology for Quantifying the Kinematic Position Errors due to Manufacturing and Assembly Tolerances

467

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 468-476 DOI:10.5545/sv-jme.2009.026

Paper received: 25.2.2009 Paper accepted: 23.3.2011

A Secondary Source Configuration for Control of a Ventilation Fan Noise in Ducts Prezelj, J. ‒ Čudina, M. Jurij Prezelj* ‒ Mirko Čudina University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

The main noise source in heating, ventilation, and air conditioning systems is usually a ventilating fan. Noise, generated by the ventilating fan is transmitted through the duct into the living and working environment. A typical fan noise spectrum consists of a broadband noise, which is superimposed with pure tones. Different methods are available to reduce a transmission of such noise from the ventilating fan into the living and working environment. In this article it is demonstrated how a feedforward active noise control system can be implemented together with a side branch resonator. Effectiveness of the feedforward active noise control system depends on the quality of a reference signal, which should be in a perfect correlation with the primary noise. An acoustic feedback is the main problem of feedforward active noise control systems in ducts. A combined method uses a single loudspeaker to work as a dipole source and a side branch resonator to reduce the acoustic feedback. A side branch resonator reduces noise transmission in a narrowband frequency range as well. In this article, a theoretical background of a dipole source with a side branch resonator is presented, along with some measurement results and simulations of active noise control. © 2011 Journal of Mechanical Engineering. All rights reserved.

Keywords: active noise control, secondary source, ventilation duct, noise, fan 0 INTRODUCTION Noise generated by heating, ventilating, and air conditioning (HVAC) systems, is a combination of narrowband discrete frequencies and a broadband random noise. Narrowband discrete frequencies are correlated with a blade passage frequency from the ventilating fan, and an electromagnetic noise, which is correlated to a mains frequency (50 or 60 Hz), and higher harmonics. A broadband random noise is aeroacoustically generated by an intensive turbulent air flow. Several different types of silencers can be used to reduce noise propagation through HVAC systems. A reduction of noise propagation through the duct with a side branch resonator is a wellknown method [1] and [2]. With this method a significant reduction of noise propagation can be achieved, but only in a narrow frequency bands.

Active noise control (ANC) is an attractive method, which can be used for a narrowband and/ or a broadband noise control in a low frequency range. A noise-cancellation speaker emits a sound wave with the same amplitude but with an inverted phase to the primary noise. The waves combine to form a new wave, in a process 468

called interference, and effectively cancel each other out. Effectiveness of the feedforward ANC system depends on the quality of the reference signal, which should be in a perfect correlation with the primary noise. An acoustic feedback is the main problem of the feedforward ANC systems in ducts. In the available literature, only a few studies discuss the problem of reducing the acoustic feedback in the acoustical domain, as for example [3] to [5]. A method, which combines a single loudspeaker to work as a dipole secondary source with a side branch resonator to reduce the acoustic feedback, is presented in this article.

A basic depiction is given in Fig. 1. A basic Swinbanks theory [6] was used to explain how a dipole source combined with a side branch resonator forms a unidirectional secondary source for the ANC system. Swinbanks showed that two monopole sources with appropriate signal delay form a unidirectional sound source in the duct.

The source strength of the upstream monopole at the time t is q1(t) and that of the downstream source is q2(t). There will be zero output from the combination of the two monopole sources in the upstream direction if:

*Corr. Author’s Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia, jurij.prezelj@fs.uni-lj.si

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 468-476

  L q1 (t ) = −q2  t −  . (1)  c0 (1 − M ) 

In Eq. (1) L denotes the spacing between two monopole sources (Fig. 1), c0 is the speed of sound and M is the Mach number of net airflow.

If the speed of the airflow through the duct is less than 3 m/s, it has practically no influence on time delay settings, [7] and [8]. From Eq. (1) it can be derived that q1 must have the same amplitude as q2, but it must be inverted and delayed relative to q2 by L/c0 in order to minimize acoustic feedback.

A time delay τ0 should be equal to the time, which is needed for “anti-noise” to travel from the downstream source to the upstream source. This distance is marked with L and basically represents the length of the side branch resonator.

τ0 =

L . (2) c0 (1 − M )

To satisfy conditions in Eqs. (1) and (2), two monopole sources with the same amplitude and reverse phase have to be provided. Two sets of loudspeakers with two different amplifiers and a unit for adding appropriate time delay into the signal are typically used. The basic idea of the proposed design is to use sound, which is generated on the backside of the loudspeaker cone, for acoustic feedback reduction [10]. Sound from the front side of the loudspeaker, which is propagating upstream, is cancelled by the sound generated on the rear side of the loudspeaker.

An acoustical short circuit is used for acoustic feedback reduction and the side branch resonator (SBR) acts like a transmission line for frequency response improvement of the secondary source in band pass frequency range. Swinbanks method is actually transferred into the pure acoustic domain, Fig. 2 [11]. Dipole source can also be used for improvement of the insertion loss on walls.

Insertion loses can be improved by using dipole sources, however, a performance of the ANC system with a dipole source significantly depends on the acoustic setup [12]. A systematic analysis of the acoustic setup is therefore needed.

A block chart is presented in Fig. 2 and it explains a proposed system. P(z) denotes a primary acoustic path from the reference microphone position to the error microphone position. C(z) denotes a transfer function of

all controller elements (microphone, AD/DA converter, preamplifier, compensation filters, and power amplifier); SF(z) presents a transfer function of sound radiation from the loudspeaker front side, including the acoustic path to the error microphone position; SSBR(z) presents a transfer function of sound radiation from the loudspeaker rear side, including the acoustic path through the side branch resonator and through the main duct to the error microphone position; FF(z) presents a transfer function of sound radiation from the loudspeaker front side, including the acoustic path to the reference microphone position through the main duct; FSBR(z) presents a transfer function of sound radiation from the loudspeaker rear side, including the acoustic path through the side branch resonator to the reference microphone position. The negative sign denotes an inversed phase between the loudspeaker rear side and front side sound radiation.

Fig. 1. A dipole source in a side branch resonator From the block chart it can be deduced that a transfer function of sound generation at the front side of the loudspeaker must be equal to the transfer function of sound generation at the rear side of the loudspeaker including acoustic paths (FSBR(z) = FF(z)), in order to minimize the acoustic feedback. A typical loudspeaker is a perfect dipole source in the discussed frequency range. Only the geometrical configuration of the secondary source affects the properties of two acoustical paths and consequently the acoustic feedback. Therefore, a special attention should be paid to the geometry design of the secondary source.

1 ONE DIMENSIONAL LINEAR MODEL An additional acoustical element erected near the secondary source (loudspeaker), affects the impulse response of the ANC system in both duct directions. A frequency response of the new

A Secondary Source Configuration for Control of a Ventilation Fan Noise in Ducts

469

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 468-476

Fig. 2. A block chart of the proposed feedforward ANC configuration secondary source configuration is not easy to determine due to some factors in the acoustical domain, which have to be taken into consideration.

The frequency response of the proposed secondary source configuration strongly depends on the geometry of the setup. When combining the SBR with a dipole source, some parameters have to be carefully considered: • the position of the loudspeaker in the SBR (LD) relative to the SBR opening, • the length of the SBR (L), • the ratio between a SBR cross section (A2) and the main duct cross section (A1), • the ratio between a SBR opening (A3) and the main duct cross section (A1), and • damping in the SBR (α).

ANC systems are used for noise control in the low frequency range. A theoretical model was therefore developed for one-dimensional wave propagation in an infinite duct. A proposed configuration of the secondary source is presented in Fig. 1. In such configuration four sound pressure fields (p1, p2, p3 and p4) and four related velocity fields (u1, u2, u3 and u4) are developed. Each pressure field is described by one wave equation.

A sound pressure field denoted with p1, presents a wave propagation from the SBR opening, back to the primary noise source (x = 0). Since there is no propagation of sound pressure from the left side of the duct, a solution of the wave equation, Eq. (4) has only one part. C1 presents the amplitude of sound that is propagating in the upstream direction. C1 presents the acoustic feedback and its value should be as low as possible. Both parts of the solution are included in equations for sound 470

pressure field p2 and for p3. A standing wave is formed in the main and in the side channel between loudspeaker (x = LD) and opening (x = 0).

A Sound pressure field denoted with p4 presents a useful part of sound generated by the secondary source in the downstream direction. C6 should be larger than C1 in order to achieve directivity and consequently, a reduction of the acoustic feedback.

Six unknowns (C1 to C6) describe the amplitude of four sound fields. Six independent boundary conditions must be fulfilled in order to solve the system. Boundary conditions are known at two locations in the duct, denoted with a gray spot at x = 0 and at x = L (Fig. 1).

A first set of boundary conditions is given at the SBR opening (x = 0). At this position two channels of the secondary source are merged and a cross section of the main duct is changed.

Sound pressure of three sound fields at this point must be the same in order to prevent net flow due to the sound pressure field. Conservation of mass flow at this point (x = 0) can be used as a third boundary condition. A second set of known boundary conditions can be obtained at the loudspeaker position (x = L). A loudspeaker is regarded as a moving surface S with a known surface velocity. Volume flow at the backside of a loudspeaker cone is exactly the same as the volume flow on the front side of the loudspeaker cone. Volume flow generated on the backside of the loudspeaker cone is distributed across SBR cross section (A2). Volume flow generated at the front side of the loudspeaker cone is distributed in two directions of the main duct (2A1). In Eqs. (7) and (8), u0 represents a surface velocity of

Prezelj, J. ‒ Čudina, M.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 468-476

the loudspeaker cone. Sound pressure to the right of the loudspeaker is exactly the same as sound pressure to the left of the loudspeaker at the front side of loudspeaker (x = L) in the main duct.

These six boundary conditions are given in Eqs. (3) to (8):

p1(x = 0) = p2 (x = 0) ,

(3)

p2(x = 0) = p3 (x = 0) ,

(4)

SBR cross-section. A system of 6 equations with 6 unknowns written in Eq. (9) was also numerically solved for discrete values of kL.

A1u1(x = 0) + A2u3(x = 0) + A1u2(x = 0) = 0 , (5) p2(x = L) = p4 (x = L) ,

(6)

A2u3(x = L) = Sv0 ,

(7)

A1u2(x = L) + Sv0 = A1u4(x = L) .

(8)

Coefficients C1 and C6 present the amplitude of sound generated by the secondary source in two opposite directions. C1 presents a part of the sound field, which propagates into a negative direction towards the reference microphone and towards the primary noise source.

Coefficient C6 presents a useful part of the sound field, which is propagating in a positive direction and is used for noise control.

These two coefficients can be simply obtained by solving the linear system given in Eq. (9). The results are given in Eqs. (10) and (11).  1  1   − A1   0  0   0

−1 0 − A1 eikL

−1 0 A1 e − ikL

0 A1eikL

0 − A1e −ikL

0 −1 − A2

0 −1 A2

0 A2 eikL 0

0 − A2 e −ikL 0

0   C1  0  C2  0   C3    = −eikL  C4  0  C5    − A1eikL  C6 

 0   0     0  = .  0   ρ cu S  0    − ρ cu0 S 

C1 = − ρ cu0 S

1  e −ikL − eikL  A2  3 + e 2ikL

(9)

  , (10) 

 1 1   e −ikL − eikL C6 = ρ cv0 S  +  2 ikL  A1 A2   3 + e

  . (11) 

Coefficients C1 and C6 depend on the kL value. Coefficients C1 and C6 also depend on the ratio between the main duct cross-section and the

Fig. 3. A frequency spectrum of the feedback reduction for three different lengths of the side branch; cross section ratio A1/A2 was set to 8

Fig. 4. A frequency spectrum of the feedback reduction for four different cross section ratios; a length of the side branch was set to 0.4 m The difference between the sound pressure level in an upstream direction and the sound pressure level in a downstream direction can be considered as a feedback reduction. In Figs. 3 and 4, a feedback reduction for different geometries of the SBR is presented. In Fig. 3, a frequency response of feedback reduction is presented for different lengths of side branch resonator. The shorter the side branch resonator, the higher is the central frequency of feedback reduction and the broader is its useful frequency range. In Fig. 4 a frequency spectrum of feedback reduction is presented for different cross section ratios.

A1 presents the cross section of the main duct

A Secondary Source Configuration for Control of a Ventilation Fan Noise in Ducts

471

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 468-476

Fig. 5. Experimental setup and A2 presents the cross section of side branch resonator. Sound pressure, generated in the upstream direction, is proportional to 1/A1. Sound pressure, generated in the downstream direction is proportional to 1/A1+1/A2. The higher the ratio A1/A2, the better is the feedback reduction.

2 EXPERIMENTAL SETUP AND MEASUREMENTS RESULTS The experimental setup is shown in Fig. 5.

Geometry of the secondary source and damping in the SBR were determined according to the theoretical results. L was shortened to 0.4 m and cross section ratio was set to A1/A2 = 1. SBR was partially filled with damping material, as shown in Fig. 5. The duct was made from ¾″ thick plywood.

It was covered with hard plastic to ensure a low coefficient of sound absorption in the main duct.

The experimental set-up was built in order to measure a directivity of the new secondary source configuration, to determine the influence of damping in the SBR on the directivity, and to measure impulse responses of all acoustic paths and impulse responses of all electro acoustic elements.

A sound field in the duct was measured by a moving microphone. Panasonic omni directional Back Electret Condenser Microphone WM-61A was mounted on the pulley. The position of the microphone was controlled with a low-noise electric motor with a reduction gear system mounted on the duct using vibration insulation.

The moving microphone was traversed with the speed of 1.5 cm/sec. Measurements were performed using a pink noise excitation. The pink 472

noise was used as a test signal because it ensures a good signal to noise ratio in low-frequency range.

Measurement results of the sound pressure level field in the duct with anechoic termination are presented in Fig. 6. The sound pressure level field was generated by the secondary source with a SBR configuration as shown in Fig. 5. The sound field is not homogeneous. The secondary source has its own frequency response and a standing wave appears. The most important result is that the new secondary source generates much higher sound pressure levels to the right side (brighter spectrogram) than to the left (darker spectrogram).

Fig. 6. Measured sound field in the duct, generated by the SBR secondary source; a duct had anechoic termination A secondary path S(n), a primary path P(n), and a feedback path F(n) were identified using impulse response measurement for different configurations of the secondary source. The secondary path and the feedback path impulse responses are presented in Figs. 7 and 8. These impulse responses are required for a valid and credible simulation of adaptive ANC system.

Prezelj, J. ‒ Čudina, M.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 468-476

Acoustic and electro-acoustic impulse responses used in ANC simulations were measured on real subsystems forming the HVAC model with an ANC system.

Fig. 7. Impulse responses of a secondary path and an acoustic feedback, measured with the classic monopole configuration of the secondary source

Fig. 8. Impulse responses of a secondary path and an acoustic feedback, measured with the new SBR configuration of the secondary source The impulse response of the secondary path SMONOPOLE(n), obtained with the classical monopole secondary source is compared with the impulse response of the feedback path FMONOPOLE(n), obtained with the classical monopole secondary source in Fig. 7. The delay of the feedback is approximately 7 ms. The amplitude of the feedback impulse is 2/3 of the secondary path impulse. Acoustic feedback is significant.

The impulse response of the secondary path SSBR(n), obtained with the SBR secondary source is presented in Fig. 8, together with the impulse response of the acoustic feedback FSBR(n), obtained with the SBR secondary source. The impulse response of the secondary source with SBR configuration has a pronounced negative peak at 2 ms. The ratio between the amplitude of the secondary path impulse response SSBR to the amplitude of the feedback impulse response FSBR is improved from 1.4 to 4.2. Acoustic feedback is three times smaller.

Impulse responses obtained with the classical monopole secondary source exhibit lowfrequency oscillations, which are more pronounced than in impulse responses obtained with the SBR secondary source. The SBR secondary source becomes stable much faster than the classical secondary source, and is therefore more suitable for control purposes. Also, a shorter FIR filter is needed for modeling the SBR secondary source than for modeling the classical secondary source.

One of the objectives of the secondary source design was to achieve better performance of the ANC system itself. Therefore, a few simulations of the ANC process were performed in order to determine the effects of the added SBR on the ANC process. Simulations of ANC were based on the measurement and analysis of the impulse responses. Simulations were performed with MathWorks Simulink. The adaptive LMS algorithm in a signal processing toolbox was rearranged in order to simulate the FX-LMS algorithm described in [9].

During the simulations of ANC for different secondary sources only the measured impulse responses representing different acoustical paths were changed in the model. The step size of the adaptive algorithm was accordingly adjusted. The maximum step size for the FX-LMS algorithm depends on the RMS of the reference signal.

Therefore, simulations were performed with a maximum step size, which was determined for both configurations. When using a classic monopole as a secondary source, a maximum step size was μmax,MONOPOLE = 0.000018. When using the secondary source with the SBR configuration a maximum step size was μmax,SBR = 0.000071. All other settings in the model were kept constant.

A Secondary Source Configuration for Control of a Ventilation Fan Noise in Ducts

473

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 468-476

The results of ANC simulations are presented in Figs. 9 to 11. Two spectrograms of residual noise at error microphone position are presented in Figs. 9 and 10. In Fig. 9 a performance of the ANC system with the classical monopole secondary source is presented. In Fig. 10 a performance of the ANC system with the SBR secondary source is presented.

By comparing Fig. 9 with Fig. 10 it can be clearly seen that the convergence of the ANC system is faster with the SBR secondary source. Levels of residual noise after the finished convergence are also lower. Improvements can be seen in a broad frequency range. In comparison to the classical ANC system residual noise is higher only in the extremely low narrowband frequency range below 40 Hz. Four frequency spectra of primary noise and residual noise at error microphone position are presented in Fig. 11. A thick gray line presents primary noise spectra at the error microphone position if a classical monopole secondary source is attached to the duct. A thick black line presents primary noise spectra at the error microphone position if the SBR secondary source is attached to the duct. Even with the SBR secondary source switched off, some attenuation of primary noise is achieved at error microphone position. According to our predictions, primary noise is reduced around the SBR secondary source eigen-frequencies.

Fig. 9. Residual noise at the error microphone position obtained with an ANC system using a classical monopole secondary source 474

Insertion loss of the switched off SBR secondary source is depicted in Fig. 12. In a narrowband frequency range the insertion loss is over 3 dB. It could be much higher, if less damping material would be inserted in the SBR. However, with less damping material in the SBR the directivity of the secondary source would not be achieved and acoustic feedback would reduce the performance of the ANC system.

Measurement results differ from pure theoretical expectations. The main reason for the deviation of the measured results from the theoretical ones is a significant amount of damping material, which was inserted in the SBR. In former studies [11] no damping was used in order to compare measurement results with the simulation results and to establish proper finite element model.

An additional reason for the deviation of the measured result from the theoretical one is that a loudspeaker acts like a mechanical resonator with own its mass, spring and damping. Furthermore, it has its own resonant frequency and acts like a membrane absorber. A resonant frequency of the loudspeaker used in the experiment was around 80 Hz. Just around this frequency deviation of the measured results most significantly differs from the theoretical ones.

Two frequency spectra of residual noise at error microphone position with working ANC (thin

Fig. 10. Residual noise at the error microphone position, obtained with an ANC system using a secondary source with the SBR configuration

Prezelj, J. ‒ Čudina, M.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 468-476

Fig. 11. Frequency spectra of ventilating fan noise and residual noise at error microphone position; with switched off ANC system (thick lines) and with switched on ANC system (thin lines); black lines for the SBR secondary source and gray lines for the classical monopole source lines) are also presented in Fig. 11. Both spectra present residual noise after the adaptive algorithm finished the convergence. A thin gray line presents residual noise spectra of ANC system using classical monopole secondary source. A thin black line presents residual noise spectra of ANC using the SBR secondary source. The ANC system with the SBR secondary source achieves lower residual noise in the broad frequency range. A frequency spectrum of the residual noise is smoother and an improvement is more than 5 dB in two frequency ranges (from 100 to 170 Hz and above 420 Hz).

Active noise control works much better if SBR secondary source is used.

Fig. 12. Insertion loss of the switched off SBR secondary source A sound pressure level across the experimental duct is presented in Fig. 13. It is

presented for a frequency range around 120 Hz. Results indicate to a standing wave which forms between the secondary and primary noise source due to the acoustic feedback. The node of this standing wave can be observed as a local minimum at 1.5 m. This minimum (-10 dB) is pronounced only for the monopole type of the secondary source. When using the secondary source with the SBR configuration this minimum is only around -3 dB.

Fig. 13. Sound pressure level across the experimental setup 3 CONCLUSIONS In this article it was demonstrated that a loudspeaker, working as a dipole source in a

A Secondary Source Configuration for Control of a Ventilation Fan Noise in Ducts

475

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 468-476

side branch resonator (SBR), can be used as a secondary source for ANC systems. The SBR has three different tasks in the proposed configuration.

First, it prevents the acoustic short circuit in one direction. At the same time it allows an acoustic short circuit to happen in the other direction.

Finally, the SBR acts like a resonator, enabling additional transmission loss of the system. This results in significant directivity of the secondary source and improved transmission loss in the desired frequency range. During the design process of the SBR secondary source special attention needs to be paid to proper selection of the SBR length, damping and opening cross section.

Therefore, a simple analysis of the proposed configuration was carried out. The results of the analysis show that significant directivity can be achieved at low kL values. This is in accordance with the purpose of active noise control to work in low frequency range. The distance between a loudspeaker and the opening from SBR should be short. A shorter distance between the loudspeaker and the SBR opening provides a flat frequency response of the secondary source on one hand, but reduces the efficiency of the secondary source to generate high sound pressure level on other.

Damping in the side branch resonator should be optimized. Too much damping reduces a SBR to work as a resonator. If there is no damping, the SBR secondary source would not have an unidirectional characteristic. Measurements of impulse responses enabled the simulations of the adaptive ANC. ANC simulations demonstrated that the ANC system with the SBR secondary source performs much better than the ANC system with the classical monopole secondary source. The acoustic feedback in feedforward ANC system is reduced when using the secondary source with SBR configuration. The convergence of the adaptive FX-LMS algorithm can be faster with the SBR secondary source, and residual noise is lower than with the classical monopole secondary source.

In future work a multi objective optimization could be performed. Optimization could involve a cross section ratio between the duct and the SBR, a damping, a distance between the loudspeaker and the SBR opening and a SBR length. After the optimization an additional increase of the system performance is expected.

476

4 REFERENCES [1] Munjal, M.L. (1987). Acoustics of ducts and mufflers. John Wiley & Sons, New York.

[2] Field, C.D., Fricke, F.R. (1998). Theory and applications of quarter wave resonators: A prelude to their use for attenuating noise entering buildings through ventilation openings. Applied Acoustics, vol. 53, no.

1-3, p. 117-132.

[3] Romeu, J., Saluena, X., Jimenez, S., Capdevila, R., Coll, L. (2001). Active noise control in ducts in presence of standing waves. Its influence on feedback effect.

Applied Acoustics, vol. 62, p. 3-14.

[4] Geddes, E.R. (1994). Dual Bandpass Secondary Source, United States Patent, Patent No. 5,319,165.

[5] Chen, K.T., Chen, Y.H., Hsueh, W.J., Wang, J.N., Liu, Y.H. (1998). The study of an adaptively active control on the acoustic propagation in a pipe. Applied Acoustics, vol. 55, p. 53-66.

[6] Swinbanks, M. (1973). The active control of sound propagation in long ducts. Journal of Sound and Vibration, vol. 27, no. 3, p. 411436.

[7] Tang, S.K., Cheng, J.S.F. (1998). On the application of active noise control in an open end rectangular duct with and without flow.

Applied Acoustics, vol. 53, no. 1-3, p. 193210.

[8] Nelson, P.A., Elliot, S.J. (1995). Active control of sound. Academic Press, London.

[9] Kuo, S.M. (1996).

Active Noise Control Systems, Algorithms and DSP implementation. Wiley intersc., New York.

[10] Prezelj, J., Čudina, M. (2007). Dipole like secondary source for restrain of acoustical feedback in active noise control systems.

Internoise 2007, p. 479-489.

[11] Prezelj, J., Čudina, M. (2007). Dipole in orthogonal direction as a secondary source for active noise control in ducts. Acta Acustica united with Acustica, vol. 93, no. 1, p. 63-72.

[12] Tarabini, M., Roure, A., Pinhede, C. (2009).

Active control of noise on the source side of a partition to increase its sound isolation.

Journal of Sound and Vibration, vol. 320, no. 4-5, p. 726-743.

Prezelj, J. ‒ Čudina, M.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6,477-484 DOI:10.5545/sv-jme.2010.128

Paper received: 08.06.2010 Paper accepted: 09.09.2010

Fault Detection of an Industrial Heat-Exchanger: A Model-Based Approach Dragan, D. Dejan Dragan* University of Maribor, Faculty of Logistics, Slovenia

One of the key issues in modelling for fault detection is how to accommodate the level of detail of the model description to suit the diagnostic requirements. The paper addresses a two-stage modelling concept to an industrial heat exchanger, which is located in a tyre factory. Modelling relies on both, prior knowledge and recorded data. During the identification procedure, the estimates of continuous model parameters are calculated by the least squares method and the state variable filters (SVF). It is shown that the estimates are largely invariant of the bandwidth of the SVFs. This greatly reduces the overall modelling effort and makes the whole concept applicable even for less experienced users. The main issues of the modelling procedure are emphasized. Based on the process model, a simple detection system is derived. An excerpt of the results obtained on operating records is given. ©2010 Journal of Mechanical Engineering. All rights reserved.

Keywords: industrial heat exchanger, fault detection, condition monitoring, model-based detection, modelling, identification 0 INTRODUCTION Model-based condition monitoring of industrial processes aims at early revelation of degradations in process equipment and instrumentation. A sensible process model acts as an additional virtual instrument, which contributes to a higher quality of production and better safety.

There are many papers and books dealing with model-based techniques for detecting, isolating and identifying faults [1] to [6]. However, in many real applications deriving a proper model still takes a bulk of overall design effort. Moreover, proper shaping of the model precision with respect to the diagnostic requirements remains to be rather an art.

This paper deals with the design of a fault detector for a heat exchanger as a possible alternative to some other approaches [7] to [11].

The work represents part of the prototyping design of a condition monitoring system for the process of incineration of vulcanisation gasses located in a tyre factory [6].

A brief idea of the process can be grasped from Fig. 1. Vulcanisation gas (VU gas), which is one of hand products of vulcanisation, is generated on vulcanisation lines. Prior to the emission into the atmosphere, all the carbon particles contained in the gas need to be destroyed by incineration in a combustion chamber.

The entire system consists of three major parts [6]: • pre-heating of the VU gas in a gas-gas heat exchanger; • incineration of the VU gas in a combustion chamber; • transportation of the flue gasses to the chimney.

The focus of this paper is on model-based condition monitoring of the cold part of heat exchanger. The goal is to improve the support to the maintenance team through permanent monitoring of the condition of sensors and detection of fouling. The project aims at designing a detector with the highest precision possible. For that purpose, the extra redundancy is achieved by means of an analytical model of the plant.

The modelling concept consists of two stages in which prior knowledge and recorded data are combined (grey-box modelling concept).

In the first stage, the model structure is derived up to unknown parameters by strongly relying on reasoning from first principles. After taking the available instrumentation into consideration, the set of prior assumptions and the diagnostic requirements the modelling procedure ended up with a continuous-time model linear in parameters.

For the purpose of parameter estimation, the least squares method (LSM) combined with state variable filters (SVF) is adopted in the second

*Corr. Author’s Address: University of Maribor, Faculty of Logistics, Mariborska c. 7, 3000 Celje, Slovenia, dejan.dragan@fl.uni-mb.si

477

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 477-484

stage. It is shown that the use of LSM leads to the identification results, which are largely invariant of the bandwidth of the SVFs. This observation deserves attention since the choice of bandwidth, as a design parameter, turns out to be quite an easy task. This is believed to be one of the contributions of the paper.

Fig. 1. The incineration system The derived model of the cold part of an exchanger employs the temperatures (Tco, Tho) and flow of the VU gas (ΦVU). The measured signals (c.f. Fig. 1) are collected, displayed and stored by the FactoryLinkTM SCADA system (USDATA Corp.)

Finally, the model-based diagnostic algorithm is designed to run on-line as an external C module of SCADA. The proposed design of the condition monitoring system is very simple and the suggested detector is a helpful indicator for the operator to take corrective action.

The derivation of the model structure and the parameter estimation approach is described in the first section. An excerpt of the experimental results is given in the second section. Finally, the diagnostic procedure is overviewed in the third section.

1 SYSTEM IDENTIFICATION 1.1 Determination of Model Structure The purpose of this section is to emphasize the importance of prior knowledge in deriving the model structure. This knowledge is essential 478

in early modelling steps, in particular in defining causal relationships between process variables.

Depending on the diagnostic requirements, these relationships can range from loose (qualitative) to precise (quantitative) descriptions. In case of poor prior knowledge and/or loose diagnostic requirements, model precision can be restricted to qualitative relationships defined on sign, interval or fuzzy sets. Perfect prior knowledge allows for precise expressions that are fully defined on the set of real numbers. Since in many practical cases prior knowledge is incomplete, process data that represent the carrier of additional information needed to complete the model description need to be employed. Here, the finest level of detailed description is observed.

The process of the heat exchange between (cool) VU gas and (hot) flue-gas is illustrated in Fig. 2.

The modelling procedure starts with setting the energy balance equation for an infinitesimally small piece of the VU gas channel and pipe wall.

This results in the following partial differential equations respectively [1]:

∂T ( x, t ) ∂TC ( x, t ) cVU + ⋅ ΦVU (t ) ⋅ C = ∂t ∂x AC1

cVU ⋅

o2 = ⋅ α 2 (t ) ⋅ [TW ( x, t ) − TC ( x, t ) ] , ρVU ⋅ AC1

∂T

( x, t )

∂t

W

−

=

(1)

o1 α1 (t ) ⋅ TH ( x, t ) − T ( x, t )  − W   (2) ρW AW cW

o2 ⋅ α 2 (t ) ⋅ [TW ( x, t ) − TC ( x, t ) ] , ρW AW cW

where TC(x,t), TH(x,t) and TW(x,t) represent spacetime behaviour of the temperatures of the cold part of the exchanger, hot part and pipe wall, respectively, while (o1 = 2πr1, o2 = 2πr2).

In order to identify the relationship between the measured variables (Tco, Tho, ΦVU), the distributed parameter models in Eqs. (1) and (2) need to be converted into the lumped models.

The lumping procedure is based on the following set of assumptions: 1. specific mass (ρVU) and specific heat (cVU) of the VU gas are assumed to be constant, 2. flow of the VU gas is space independent (ΦVU(x,t) = ΦVU(t)),

Dragan, D.

StrojniĹĄki vestnik - Journal of Mechanical Engineering 57(2011)6, 477-484

Fig. 2. Illustration of the heat exchange process 3. convective heat exchange coefficients are independent of time and space (Îą1(x,t) = Îą1, Îą2(x,t) = Îą2), 4. as the wall is only 4 mm thin Aw ≈ 0 and ∂TW(x,t) / ∂t ≈ 0 can be assumed.

5. The retention time of vulcanisation gasses in the heat exchanger is fairly low, as the speed of VU gasses is relatively high while the length of the heat exchange channel is quite short. This implies an almost momentary formation of temperature profiles along the heat exchange channel.

Assumption 4 implies short time constant for the dynamics of the wall temperature TW(x,t) in Eq. (2). Transients can be neglected so that the following static relation emerges:

TW ( x, t ) = m1 ⋅ TH ( x, t ) + (1 − m1 ) ⋅ TC ( x, t ),, m1 =

(3)

o1 â‹… Îą1 . o1 â‹… Îą1 + o2 â‹… Îą 2

In the next step let us combine Eqs. (3) and (1) at x = L, which results in:

|

|

∂TC ( x, t ) ∂T ( x, t ) 1 = + â‹… ÎŚ (t ) â‹… C x = L AC1 VU x=L ∂t ∂x

. b1' T co (t )

(4) o2 â‹… Îą 2 â‹… m1 = â‹… [TH ( L, t ) − TC ( L, t ) ].

Ď â‹… AC1 â‹… cVU VU

T t T − ( t ) ( ) ho c o a1

Then, according to assumption 5, the temperature profiles along the heat exchange channels exhibit almost static behaviour. From the steady-state condition at x = L, Eq. (4) results in the following expression:

|

∂TC ( x) a1 = â‹… (Tho − Tco ) = Îľ â‹… (Tho − Tco ) , ' (5) ∂ x x = L b â‹… ÎŚVU 1

Îľ

where bars over symbols denote stationary values.

However, a number of identification runs, carried out on model structures based on Eq. (5), turned to produce one-step-ahead predictor with relatively poor performance. Careful examination of the unmodelled effects related to the derivation of Eq. (5), suggests the approximation of the gradient ∂TC(x,t) / ∂x at x = L by a richer structure in order to improve the predictive power of the final process model. It can be shown that the gradient is related with the measured input and ouput temperatures and flow in a very complicated manner. Instead of an exact solution a black-box structure is sought so that experimental data are fitted as well as possible while keeping the number of unknown parameters at a minimum. Among many candidates (polynomials, neural networks) it turned out that already a simple structure with only two free parameters can significantly raise the quality of fit of the model Eq. (4). The approximation reads very similarly to Eq. (5) i.e.

|

∂TC ( x ,t ) ≈ Îľ 1 â‹… Tho ( t ) − Îľ 2 â‹… Tco ( t ). (6) x=L ∂x

If Eq. (6) is entered to Eq. (4), the following expression is obtained:

.

T co (t ) + b 1'â‹… ÎŚVU (t ) â‹… Îľ1 â‹… T (t ) − Îľ 2 â‹… T (t ) =

(

ho

= a 1⋅ (Tho (t ) − Tco (t ) ) ,

co

)

(7)

i.e.

Fault Detection of an Industrial Heat-Exchanger:A Model-Based Approach

479

StrojniĹĄki vestnik - Journal of Mechanical Engineering 57(2011)6, 477-484

.

T co (t ) + b1' â‹… Îľ1 â‹… [ ÎŚVU (t ) â‹… Tho (t ) ] +

b1 − F1 (t )

(8) + b1' â‹… Îľ 2 â‹… [ âˆ’ÎŚVU (t ) â‹… Tco (t ) ] = a1 â‹… [Tho (t ) − Tco (t ) ].

b2 − F2 (t ) Eq. (8) represents the lumped model of the cold part of the heat exchanger. The same modelling procedure can be applied to derive a model of the hot part. Unfortunately, the task is not that easy due to a lack of a flow sensor in the flue-gas channel (hot part). Nevertheless, a careful analysis indicates that the unmeasurable flow mostly depends on flow of the vulcanisation gasses, roughly in a linear way. With this in mind a model of the hot part being entirely similar to that of the cold part is achieved. Due to a lack of space the hot part will not be treated here. In the sequel emphasis is on the cold part only.

After filtering all the signals in Eq. (8) and after transformation of filtered data, an appropriate form for identification procedure is obtained. For the data collected by the SCADA system, an overdetermined system of equations, which can be represented in the following form, is obtained: (10)

Y = Ψ ¡ θ + Δnf1 , where:  ∆Tco _ mf (1) − ∆Tho _ mf (1)    Y= ď Œ ,  ∆Tco _ mf ( N ) − ∆Tho _ mf ( N )   

.  ∆F2 _ mf (1)  −∆ T co _ mf (1) ∆F1 _ mf (1) Ψ =  ... ... ...  .   −∆ T co _ mf ( N ) ∆F1 _ mf ( N ) ∆F2 _ mf ( N ) 1 θ=  a1

b1 a1

b2 a1

 1 ... ,  1 

T

K  (11)  , a1 

 ∆n f 1 (1)    ∆n f 1 =  ď Œ .  ∆n ( N )    f 1

1.2 Parameter Estimation

In order to identify the model parameters batch the least squares method is chosen.

To estimate the parameters of the continuous time model in Eq. (8), the derivative of Tco is needed. As direct differentiation is prone to significant errors due to measurement noise, the problem is alleviated by using state variable filters. In this case, any stable filter with relative degree ≼ 1 and appropriate bandwidth would suit.

A simple transfer function is employed to filter the signals in Eq. (8):

The unknown constant K is added in order to take into account the bias in the prediction error. In case of perfect model structure (i.e. free of modelling errors) the estimated K should be zero. Vector Δnf1 is added to encounter the noise effects.

The unknown parameters of the system Eq. (10) result as follows [12] and [13]:

G f (S ) =

1 , Ď„ â‹… s +1

(9)

where 1/Ď„ is the bandwidth of filter (Ď„ is the time constant). If Eq. (9) is applied to both sides of Eq. (8), filtering preserves the original model structure.

To improve the numerical properties of the algorithm, the filtered signals from Eq. (8) are transformed in the next step as for example:

ΔTco_mf (t) = Tco_mf (t) ‒ Tco_mf (t-1) ,

where index “mf� refers to the value of the filtered signal. Differentiated data preserve the original model structure while eliminating the problem of offsets in prediction error.

480

∧

−1

θ =  Ψ T ⋅ Ψ  ⋅ Ψ T ⋅ Y.  

(12)

2 PRACTICAL IDENTIFICATION RESULTS Process identification was carried out on a batch of 14000 samples taken during normal operating regime (interval [1, 14000] [min]).

Outliers and intervals with non-informative data are carefully eliminated from further processing.

The results of the estimation achieved at various bandwidths of SVF’s (1/τ) are shown in Fig. 3. It can be seen that the results of estimation do not vary significantly with respect to SVF bandwidth changing over two decades. This statement includes the variations of parameter (τ ) a (τ ) , which are small compared to K 1 the elements of vector Y (Eq. (11)) and can be

Dragan, D.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 477-484

neglected. Identification results largely invariant to pre-filtering imply great freedom in choosing the SVF bandwidth. This is an advantage for every practitioner.

The purpose of the designed model is to predict the difference between temperatures of cold and hot part at the end of exchanger.

Thus, by taking into account Eq. (8) process output can be represented as follows:

yf (t) = Tco_mf (t) ‒ Tho_mf (t) ,

(13)

and the predicted output as:

. y (t ) =  1  ⋅  − T co _ mf (t )  +  b1  ⋅ F (t ) +    1_ mf f      a1   a1   (14)

 b 2  y , +   ⋅ F2 _ mf (t ) + K f  a 1   

respectively. Similarly to Eq. (11), estimated  y is added to avoid bias in modelling constant K f error.

A comparison between process output Eq. (13) and predicted output Eq. (14) on validation set containing 5000 samples (interval [14000, 19000] [min]) is shown in Fig. 4. The model fits the process reasonably well so that the prediction error does not exceed 10% of the dynamic range of the signal Eq. (13). In other words, the underlying mathematical model can be viewed as an additional (virtual) instrument, which obviously brings extra redundancy into the system.

3 DIAGNOSTIC RESULTS Inference about faults is made on the basis of the residual signal defined as follows:

Fig. 3. The estimated parameters in dependence of different bandwidths (1/τ) of the SVF

Fig. 4. Model validation: measured output, predicted output and prediction error; a) yf(t); b) prediction error Fault Detection of an Industrial Heat-Exchanger:A Model-Based Approach

481

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 477-484

r (t ) = y f (t ) − y f (t ),

(15)

where yf (t) and y f (t ) are process output Eq. (13) and predicted output Eq. (14), respectively.

Based on the residual Eq. (15) the presence of a fault can be detected. This means that if the residual is near zero, there is no evidence that a fault is present. On the contrary, if the residual departs significantly far from zero, the presence of a fault can be inferred. Generally, purely on the basis of the residual Eq. (15) it is not possible to determine the location of the fault. The exception is discussed in the literature [3] and [5]. Indeed, provided the parameters θ of the process model bear physical meaning, the regressor form of the model Eq. (10) reads:





Y = Ψ · θ + Δnf1 and Y = Ψ ⋅ θ ,

for the true process and the mathematical model. Symbols Y and Y denote true and predicted outputs, while θ and θ denote true and estimated process parameters, respectively. By differentiating the two equations the following is obtained:

e = Y − Y = Ψ ⋅ θ + ∆n f 1 − Ψ ⋅ θ = = Ψ ⋅ (θ − θ ) + ∆n = Ψ ⋅ ∆θ + ∆n , f1

f1

where e denotes the vector of residuals. Since e results in a straightforward manner from the recorded data and the nominal process model, the estimate of the vector Δθ can easily be calculated Any non-zero term in Δθ indicates a variation in the process parameters due to the fault. Knowing the physical origin of such a term in the vector θ the position of the fault can be inferred. An important technical requirement is that the data matrix Ψ is full rank (persistent excitation).

However, the idea is not applicable in the present case for three reasons: 1. The temperature signals Tho and Tco are generally too poor from the point of view of information content, which means that with recursive parameter estimation it would not be possible to unambiguously estimate all the parameters in Eq. (14); 2. Signal to noise ratio in certain intervals of process operation can be rather low.

3. Not all the parameters in Eq. (14) reflect the physical properties of the system (the model 482

Eq. (8) is semi-physical); consequently such parameteres would be of little use in fault localisation.

Having residual Eq. (15) it is neccessary to draw the decision about alarm. If pure Boolean logic were applied, then frequent, even small deviations in residuals in the vicinity of the threshold value would lead to large variations in the diagnostic results [14] and [15]. In order to smooth the diagnostic output, approximate reasoning techniques seem to be a better alternative. A residual is no longer qualified as zero (0) or non-zero (1) but is associated a degree of being zero, which is a number between 0 and 1. In this way, incremental changes in the residual Eq. (15) result in incremental changes in the diagnostic results.

For the sake of detection the belief mass (Fig. 5) is introduced, which can be represented in the following form [14]:

bel (r ) =

1 , 1 − δ d h 2γ d 1+ ( ) δd r

(16)

h and γd are threshold and the adjustable smoothing parameter respectively, while ( γd = bel(r=h) ).

Fig. 6 shows the response of the detector, when two consecutive faults are injected into the system. Firstly, the temperature sensor related to Tho is stuck high. This fault is emulated by fixing the sensor output to 560 °C for the period of 700 minutes. The next fault is emulated drift in the sensor related to Tco. Obviously, the detector is quite sensitive to the occurrence of both faults.

Fig. 5. Belief assignment function Since belief in a non-zero residual also significantly increases, the detector provides clear evidence of the presence of both faults (c.f. Fig. 6).

Though obvious, the first fault remains undetected

Dragan, D.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 477-484

by the classical alarm system. Indeed, the alarm is set at 600 °C as lower values can be easily reached during normal operating conditions (not presented in Fig. 6). On the contrary, the model-based detector reacts quite quickly and accurately. This detector reacts relatively promptly in the case of a drift type of fault as well. For example, it reports the presence of fault at approx. t = 4100 min with 70% belief. The classical alarm system would react, but much later (at approx. 5200 min).

To sum up, by running the suggested detector on the recorded data sets it has been possible to reveal several temporary erroneous sensor readings during operation of the real plant.

These were completely overlooked by the existing alarm system (missed alarms).

model structure, while data are used to identify unknown model parameters.

The paper makes a significant contribution in two ways. Firstly, a set of prior (heuristic) assumptions provides a means for determining the model structure. Secondly, it is shown that the results of estimation, obtained using the least squares method, are largely invariant of the bandwidth of the SVF. This greatly reduces the overall modelling effort and is an advantage for every practitioner.

The suggested detection algorithm is very simple for execution in real time. The diagnostic results show that the module is able to accurately indicate the presence of incipient faults and thus facilitate timely on-condition maintenance.

Traditional alarm systems based on thresholding are insensitive to such faults, i.e. do not react until large deviations and failures occur.

5 REFERENCES

Fig. 6. Detection of sticking high of sensor Tho and detection of drift in sensor Tco 4 CONCLUSIONS A two-stage modelling procedure is presented and applied to a heat exchanger in an incineration unit. It relies on blending prior knowledge with the information contained in data records. Prior knowledge serves to derive the

[1] Bogaerts, P., Castillo, J., Hanus, R. (1997).

Analytical solution of the non uniform heat exchange in a reactor cooling coil with constant fluid flow. Mathematics and Computers in Simulation, vol. 43, no. 2, p.

101-113.

[2] Weyer, E., Hangos, K.M. (1997). Grey box fault detection in heat exchanger networks.

Prepr. IFAC Symp. Safeprocess, vol. 1, p.

187-192.

[3] Gertler, J. (1998). Fault detection and diagnosis in engineering systems. Marcel Dekker, New York.

[4] Krishnan, R.A., Pappa, N. (2005). Real time fault diagnosis for a heat exchanger - a model based approach. INDICON, Annual IEEE, p.

78-82.

[5] Klančar, G., Juričić, Đ., Karba, R. (2002).

Robust fault detection based on compensation of the modelling error. International Journal of Systems Science, vol. 33, no. 2, p. 97-105.

[6] Dragan, D., Juričić, Đ., Strmčnik, S. (2000). Modelling for condition monitoring application to a heat transfer process. Process control qual., vol. 11, p. 419-431.

[7] Lalot, S., Mercère, G. (2008). Detection of fouling in a heat exchanger using a recursive subspace identification algorithm.

Fault Detection of an Industrial Heat-Exchanger:A Model-Based Approach

483

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 477-484

Proceedings of the 19th International Symposium on Transport Phenomena, Paper #37.

[8] Jonsson, G.R., Lalot, S., Palsson, O.P., Desmet, B. (2007). Use of extended Kalman filtering in detecting fouling in heat exchangers. International Journal of Heat and Mass Transfer, vol. 50, no. 13-14, p.

2643-2655.

[9] Ingimundardóttir, H., Lalot, S. (2009).

Detection of fouling in a cross-flow heat exchanger using wavelets. Proceedings of International Conference on Heat Exchanger Fouling and Cleaning, p. 484-491.

[10] Ramasamy, M., Shahid, A., Zabiri, H. (2008). Drift analysis on neural network model of heat exchanger fouling. Journal of Engineering Science and Technology, vol. 3, no. 1, p. 40-47.

[11] Juan, C., Martínez, T., Morales-Menendez, R., Garza-Castañón, L.E. (2010). Fault diagnosis

484

in a heat exchanger using process history based-methods, computer aided chemical engineering. 20th European Symposium on Computer Aided Process Engineering, Pierucci, S., Buzzi, G. (eds.), Elsevier B.V., vol. 28, p. 169-174.

[12] Garnier, H., Wang, L. (2008). Identification of continuous-time models from sampled data.

Springer-Verlag, London.

[13] Ljung, L. (1999). System identification: theory for the user. 2nd ed., Prentice Hall, New York.

[14] Rakar, A., Juričić, Đ., Balle, P. (1999).

Transferable belief model in fault diagnosis.

Engineering Applications of Artificial Intelligence, vol. 12, p. 555-567.

[15] Rakar, A., Juričić, Đ. (2002). Diagnostic reasoning under conflicting data: the application of the transferable belief model.

Journal of Process Control, vol. 12, p. 55-67.

Dragan, D.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 485-494 DOI:10.5545/sv-jme.2010.212

Paper received: 07.10.2010 Paper accepted: 17.02.2011

Numerical Methods for TMF Cycle Modeling Zaletelj, H. – Fajdiga, G. – Nagode, M. Henrik Zaletelj* – Gorazd Fajdiga – Marko Nagode University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Experimental tests for the endurance evaluation of the machine parts that are exposed to thermomechanical fatigue (TMF) require advanced and expensive testing machines. Numerical methods for the determination of stress-strain material behavior have become very frequent and known due to lower costs. There are several different approaches for the determination of stress-strain behavior. In the article three different numerical methods and their results are presented. The numerical results for different load conditions are compared with the experimental results and the accuracy of the methods can be compared. The Chaboche, Skelton and Prandtl operator approaches are presented, presuming a stabilized elastoplastic response and not including creep. The properties of the model, their weaknesses and possible improvements are also studied in the paper. ©2011 Journal of Mechanical Engineering. All rights reserved.

Keywords: cyclic loading, elastoplasticity, kinematic hardening, stress-strain trajectory, thermomechanical fatigue 0 INTRODUCTION Many components such as internal combustion engines, turbines, nuclear reactors, etc., are subjected to thermo-mechanical fatigue (TMF) [1] and [2]. Fatigue life depends primarily on loads, material, geometry and environmental effects. Its evolution is generally based on tests of three forms [3] and [4]: • isothermal strain-controlled low cycle fatigue (LCF) tests, • TMF tests on specimens and components, and • thermal shock tests.

In view of their relative simplicity [5], LCF tests are often favored. The data from LCF tests conducted on servo-controlled uniaxial testing machines have been collected and tabulated for many years [3]. The key idea is, therefore, to predict fatigue life by avoiding expensive TMF and thermal shock tests. The present work is concerned with the cyclic stress-strain response for variable temperatures. Creep and transient effects, such as cyclic hardening and cyclic softening are not considered in this paper. Thus, hysteresis loops are supposed to be stabilized and constitutive equations for elastoplasticity are applicable for the stress-strain behavior modeling.

To analyze the response of different calculating models, the material 9Cr2Mo alloy is used where the material parameters are presented in [5].

The paper presents the predicted TMF cycles based on the three different approaches.

The Skelton, Chaboche and spring-slider model with the temperature dependant Prandtl densities [7] are verified with the experimental results. The strain and temperature are controlled.

The paper is structured as follows. After the explanation of the material and TMF tests, the constitutive equations of the Chaboche model and the definition of the parameters are presented.

The following section introduces the spring-slider model with the Prandtl densities. Then, the review of different model results and verification with several TMF tests follow. Finally, the final section lists the conclusions where the characteristics of the Prandtl and Chaboche models are introduced.

1 MATERIAL AND TYPES OF TMF TESTS 1.1 Material The material used to analyze the efficiency of the predicted stress-strain curves is advanced ferritic-martensitic steel EM 12 (9% Cr, 2% Mo).

It is used as material in conventional thermal plants operating at temperatures up to 600 °C [5].

The material belongs to the cyclic softening class of alloys. Further details can be found in [5].

*Corr. Author’s Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia, henrik.zaletelj@fs.uni-lj.si

485

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 485-494

1.2 TMF Cycles The cycle time for each loop was five minutes, i.e. at the strain rate of 4 × 10-5 s-1 for 0.6% strain range. Fig. 1 presents TMF cycles, where it is convenient to plot strain on the vertical axis and temperature on the horizontal axis.

The strain range is 0.6% and temperature varies between 270 and 570 °C. Using the paths in Fig. 1, nine TMF types of cycles were performed to compare the stress-strain hysteresis curves of numerical models. Path PXRM represents a 45° kite cycle formed in the anticlockwise direction, while the corresponding clockwise cycle is taken in the PMRX order. A similar scheme applies in the 135° kite cycle PKRZ. The parallelogramshaped PXRZ and PKRM mark the 45° zero strain and the 135° zero strain, respectively. Finally, complex cycle (dashed line) was considered, which shows a cycle of industrial gas turbine blades. A detailed explanation and the reasons for choosing these types of TMF loops are given in [5].

where the corresponding total strain range, Δε, at any temperature is given by: 1

∆σ  ∆σ  β ∆ε = +  , E  A  while Δσ stands for the stress range.

(1)

2 THE CHABOCHE MODEL Most metals approach a cyclically stable state after a certain number of cycles. Cyclically stable or half-life material properties are usually used in fatigue analysis [7]. To describe the stable state of material, the Chaboche model of kinematic hardening can be used [8] and [9].

To the Chaboche model considerable attention has been paid due to its capacity of modeling a wide range of inelastic material behavior such as cycling hardening/softening, the Bauschinger effect, stress relaxation and creep for a range of materials [10].

Cyclic stress-strain curves are modeled with the uniaxial form of the Chaboche model.

Stabilized cyclic curves are defined with kinematic hardening, which corresponds to the movement of the loading surface, where σ is stress at each moment and k is the yield stress. The hardening variable χ (back-stress) indicates the present position of the loading surface. Back stress χ also indicates the directional dependent effects, such as the Bauschinger effect. The criterion and the equation of flow and hardening can be expressed in the form [11]:

f(σ, χ, k) = |σ – χ| – k = 0 .

(2)

The evolution equation of the backstress for non-linear kinematic hardening used in the Chaboche material model was originally introduced by Armstrong and Frederick [12]:

Fig. 1. TMF cycles For the observed (experimental) values of stress-strain curves, the Ramberg-Osgood parameters were used. The parameters of cyclic hardening coefficient A and cyclic hardening exponent β are examined for isothermal loops for the temperature span of 270 to 570 °C [5].

Experimental curves were drawn for Eq. (1) 486

χ i = Ciε p − γ i χ i p .

(3)

The Eq. presents the nonlinear kinematic hardening where p is the accumulated plastic strain, i = 1, 2, χ = χ1+χ2. Ci and γi are temperature dependant material parameters. If γi = 0, Eq. (3) presents the model of the linear kinematic hardening (Prager’s kinematic hardening law) [11]. The essence of the model is in the velocity of plastic deformation ε p and in the velocity of accumulated plastic deformation p :

Zaletelj, H. – Fajdiga, G. – Nagode, M.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 485-494

p = ε p . (4)

The model in Eq. (3) describes kinematic hardening within one load cycle as well as kinematic hardening of the accumulated plastic deformation developed over more cycles to the saturation condition. The integration of Eq. (3) gives:

χi =

Ci 1 − exp ( −γ iε p ) , (5) γi

(

)

and for i = 1, 2, χ=

2

Ci

∑ γ (1 − exp ( −γ ε ) ) and χ = χ1+χ2. (6) i =1

i

p

i

The first part of kinematic hardening χ1 describes the transition area of inelastic deformation, while the second part χ2 describes the behavior at higher inelastic deformations after χ1 reaches saturation value C1/γ1. In tensioncompression, and more generally, in proportional loading, the evolution equation of hardening can be integrated analytically to give [11]:

C  C  χ i = ν i +  χ 0 −ν i  exp −νγ i ( ε p − ε p 0 ) , (7) γi  γi 

(

)

where ν = ±1 according to the direction of the flow, εp0 and χ0 denote the initial values, for example at the beginning of each plastic flow. It is not necessary to update variables εp0 and χ0 from the previous flow. At each moment the stress is given by:

σ = χ + νk .

(8) 2.1 Parameter Estimation To model the cyclic curves, kinematic hardening variable of transition area χ1 is used.

With regard to the strain magnitude, the saturated value of χ1 is not reached, so parameters C1 and γ1 have to be defined. The presented model, Eqs. (7) and (8), contains four material parameters. These are Young’s modulus E, kinematic hardening parameters C1 and γ1 and the initial size of yield surface k.

Young’s modulus is presented in [5] as a function of temperature:

E = a – bT , (9)

where a = 2.08 × 105 MPa and b = 97.5 for T in °C. Parameters C1 and γ1 can be estimated from the tension part of the cyclically stable rising hysteresis branch. They can also be estimated on the first cyclically stable stress-strain curve. The latter option is used in this paper. The estimation is made at a low level of plastic strain, where the transition kinematic hardening χ1 is more obvious.

As σ > χ it follows: n

σ − χ − R−k  ε p =   , (10) Z  

where σ presents total stress, χ kinematic hardening, R isotropic hardening, Z and n viscous parameters and k yield surface. Viscosity stress is defined as: (11)

σν = σ – χ – R – k, As the stabilized cycle is observed where

R = 0 by considering the constant k parameter

and viscosity stress σν = 0, the derivate of Eq. (11) is presented as:

∂σ ∂χ = . ∂ε p ∂ε p

(12)

Considering Eq. (3), the logarithm of Eq. (12) is expressed as:

 ∂σ ln   ∂ε p 

  = ln(C1 ) − γ 1ε p . 

(13)

Parameters C1 and γ1 are determined from the linear regression. Parameter –γ1 presents the line slope and parameter ln(C1) the intersection with the ordinate axis. σ and εp are evaluated with the Ramberg-Osgood equation with the parameters that are presented in the article [5].

The parameters are evaluated for the temperatures ranging from 270 to 570 °C with the interspaces of 30 °C. To obtain the material parameters that were not measured, the linear parameter interpolation is used.

The estimated values are used as initial values in the optimization process, where the parameters were finally fitted on the stabilized cycle loop. The aim of the optimization is to find the minimum difference between the back-stress values of the first cycle and the back-stress of the stabilized cycle. The parameters are presented in Table 1.

Numerical Methods for TMF Cycle Modeling

487

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 485-494

Table 1. Chaboche parameters Temperature [°C] 270 300 330 360 390 420 450 480 510 540 570

C1 [MPa] 104541 101844 101344 113963 92139 97456 89232 71252 74475 59331 63820

and γ1

192

Ramberg-Osgood

E [MPa] 181675 178750 175825 172900 169975 167050 164125 161200 158275 155350 152425

A [MPa] 1611 1554 1534 1763 1423 1505 1395 1133 1165 964 990

β 0.135 0.127 0.123 0.147 0.115 0.126 0.121 0.095 0.108 0.091 0.108

2.2 Temperature Dependence in the Back-Stress Evolution Equation The response and evolution of kinematic hardening is dependent on temperature.

Temperature influence is considered in parameter dependence; besides, the changing of temperature vs. time is taken into account. If the temperature changing vs. time is fast, the influence is high.

The influence is negligible for long periods.

Fig. 2 presents the graph where the stress vs. strain is dependent on temperature and time. The temperature at ε = 0 was 270 °C and at maximum load ε = 0.006 it was 570 °C. Eq. (14) represents the back-stress dependence of temperature (only one back-stress is considered here). As compared to Eq. (3), temperature influence is introduced directly by the variation of parameter Ci:

χ i = Ci (T )ε p − γ i χ i p +

1 ∂Ci  χ iT . (14) Ci (T ) ∂T

The main advantage of the model is that the equation does not contain new material parameters. It uses the parameters in dependence on the temperature where the partial derivatives upon temperature can be estimated from.

2.3 Relaxation of Mean Stress If the load is not purely alternating, additional effects can occur. In a strain-controlled test, when the mean strain is not zero, the phenomena of the relaxation of the mean stress appear. The initial asymmetry of the stress 488

Fig. 2. Stress-strain dependence on temperature and time

Fig. 3. Relaxation of mean stress

disappears progressively in the first few cycles, Fig. 3.

The temperature changing inside the load cycle defines the level of mean stress. Temperature load cycle defines the shape of the hysteresis curve, maximum and minimum stress. In Fig. 4, the position of a stabilize curve is shown for the changing the temperature between 270 and 570 °C. The mean stress of the stabilized curve where the temperature is constant equals zero, Fig. 4.

Zaletelj, H. – Fajdiga, G. – Nagode, M.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 485-494

 ε (ti ) − q j ,  . (15) min {ε (ti ) + q j , ε α j (ti −1 )}

ε α j (ti ) = max 

Fig. 5. Rheological spring-slider model [7]

Fig. 4. Stabilized curves and temperature variation

3 PRANDTL OPERATOR APPROACH It is standard practice in the isothermal strain-life approach to use cyclic material properties together with the Masing and Memory models to define the cyclic uniaxial response of the material, which determines the stress and strain range of the closed hysteresis loop and the mean stress associated with each loop. It has been shown in [13] that the Masing and Memory models are not valid if the temperature varies during the cycle. Masing based his finding on the rheological spring-slider model and assumed that the model parameters are time independent.

As the spring-slider model supports the modeling of elastoplastic hardening, it has been adapted for variable temperatures [7]. The model is capable of modeling elastoplastic hardening solids and nonlinear kinematic hardening under strain control (see Fig. 5). The stress controlled model is given in [14].

From the equilibrium in a single springslider segment, total strain ε is obtained ε = εqj + εαj where slider strain | εqj | can never exceed fictive yield strain qj that is also known as the half-width of the play operator. Spring strain εαj can now be expressed as the play operator with general initial values [7]:

. Thus, current strain for 0 ≤ t1 ≤ t2 ≤ ... ≤ ti ≤ ... state εαj(ti) depends on the previous εαj(ti–1) called the memory point. Presumably, there is no residual strain initially, so εαj(0) = 0 and σj(0) = 0.

Determination of the parameter nq is thoroughly described in [26]. The stress in the spring-slider segment is then:

σ j (ti ) = E j (Ti )ε α j (ti ) = α j (Ti )ε α j (ti ) , (16)

where Ti = T(ti) and αj(Ti) is the Prandtl density.

Adding the spring-slider stresses results in total stress in the form known as the operator of the Prandtl type [7]: nq

σ (ti ) = ∑ α j (Ti )ε α j (ti ), (17) j =1

with temperature-dependant Prandtl densities. The play operator given in Eq. (16) is independent of time and temperature. Therefore, it is modified [7] to assure equilibrium in the spring-slider:

 ε (ti ) − q j ,    ε (ti ) + q j ,  ε α j (ti ) = max   . (18)  min  α j (Ti −1 ) ε (t )   α (T ) α j i −1     j i  When Eq. (18) is inserted into both Eqs. (17) and (19), temperature-dependant stress-strain behavior can be modeled because the temperaturemodified play operator guarantees equilibrium in the spring-slider segments at any time and temperature:

σ j (ti −1 ) = α j (Ti −1 )ε (ti −1 ). (19)

Numerical Methods for TMF Cycle Modeling

489

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 485-494

The Prandtl densities can be calculated from the temperature-dependent Ramberg-Osgood curves. For a further explanation, interested readers should see papers [13] to [21]. In newer publications index j in Eq. 17 starts with 1 instead of 0 to simplify the notation. The Skelton approach is thoroughly discussed in [3], [5] and [23].

4 VERIFICATION OF MODELS The Chaboche, Skelton and Prandtl operator approaches have been compared to

several TMF tests conducted by Skelton [5] at the total strain range of 0.6 % on the 9Cr2Mo alloy.

The paper is concerned with the paths given in Fig. 1.

The observed hysteresis loops from the tests are plotted as crosses in Figs. 6 to 15. The circles and the dot line denote the stress-strain trajectories modeled by the Skelton and the Prandtl operator approach, respectively. The thin solid line denote the stabilized cycle of the Chaboche non-linear kinematic hardening model. The thick solid line presents the shifted Chaboche hysteresis curve

Fig. 6. Shift of the Chaboche hysteresis curve (load case PZRKP)

Fig. 8. TMF cycle PMRXP

Fig. 7. TMF cycle PZRKP

Fig. 9. TMF cycle PXRMP

490

Zaletelj, H. – Fajdiga, G. – Nagode, M.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 485-494

(thin line) that is fitted to the observed results, Fig. 6. The shift is required due to unwanted ratcheting effect built into the Chaboche model.

The ratcheting effect is noticed at different load conditions. It happens due to a small amount of plastic strain in each cycle, which leads to unacceptable accumulated strain. This is true even for the material that does not intrinsically present a risk of ratcheting [24]. The ratcheting effect is dependent on the TMF load as well as the changing of the temperature vs. time. The presented Chaboche model does not consider

the elimination of the ratcheting effect, so only the shape of the curve can be observed. To take into consideration the effect of ratcheting, several kinematic hardening parameters have to be defined requesting additional work and calculations. Figs.

7 to 15 present the result agreement of different numerical models to the observed values.

It can be seen from the figures that the results of the Prandtl operator approach fit the Chaboche model well. A better agreement of the Prandtl operator results is noticed with the Skelton model. The stress-strain behaviors of all models

Fig. 10. TMF cycle PKRZP

Fig. 12. TMF cycle PKRMP

Fig. 11. TMF cycle PXRZP

Fig. 13. TMF cycle PZRXP

Numerical Methods for TMF Cycle Modeling

491

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 485-494

are in good agreement with the observed values.

The shapes of the stress-strain curves for the TMF loops are predicted well, as well as the minimum and maximum value of stress.

The advantage of the Prandtl operator approach is the fact that it does not take into account the effect of ratcheting, which is a benefit as compared to the Chaboche model. For more precise results of the Chaboche model, the elimination of ratcheting should be included.

A comparison of the results is also applied for the complex cycle, Fig. 15. The shapes of the curves deviate from the observed values. It should be noted that experimental testing cannot perform the changing of strain vs. temperature as linearly as it can in numerical calculation.

5 CONCLUSIONS The results of the three models are introduced and compared to the experimental TMF cycles. The Prandtl operator approach is compared to the Skelton and Chaboche models.

The non-linear kinematic hardening model is used in the framework of time-independent plasticity to model the stabilized curves. The influence of temperature is taken into account in all models. Temperature influence is introduced as parameter dependence as well as changing temperature vs. time. The results of the Chaboche model indicate the ratcheting effect, which is

Fig. 14. TMF cycle PMRKP 492

difficult to eliminate. Its effect depends on the TMF cycle.

The classical Chaboche constitutive equations do not describe the ratcheting effect correctly, especially the ones observed when the mean-stress is significantly lower than the stress amplitude [24]. The non-linear kinematic model greatly over predicts ratcheting when its identification is performed for normal monotonic and reversed cyclic conditions. In [24] a set of modified kinematic rules for the elimination of ratcheting is introduced. The non-linear kinematic model with a threshold presents the best choice to describe both the normal cyclic condition and the ratcheting condition [24].

The comparison of the results of different numerical models shows good agreement with the observed values. The ratcheting effect can cause higher deviation from the predicted results and for this reason the Prandtl operator approach is preferred to the Chaboche model. For precise results of the Chaboche model the correction of the ratcheting effect should be considered.

6 REFERENCES [1] Constantinescu, A., Charkaluk, E., Lederer, G., Verger, L. (2004). A computational approach to thermomechanical fatigue.

International Journal of Fatigue, vol. 26, p.

805-818.

Fig. 15. TMF cycle – dashed line

Zaletelj, H. – Fajdiga, G. – Nagode, M.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 485-494

[2] Muhič, M., Tušek, J., Kosel, F., Klobčar, D. (2010). Analysis of Die Casting Tool Material.

Strojniški vestnik - Journal of Mechanical Engineering, vol. 56, no. 6, p. 351-356.

[3] Skelton, R.P., Webster, G.A. (1996). History effects in the cyclic stress-strain response of a polycrystalline and single crystal nickel-base superalloy. Mater. Sci. Engen., vol. A 216, p.

139-154.

[4] Charkaluk, E., Bignonnet, A., Constantinescu, A., Dang, V.K. (2002). Fatigue design of structures under thermomechanical loadings.

Fatigue Fract. Engng. Mater. Struct., vol. 25, p. 1199-1206.

[5] Skelton, R.P. (2004). Hysteresis, yield, and energy dissipation during thermo-mechanical fatigue of ferritic steel. International Journal of Fatigue, vol. 26, p. 253-264.

[6] Urevc, J., Koc, P., Štok, B. (2009). Numerical simulation of stress relieving of an austenite stainless steel. Strojniški vestnik – Journal of Mechanical Engineering, vol. 55, no. 10, p.

590-598.

[7] Nagode, M., Fajdiga, M. (2005).

Temperature-stress-strain trajectory modeling during thermo-mechanical fatigue. Fatigue Fract. Engng Mater. Struct., vol. 29, p. 175182.

[8] Chaboche, J.L. (1989). Constitutive equations for cyclic plasticity and cyclic viscoplasticity.

International Journal of Plasticity, vol. 5, p.

247-302.

[9] Chaboche, J.L. (2008). A review of some plasticity and viscoplasticity constitutive theories. International Journal of Plasticity, vol. 24, p. 1642-1693.

[10] Tong, J., Vermeulen, B. (2003). The description of cyclic plasticity and viscoplasticity of waspaloy using unified constitutive equations. International Journal of Fatigue, vol. 25, p. 413-420.

[11] Lemaitre, J., Chaboche, J.L. (1990).

Mechanics of solid materials. Cambridge University Press, Cambridge.

[12] Armstrong, P.J., Frederick, C.O. (1966). A mathematical representation of the multiaxial Bauschinger effect. CEGB Report RD/B/ N731.

[13] Nagode, M., Zingsheim, F. (2004). An online algorithm for temperature influenced fatigue-

life estimation: Strain-life approach. Int. J. Fatigue, vol. 26, p. 155-161.

[14] Nagode, M., Hack, M. (2004). An online algorithm for temperature influenced fatiguelife estimation: Stress-life approach. Int. J. Fatigue, vol. 26, p. 163-171.

[15] Conle, A., Oxland, T.R., Topper, T.H. (1988). Computer-based prediction of cyclic deformation and fatigue behaviour. Low Cycle Fatigue, Solomon, H.D., Halford, G.R., Kaisand, L.R., Leis, B.N. (Eds.). ASTM STP 942, p. 1218-1236.

[16] Brokate, M., Sprekels, J. (1996). Hysteresis and Phase Transition. Applied Material Science 121. Springer Verlag, New York.

[17] Nagode, M., Fajdiga, M. (2006). ThermoMechanical Modeling of Stochastic StressStrain States. Strojniški vestnik – Journal of Mechanical Engineering, vol. 52, no. 2, p.

74-84.

[18] Brokate, M., Rachinskii, D. (2005). On global stability of the scalar Chaboche models.

Nonlinear Anal.: Real World Appl., vol. 6, p.

67-82.

[19] Hack, M. (1998). Schädigungsbasierte Hysteresefilter. Shaker Verlag, Aachen.

[20] Nagode, M., Hack, M., Fajdiga, M. (2009).

Low cycle thermo-mechanical fatigue: damage operator approach. Fatigue Fract Engng. Mater. Struct., vol. 33, p. 149-160.

[21] Nagode, M., Längler, F., Hack, M. (2011) A time-dependant damage operator approach to thermo-mechanical fatigue of Ni-resist D-5S. International Journal of Fatigue, vol. 33, p.

692-699.

[22] Jovičić, G., Živković, M., Jovičić, N. (2009).

Numerical simulation of crack modeling using extended finite element method.

Strojniški vestnik – Journal of mechanical engineering, vol. 55, no. 9, p. 549-554.

[23] Skelton, R.P., Webster, G.A., de Mestral, B., Wang, C.Y. (2000). Modeling thermomechanical fatigue hysteresis loops from isothermal cyclic data. Thermo-Mechanical Fatigue Behaviour of Materials: Sehitoglu, H. Maier, H.J. (Eds.). ASTM STP 1371, p. 69-84.

[24] Chaboche, J.L. (1991).

On some modifications of kinematic hardening to improve the description of ratcheting effects.

Numerical Methods for TMF Cycle Modeling

493

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 485-494

International Journal of Plasticity, vol. 7, p.

661-678.

[25] Stamenković, D., Maksimović, K., NikolićStanojević, V., Maksimović, S., Stupar, S., Vasović, I. (2010). Fatigue life estimation of notched structural components. Strojniški

494

vestnik – Journal of mechanical engineering, vol. 56, no. 12, p. 846-852.

[26] Laug, P., Borouchaki, H. (2004). Curve linearization and discretization for meshing composite parametric surfaces. Commun.

Numer. Meth. Engng., vol. 20, p. 869-876.

Zaletelj, H. – Fajdiga, G. – Nagode, M.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 495-502 DOI:10.5545/sv-jme.2010.077

Paper received: 13.04.2010 Paper accepted: 12.01.2011

Criticality Analysis of the Elements of the Light Commercial Vehicle Steering Tie-Rod Joint Ćatić, D. – Jeremić, B. – Djordjević, Z. – Miloradović, N. Dobrivoje Ćatić* – Branislav Jeremić – Zorica Djordjević – Nenad Miloradović University of Kragujevac, Faculty of Mechanical Engineering, Serbia

The introduction of the paper gives the basic concepts of Failure Modes, Effects and Criticality Analysis - FMECA. Features of elements of mechanical systems regarding failure intensity demand a special approach of quantitative FMECA. The paper presents this approach, applied to the elements of mechanical systems and used for the design of a software package. Criticality analysis of failure modes of light commercial vehicle’s steering tie-rod joint elements was conducted based on the exploitation results and with the use of the previously mentioned method and program. In conclusion, the possibilities of application of the obtained results are presented. © 2011 Journal of Mechanical Engineering. All rights reserved.

Keywords: reliability, FMECA, steering system of a light commercial vehicle, tie-rod joint 0 INTRODUCTION According to the IEC standard [1], Failure Modes and Effects Analysis (FMEA) is a method for analysis of technical systems reliability.

FMEA may be defined as a systematic set of data intended for [2] and [3]: identification and assessment of potential product failures and their effects; determination of measures and activities for elimination or reduction of the possibility for failure to occur and documentation of the previous two procedures.

FMEA was developed for USA military purposes as a technique for the assessment of reliability through the determination of effects of different failure modes of technical systems.

This method dates from November 9th, 1949, as an official document in a form of an American military standard, denoted as MIL-P 1629 and named as “Procedure for conduction of analysis of modes, effects and acuteness of failures” [3].

Application of FMEA in automotive industry projects followed no sooner than in the second half of the 1980’s and it was related with the introduction of quality regulations Q-101 by American Ford Company. Different extensions of FMEA and customizations of the FMEA method for application in automotive industry were conducted within these activities.

FMEA is a procedure for the evaluation of reliability of a technical system that may be applied in all phases of its lifetime [4]. FMEA is generally an inductive method. It is based

on the consideration of all potential failures of constitutive parts of the system and effects they have on the system. Criticality Analysis (CA) is a procedure for the evaluation of criticality rating for all constitutive parts, where, by criticality, a relative measure of item failure modes influence on reliable and safe operation of the system is meant. Joint FMEA and CA analysis are called Failure modes, effects and criticality analysis FMECA. According to previous considerations, the application of FMECA based on exploitation data is founded on the assumption that the intensity of all failure modes of system elements is constant, which is valid for electronic systems [5] and [6]. This assumption considerably simplifies the procedure for criticality assessment. However, the application of this methodology in cases when failure intensity is a function of time may lead to distortion of the real picture of elements’ criticality. A proposal for the procedure of quantitative FMECA of machine system elements, originating from modification of the existing method, is given in book [7].

Element criticality analysis is extremely important for the systems with serial connection between elements (as in the vehicle steering system), where failure of any element leads to a failure of the entire system. The steering system is one of the vital parts of a motor vehicle complex mechanical system [8]. Together with the braking system and the tires, it has a crucial significance for safety of motor vehicles and people in traffic.

Thus, great attention is given to the demands

*Corr. Author’s Address: University of Kragujevac, Faculty of Mechanical Engineering, 6 Sestre Janjic Street, 34000 Kragujevac, Serbia, caticd@kg.ac.rs

495

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 495-502

that are set before the steering system regarding reliability.

1 QUANTITATIVE FMECA PROCEDURE FOR MACHINE SYSTEMS’ ELEMENTS

The calculation of Ci( k ) enables the isolation of the most important elements whose failures lead to certain categories of effects.

3. Determination of “absolute criticality” of element i according to:

Depending on the requirements and the possibilities for supplying the corresponding data, FMECA may be performed quantitatively and qualitatively [5]. Uniqueness of the machine system elements regarding failure intensity and objective impossibility to determine failure intensity for every possible element failure as the function of time, require special treatment during quantitative FMECA. Quantitative FMECA of machine system elements is defined in four steps: 1. Determination of criticality of failure mode j of element i is to be done by categories of failure effects k (k = 1, ..., 4), using:

Ci( k ) = ∑ Cij( k ) . (2) j

Ci = a1Ci(1) + a2Ci( 2 ) + a3Ci(3) + a4Ci( 4 ) , (3)

where αij is a relative rate (frequency measure) of failure mode j of element i (0 ≤ αij ≤ 1, ∑ α ij = 1 ),

where ak is “weight” of the kth category of effects (values may be determined using subjective evaluation of the effect’s “weight” for each case, (k ) from the interval between 0 and 1) and Ci is the ith element criticality for the kth category of effects.

System element criticality rate may be evaluated indirectly, by ranking of acquired values; there is no need for additional complicated analysis when safety and duration aspects are in scope.

4. Determination of criticality of the kth category of the system’s effects, by summation of criticalities of all elements failure modes for the specified effect category:

βij( k ) is conditional probability that failure mode

(k ) ij

C

=

α ij ⋅ βij( k ) ⋅ ti tsri

, (1)

j

j of element i will cause category k failure effect according to the adopted classification (values are taken from Table 1, according to recommendations from [5] and [6]), ti is operating time of element i and tsri is mean operating time until failure of element i occurs.

Table 1. Values of conditional probabilities Degree of occurrence of the kth failure effect category Certain event Probable event Most probably would not happen Practically does not happen

βij( k ) [-]

1 0.1 ... 1 0 ... 0.1 0

The calculated values of Cij( k ) are the initial basis for determining other quantitative properties of element criticality. They make it possible to rank element failure modes according to effects in order to evaluate the most critical system’s failure modes from the aspect of safety.

2. Determination of failure criticality of element i, which causes the kth category of failure effects: 496

Ck = ∑ ∑ Cij( k ) . (4) i

j

Calculated values of Ck are statistical indicators of the representation rating of the individual category of effects.

According to the suggested methodology, mean operating time until element failure occurs is one of the basic parameters for determining element criticality. Differences between machine elements in regard to reparability, the percentage of failure occurrence in the system’s total operating time, belonging to appropriate structural set, etc., require a special approach in the definition of mean operating time until failure; the calculated element criticalities according to this parameter will then be comparable [7].

Fortran 77 software package was developed for FMECA of machine system elements. This software consists of a routine for calculation of failure modes’ criticality and effect categories and their ranking, a subroutine for the transformation of elements of arbitrary array into descending series and a routine for forming of input files.

Ćatić, D. – Jeremić, B. – Djordjević, Z. – Miloradović, N.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 495-502

2 TIE-ROD JOINT AS A COMPONENT PART OF MOTOR VEHICLE’S STEERING SYSTEM The motor vehicle steering system is a mechanical system that has to meet high demands of reliability [8]. Importance of the motor vehicle steering system for human safety requires a detailed analysis of structural components in view of occurrence of their failure during exploitation.

The steering system of the wheeled vehicle contains two basic subsystems: the steering mechanism (group of steering wheel’s column) and the steering linkage (group of tie-rods and steering arms).

In addition to connecting the steering mechanism and the steered wheels, steering linkage has a very important task to provide proper kinematics of the wheel turn. This means that the steering linkage must be completely in accord with the suspension system of the steered wheels, so the motion of the wheels relative to the vehicle frame, does not influence the safety of steering. The previously given task is obtained by designing the linkage system in the form of trapeze.

The steering linkage of a light commercial vehicle’s steering system with one-piece cross tierod is presented in Fig. 1 [9].

clamps (3) and cross tube (8)), to the identical link at the right wheel spindle. The front axle, wheel arms (6) and the cross tube form the steering trapeze.

In order to achieve the basic function of the steering system - turning the wheels at a given angle, the linkage mechanisms must never be rigid constructions. Spherical joints are the most convenient links between the elements due to a complex relative motion of the elements of the steering linkage system during turn.

Spherical joints provide mobility in all three planes. There must not be any clearance in the tie-rod joint in order to preserve proper steering kinematics. Cancelation of clearance is achieved by designing the joint cup in two parts, so the upper moving part of the cup presses the ball pin sphere with the help of a spring.

The elements of the tie-rod joint used in the light truck steering systems are shown in Fig. 2.

Fig. 2. Tie-rod joint: 1 - Joint’s body, 2 - Ball pin, 3 - Cup, 4 - Spring, 5 - Cover, 6 - Sealing cap, 7 - Nut

Fig. 1. Steering linkage of light truck’s steering system Torque is transmitted from the output shaft of the steering gear, through the steering arm (1) and the drag arm (consisting of a drag arm joint (2), a clamp (3) and a drag link with joint (4)), to a drag link steering arm (5) on the front left wheel.

Drag link steering arm is connected to the wheel spindle by bolts. On the lower side of the left wheel spindle, there is an arm (6) that transfers the force through a cross-link (consisting of joints (7),

By analysis of modes, effects and criticality of failures of the steering system elements built in light commercial vehicles, it has been established that the tie-rod joints are the most critical elements from the aspect of reliability and safety [7].

3 CRITICALITY DEGREE ANALYSIS OF THE TIE-ROD JOINT’S ELEMENTS Due to indisputable importance that the tie-rod joint has in a reliable and safe operation of the motor vehicle steering system, quantitative criticality analysis of this structure was conducted starting at the level of elements. The basis for this analysis was a complex data structure in the form of a table data sheet. The procedure for the acquisition of data needed for quantitative FMECA

Criticality Analysis of the Elements of the Light Commercial Vehicle Steering Tie-Rod Joint

497

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 495-502

of the tie-rod joint of the light commercial vehicle steering system consisted of the following steps: 1. Structural system division, identification and coding of constitutive elements of the tie-rod joint was done within the structural division of the group of tie-rods and steering arms of light commercial vehicle steering system (Fig. 3); 2. Identification and recording of the most tie-rod joint elements failure modes were performed by forming the fault tree shown in Fig. 4; 3. Determination of relative share of individual element failure modes; 4. A category definition of final failure effects (all failure mode effects of the steering system tie-rod joint elements are classified in four categories: the first category is of the highest rank, and the fourth category is of the lowest rank);

5. Categorization of element failure modes according to effects and determination of conditional probabilities of final effects occurrence; 6. Determination of mean operating time until element failure occurs and 7. The calculation of total operating time of an element (equal for all tie-rod joint elements).

The fault tree of the tie-rod joint presented in Fig. 4 was obtained using symbols for events and logical gates [2]. A rectangle represents the peak or intermediary event in the fault tree, a triangle - primary basic event and a rhomb secondary basic event. Of all logical gates, only a symbol for logical gate OR was used, which produces an output event if one or more input events occur.

Systems and elements of a motor vehicle are loaded with variable loads in the course of time. The total number of load variation cycles

Fig. 3. Division layout of tie-rod and arms group of the light commercial vehicle’s steering system 498

Ćatić, D. – Jeremić, B. – Djordjević, Z. – Miloradović, N.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 495-502

Fig. 4. Independent fault tree of the light commercial vehicle’s steering tie-rod joint is proportional to the distance passed. Thus, time until failure of the most elements of motor vehicles occurs is measured in kilometres of distance passed. Tie-rod joints belong to the group of machine elements whose durability is limited by durability of the critical elements in the structure. Durability of the tie-rod joint is limited by durability of the sliding surfaces of the ball pin and the cup. The estimated mean operation time until failure occurs is 100,000 km [10].

Elements of the tie-rod joint belong to the group of machine elements in which failure modes occur only in a certain number of them. In that case, it is assumed that the rest of elements that did not fail have the mean time before the failure occurs equal to the mean time until failure of structure occurs. Eq. (5) was used for the calculation of mean time until failure of the tierod joint element occurs in kilometres of distance travelled [7].

ssr =

n    1  n ⋅  ∑ p j ssrj + 100 − ∑ p j  ⋅ ssrs  , (5) 100  j =1 j =1   

where ssrj is mean operation time of the tie-rod joint elements that have failed in the jth mode, n is the number of different element’s failure modes, pj is percentage rate of the jth failure mode in total number of structure elements failures, ssrs is mean time before structure failure occurs.

Operation time, ti, from Eq. (1), is usually expressed in hours. Since criticality is a nondimensional value, it is necessary to express the mean time until element failure occurs in the same units as ti. Transformation of mean time until failure of the tie-rod joint elements occur expressed in kilometres of the distance travelled into mean operation time in hours is conducted by the adoption of mean vehicle velocity of 60 km/h.

Table 2 contains data used to form input files for FMECA.FOR program. To calculate the elements absolute criticality, the following weighting factors of effect categories are adopted: a1 = 1; a2 = 0.5; a3 = 0.3; a4 = 0.2. Weighting factors were adopted by subjective assessment of the experts from the subject area.

Designation Q in Table 2 is for quantity or number of identical elements within the scope of discussed object of analysis.

By processing the acquired data and by using the computer program, output lists are gained for the degree of criticality of the tie rod joint elements failure modes without taking into account the effects (Table 3), the degree of criticality of the tie rod joint elements with taking into account the effects (Table 4), the degree of absolute criticality of the tie rod joint elements (Table 5) and the degree of criticality of final failure effects of the tie rod joint elements (Table 6).

Criticality Analysis of the Elements of the Light Commercial Vehicle Steering Tie-Rod Joint

499

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 495-502

Table 2. Basics of the FMECA procedure for elements of the tie rod joint for light commercial vehicles Element’s Elem. Q name code [-] Joint’s body Ball pin

Cup Spring Cover Sealing cap

Failure mode

62211 1 Deformation Tearing of joint’s body Thread damage 62212 1 Rupture of ball pin Thread damage Pluck of ball pin Wear 62213 1 Cup’s rupture Wear 62214 1 Spring rupture Spring force attenuation 62215 1 Deformation Cover falling off 62216 1 Total rupture Falling off Appearance of small rifts

Failure mode code N.02 N.06 N.07 N.06 N.07 N.29 N.77 N.06 N.77 N.06 N.12 N.02 N.29 N.06 N.29 N.37

Rel. Loss rate prob. αij [-] βij [-] 0.0001 0.3 0.0007 1.0 0.003 0.8 0,0007 1.0 0.0030 0.9 0.0002 1.0 0.9961 1.0 0.003 0.7 0.997 1.0 0.0008 1.0 0.001 0.8 0.0002 0.7 0.0004 1.0 0.002 1.0 0.007 0.9 0.004 0.2

Final effect k.3 k.1 k.2 k.1 k.2 k.1 k.3 k.3 k.3 k.3 k.3 k.3 k.3 k.4 k.4 k.4

ssri ×103 [km] 200 200 100 200 100 100 100 150 100 200 20 100 200 30 150 20

ssr ×103 [km] 100

tsr ×103 [h] 1.67

100

1.67

100

1.67

100

1.67

100

1.67

99.89

1.66

Table 3. Criticality of elements’ failure modes without taking into account the effects No. Code 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

62213 62212 62216 62212 62211 62213 62216 62216 62214 62214 62212 62211 62215 62212 62215 62211

Element’s name Cup Ball pin Sealing cap Ball pin Joint’s body Cup Sealing cap Sealing cap Spring Spring Ball pin Joint’s body Cover Ball pin Cover Joint’s body

Eff. name Wear k.3 Wear k.3 Falling off k.4 Thread damage k.2 Thread damage k.2 Cup’s rupture k.3 Total rupture k.4 Appear. of small rifts k.4 Spring rupture k.3 Spring force attenuation k.3 Rupture of ball pin k.1 Tearing of joint’s body k.1 Cover falling off k.3 Pluck of ball pin k.1 Deformation k.3 Deformation k.3 Failure mode

The total number of 16 different failure modes was discussed during estimation of criticality degree of the failure mode of tierod joint elements. Based on Table 3, the most critical failure mode of the tie-rod joint elements regardless the effects is wear of the cup and the ball pin. Its criticality is almost 160 times higher 500

α [-] 0.9970 0.9961 0.0070 0.0030 0.0030 0.0030 0.0020 0.0040 0.0008 0.0010 0.0007 0.0007 0.0004 0.0002 0.0002 0.0001

β [-] 1.0 1.0 0.9 0.9 0.8 0.7 1.0 0.2 1.0 0.8 1.0 1.0 1.0 1.0 0.7 0.3

tsr ti [h] [h] 1.67E+03 1 1.67E+03 1 1.66E+03 1 1.67E+03 1 1.67E+03 1 1.67E+03 1 1.66E+03 1 1.66E+03 1 1.67E+03 1 1.67E+03 1 1.67E+03 1 1.67E+03 1 1.67E+03 1 1.67E+03 1 1.67E+03 1 1.67E+03 1

Criticality Cij( k ) [-] 0.5974E-03 0.5965E-03 0.3795E-05 0.1617E-05 0.1437E-05 0.1258E-05 0.1205E-05 0.4819E-06 0.4799E-06 0.4799E-06 0.4192E-06 0.4192E-06 0.2400E-06 0.1198E-06 0.8398E-07 0.1796E-07

than criticality of the next failure mode in the descending order from the Table 3.

Based on criticality degree of the tie-rod joint elements with taking into account the effects (Table 4), it may be seen that, in severe categories of effects, only failure modes of the ball pin and the joint’s body occur, but with relatively small

Ćatić, D. – Jeremić, B. – Djordjević, Z. – Miloradović, N.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 495-502

values of criticality. For effect k.3, the cup and the pin-ball have criticality degree of order 10-3, while criticality degrees of other elements have orders of 10-6 and less. Generally, element failure modes with effect k.4, regardless of occurrence, are not authoritative for determining the most critical parts of the observed object.

Table 4. Criticality of elements with taking into account the effects No. Code Element’s name a) Criticality by effects k.1 1 62212 Ball pin 2 62211 Joint’s body b) Criticality by effects k.2 1 62212 Ball pin 2 62211 Joint’s body c) Criticality by effects k.3 1 62213 Cup 2 62212 Ball pin 3 62214 Spring 4 62215 Cover 5 62211 Joint’s body d) Criticality by effects k.4 1 62216 Sealing cap

Ci( k ) [-]

0.5389E-06 0.4192E-06 0.1617E-05 0.1437E-05 0.5986E-03 0.5965E-03 0.9598E-06 0.3239E-06 0.1796E-07 0.5482E-05

Table 5. Absolute criticality of elements No. 1 2 3 4 5 6

Code 62212 62213 62211 62216 62214 62215

Element’s name Ball pin Cup Joint’s body Sealing cap Spring Cover

ci [-] 0.1803E-03 0.1796E-03 0.1143E-05 0.1096E-05 0.2879E-06 0.9718E-07

Table 6. Criticality of final failure effects No. Final effect 1 k.3 2 k.4 3 k.2 4 k.1

Ck [-] 0.1196E-02 0.5482E-05 0.3054E-05 0.9581E-06

followed by the joint’s body, the sealing cap, the spring and the cup.

The other way of determining the most critical elements of the steering tie-rod joint is comparative analysis of Tables 6 and 4. In Table 6, there is obvious predominant occurrence of elements’ failure modes with category of effect equal to three. 99.21% of total sum of elements’ criticality are elements’ failure modes with third category of effects. Table 4 contains the elements ranked by criticality and by category of effects.

For effect k.3, the most critical elements are the cup and the ball pin of the tie-rod joint. Fig. 5 shows distribution of criticality degree of the tierod elements for effect category k.3.

Rel. crit. [%] 99.21 0.45 0.25 0.08

Table 5 contains elements of the tie-rod ranked by absolute criticality. The ball pin and the joint’s cup have the highest absolute criticality,

Fig. 5. Distribution of degree of criticality for elements of the steering system’s tie rod joint for category of effects k.3 As it may be seen in Fig. 5, failure of the tie-rod joint in the largest number of cases appears due to failure of the cup or the ball pin, while only 0.11% is due to other elements. A similar conclusion may be reached by analysis of absolute criticality of elements (Table 5).

An analysis of failure causes of steering system tie-rod joint elements may give directions for taking appropriate measures for minimization or total elimination of failure causes, or for failure effects reduction. In this way, by using other tools and techniques of the quality system, we get the basis for continuous improvement of the product quality and manufacturing process [11]. Increased clearance in tie-rod joints most frequently occurs in aging period, due to wear of sliding surfaces of the ball pin and the cup. An influence may be exerted on the increase of the mean operating time until the increased clearance in the joint occurs or on the reduction of tie-rod joint criticality by the increase of material or surface layer resistance to wear, by better lubrication and by better protection from the influence of the environment.

Criticality Analysis of the Elements of the Light Commercial Vehicle Steering Tie-Rod Joint

501

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 495-502

4 CONCLUSIONS A lot of the information necessary for taking measures in order to eliminate detected defects may be obtained by forming the databases of machine system element failures during exploitation and by their processing. Determining critical elements of mechanical systems that limit reliable and safe operation of the system and taking corrective measures in order to reduce the acuteness, present the fastest and the cheapest way to increase reliability and, accordingly, the product‘s quality.

The use value of FMECA results is in proportion to volume and credibility of the initial data. This points to the need that every company should form an information system for the acquisition and processing of data on errors, defects and failures of company‘s products. An organised system for data acquisition must provide continuous flow of data, their processing and availability. In order to get complex and credible database on modes, causes, effects and operation periods before failure of mechanical systems or elements occurs, data must be acquired in the design phase, development phase and in product‘s exploitation.

Parameters of durability (relative frequency of failure occurrence and mean operation time until failure) and parameters of safety (probability of failure effect occurrence - quantitative in nature and failure effect categories - qualitative in nature) have an influence on the criticality of machine system elements in exploitation. As far as quantitative indices are concerned, by analysis of intervals of possible parameter values, it may be concluded that parameters of durability have a dominant influence on the criticality of elements. This is another reason for the criticality of elements, defined according to the given methodology, to be the basis for determination of critical elements that have limiting effect on machine system’s reliability.

Generally, the machine system level of reliability can be increased by increasing the reliability of constitutive components or by introducing the parallel connections. Due to space limitations in motor vehicles steering systems, it is not possible to introduce parallel connections, so the only possibility to increase the 502

system’s reliability is through the increase of each component’s reliability.

5 REFERENCES [1] IEC 60812. (2006). Analysis techniques for system reliability - Procedure for failure mode and effects analysis (FMEA). 2nd ed., Geneva, International Electromechanical Commission.

[2] Lazor, D.J. (1995). Failure mode and effects analysis (FMEA) and fault tree analysis (FTA). Handbook of Reliability Engineering and Management, McGraw-Hill, New York, p. 6.1-6.46.

[3] Stamatis, D.H. (2003). Failure mode and effect analysis. 2nd ed., American Society for Quality, Milwaukee.

[4] Popović, V., Vasić, B., Petrović, M. (2010).

The possibility for FMEA method improvement and its implementation into bus life cycle. Strojniški vestnik - Journal of Mechanical Engineering, vol. 56, no. 3, p.

179-185.

[5] Military standard MIL-STD-1629A (1980).

Procedures for Performing a Failure Mode.

Effects and Criticality Analysis Department of Defense, Washington.

[6] Mtain Inc. Reliability, maintainability, logistics support, engineering services, reliability failure modes, effects and criticality analysis, from http://www.mtain.com/relia/relfmeca. htm, accessed on 2010-06-08.

[7] Ćatić, D. (2005). Development and application of reliability theory methods, Faculty of Mechanical Engineering, Kragujevac.

[8] Janićijević, N., Janković, D., Todorović, J. (1998). Design of motor vehicles. Faculty of Mechanical Engineering, Belgrade.

[9] Service manuals. (1991). Zastava Iveco trucks, Kragujevac.

[10] Ćatić, D., Krstić, B., Miloradović, D. (2009).

Determination of reliability of motor vehicle’s steering system tie-rod joint. Journal of the Balkan Tribological Association, vol. 15, no.

3, p. 309-322.

[11] Soković, M., Jovanović, J., Krivokapić, Z., Vujović, A. (2009). Basic quality tools in continuous improvement process. Strojniški vestnik - Journal of Mechanical Engineering, vol. 55, no. 5, p. 333-341.

Ćatić, D. – Jeremić, B. – Djordjević, Z. – Miloradović, N.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516 DOI:10.5545/sv-jme.2010.182

Paper received: 04.08.2010 Paper accepted: 05.01.2011

Mathematical Model of an Autoclave

Preglej, A. ‒ Karba, R. ‒ Steiner, I. ‒ Škrjanc, I. Aleksander Preglej1,* ‒ Rihard Karba2 ‒ Igor Steiner1 ‒ Igor Škrjanc2 1 INEA d.o.o., Slovenia 2 University of Ljubljana, Faculty of Electrical Engineering, Slovenia This paper presents the mathematical modelling of the following autoclave processes: heating, cooling and pressure changes. An autoclave is a pressure vessel of a cylindrical form where the composite semi-products are placed on a metal plate above electrical heaters and heated at selected temperatures and under a higher pressure. The purpose of the modelling is to build a mathematical model with which the behaviour of the processes can be simulated and the temperature and pressure control in the autoclave can be improved. Furthermore, using this mathematical model we intend to test advanced uni- and multi-variable control algorithms. The mathematical model is built on the basis of the heat-transfer and pressure-changing theories. While the pressure-changing process is not very complex, the heating and cooling processes involve complex phenomena of heat conduction and convection. In the mathematical model some simplifications were considered and so the heat-transfer correlations past flat plates were used. Most of the data are real and obtained from the autoclave manufacturer, but where not possible, the method of the model’s response fitting to the measured data with the criterion function of the sum of squared errors was used. In this way, to a great extent simulated similarly to the real process responses were obtained. It can be concluded that the obtained mathematical model is usable for the design of a variety of process-control applications. ©2011 Journal of Mechanical Engineering. All rights reserved.

Keywords: autoclave, mathematical model, heat transfer, convection, conduction, temperature, pressure 0 INTRODUCTION In the paper, the mathematical model of an autoclave development is presented. The control mechanism was already designed although it was not working well because the parameters of the controllers were not well tuned. The time constants of the process are very long and so the tests of the controller parameter settings on the real-time process take a long time. That is why it is reasonable to build a mathematical model with which the control of the process in the Matlab environment can be simulated, where the execution time is very short which enables quick and optimal settings of the controller parameters.

Furthermore, using the developed mathematical model we also intend to test advanced uni- and multi-variable control algorithms.

The basic principles of dynamic modelling are described in [1]. The main problem in the mathematical model of the autoclave is heat transfer, which has been extensively studied in many books, like [2] to [5], describing basic theories and theoretical models regarding various types of heat transfer. A more restricted theory of

forced convection is treated in [6], where heattransfer correlations for the flow in pipes, past flat plates, single cylinders, single spheres and for the flow in packed beds and tube bundles are described. Most of the data are real and obtained from the autoclave manufacturer. However, in cases where this was not possible, the method of the model’s response fitting to the measured data with the criterion function of the sum of squared errors [7] was used. The other mathematically treated process is pressure changing the basic principles of which can be found in [8]. Specific theories about dimensionless numbers like the Nusselt, Reynolds and Prandtl numbers can be found in [3] to [5] and [8] all various special heattransfer coefficients are listed.

Some papers proceed from basic heat transfer equations and deal with heat transfer coefficients, heat flow, conduction, convection, thermal resistance, Nusselt numbers, etc. in iceslurry flow [9], in the thermoregulatory responses of the foot [10] and during the gas quenching process [11]. While some papers like [12] to [14] studied similar heat-transfer processes inside autoclaves, their main focus was heat transfer

*Corr. Author’s Address: INEA d.o.o., Stegne 11, 1000 Ljubljana, Slovenia, aleksander.preglej@inea.si

503

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

and distribution within the composite material and determining the optimal temperature profile, otherwise known as the cure cycle. On the other hand, our focus was the process inside the autoclave, which can be more simply described as heating, cooling and changing the pressure.

Similar work with heating and cooling processes was reported in [15], where the convection coefficients were estimated experimentally.

The radiation heat transfer was considered separately, which is not neglected in the presented mathematical model, but considered in the Nusselt number coefficients.

The definition of the modelling purpose is highly significant [1] in the process of model development. In this case it is to gain more accurate data and improve the temperature and pressure control in the autoclave. As temperature and pressure are mutually closely connected by physical laws, we would like to consider them in a multi-variable manner which indicates interactions between them will have to be taken into account.

However, at the moment temperature and pressure control are treated as two independent control loops.

The temperature is controlled continuously with two predictive functional controllers (PFC) and pulse-width modulation of heating with the electrical heaters and cooling with the water cooler and the analog valve.

The pressure is discretely controlled with pressure increasing through the on-off valve and pressure decreasing through two on-off valves of different sizes.

The paper is organized in the following way: in Section 1 the technological data of the autoclave are described. In Section 2 and 3 the modelling of the autoclave heating and cooling is presented and in Section 4 the modelling of the pressure changes is given. The results of the modelling are collected in Section 5, while the model validation is depicted in Section 6.

The optimization experiment is presented in the Appendix A. 1 AUTOCLAVE TECHNOLOGICAL DATA An autoclave is a pressure vessel of a cylindrical form shown in Fig. 1, where composite 504

semi-products are placed on a metal plate above electrical heaters and heated at selected temperatures and under a higher pressure. These semi-products like boat moulds, kiosks, plane and automobile parts, children’s playthings, flower pots, etc. are composed of composite materials like resin, metal, ceramics, glass, carbon, etc. which under the applied conditions become harder and therefore of a higher quality. In the autoclave the working pressure is up to 7 bar and the working temperature is up to 180 °C. The autoclave is made of stainless steel and isolated with mineral wool and an isolating aluminium coat. The length of the cylindrical coat is 2850 mm, where the useful length is only 2600 mm, the inner diameter is 1500 mm and the thickness of the metal coat is 100 mm. The volume of the autoclave is 5600 litres.

Fig. 1. The treated autoclave The autoclave is heated with electrical heaters of power up to 110 kW (the temperature gradient is up to 3 °C/min) and cooled with an inner cooler of power up to 73 kW (the temperature gradient is up to -2 °C/min), where the cooling medium is water with a temperature of 15 °C. The pressure in the autoclave is increased by a compressed air flow up to 100 kg/h and decreased by the air flow up to 100 kg/h.

A centrifugal ventilating fan on the back of the autoclave with a water-cooled mechanical axle washer and an electromotor drive outside the autoclave of power up to 11 kW provide the air circulation.

The cooperative energy sources are compressed air of 7 bar pressure and a 300 kg/h

Preglej, A. ‒ Karba, R. ‒ Steiner, I. ‒ Škrjanc, I.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

flow, cooling water of pressure from 3 to 6 bar and 5 m3/h flow, and an electrical current at a 380 V voltage and 115 kW of attachable power.

2 MODELLING OF THE AUTOCLAVE HEATING 2.1 Description of the Process The process can be presented as cylindrical vessel seen in Fig. 2. The wall is composed of the inner metal coat, the isolation with mineral wool and the exterior metal coat. On the back of the autoclave, where the ventilating fan is mounted, there is just a layer of the exterior metal coat without isolating material as seen in Fig. 2. The cooler, the ventilating fan and all the other metal parts inside the vessel can be approximated as one vertical metal block. The composite material, which is inserted into the autoclave, can be represented as a horizontal block.

•

Wen1 is the heat flow from the metal coat to the environment over the isolation and • Wen2 is the heat flow from the metal coat to the environment over the non-isolated metal.

Joining the heat flows W3 and Wen1 and eliminating the coat temperature ϑ3 was also proposed. However, it did not work well, because in that way quite a lot of the mass of the metal coat was not taken into account.

2.2 The Mathematical Model The heat flows [2] are as follows: W1 = Qel , (1)

ϑ1 − ϑ2

W2 = K ame S ame (ϑ1 − ϑ2 ) =

W3 = K ac S ac (ϑ1 − ϑ3 ) =

ϑ1 − ϑ3

W4 = K am S am (ϑ1 − ϑ4 ) =

ϑ1 − ϑ4

Wen1 = K ce S ce (ϑ1 − ϑen ) =

ϑ3 − ϑen

Wen 2 = K nim S nim (ϑ1 − ϑen ) =

, (2)

Rame Rac

, (3)

Ram Rce

, (4) , (5)

ϑ1 − ϑen Rnim

. (6)

Energy balance Eqs. [2] are the following: Fig. 2. Scheme for the heating process modelling In Fig. 2 the following notations are presented: • ϑ1 [K] is the temperature of the air in the autoclave, • ϑ2 is the temperature of the metal, • ϑ3 is the temperature of the metal coat, • ϑ4 is the temperature of the composite material, • ϑen is the temperature of the environment, • W1 [W] is the heat flow from the heaters to the air in the autoclave, • W2 is the heat flow from the air in the autoclave to the metal, • W3 is the heat flow from the air in the autoclave to the metal coat, • W4 is the heat flow from the air in the autoclave to the composite material,

ϑ − ϑ2 ϑ1 − ϑ3  − − W1 − 1  ma ca  Rame Rac (7) ϑ1 − ϑ4 ϑ1 − ϑen   − −  = ϑ1 , Ram Rnim 

1

 ϑ1 − ϑ2     = ϑ2 , (8) mme cme  Rame  1

 ϑ1 − ϑ3 ϑ3 − ϑen −  mc cc  Rac Rce 1

   = ϑ3 , (9) 

 ϑ1 − ϑ4   (10)   = ϑ4 . mm cm  Ram  1

In Eqs. (1) to (10) (some notations are presented in Fig. 2) the following notations are included:

Mathematical Model of an Autoclave

505

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

• • • • • • • • • • • • • • • •

• • • • • • 506

Qel [W] is the electrical heaters power, Kame [W/(m2K)] is the heat-transfer coefficient between the air in the autoclave and the metal, Same [m2] is the area between the air in the autoclave and the metal, Rame [K/W] is the resistance of the thermal conductivity between the air in the autoclave and the metal, Kac is the heat-transfer coefficient between the air in the autoclave and the metal coat, Sac is the area of the thermal conductivity between the air in the autoclave and the metal coat, Rac is the resistance of the thermal conductivity between the air in the autoclave and the metal coat, Kam is the heat-transfer coefficient between the air in the autoclave and the material, Sam is the area of the thermal conductivity between the air in the autoclave and the material, Ram is the resistance of the thermal conductivity between the air in the autoclave and the material, Kce is the heat-transfer coefficient between the metal coat and the environment, Sce is the area of the thermal conductivity between the metal coat and the environment, Rce is the resistance of the thermal conductivity between the metal coat and the environment, Knim is the heat-transfer coefficient between the air in the autoclave and the environment over the non-isolated metal, Snim is the area of the thermal conductivity between the air in the autoclave and the environment over the non-isolated metal, Rnim is the resistance of the thermal conductivity between the air in the autoclave and the environment over the non-isolated metal, ma [kg] is the mass of the air in the autoclave, ca [J/(kgK)] is the specific heat capacity of the air in the autoclave, mme is the mass of the metal, cme is the specific heat capacity of the metal, mc is the mass of the metal coat, cc is the specific heat capacity of the metal coat,

• •

mm is the material mass and cm is the specific heat capacity of the material.

2.3 Calculation of the Parameters In addition to the influence of the conductance on the heat transfer, forced convection [3] is also significant. The air in the autoclave namely circulates as shown in Fig. 3.

Fig. 3. Scheme for the air circulation modelling In the simplified case it can be presumed that the air flow in every part of the autoclave is the consequence of forced convection (in Figs. 4 to 6 marked with straight lines). Also the conductance through the layer of metal and material (in Figs.

4 to 6 marked with wavy line and the letter l) is assumed.

The cylindrical metal coat can be represented as flat plates, as seen in Fig. 4. The part without the isolation also has different convection, but the conductance is the same in the whole metal coat.

Fig. 4. Simplified air circulation

Preglej, A. ‒ Karba, R. ‒ Steiner, I. ‒ Škrjanc, I.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

The air flow next to the metal is considered only on the left-hand side and so the upper, lower and right-hand air flows are neglected, but the conductance is present in the whole metal, as seen in Fig. 5.

Fig. 5. Simplified air flow next to the metal The air flow next to the material is considered only on the upper side, because the material is placed on a straight basis. The side air flows are neglected, but conductance is again present in the whole material as seen in Fig. 6. In Figs. 4 to 6 a, b, d, n and j are dimensions needed in the below equations.

Fig. 6. Simplified air flow next to the material We have to calculate the heat transfer coefficients [4], which are the inverse values of sums of the conductance and the convection inverses need to be calculated:

K ame = 1

K ac = 1

( lme / λme + 1 / hame )

K ce = 1

, (12)

( lc / λc + lw / λw + 1 / hce )

K am = 1

K nim = 1

( lc / λc + 1 / hac )

, (11)

( lm / λm + 1 / ham )

, (13)

, (14)

( lnim / λnim + 1 / hanim + 1 / hnime )

. (15)

In Eqs. (11) to (15) the following notations are included: • lme [m] is the metal thickness,

λme [W/(mK)] is the thermal conductivity of the metal, • hame [W/(m2K)] is the convection coefficient between the air in the autoclave and the metal, • lc is the metal coat thickness, • λc is the metal coat thermal conductivity, • hac is the convection coefficient between the air in the autoclave and the metal coat, • lw is the mineral wool thickness, • λw is the mineral wool thermal conductivity, • hce is the convection coefficient between the metal coat and the environment, • lm is the material thickness, • λm is the material thermal conductivity, • ham is the convection coefficient between the air in the autoclave and the material, • lnim is the non-isolated metal thickness, • λnim is the non-isolated metal thermal conductivity, • hanim is the convection coefficient between the air in the autoclave and the non-isolated metal and • hnime is the convection coefficient between the non-isolated metal and the environment.

Furthermore, we must calculate the convection coefficients must be calculated : •

hame = λa Nume Lme , (16)

hac = λa Nuac Lci , (17)

hce = λa Nuce Lce , (18)

ham = λa Nuam Lm , (19)

hanim = λa Nuanim Lnim , (20)

hnime = λa Nunime Lnim . (21)

In Eqs. (16) to (21) the following notations are included: • λa is the air thermal conductivity, • Lme [m] is the length of the characteristic metal, • Nume is the Nusselt number for the convection between the air in the autoclave and the metal, • Lci is the characteristic inner metal coat length, • Nuac is the Nusselt number for the convection between the air in the autoclave and the inner metal coat, • Lce is the characteristic exterior metal coat length,

Mathematical Model of an Autoclave

507

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

•

Nuce is the Nusselt number for the convection between the exterior metal coat and the environment, • Lm is the length of the characteristic material, • Nuam is the Nusselt number for the convection between the air in the autoclave and the material, • Lnim is the length of the characteristic nonisolated metal, • Nuanim is the Nusselt number for the convection between the air in the autoclave and the non-isolated metal and • Nunime is the Nusselt number for the convection between the non-isolated metal and the environment.

The Nusselt numbers are calculated as follows: z

y

Nume

 ρ ⋅ Lme ⋅ u   ca ⋅ µ  = x ⋅    , (22) µ    λa  y

 ρ ⋅ Lci ⋅ u   ca ⋅ µ 

z

   , (23)   λa  w  g ⋅ (ϑce − ϑen ) ⋅ Lce 3  Nuce = q ⋅   , (24) v ⋅ ϑa.abs   z y  ρ ⋅ Lm ⋅ u   ca ⋅ µ  Nuam = x ⋅     , (25)  µ   λa  Nuac = x ⋅ 



µ

y

z

 ρ ⋅ Lnim ⋅ u   ca ⋅ µ  Nuanim = x ⋅     , (26) µ    λa  w  g ⋅ (ϑnim − ϑen ) ⋅ Lnim 3  Nunime = q ⋅   . (27) v ⋅ ϑa.abs   In Eqs. (22) to (27) coefficients x, y, z, q and w are defined experimentally and so they are unique for every mathematical model [5]. For the presumed theory of flat plates the recommended values for forced convection are x = 0.664, y = 0.5 and z = 0.333, and for natural convection q = 0.478 and w = 0.25 [6]. For the coefficients y, z and w we used recommended values, while for the coefficients x and q the recommended values were not usable. Therefore, the model’s response fitting to the measured data described in Eq. (34) was used to obtain x = 431.6 and q = 310.7. It can be presumed that these values also consider the radiation heat transfer.

508

In these Eqs. also the following notations are included: • ρ [kg/m3] is the air density, • u [m/s] is the velocity of the air circulation in the autoclave, • µ [kg/(ms)] is the air viscosity, • g [m/s2] is the gravitational acceleration, • ϑce is the exterior metal ct’s temperature, which is simplified ϑ3, • v is the velocity of the air circulation in the environment, • ϑa.abs is the absolute air temperature, which is the same as ϑen, and • ϑnim is the non-isolated metal temperature, which is simplified ϑ3.

The air density in the Nusselt numbers is changing as follows:

ρ= p

( R ϑ ) . (28) g

1

In Eq. (28) Rg is the gas constant [J/(kgK)].

The air density depends on the pressure p [kg/ (ms2)] and the temperature ϑ1 in the autoclave, so the Nusselt numbers are constantly changing.

Finally, the characteristic lengths must be assumed and calculated for all cases where we use the length of a flat plate. The material data is not yet defined, because no material was placed in the autoclave (in the below equations marked with not def.). Therefore, the data were set in a way to avoid the problems with zero division. Material surface was set to zero so that multiplication returned zero.

Lme= n= 1, (29)

Lci = 2 ( a + b ) − d =

= 2 ( 2.85 + 1.5 ) − 0.5 = 8.2,

Lce = 2 ( e + f ) − d =

(30)

= 2 ( 3.09 + 1.74 ) − 0.5 = 9.16,

Lm=

(31)

j= not def . (32)

Lnim= d= 0.5. (33)

In Eqs. (29) to (33) the meaning of the coefficients a, b, d, n and j is evident from Figs.

4 to 6. The values of the parameters Lme (n) and Lnim (d) were assumed and other values were calculated. The coefficients e and f are lengths a and b with the added mineral wool and exterior metal coat thickness.

Preglej, A. ‒ Karba, R. ‒ Steiner, I. ‒ Škrjanc, I.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

Some parameters were optimized with the method of the model response fitting to the measured data with the criterion function of the sum of squared errors [7], described symbolically as follows:

θ p.set = argmin

(∑ ( y

process

− ymodel )

2

) . (34)

In Eq. (34) the following notations are included: • θp.set is the set of parameters, • yprocess is the real process output and • ymodel is the mathematical model output.

The experiments with the mentioned optimization method are depicted in greater detail in the Appendix A.

obtained from the various theories mentioned and from the physical equations of the process.

By the real process step response the temperature rises after 30 s and this dead time was also considered in the simulations.

3 MODELLING OF THE AUTOCLAVE COOLING 3.1 Description of the Process The cooling process is very similar to the heating one. The only difference is the source, which is represented here by the cooler with its own heat flow as seen in Fig. 7. All the other heat flows are the same as presented in Fig. 2.

2.4 Defined or Estimated Data In the real process of autoclave heating the pressure was approximately 3.23 bar (p = 323000 kg/(ms2)), the power of the heaters was at 3% of the maximum value (W1 = 3300 W), the environment temperature was at room temperature (ϑen = 23 °C) and the initial air temperature in the autoclave was ϑin = 61.3 °C. Other data values are: • The specific heat capacities: ca = 725, cme = 510, cc = 510, cm = not def.

• The gas constant: Rg = 287.05.

• The autoclave volume: V = 5.6 m3.

• The thicknesses: lem = 0.5, lc = 0.01, lw = 0.1, lm = not def., lnim = 0.01.

• The surfaces: Same = 3, Sac = 17, Sam = not def., Sce = 20, Snim = 0.75.

• The masses: ma = ρa⋅V, mme = 1208, mc = 1198, mm = not def.

• The thermal conductivities [8]: λa = 0.025, λw = 0.04, λme = 16.3, λc = 16.3, λm = not def.

• The air circulation velocity by forced convection is u = 3 and by natural convection is v = 0.3.

• The air viscosity: µ = 2.484⋅10-5.

• The acceleration due to gravity: g = 9.81.

The metal and coat masses were first estimated at 1500 kg and then optimized with the above mentioned method of the model’s response fitting to the measured data. Additionally, the values of the parameters lme, lw, Same, u and v were first assumed and finally optimized with the above mentioned method. Other parameters were

Fig. 7. Scheme for the cooling process modelling In Fig. 7 the following notations are presented: • ϑ1 is the temperature of the air in the autoclave, • ϑcw is the temperature of the cooling water, • Φcwi [m3/s] is the volume flow of the cooling water, • Wcw is the heat flow between the cooler and the air in the autoclave, • ϑin is the entry temperature of the cooling water and • ϑout is the exit temperature of the cooling water.

3.2 The Mathematical Model The volume flow of the cooling water Φcwi is controlled by the entry valve and therefore, the cooler’s heat flow [2] is as follows:

Mathematical Model of an Autoclave

509

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

Wcw = K cwa S cwa (ϑ1 + ϑcw ) =

ϑ1 + =

ϑ1 + ϑcw Rcwa

=

(35)

Wcw cw ⋅ Φcwi Rcwa

=

ϑ1 Rcwa

+

Wcw Rcwa ⋅ cw ⋅ Φcwi

.

In Eq. (35) the temperatures are summed because the cooler’s heat flow is given as a negative value.

The energy balance equation [2] is the following:

 ϑ1 − ϑ2 ϑ1 − ϑ3 ϑ1 − ϑ4 − − − − ma ca  Rame Rac Ram

(36) ϑ1 − ϑen Wcw ϑ1   − − −  = ϑ1.

Rnim Rcwa Rcwa ⋅ cw ⋅ Φcwi  1

In Eqs. (35) and (36) (some meanings have already been presented in Figs. 2 and 7 and in Eqs. (7) to (10)) the following notations are included: • Kcwa is the heat-transfer coefficient between the cooling water and the air in the autoclave, • Scwa is the area of the thermal conductivity between the cooling water and the air in the autoclave, • Rcwa is the resistance of the thermal conductivity between the cooling water and the air in the autoclave and • cw is the water’s specific heat capacity.

3.3 Calculation of the Parameters The heat-transfer coefficient [4] can be calculated similarly as presented for the heating process:

K cwa = 1

( lw / λw + 1 / hcwm + 1 / hma ) . (37)

In Eq. (37) the following notations are included: • lw is the thickness of the cooler filled with the cooling water, • λw is the thermal conductivity of the water, • hcwm is the convection coefficient between the cooling water and the metal and • hma is the convection coefficient between the metal and the air in the autoclave.

Below let us calculate the convection coefficients: 510

hcwm = λw Nucwm Lme , (38)

hma = λa Numa Lme . (39)

In Eqs. (38) and (39) (some meanings have already been presented in Eqs. (16) to (21)) the following notations are included: • Lme is the characteristic length of the metal length, • Nucwm is the Nusselt number for the convection between the cooling water and the metal and • Numa is the Nusselt number for the convection between the metal and the air in the autoclave.

Finally, the Nusselt numbers must be calculated as follows: y

Nucwm

y

z

 ρ ⋅ L ⋅u   c ⋅ µ  = x ⋅  w me w   w w  , (40) µw    λw 

Numa

z

 ρ ⋅ L ⋅u   c ⋅ µ  = x ⋅  a me   a  . (41) µ    λa 

In Eqs. (40) and (41) (some meanings have already been presented in Eqs. (7) to (10), (22) to (27) and (36)) the following notations are included: • ρw is the density of the water, • uw is the velocity of the water motion and • µw is the viscosity of the water.

However, for the cooling process there is much less disposable data than for the heating one. For the given modelling purposes it is not significant how the heat-transfer coefficient between the cooling water and the air in the autoclave is calculated. We decided to use the method of model’s response fitting to the measured data described in (34).

3.4 Defined or Estimated Data In the real process of autoclave cooling the pressure was approximately 1.3 bar, the cooler’s heat flow was at 20% of the maximum value (Wcw = -14600 W), the environment temperature was at room temperature (ϑen = 23 °C) and the initial air temperature in the autoclave was ϑin = 135.1 °C. Other data values are: • The water’s specific heat capacity: cw = 4181.3.

Preglej, A. ‒ Karba, R. ‒ Steiner, I. ‒ Škrjanc, I.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

•

The volume flow of the cooling water: estimated as Φcwi = 0.011.

• The cooler surface: estimated as Scwa = 0.31.

The heat-transfer coefficient between the cooling water and the air in the autoclave was estimated using the already-mentioned method: Kcwa = 1905.

Because in Kcwa also some amount of the cooler’s metal was taken into account, which in the heating process was considered with all the other metal in the autoclave, the values of the surface Same and the metal thickness lme, which were defined by the heating process, must be correspondingly reduced. The new values were estimated as Same = 0.312 and lme = 0.002.

By the real process step response the temperature falls after 30 s and this dead time was also considered in simulations.

4 MODELLING OF THE PRESSURE CHANGES

• • • • • • • • • •

p is the pressure in the autoclave, ϑ1 is the temperature in the autoclave, ρ is the air density, pin is the entry pressure, Sin is the entry valve cross-section area, ϕin [kg/s] is the entry mass flow of air, pout is the exit pressure, Sout is the exit valve cross-section area, ϕmout is the exit mass flow of air and V is the autoclave volume.

4.2 The Mathematical Model The mass balance equation is described [8] as follows:

φmin − φmout = V ρ . (42) The air density described in Eq. (28) is pressure and temperature dependent, and so its derivative is described as follows:

ρ =

4.1 Description of the Process The pressure in the autoclave is increased with compressed air through the entry on-off valve and decreased by letting the air out through two exit on-off valves of different sizes. Valves are modelled as analog ones, where both exit valves are considered as a single valve with a larger dimension. Fig. 8 shows the pressure changing situation.

∂ρ dp

∂ρ dϑ1

. (43) ∂p dt ∂ϑ1 dt Furthermore, the partial derivatives are: ∂ρ ∂p = 1 Rgϑ1 , (44)

∂ρ ∂ϑ1

form:

=

+

2

− p Rgϑ1 . (45)

Then the mass flows can be given in the

φmin = K in Sin pin , (46)

φmout = K out Sout

pout ( p − pout ) . (47)

In Eqs. (46) and (47) the following notations are included:  Kin [s/m] is the entry valve constant and  Kout is the exit valve constant.

The final Eq. is given in the form: p =

Fig. 8. Scheme for the pressure changing modelling In Fig. 8 the following notations are presented:

Rgϑ1 V

( Kin Sin pin −

)

− K out Sout pout ( p − pout ) + K nl

p

ϑ1

ϑ1.

(48)

In Eq. (48) Knl is the nonlinearity, which considers interactions between the temperature and the pressure. Knl was estimated with alreadymentioned method of model’s response fitting to the measured data.

Mathematical Model of an Autoclave

511

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

With the increasing pressure the temperature in the autoclave increased by approximately 5 °C, from an initial 51 to 56 °C, and with the decreasing pressure the temperature in the autoclave dropped by approximately 3.5 °C, from an initial 47 to 43.5 °C. The other data values are: • The nonlinearity: estimated as Knl = 1.97.

• The compressor entry pressure is pin = 7 bar, while the exit pressure is almost a vacuum pout = 0.015 bar.

• The valve cross-section areas: Sin = π(0.025 m/2)2 = 4.91⋅10-4 and Sout = π((0.032 m + 0.015 m)/2)2 = 17⋅10-4.

• The valve constants: estimated as Kin = 1.06⋅10-3 and Kout = 50.5⋅10-3.

By the real process step response the pressure rises or falls after 1 s and this dead time was also considered in the simulations.

5 RESULTS AND DISCUSSION 5.1 Comparison of the Heating Responses

10, and in the middle, but responses do not differ more than 2 °C. 63 62.8

temperature [°C]

4.3 Defined or Estimated Data

62.6 62.4 62.2 62 61.8 61.6 61.4 0

50

100

150

time [s]

200

250

300

Fig. 10. A more detailed comparison of the heating responses: real process (solid line) and mathematical model (dashed line) 5.2 Comparison of the Cooling Responses Figs. 11 and 12 represent a comparison of the mathematical model and the real process of the autoclave cooling responses at the given conditions.

140 130 120

temperature [°C]

Figs. 9 and 10 represent a comparison of the mathematical model and the real process autoclave heating responses at the given conditions.

110 100 90 80 70 60 50

180

40

0

temperature [°C]

160 140

time [s]

10000

15000

Fig. 11. Cooling responses comparison: real process (solid line) and mathematical model (dashed line)

120 100 80 60 0

0.5

1

1.5

2

2.5

time [s]

3

3.5

4

4.5

5

x 10

5

Fig. 9. Heating responses comparison: real process (solid line) and mathematical model (dashed line) Both responses fit very well, as seen in Fig. 9. The real process response has more nonlinearities, which are not seen in the mathematical model response because of unmodelled dynamics. These differences are the most noticeable at the beginning, as seen in Fig. 512

5000

Both responses again fit well as seen in Fig. 11. The fitting is slightly worse than for the heating, which could be ascribed to the lack of real data of the autoclave cooling system, and for this reason used method of model’s response fitting to the measured data. Because the cooling response has a similar course as the heating one, the differences the most noticeable at the beginning, as seen in Fig. 12, but responses do not differ for more than 5 °C. The error in the steady state is less than 0.53 °C.

Preglej, A. ‒ Karba, R. ‒ Steiner, I. ‒ Škrjanc, I.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

13 and 14. Smaller deviations, which can be again ascribed to unmodelled dynamics, can be seen.

135

temperature [°C]

130

6 MODEL VALIDATION

125 120

6.1 The Heating Model Validation

110 250

300

350

400

450

500

time [s]

550

600

650

700

750

Fig. 12. A more detailed comparison of the cooling responses: real process (solid line) and mathematical model (dashed line) 5.3 Comparison of the Pressure Changing Responses Figs. 13 and 14 show a comparison of the mathematical model and the real process autoclave pressure changing responses at the given conditions.

3

pressure [bar]

140 120 100 80 60

20 0

2

0.5

1

1.5

2

time [s]

2.5

3

3.5 5 x 10

Fig. 15. Validation of the mathematical model of the heating under different conditions: real process (solid line) and mathematical model (dashed line)

1.5 1 0.5

50

100

150

time [s]

200

250

300

Fig. 13. Pressure increasing responses comparison: real process (solid line) and mathematical model (dashed line) 3.5 3

pressure [bar]

160

40

2.5

0 0

The mathematical model of the heating was validated under different conditions as presented in Fig. 15. The pressure was approximately 1 bar, the power of the heaters was at 2% of the maximum value (W1 = 2200 W), the environment temperature was at room temperature (ϑen = 23 °C) and the initial air temperature in the autoclave was ϑin = 24.5 °C.

temperature [°C]

115

2.5

In Fig. 15 it can be seen that both responses have very similar courses, however the fitting is worse than in Fig. 9, which can be the consequence of some simplifications with different working conditions, the above mentioned unmodelled dynamics and interactions between the temperature and the pressure. The steady state of both responses differs by approximately 3.5 °C.

2

6.2 The Cooling Model Validation

1.5 1 0.5 0 0

50

time [s]

100

150

Fig. 14. Pressure decreasing responses comparison: real process (solid line) and mathematical model (dashed line) The responses of the increasing and decreasing pressure fit very well, as seen in Fig.

The mathematical model of the cooling was also validated under different conditions as presented in Fig. 16. The pressure was approximately 3 bar, the cooler’s heat flow was at 18% of the maximum value (Wcw = -13140 W), the environment temperature was at room temperature (ϑen = 23 °C) and the initial air temperature in the autoclave was ϑin = 151 °C.

Mathematical Model of an Autoclave

513

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

Figs. 17 and 18 show that both responses fit relatively well. However, the fitting is (especially for the pressure increasing) worse than in Figs.

13 and 14, what can again be ascribed to some simplifications with different working conditions, to unmodelled dynamics and to interactions between the temperature and the pressure.

160

120 100 80 60 40 0

5000

10000

time [s]

15000

Fig. 16. Validation of the mathematical model of the cooling under different conditions: real process (solid line) and mathematical model (dashed line) In Fig. 16 it can be seen that both responses again have a similar course, but fitting is logically worse than in Fig. 11, which can also be contributed to some simplifications with different working conditions, to unmodelled dynamics and to interactions between the temperature and the pressure. The steady state of both responses differs by approximately 5 °C. 6.3 The Pressure Changing Model Validation The mathematical model of the increasing and decreasing pressure was again validated under different conditions as presented in Figs. 17 and 18. With the increasing pressure, the temperature in the autoclave increased by approximately 4 °C, from an initial 48 to 52 °C, and with the decreasing pressure the temperature in the autoclave dropped by approximately 6.5 °C, from an initial 53 to 46.5 °C. 6

pressure [bar]

5 4 3 2 1 0 0

50

100

150

200

250

time [s]

300

350

400

450

500

Fig. 17. Validation of the mathematical model of pressure increasing under different conditions: real process (solid line) and mathematical model (dashed line) 514

5

4

pressure [bar]

temperature [°C]

140

3

2

1

0 0

20

40

60

80

time [s]

100

120

140

160

Fig. 18. Validation of the mathematical model of pressure decreasing under different conditions: real process (solid line) and mathematical model (dashed line) 7 CONCLUSIONS For the needs of mathematical modelling of the autoclave processes (heating, cooling and pressure changing) first all the responses were recorded, then the detailed mathematical models with physical descriptions were developed and finally simulated. Considering some simplifications and using curve fitting procedure very similar simulated and real process responses were obtained, what means that the designed model is usable for the design of a variety of process control, including advanced uni- and multi-variable control algorithms.

In the future also interactions between the temperature and the pressure will have to be taken into account to show whether the autoclave should be controlled as two independent uni-variable processes or as one multi-variable process.

In spite of the fact that the developed model works well for the given conditions, it will have to be additionally validated also for the other real operating conditions. Due to the very different regimes of operation of multifaceted modelling including fuzzy approaches can be expected.

Preglej, A. ‒ Karba, R. ‒ Steiner, I. ‒ Škrjanc, I.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

8 ACKNOWLEDGEMENTS The operation part was financed by the European Union, European Social Fund.

Operation implemented in the framework of the Operational Programme for Human Resources Development for the Period 2007-2013, Priority axis 1: Promoting entrepreneurship and adaptability, Main type of activity 1.1.: Experts and researchers for competitive enterprises.

APPENDIX A: OPTIMIZATION EXPERIMENT The fitting of the parameters using Eq. (34) is very critical to the success of the model. We used this method for several parameters, but not for all at the same time, because using a lot of the parameters results in a lot of the model variations.

For the useful results of the optimization also the initial values of the parameters are very important. We have reasonably chosen a few of the parameters at a time, then logically set their assumed initial values and started the optimization method.

It took a lot of time, effort and performed optimization experiments to obtain the right values of the parameters that gave satisfying mathematical model responses. We used environment Matlab and its function fminsearch.

The goal of the optimization is to minimize the criterion function ISE (integral square error) described as: ∞

ISE =

∫ ( y process (t ) − ymodel (t ) )

2

better response presented in Fig. A2 with value of the criterion function Eq. (A1) 3.478⋅104.

Fig. A1. Initial experiment of the fitting process: real process (solid line) and mathematical model (dashed line)

Fig. A2. Operation of the fitting process: real process (solid line) and mathematical model (dashed line) At the end the optimization returned optimal values Same = 0.312, lme = 0.002 and Kcwa = 1905 with minimal value of the criterion function 268.62. That results returned already presented responses in Figs. 11, 12 and 16.

dt . (A1)

9 REFERENCES

0

The example of the fitting process of last three autoclave cooling parameters Kcwa, Same and lme is presented in Figs. A1 and A2.

The initial values Same = 3 and lme = 0.5 were set from the autoclave heating and initial value Kcwa = 500 was assumed, which returned the response presented in Fig. A1.

The calculated value of the criterion function Eq. (A1) in the initial fitting process experiment was 2.498⋅105.

Somewhere in the operation the optimization process returned values Same = 1.1, lme = 0.6167 and Kcwa = 750, which resulted in the

[1] Shearer, J.L., Kulakowski, B.T. (1990).

Dynamic modelling and control of engineering systems. Macmillan Publishing Company, New York.

[2] Russell, L.D., Adebiyi, G.A. (1993). Classical Thermodynamics, International Edition.

Saunders College Publishing, Philadelphia.

[3] Incropera, F.P., De Witt, D.P. (1990).

Fundamentals of Heat and Mass Transfer, 3rd ed. John Wiley & Sons, New Jersey.

[4] Kutateladze, S.S., Borishanskii, V.M. (1966).

A Concise Encyclopedia of Heat Transfer.

Pergamon Press, London.

Mathematical Model of an Autoclave

515

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 503-516

[5] Perry, R.H., Green, D.W. (1997). Perry’s Chemical Engineers’ Handbook, 7th ed.

McGraw-Hill, New South Wales.

[6] Whitaker, S. (1972). Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres, and for flow in packed beds and tube bundles.

AIChE Journal, vol. 18, no. 2, p. 361-371.

[7] Isermann, I., Lachmann, K.H., Matko, D. (1993). Adaptive Control Systems. Prentice Hall, New York.

[8] Sears, F.W., Zemansky, M.W., Young, H.D. (1991). College physics, 7th ed. Addison Wesley, Massachusetts.

[9] Šarlah, A., Poredoš, A., Kitanovski, A., Egolf, P. (2005). Heat Transfer in an Ice-Slurry Flow.

Strojniški vestnik - Journal of Mechanical Engineering, vol. 51, no. 1, p. 3-12.

[10] Babič, M., Lenarčič, J., Žlajpah, L., Taylor, N.A.S., Mekjavić, I.B. (2008). A Device for Simulating the Thermoregulatory Response of the Foot: Estimation of Footwear Insulation and Evaporative Resistance. Strojniški vestnik - Journal of Mechanical Engineering, vol. 54, no. 9, p. 628-638.

[11] Narazaki, M., Kogawara, M., Qin, M., Watanabe, Y. (2009). Measurement and

516

Database Construction of Heat Transfer Coefficients of Gas Quenching. Strojniški vestnik - Journal of Mechanical Engineering, vol. 55, no. 3, p. 167-173.

[12] Zeng, X., Raghavan, J. (2010). Role of toolpart interaction in process-induced warpage of autoclave-manufactured composite structures. Composites: Part A, doi: 10.1016/j.compositesa.2010.04.017.

[13] Dufour, P., Michaud, D.J., Toure, Y., Dhurjati, P.S. (2004). A partial differential equation model predictive control strategy: application to autoclave composite processing.

Computers & Chemical Engineering, vol. 28, p. 545-556.

[14] Razak, A.A., Salah, N.J., Majdi, H.S. (2007).

Mathematical model of autoclave curing of epoxy resin based composite materials.

Engineering & Technology, vol. 25, no. 7, p.

828-835.

[15] Monaghan, P.F., Brogan, M.T., Oosthuizen, P.H. (1991). Heat transfer in an autoclave for processing thermoplastic composites.

Composited Manufacturing, vol. 2, no. 3-4, p.

233-242.

Preglej, A. ‒ Karba, R. ‒ Steiner, I. ‒ Škrjanc, I.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 517-525 DOI:10.5545/sv-jme.2009.151

Paper received: 03.11.2009 Paper accepted: 23.03.2011

Theoretical Analysis of Stiffness Constant and Effective Mass for a Round-Folded Beam in MEMS Accelerometer Wong, W.C. ‒ Azid, I.A. ‒ Majlis, B.Y. Wai Chi Wong1,2,* ‒ Ishak Abdul Azid1 ‒ Burhanuddin Yeop Majlis3 1School of Mechanical Engineering, Sains University Malaysia 2Department of Mechanical Engineering, College of Engineering, Tenaga Nasional University, Malaysia 3Institute of Microengineering and Nanoelectronic, Kebangsaan University Malaysia

In this paper, the governing equations of stiffness constant and effective mass for a round folded suspension beam in Micro-Electro Mechanical System (MEMS) accelerometer are derived and solved.

The stiffness constant is determined by the strain energy and Castigliano’s displacement theorem, whereas the effective mass is determined by the Rayleigh principle. The stiffness constant and the effective mass are solved separately by components and then combined by using the superposition method. The results obtained by the derived equations agree well when compared with the finite element results for several thickness values. The governing equations derived in this paper can be used to predict the natural frequencies and sensitivity of the MEMS-accelerometer. ©2011 Journal of Mechanical Engineering. All rights reserved.

Keywords: effective mass, folded beam, MEMS-accelerometer, stiffness constant, strain energy 0 INTRODUCTION The development of MEMS devices has become increasingly important since the beginning of the 1990s. MEMS-accelerometers are the most commercially successful MEMS devices [1] and [2]. Among these, the comb finger type capacitive accelerometer is the most successful type of MEMS accelerometer due to its high sensitivity, low drift, stable dc-characteristics, low power dissipation, high bandwidth, simplicity of the fabrication process, absence of exotic materials, and low temperature sensitivity [2] to [5].

Stiffness constant and resonant frequency, which are the topics of this research, are two of the most important functionalities required in the design of any MEMS devices with moving parts. Sensors and actuators in particular, often require a specific stiffness constant and resonant frequency to guarantee successful and repeatable performance. The stiffness constant and effective mass, which incorporate both material properties and physical geometry, characterize the sensitivity and resonant frequency of the MEMS accelerometer. The sensitivity of the accelerometer is a measure of displacement with respect to acceleration. In contrast, resonant frequency characterizes the bandwidth of the accelerometer. The suspension beam can be of

different designs depending on the application of the MEMS accelerometer. However, the study on the structural analysis of suspension beam in the comb finger type accelerometer has not been established well yet. Analytical derivation of these two parameters is only available for simple designs, such as straight beams [5] to [11].

For different geometric shapes, Legtenberg et al. [8] and Zhou and Dowd [9] determined the material properties of the suspension beam and the stiffness constant by using Hooke’s law and the total potential energy, respectively. Tay et al. [10] used the Rayleigh’s energy principle for the determination of resonant frequency as a function of effective mass. Wittwer and Howell [11] used Castigliano’s displacement theorem for the analysis of the vertical deflection due to an applied moment or shear force and the geometric shape. As far as the authors know, the formula for the round folded beam has not yet been derived.

Therefore, there is a need to derive the stiffness constant theoretically for a round folded beam to predict the performance of this type of design.

In this paper, a detailed derivation of the stiffness constant, as well as the effective mass for the round folded beam as used for the suspension beam in MEMS accelerometer, is carried out. The principle of the strain energy and Castigliano’s displacement theorem is extensively used in

*Corr. Author’s Address: School of Mechanical Engineering, Engineering Campus, Universiti Sains Malaysia, 14300 Nibong Tebal, Seberang Perai Selatan, Pulau Pinang, Malaysia, WaiChi@uniten.edu.my

517

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 517-525

the derivation. From the derived equations, the stiffness constant and effective mass for various parameters are determined and compared with the finite element simulation using ANSYS® 8.1.

1 EQUIVALENT STIFFNESS CONSTANT OF THE SUSPENSION BEAM IN MEMS ACCELEROMETER A schematic design of MEMS accelerometer with the round folded beam is shown in Fig. 1a. The close up of the round folded beam with the symbol used in derivation is shown in Fig. 1b, where L is the length of the beam, r is the radius of the round part, and w is the width of the beam.

In Fig. 1a, the proof mass is suspended by four suspension beams symmetrically at the four edges. The proof mass can be approximated by a central proof mass supported by four springs. The free body diagram of the typical arrangement of an accelerometer can then be approximated by a system of mass and spring as shown in Fig. 2.

In Fig. 2, m is mass of the proof mass; k1, k2, k3, and k4 are the stiffness constants of each suspension beam; and x is the displacement. In this spring-mass system, the mass is supported by four springs, thus the external forces are balanced by the four springs evenly and stored as the strain potential energy. The equivalent stiffness constant of the spring-mass system as shown in Fig. 2 can be determined by the following equation of equilibrium:

∑F

x

− ( k1 + k2 ) x − ( k3 + k4 ) x = mx,

Fig. 1. Round folded beam as the suspension beam in accelerometer, a) a 2D schematic diagram of MEMS accelerometer, b) the round folded beam of suspension beam

= mx,

(1)

mx + ke x = 0, (2)

where  x is the acceleration and Fx is the force.

Therefore, the equivalent stiffness constant, ke = k1 + k2 + k3 + k4.

Since the four suspension beams are in the same dimension and of the same material, then (3) k1 = k2 = k3 = k4 = k1/4 , ke = 4k1/4 , where k1/4 is the stiffness constant of a quarter system.

In obtaining the governing Eqs. of the suspension beam for the stiffness constant, the following assumptions are made: • The proof mass and comb fingers of the structure are rigid.

• The flexible members are attached to perfectly rigid supports.

• The device only vibrates in the sensing axis.

• Young’s modulus is constant.

• Damping is ignored.

• 3D effects such as fringing field and comb finger end effects are neglected [12] to [14].

• Euler–Bernoulli beam model assumption is applied.

2 DETERMINATION OF STIFFNESS CONSTANT IN ROUND FOLDED BEAM

Fig. 2. Free body diagram of typical arrangement of an accelerometer 518

The resolved components of the round folded beam are shown in Fig. 3. The quarter

Wong, W.C. ‒ Azid, I.A. ‒ Majlis, B.Y.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 517-525

Fig. 3. Resolved components of theround folded beam model of the suspension beam together with its boundary condition is shown in Fig. 3a, and its free body diagram is shown in Fig. 3b. In the analysis, the round folded beam can be resolved into three components, with two models of half fixed-fixed beam (Fig. 3c and d) and a model of half ring (Fig. 3e).

Since the device is assumed to vibrate only in the sensing axis, the total displacement of the quarter model δ1/4 is equal to the sum of the displacement of each component, δ1/4 = δc1 + δc2 + δc3 .

Based on Hooke’s law (i.e., F = kδ), there is k ∝

1 . In this context, the stiffness constant of δ

the quarter model can be given in a complementary form by: 1 1 1 1

= + + , (4) k1 4 kc1 kc 2 kc3 where kc1, kc2, and kc3 are the stiffness constants for components 1 to 3, respectively.

2.1 The Stiffness Constant for the First and Third Components The free body diagram of the first and third components are similar to the model of half fixedfixed beam subjected to the transverse loading,

as shown in Fig. 4. Fig. 4a shows a fixed-fixed beam with length 2L subjected to a transverse load F at the midspan of the beam. This force causes bending, which could result in reactions at both fixed ends consisting of forces and moments.

The maximum displacement δmax occurs at the midspan of the beam. If this model is cut through its midspan, this part can be modeled as a half fixed-fixed beam. The reactions at both fixed ends are the bending moment M0, shear reaction force in y-direction Ry, and the axial reaction force in x-direction Ra, as shown in Fig. 4b. Since the load is transverse to the axis of the beam, the axial reaction force Ra is extremely small compared to the bending moment and shear force. Therefore, Ra is ignored in the calculation. The shear reaction force Ry, and bending moment M0 for the model of the half fixed-fixed beam is obtained as Ry = F/2 and M0 = FL/4.

The maximum displacement in the model of the half fixed-fixed beam δmax is caused by both displacement due to bending moment δbm , and the displacement due to shear δs , or δmax = δbm + δs.

Thus, the stiffness constant of the half fixed-fixed beam is taken as:

1 khalp

=

1 kbm

+

1 . (5a) ks

Therefore, the stiffness constant for components 1 and 3 is equal to the stiffness

Theoretical Analysis of Stiffness Constant and Effective Mass for a Round-Folded Beam in MEMS Accelerometer

519

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 517-525

constant of the half fixed-fixed beam as given in Eq. (5a) as:

1 1 1 1 1 = = = + . (5) kc1 kc 3 khalp kbm k s

2.1.1 The Stiffness Constant Due to Bending Moment

where G is shear modulus, G =

E and μ is 2 (1 + µ )

the Poisson’s ratio. By knowing Fs = F/2 and by replacing G and Fs in δs, the maximum deflection (at middle point) due to the shear is:

δs =

6 (1 + µ ) FL . (10) 5 bdE

Since ks = F/δs, stiffness constant due to shear ks in the complementary form is given as

1 δ s 6 (1 + µ ) L = = . (11) ks F 5 bdE

Fig. 4. Fixed-fixed beam For a fixed-fixed beam (Fig. 4), the maximum deflection due to a bending moment occurs at the middle of the beam. It is given by [15]:

3

δ bm =

F ( 2L ) 192 EI

=

FL3 (6) .

24 EI

The stiffness constant due to the bending moment kfull for the full model of the fixed-fixed beam is then derived as:

k full =

F 24 EI = 3 . (7) δ bm L

Thus, the stiffness constant due to the bending moment kbm for the half model of a fixedfixed beam is:

1 kbm

=

2.2 The Stiffness Constant for the Second Component The second component is approximated to be the model of a half ring with simple support on the left, transferred from the first and third components. The free body diagram is shown in Fig. 5a. This component is subjected to two forces, the transverse force Ry and bending moment M. The stiffness constant of the second component kc2 is the combination of the stiffness constant due to transverse force kt (Fig. 5b), and stiffness constant due to bending moment km (Fig. 5c). kt is approximated to be half of a stiffness constant in the ring kc due to the same value of transverse force, as shown in Fig. 6.

L3 , (8) 12 EI

where E is the Young’s modulus, I is the second moment of the cross sectional area, and L is the length of the half model of the fixed-fixed beam.

2.1.2 The Stiffness Constant Due to Transverse Force

a) b) c) Fig. 5. Free body diagram of the second component of round folded beam

For a rectangular cross sectional area of width b and depth d, with the total beam length L, and with the applied transverse load of F/2, as shown in Fig. 4b, the maximum deflection (at middle point) due to the shear is given as [15]:

520

δs =

3 Fs L , (9) 5 bdG

Fig. 6. Transverse force in the second component

Wong, W.C. ‒ Azid, I.A. ‒ Majlis, B.Y.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 517-525

2.2.1 The Stiffness Constant Due to Transverse Force Acting on Half a Ring The stiffness constant of the ring component (Fig. 5b) due to transverse force can be determined by using Castigliano’s displacement theorem. The theorem states that deflection at a point on a member in the direction of a force applied at the specific point is given by the partial derivative of the complementary energy with respect to the external force at the point, or δc = ∂U/∂R, , where U is the internal strain energy given by U =

2

1σ dV . For a beam subjected 2 E

∫

V

to bending, the direct stress is σ = My / I , where M is the bending moment, y is the distance from the neutral axis to the point in question, and I is the second moment of the cross sectional area I = y 2 dA .

∫

A

Thus, U =

M2

L

∫∫ 0

A

2 EI 2

y 2 dAdx =

∫

L

0

M2 dx , 2 EI

where L is the length of the beam. Accordingly, ∂U ∂ = ∂R ∂R

δc =

L

∫

0

M2 dx .

2 EI

∫

2π

0

rdφ = 4

∫

π

0

2

=4

∫

π

2

0

the ring, and ϕ is the angle at any point within the

∫

2π

0

∫

π

0

2

 M c2 r dφ  =  2 EI  (12)

M c ∂M c r dφ .

EI ∂Ry

where M0, the imaginary bending moment, is unknown but may be obtained from the strain energy.

The strain energy for the quadrant of the ring is: π 2 1 Uq = M c2 r dφ = 0 2 EI (14)

2 π Ry r  2 1  = (1 − cos φ ) r dφ . M 0 − 0 2 EI 2  

∫ ∫

The total strain energy for the ring is given

by:

π

2

1 2 EI

2

Ry r   (1 − cos φ ) r dφ , (15) M 0 − 2  

owing to the symmetry at ϕ = 0, ∂U/∂M0 = 0.

Therefore, 1 EI

dφ is always

zero, and in view of the symmetry of the ring, only one quadrant of the ring needs to be considered as shown in Fig. 7.

  4 

By cutting the ring at any section within the quadrant of the ring, the bending moment at any point c with an angle of ϕ is given as [15]: Ry r

Mc = M0 − (1 − cos φ ) , (13) 2

0

rdφ , where r is the radius of

ring. Since the integration of

∂U c ∂ = ∂Ry ∂Ry

U c = 4U q = 4 ∫

For a full ring, the total length is given as L=

δc =

∫

π

0

Ry r  (1 − cos φ ) r dφ = 0. (16) M 0 − 2  

2

Since 1 / EI ≠ 0 and r ≠ 0,

∫

π

0

Ry r Ry r  cos φ  dφ = 0 + M 0 − 2 2  

2

 π  Ry r M0   = 2 2

solved

as

π   2 − (1 − 0 )  . Therefore,  

1 1 (17) M 0 = Ry r  −  . 2 π  Substituting Eq. (17) into Eq. (13), the bending moment at any point c within the full ring, is:

Fig. 7. Transverse forces at a ring According to Castigliano’s displacement theorem, deflection of the ring at any point c is:

1 1 M c = Ry r  cos φ −  . (18) π 2

Theoretical Analysis of Stiffness Constant and Effective Mass for a Round-Folded Beam in MEMS Accelerometer

521

1 σ2 dV V 2 E Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 517-525 2 π2 M2 2 1 = y dA ⋅ rdφ = 2 E 0 A I c22 E U=

∫

∫ ∫

By partial derivative with respect to Ry:

∂M c 1 1 = r  cos φ −  . (19) π ∂R y 2

Substituting Eqs. (18) and (19) into Eq.

(12), δc = 4

= =

∫

π

M c ∂M c r dφ EI ∂Ry

2

0

4 EI

∫

π

0

4 Ry r 3 EI

2

1 1 1 1 Ry r  cos φ −  ⋅ r  cos φ −  rdφ (20) π  2 π 2

∫

π

0

2

1 1  2 cos φ − π  dφ ,  

2

4 Ry r 3  π 1 1  δc = − + . (21)  EI  16 π 2π 

δc =

1 4 Fr 3  π 1 − +  EI  16 π 2π

  . (22) 

The stiffness constant of the ring is then given by: 1 δ 4r 3  π 1 1  = = − + . (23)  kc F EI  16 π 2π 

Since the stiffness constant for the second component is equal to the half of the stiffness constant of the full ring, kt = ½ kc . Therefore, the stiffness constant for the second component due to transverse force is given by: 1 2 8r 3  π 1 1 = = − +  kt kc EI  16 π 2π

 (24) . 

2.2.2 The Stiffness Constant Due to Bending Moment The stiffness constant due to the bending moment for this component (Fig. 5c) can be determined by using the energy method. The internal strain energy is:

2

1σ dV 2 E 2 π2 M2 2 1 = y dA ⋅ rdφ = 2 2E 0 A Ic2 E

U=

∫

= U=

522

U=

∫

π

0

2

M2 rdφ Ic2

2

0

M2 rdφ Ic2

M r π ⋅ , EI c 2 2

(25)

π M 2r .

2 EI c 2

Since the total internal strain energy is equal to the external work performed, U = ½ Fδ , 2 then, π M r = 1 F δ .Substituting M = FL/2 into 2 EI c 2 2 2

1  FL  π r = Fδ . the equation,    2  2 EI c 2 2 By rearranging the above equation, the deflection due to the bending moment is:

δ=

π FL2 r . (26) 4 EI c 2

Therefore, the stiffness constant due to the bending moment is given by: 1

kbm 2

=

δ π L2 r = .

F 4 EI c 2

Since the width and depth of the second component is equal to the width and depth of the first component, Ic2 = Ic1 = I, thus

1 kbm 2

=

π L2 r . (27) 4 EI

2.3 The Effective Stiffness Constant of the Round Folded Beam The effective stiffness constant of the round folded beam can then be determined from Eq. (3): 1 1 11 = = , ke 4k 4  k 

1 1 1 1 1  =  + + , ke 4  kc1 kc 2 kc3  1 1 1 1 + = = + ke 4kc1 4kc 2 4kc 3 1 1 1  1 1 1 +  +  + 4  kbm k S  4  kt kbm 2 1 1 1 1 1 = + + + . ke 2kbm 2k S 4kt 4kbm 2 =

V

∫ ∫

=

π

2

Since Ry = F,

∫

(28)  1 1 1  + , +  k k 4 S   bm 

2

M r π ⋅ , EI c 2 2 2

πM r .

2 EI c 2

Substituting Eqs. (8), (11), (24), and (27) into Eq. (28): Wong, W.C. ‒ Azid, I.A. ‒ Majlis, B.Y.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 517-525

 1  6 (1 + µ ) L   +  +  2  5 bdE  1   1  π L2 r  1  8r 3  π 1 +  − +  +   4  EI  16 π 2π   4  4 EI  (29) 1  π L2 r 3 (1 + µ ) L 2r 3  π 1 L3 = + , + − + + EI  16 π 2π  16 EI 24 EI 5 bdE 3 (1 + µ ) L 24r 3  π 1 1  3π L2 r  1 1  L3 + 3  − + = .  3+ + ke Et  2 w 5w wr  16 π 2π  4 wr3 

1 1  L3 =  ke 2  12 EI

mb,e = ρ

dδ ( x )

δ(x) = N(x) δmax and

dt

= N ( x)

d δ max . (30) dt

The effective mass is given by:

me = ρ

∫

L

0

N 2 ( x ) A ( x ) dx. (31)

3.1 Effective Mass for Half Model of Fixedfixed Beam Since the distribution function is independent of the applied force, the distribution function can be determined by assuming the half model of the fixed-fixed beam is deflected under a concentrated force F. Displacement at any point within the beam is given by James et al. [16] as: F 

δ ( x) = 3Lx 2 − 2 x3  . (32)  12 EI  thus,

The maximum displacement occurs at x=L

δ max

FL3 = . (33) 12 EI

Consequently, the distribution function is:

N ( x) =

δ ( x) δ max

=

3Lx 2 − 2 x3 L3

. (34)

Effective mass of half model of fixed-fixed beam under bending moment is then given as:

N 2 ( x ) A ( x ) dx =

0

= ρA

3 DETERMINATION OF EFFECTIVE MASS IN ROUND FOLDED BEAM The effective mass of the folded beam is determined by using the Rayleigh principle. By taking a beam model with a cross sectional area A, length L, the displacement at any point x to be δ(x), and velocity at any point x to be dδ(x)/dt, the maximum displacement δmax is related to the distribution function N(x) as:

∫

L

∫

L  3Lx 2

0

2

− 2 x3   dx = L3 

 

L

mb,e

ρ A  9 L2 x5 12 Lx 6 4 x 7  = 6  − +  = 6 7  L  5 0 ρ A 9 4 = 6  − 2 +  × L7 , 7 L 5 13 = ρ AL .

35

(35)

Thus, the effective mass of the first component m1 is equal to that of the third component m3 or: 13

m1 = m3 = mb,e = ρ AL . (36) 35 3.2 Effective Mass for Half Model of Ring For the half model of ring under transverse force (Fig. 5), the displacement at any point within the beam is given by Eq. (20) as:

δ ( x) =

4 Ry r 3 EI

∫

0

2

1 1  2 cos φ − π  dφ ,  

φ

or by expanding: δ ( x) =

4 Ry r 3  1  φ sin 2φ  1 φ  − sin φ + 2  . (37)   +  4  π EI  4  2 π 

The maximum displacement is given by 4 Ry r 3  π 1 1  − + .

EI  16 π 2π  The distribution function:

Eq. (21) as δ max =

 1  φ sin 2φ  1 φ  − sin φ + 2  +    δ x 4  π π  N ( x ) = ( ) =  4  2 1  δ max π 1  16 − π + 2π    is solved by: 1 φ 1 1 φ + sin ( 2φ ) − sin (φ ) + 2 8 16 π π . (38) N ( x) = 1 1 π− 16 2π

Theoretical Analysis of Stiffness Constant and Effective Mass for a Round-Folded Beam in MEMS Accelerometer

523

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 517-525

Effective mass of the half model of the ring under transverse force is then given as: ma ,e = ρ

∫

L

0

= 2ρ A

N 2 ( x ) A ( x ) dx =

∫

π

0

Elastic modulus

1 φ 1 1  φ + 16 sin ( 2φ ) − π sin (φ ) + 2 2 8 π  1 1  π−  16 2 π 

2

   rdφ .   

As a result, the effective mass of the second component m2 is:

  ρ Ar  2π 7 + 47π 5 − 32π 3 − 6144π  . (39) 2 4 2  6  π π − 16 π + 64  

(

Table 1. Material properties and physical geometries for accelerometer

)

Thus, the effective mass of the device is me = m1 + m2 + m3 + m pm + n m f = (40) = 8mb,e + 4ma ,e + m pm + n m f ,

where mpm is the mass of proof mass, mf is the mass of finger, and n is the number of fingers.

4 COMPARISON RESULTS WITH THE FINITE ELEMENT SIMULATION The round folded beam in MEMS accelerometer is further analyzed by using the finite element (FE) package, ANSYS® 8.1.

The three-dimensional 20-node structural solid element, SOLID 186, has been used in this analysis. The material properties and the physical geometries used in the FE simulation are shown in Table 1. The FE model meshed by the free meshing is shown in Fig. 8. The simulation result will be used for a comparison with the analytical results obtained by Eqs. (29) and (40).

The resonant frequency is obtained directly from the modal analysis. The stiffness constant is determined by using a static analysis.

Poison’s ratio Density Length of suspension beam (L) Width of suspension beam (w) Radius of ring, r Length of comb finger (lfinger) Width of comb finger (wfinger) Number of comb finger Sensing gap distance (d0)

127×103 kg/µm s2 0.27 2330×10-18 kg/µm3 803 mm 11 mm 50 mm 650 mm 15 mm 56 pairs 2 mm

Table 2 shows the result of the comparison between the analytical and simulation analysis for stiffness constant and effective mass for several thicknesses (t). From Table 2, it can be seen that the stiffness constant and the effective mass obtained from the derived equations based on the analytical analysis show good agreement with those obtained from the finite element simulation.

The difference between analytical and simulation results is below 5%. Based on this, it can be concluded that the derived analytical formulas are capable of obtaining the effective mass and the stiffness constant of the round folded beam of the suspended comb finger type accelerometer.

Therefore, they are capable of predicting the resonant frequency and the sensitivity of the accelerometer.

5 CONCLUSIONS The stiffness constant and the effective mass of round folded beam in MEMS accelerometer have been derived successfully.

The derivation of the stiffness constant is obtained by using the strain energy method and the Castigliano’s displacement theorem, while the derivation of the effective mass is determined by using the Rayleigh principle. The result obtained from these derived formulas agree well with the results obtained from FE software ANSYS. 6 ACKNOWLEDGEMENT

Fig. 8. Meshed model 524

This work was carried out under the support of the short-term and the E-science grants Wong, W.C. ‒ Azid, I.A. ‒ Majlis, B.Y.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 517-525

Table 2. Comparison of stiffness constant and effective mass between analytical result and simulation (ANSYS) result t [mm] 120 80 60 40

Analytical 59.79 39.86 29.89 19.93

k [mN/mm] ANSYS 62.01 40.46 30.39 19.85

% diff 3.59 1.50 1.64 -0.42

provided by the Universiti Sains Malaysia and the Malaysian Government.

7 REFERENCES [1] Bernstein, J. (2003). An overview of MEMS inertial sensing technology. Sensors Magazine Online, vol. 20, no. 2, p. 14-21.

[2] Yazdi, N., Ayazi, F., Najafi, K. (1998).

Micromachined inertial sensors. Invited Paper, Special Issue of IEEE Proc., vol. 86, no. 8, p. 1640-1659.

[3] Amini, B.V., Ayazi, F.A. (2004). 2.5V 14bit ΣΔ CMOS SOI capacitive accelerometer.

Journal of Solid-State Circuits, vol. 39, no.

12, p. 2467-2476.

[4] Rödjegård, H., Lööf, A. (2005). A differential charge-transfer readout circuit for multiple output capacitive sensors. Sensors and Actuators A, vol. 119, no. 2, p. 309-315.

[5] Borovic, B., Liu, A.Q., Popa, D., Cai, H., Lewis, F.L. (2005). Open-loop versus closedloop control of MEMS devices: choices and issues. Journal of Micromechanics and Microengineering, vol. 15, p. 1917-1924.

[6] Chae, J.S., Kulah, H., Najafi, K. (2002). A hybrid silicon on glass lateral microaccelerometer with CMOS readout circuitry technical digest. IEEE International Conf. MEMS, Las Vegas, p. 623-626.

[7] Lüdtke, O., Biefeld, V., Buhrdorf, A., Binder, J. (2000). Laterally driven accelerometer fabricated in single crystalline silicon.

Sensors and Actuators A, vol. 82, p. 149-154.

[8] Legtenberg, R. Groeneveld, A.W., Elwenspoek, M. (1996). Comb-drive actuators for large displacements. Journal of

Analytical 4.88 3.26 2.44 1.63

meff [×10-7 kg] ANSYS 4.72 3.13 2.35 1.58

% diff -3.50 -4.01 -3.93 -3.26

Micromechanics and Microengineering, vol.

6, p. 320-329.

[9] Zhou, G.Y. Dowd, P. (2003). Tilted foldedbeam suspension for extending the stable travel range of comb-drive actuators. Journal of Micromechanics and Microengineering, vol. 13, p. 178-183.

[10] Tay, F.E.H., Kumaran, R., Chua, B.L., Logeeswaran, V.J. (2000). Electrostatic spring effect on the dynamic performance of microresonators.

Technical Proc.

International Conf. Modeling and Simulation of Microsystems, San Diego, p. 454-457.

[11] Wittwer, J.W., Howell, L.L. (2004).

Mitigating the effect of local flexibility at the built-in ends of cantilever beams. Journal of Applied Mechanics , vol. 71, p. 748-751.

[12] Tang, W.C, Lim, M.G., Howe, R.T. (1992).

Electrostatic comb drive levitation control method. Journal of Microelectromechanical Systems, vol. 1, p. 170-178.

[13] Li, L. Uttamchandani, D.Ȅ. (2006). Twinbladed microelectro mechanical systems variable optical attenuator. Optical Review, vol. 13, no. 2, p. 93-100.

[14] Spengen, W.M.V., Oosterkamp, T.H. (2007). A sensitive electronic capacitance measurement system to measure comb drive motion of surface micromachined MEMS devices. Journal of Micromechanics and Microengineering, vol. 17, p. 828-834.

[15] Benham, P.P., Drawford, R.J., Armstrong, C.G. (1996). Mechanics of Engineering Materials. 2nd ed. Prentice Hall Ltd., London.

[16] James, M.L., Smith, G.M., Wolford, J.C., Whaley, P.W. (1989). Vibration of Mechanical and Structural Systems. Happer & Row Publishers, Singapore.

Theoretical Analysis of Stiffness Constant and Effective Mass for a Round-Folded Beam in MEMS Accelerometer

525

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 526-527 Instructions for Authors

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/.

Send to: Strojniški vestnik Journal of Mechanical Engineering Aškerčeva 6, 1000 Ljubljana, Slovenia Phone: 00386 1 4771 137 Fax: 00386 1 2518 567 E-mail: info@sv-jme.eu strojniski.vestnik@fs.uni-lj.si Please add 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?

5. We kindly ask you to suggest at least two reviewers for your paper and give us their names and contact information (email).

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.

526

- -

A Theory or experimental methods used.

An Experimental section, which should provide details of the experimental set-up and the methods used for obtaining the results.

- A Results section, which should clearly and concisely present the data using figures and tables where appropriate.

- A Discussion section, which should describe the relationships and generalizations shown by the results and discuss the significance of the results making comparisons with previously published work. (It may be appropriate to combine the Results and Discussion sections into a single section to improve the clarity).

- 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. Multicurve 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

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, 526-527

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.

[1] Zadnik, Ž., Karakašič, M., Kljajin, M., Duhovnik, J. (2009). Function and functionality in the conceptual design process. Strojniški vestnik Journal of Mechanical Engineering, vol. 55, no.

7-8, p. 455-471.

Journal titles should not be abbreviated. Note that journal title is set in italics.

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. 422427.

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 3.000 to 3.500 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.sv-jme.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 180.00 EUR (for articles with maximum of 6 pages), 220.00 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.

527

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6 Vsebina

Vsebina Strojniški vestnik - Journal of Mechanical Engineering letnik 57, (2011), številka 6 Ljubljana, junij 2011 ISSN 0039-2480 Izhaja mesečno

Povzetki člankov Dragica Jošt, Andrej Lipej: Napoved kavitacijske in nekavitacijske vrtinčne cevi v sesalni cevi Francisove turbine z numerično analizo toka Paulo Flores: Metodologija za kvantifikacijo kinematičnih položajnih napak zaradi proizvodnih in montažnih toleranc Jurij Prezelj, Mirko Čudina: Konfiguracija sekundarnega vira za obvladovanje hrupa prezračevalnih ventilatorjev v kanalih Dejan Dragan: Zaznavanje napak na industrijskem prenosniku toplote s pomočjo modela Henrik Zaletelj, Gorazd Fajdiga, Marko Nagode: Numerične metode za modeliranje termomehanskega utrujanja Dobrivoje Ćatić, Branislav Jeremić, Zorica Djordjević, Nenad Miloradović: Analiza kritičnosti elementov zgloba jarmovega droga lahkega gospodarskega vozila Aleksander Preglej, Rihard Karba, Igor Steiner, Igor Škrjanc: Matematični model avtoklava Wai Chi Wong, Ishak Abdul Azid, Burhanuddin Yeop Majlis: Teoretična analiza vzmetne konstante in efektivne mase za polkrožno upognjeni nosilec v pospeškomeru MEMS

SI 87 SI 88 SI 89 SI 90 SI 91 SI 92 SI 93 SI 94

Navodila avtorjem

SI 95

Osebne vesti Doktorati, magisteriji, specializacije in diplome

SI 97

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 87

Prejeto: 25.03.2010 Sprejeto: 23.03.2011

Napoved kavitacijske in nekavitacijske vrtinčne cevi v sesalni cevi Francisove turbine z numerično analizo toka Jošt, D. – Lipej, A. Dragica Jošt* – Andrej Lipej Turboinštitut, Ljubljana, Slovenija

Članek obravnava napoved pojava vrtinčne cevi v sesalni cevi Francisove turbine z numerično analizo toka. Vrtinčna cev se pojavlja predvsem pri delnih pretokih. Intenziteta vrtinca je odvisna od obratovalnega režima, specifične hitrosti turbine in predvsem od oblike kanala in lopatic gonilnika.

Posledice pojava vrtinca so oscilacije tlaka, vibracije in nihanje moči turbine. Zato je pomembno vnaprej napovedati pojav vrtinca ter velikost in frekvenco oscilacije tlaka, to pa je v fazi oblikovanja gonilnika in sesalne cevi mogoče le z numerično analizo toka. V večini člankov s tega področja je predstavljen izračun vrtinca le v eni ali največ dveh obratovalnih režimih z enim turbulentnim modelom. Računske mreže so bile pogosto zelo redke, računski čas pa prekratek, da bi dobili zanesljive rezultate. Raziskava, predstavljena v tem prispevku, je veliko bolj kompleksna. Namen članka je pokazati vpliv različnih turbulentnih modelov, gostote mreže, časovnega koraka ter časa računanja in upoštevanja kavitacije na natančnost in zanesljivost numeričnih rezultatov za različne obratovalne režime.

Numerične rezultate smo primerjali z rezultati meritev na modelu Francisove turbine. Meritve so bile opravljene v skladu z mednarodnim standardom IEC 90193. Rezultati meritev so frekvence in amplitude oscilacij tlaka in posnetki vrtinčne cevi, ki je dobro vidna le, ko je v toku zaradi kavitacije prisotna določena količina vodne pare. Ker je bila pri meritvah prisotna kavitacija, so bili numerični izračuni v drugem delu raziskave izvedeni tudi s kavitacijo.

Vsi izračuni so bili narejeni s programskim paketom ANSYS-CFX z metodo končnih volumnov.

Rezultati stacionarnih izračunov so bili uporabljeni kot začetni pogoji za nestacionarne izračune.

Raziskava je potekala v dveh delih za dve Francisovi turbini. Za prvo turbino smo izračunali tok v štirih obratovalnih točkah s turbulentnim modelom SAS-SST na različno gostih računskih mrežah z od 3,3 milijona do 25 milijoni vozlišč. Območje računanja je bila cela turbina od vstopa v spiralo do izstopa iz sesalne cevi. Računali smo s časovnimi koraki, ki so ustrezali 10, 30 in 60 vrtenja gonilnika. Za drugo turbino so bili izračuni narejeni le za eno obratovalno točko s tremi turbulentnimi modeli (SAS-SST, RSM in LES) brez modela kavitacije in s kavitacijo. Pri izračunih s kavitacijo smo uporabili homogeni dvofazni model, referenčni tlak na izstopu iz sesalne cevi pa smo dobili iz meritev. Pri turbulentnih modelih SASSST in RSM je bilo računsko območje cela turbina, mreža pa je vsebovala 5,6 milijona vozlov, od tega 3,4 milijona v sesalni cevi. Časovni korak je ustrezal 20 vrtenja gonilnika, v primeru RSM s kavitacijo pa smo ga zmanjšali na 10. Turbulentni model LES zahteva gostejšo mrežo, zato smo tu območje računanja skrčili na sesalno cev, računska mreža je vsebovala 23,5 milijona vozlov, časovni korak pa je ustrezal 0,50 vrtenja gonilnika. Na vstopu v sesalno cev smo podali komponente hitrosti, dobljene iz izračuna cele turbine.

Rezultati prvega dela raziskave so pokazali, da z numerično analizo toka lahko napovemo pojav in obliko vrtinčne cevi v sesalni cevi Francisove turbine za različne obratovalne režime. Pomembno je, da izračun traja dovolj dolgo, saj se med izračunom vrtinčna cev šele oblikuje, vrednost frekvence nihanja tlaka pa se ustali po 30 do 40 vrtljajih gonilnika, oziroma po petih vrtljajih vrtinčne cevi. Izračunana frekvenca oscilacij tlaka se od izmerjenih vrednosti razlikuje za približno 1 %. Numerični izračun pravilno napove, v kateri obratovalni točki so oscilacije tlaka največje, izračunane amplitude pa so manjše od izmerjenih, a se z zgostitvijo računske mreže natančnost napovedi zelo izboljša. Drugi del raziskave je pokazal, da so turbulentni modeli SAS-SST, RSM in LES primerni za napoved pojava vrtinčne cevi in da se rezultati glede na uporabljeni turbulentni model pri izračunih brez kavitacije bistveno ne razlikujejo.

Pri izračunih s kavitacijo se z meritvami najbolje ujemajo rezultati, dobljeni s turbulentnim modelom LES. ©2011 Strojniški vestnik. Vse pravice pridržane.

Ključne besede: vodne turbine, Francisova turbina, vrtinčna cev, kavitacija, pulzacije tlaka, turbulentni modeli *Naslov avtorja za dopisovanje: Turboinštitut, Rovšnikova 7, 1210 Ljubljana, Slovenija, dragica.jost@turboinstitut.si

SI 87

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 88

Prejeto: 16.11.2009 Sprejeto: 06.03.2011

Metodologija za kvantifikacijo kinematičnih položajnih napak zaradi proizvodnih in montažnih toleranc Flores, P. Paulo Flores* Univerza v Minhu, Oddelek za strojništvo, Portugalska

Glavni namen tega raziskovalnega dela je predstavitev splošnega in sistematičnega pristopa k kvantifikaciji kinematičnih položajnih napak zaradi toleranc pri proizvodnji in montaži.

Kinematične omejitve in enačbe gibanja sistemov več teles so zapisane v skladu z njihovimi metodologijami obravnave sistemov. Sistem je definiran z naborom posplošenih koordinat, ki predstavlja trenutni položaj vseh teles, skupaj z naborom posplošenih dimenzijskih parametrov, ki določa funkcijske dimenzije sistema. Posplošeni dimenzijski parametri upoštevajo tolerance, povezane z dolžinami.

Predstavljena je povezava med kinematičnimi omejitvami, dimenzijskimi in izhodnimi kinematičnimi parametri.

Obstajata dva glavna pristopa k preučevanju vpliva proizvodnih toleranc na kinematične položajne napake: deterministične in verjetnostne metode. Deterministična metoda uporablja fiksne vrednosti ali omejitve, s katerimi se išče natančna rešitev. Te metode se uporabljajo takrat, ko so tolerance znane in je treba določiti položajno napako v najslabšem možnem primeru. Verjetnostne ali statistične metode pa uporabljajo naključne spremenljivke, ki dajejo verjetnostni odziv. Statistični pristopi se uporabljajo, kadar imajo dimenzije naključno porazdelitev in je treba ovrednotiti verjetnost, da bo določena dimenzija v danem tolerančnem pasu. Metodologija, predstavljena v tej študiji, uporablja deterministični pristop z iskanjem najslabšega možnega primera. Posplošene kartezijske koordinate se uporabljajo za enostavno matematično formulacijo kinematičnih omejitev in enačb gibanja sistemov več teles. Sistemi so zato določeni z naborom posplošenih koordinat, ki predstavlja trenutni položaj vseh teles, skupaj z naborom posplošenih dimenzijskih parametrov, ki določa funkcijske dimenzije sistema. Ti posplošeni dimenzijski parametri med drugim upoštevajo tolerance, povezane z dolžinami, nespremenljivimi koti, premeri in razdaljami med središči. Variabilnost geometrijskih dimenzij se na osnovi teorije dimenzijskih toleranc obravnava kot tolerančni razred z intervalom, pripisanim vsaki dimenziji, in posledično variabilnost kinematične amplitude za položaj teles.

Metodologija, predstavljena v tej študiji, je prikazana na primeru kinematike drsno-ročičnega mehanizma. S tem je demonstrirana enostavnost in splošna uporabnost predlagane metodologije za študijo kinematičnih položajnih napak zaradi proizvodnih toleranc. Omeniti je treba, da so v tem članku obravnavani samo dimenzijski parametri, kot je npr. dolžina povezav. V splošno metodologijo, ki je predstavljena v tem delu, pa je možno integrirati tudi druge parametre, povezane z okroglostjo krožnih površin in zračnostjo zglobov.

Glavna prednost predlaganega pristopa je njegova splošnost in možnost vključitve v kinematično analizo ravninskih mehanizmov. Metodologija je enostavna in nezahtevna za implementacijo, hkrati pa je tudi učinkovita z računskega vidika. ©2011 Strojniški vestnik. Vse pravice pridržane.

Ključne besede: položajna napaka, proizvodne tolerance, montažni sistemi, ravninski mehanizmi

SI 88

*Naslov avtorja za dopisovanje: Univerza v Minhu, Oddelek za strojništvo, Campus de Azurem, 4800-058 Guimaraes, Portugalska, pflores@dem.uminho.pt

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 89

Prejeto: 25.02.2009 Sprejeto: 23.03.2011

Konfiguracija sekundarnega vira za obvladovanje hrupa prezračevalnih ventilatorjev v kanalih Prezelj, J. ‒ Čudina, M. Jurij Prezelj* ‒ Mirko Čudina Univerza v Ljubljani, Fakulteta za strojništvo, Slovenija

Aktivno dušenje hrupa (ADH) deluje na principu dodajanja »protihrupa« v primarni hrup.

Protihrup ima v časovnem prostoru popolnoma enako pojavno obliko kot primarni hrup, le da ima drugačen predznak. Protihrup ustvarimo s sekundarnim virom, ki je največkrat kar dinamični zvočnik.

Signal za protihrup dobimo iz referenčnega signala z algoritmi za ADH. Referenčni signal mora biti v popolni korelaciji s primarnim hrupom. Na delovanje sistema za ADH močno vpliva ravno kakovost referenčnega signala.

Za aktivno dušenje širokopasovnega in naključnega hrupa, ki je največkrat posledica naključnega turbulentnega gibanja zraka, smo danes še vedno primorani uporabljati mikrofone. Referenčni mikrofon za zaznavanje referenčnega signala se najpogosteje postavi v bližino primarnega vira hrupa. Referenčni mikrofon, ki naj bi zaznal samo primarni hrup, zazna tudi protihrup iz sekundarnega vira, s katerim smo izničili primarni hrup v predhodnem koraku. Referenčni signal za primarni hrup je tako popačen z zakasnelim signalom protihrupa, ki v povratni zanki povzroča nestabilnost celotnega sistema. Znano je, da akustična povratna zanka povzroča največ težav pri realizaciji sistema za ADH. Zato smo se pri našem delu omejili na zmanjševanje vpliva akustične povratne zanke in razvili usmerjeni sekundarni zvočni vir brez digitalne obdelave signalov.

Naš cilj je bil poiskati takšno razporeditev akustičnih elementov, ki bo omogočala močno usmerjenost sekundarnega vira in s tem posledično dobro globalno dušenje hrupa. Za doseganje usmerjenosti zvočnega vira načrtno nismo uporabili dodatne digitalne obdelave signalov, zato da bi lahko pozneje, v kombinaciji s kompleksnejšimi algoritmi, izboljšali delovanje sistema za ADH kot celote.

Analitično rešitev za novi tip sekundarnega vira smo izpeljali neposredno iz Swinbanksove teorije.

Dinamičnemu zvočniku kot dipolnemu izvoru zvoka smo prigradili enostaven prenosni kanal in akustični sistem numerično rešili. Ugotovili smo, da na delovanje sekundarnega vira najbolj vplivata dolžina prenosne linije in razmerje prerezov. Numerični rezultati potrjujejo, da sekundarni vir deluje kot prenosna linija. Izdelali smo prototip in na njem izvedli preizkuse, s katerimi smo potrdili močno usmerjenost novega sekundarnega vira. Pokazala se je tudi dodatna prednost novega tipa sekundarnega vira, da zagotavlja prenosne izgube pri resonančnih frekvencah, celo če sistem za ADH ne deluje. Na prototipu smo izmerili ustrezne impulzne odzive, ki so potrebni za algoritme ADH. Pokazalo se je, da novi tip sekundarnega vira razbremeni adaptivni algoritem.

Z novo konstrukcijo sekundarnega vira smo tako spremenili osnovni pristop k snovanju sistemov za ADH. Širjenje hrupa z diskretnimi frekvencami po kanalu se sedaj lahko omejuje preko reaktivne lastnosti samega sekundarnega vira, ki deluje kot resonator pri četrtini valovne dolžine. V preostalem frekvenčnem območju pa novi sekundarni vir deluje močno usmerjeno, in tako sistemu za ADH omogoča širokopasovno delovanje.

Predstavljeni na novo razviti sekundarni vir ima tri prednosti: • Usmerjena karakteristika zmanjšuje vpliv akustične povratne zanke, saj se protihrup širi od sekundarnega vira proč od primarnega vira.

• Prenos hrupa mimo sekundarnega vira se zmanjša tudi ko sistem za ADH ne deluje, • Dodatna digitalna obdelava signala ni potrebna. ©2011 Strojniški vestnik. Vse pravice pridržane.

Ključne besede: aktivno dušenje hrupa, sekundarni vir, prezračevalni kanal, hrup, ventilator

*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za strojništvo, Aškerčeva 6, 1000 Ljubljana, Slovenija, jurij.prezelj@fs.uni-lj.si

SI 89

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 90

Prejeto: 08.06.2010 Sprejeto: 09.09.2010

Zaznavanje napak na industrijskem prenosniku toplote s pomočjo modela Dejan Dragan* Univerza v Mariboru, Fakulteta za logistiko, Slovenija Za sodobne procesne industrije je značilna težnja po zagotavljanju visoke kakovosti proizvodov in varnosti ob čim nižjih stroških proizvodnje. Zgodnje odkrivanje nepravilnega delovanja sistemov je zato ključnega pomena za učinkovito vodenje sodobnih procesov. Pri tem sodobni nadzorni sistemi omogočajo avtomatsko nadziranje stanja opreme in lahko temeljijo tudi na modelu procesa. Le-ta predstavlja navidezni element instrumentalne opreme, s katerim se »razkrijejo« dodatne informacije o stanju procesa.

Tako je zagotovljen učinkovitejši nadzor delovanja procesa, zgodnje odkrivanje nepravilnosti pa zmanjša možnost izpadov proizvodnje.

Prispevek obravnava razvoj modela hladnejšega dela industrijskega prenosnika toplote, ki se nahaja v procesu sežiganja vulkanizacijskih plinov v gumarski industriji. Dani model naj bi služil kot pomoč detektorju pri zaznavanju različnih tipov napak v prenosniku toplote. Detektor je sestavni del nadzornega sistema za proces sežiganja, ki naj bi v celoti nadomestil obstoječi alarmni sistem in odpravil njegove pomanjkljivosti. Preprosti alarmni sistemi običajno odreagirajo šele pri večjih odpovedih, ki že lahko pripeljejo do izpada proizvodnje. Modeliranje hladnejšega dela prenosnika toplote sestoji iz dveh korakov. V prvem koraku uporabimo fizikalna znanja, s pomočjo katerih izpeljemo strukturo modela, pri čemer upoštevamo tudi razpoložljivo instrumentalno opremo, diagnostične zahteve in določen nabor predpostavk. Pri tem dobimo na osnovi nekaterih dodatnih hevrističnih predpostavk dokaj preprost model v zveznem časovnem prostoru z linearnimi parametri. V drugem koraku uporabimo za potrebe ocenjevanja parametrov procesa metodo najmanjših kvadratov in filtre spremenljivk stanj, pri čemer se ocenjeni parametri izračunajo na osnovi posnetih meritev izhodnih temperatur prenosnika ter pretoka vulkanizacijskega plina.

Namen razvitega modela je napovedovati razliko temperatur med hladnim in toplim delom na koncu prenosnika. Razlika med napovedjo modela in dejanskim izhodom procesa je osnova za učinkovito delovanje preprostega detektorja, razvitega za potrebe zaznavanja napak v hladnejšem delu prenosnika.

Ocenjeni parametri so v veliki meri neodvisni od pasovne širine filtrov stanj, kar lahko pri postavljanju modela olajša napore pri ustrezni izbiri pasovne širine filtra stanj. Pri uporabi detektorja za zaznavanje napak, ki temelji na razvitem modelu hladnejšega dela prenosnika toplote, je razvidno, da je sposoben zaznavati različne tipe napak veliko učinkoviteje od obstoječega alarmnega sistema. Tako lahko z detektorjem na osnovi zasnovanega modela pravočasno zaznamo šibke, komaj opazne napake, ki jih klasični sistem alarmiranja ne more. Napake, ki jih lahko klasični sistem sicer zazna, detektor zazna mnogo prej. Največja omejitev razvitega sistema za zaznavanje napak je v tem, da je v sedanji obliki uporaben le za hladnejši del prenosnika toplote. Sistem v prihodnosti bo potrebno razširiti na nadzor toplejšega dela prenosnika. Nadzorni sistem v sedanji obliki je primeren le za učinkovito zaznavanje napak, ne pa tudi za njihovo odkrivanje oz. izolacijo.

Pokazano je, da nabor določenih hevrističnih predpostavk pripelje do takšne strukture modela hladnejšega dela prenosnika toplote, ki kljub enostavnosti zagotavlja uporabne rezultate za potrebe zaznavanja napak. Rezultati ocenjevanja parametrov modela so v veliki meri neodvisni od pasovne širine filtra stanj, kar zmanjša napore pri modeliranju. Detektor, zasnovan na osnovi razvitega modela, je izredno učinkovit pri zaznavanju napak v hladnejšem delu prenosnka toplote. ©2011 Strojniški vestnik. Vse pravice pridržane.

Ključne besede: industrijski prenosnik toplote, zaznavanje napak, nadzor procesov, modeliranje, identifikacija, odkrivanje napak, detekcija napak na osnovi modela

SI 90

*Naslov avtorja za dopisovanje: Univerza v Mariboru, Fakulteta za logistiko, Mariborska c. 7, 3000 Celje, Slovenija, dejan.dragan@fl.uni-mb.si

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 91

Prejeto: 07.10.2010 Sprejeto: 17.02.2011

Numerične metode za modeliranje termomehanskega utrujanja Zaletelj, H. – Fajdiga, G. – Nagode, M. Henrik Zaletelj* – Gorazd Fajdiga – Marko Nagode Univerza v Ljubljani, Fakulteta za strojništvo, Slovenija

Eksperimentalno preverjanje vzdržljivosti strojnih delov, ki so izpostavljeni termomehanskim obremenitvam, zahteva napredno in drago merilno opremo. Pri določevanju napetostno-deformacijskega odziva materiala so zato vse bolj uveljavljene in uporabne numerične metode. Obstaja več različnih metod določevanja napetostno-deformacijskega odziva materiala. V članku so predstavljeni rezultati treh različnih numeričnih modelov. Predstavljeni so Chabocheov, Skeltonov in numerični model s Prandtlovimi operatorji, kjer je upoštevan elastoplastični odziv materiala ob neupoštevanju vpliva lezenja. Rezultati numeričnih modelov za različna obremenitvena stanja so primerjani z eksperimentalnimi rezultati.

Predstavljene so lastnosti modelov, njihove slabosti in možne izboljšave.

Predstavljeno delo obravnava napetostno-deformacijski odziv pri spremenljivih temperaturnih pogojih. Vplivi lezenja in prehodni vplivi, kot so utrjevanje ali mehčanje, niso upoštevani. Histerezne zanke predstavljajo stabilno stanje, kjer so enačbe za elastoplastični odziv primerne za popis napetostno-deformacijskega odziva materiala. Za popis numeričnih modelov je bil uporabljen material 9Cr2Mo. Termomehansko obremenitveno stanje je bilo kontrolirano z raztezkom in temperaturo. Čas obremenitvenega cikla je bil določen s hitrostjo spreminjanja obremenitve in velikostjo območja raztezka.

Območje raztezka je bilo 0,6 %, pri čemer se je temperatura znotraj enega obremenitvenega cikla spreminjala med 270 in 570 °C. Izvedenih je bilo devet različnih obremenitvenih primerov za primerjavo napetostno-deformacijskih histereznih krivulj. Natančno je bil obravnavan Chabocheov kinematični model, kjer so bili na osnovi eksperimentalnih podatkov določeni parametri modela pri različnih temperaturnih pogojih.

Pri določanju napetostno-deformacijskega odziva s Chabocheovim modelom se je pojavil nezaželjeni odmik histerezne krivulje ali t.i. ‘’ratcheting’’ efekt, kar gre pripisati akumuliranemu plastičnemu raztezku v posameznem obremenitvenem ciklu. Pojav odmika je odvisen tako od termomehanskega obremenitvenega cikla kot tudi od hitrosti spreminjanja temperature. Predstavljeni Chabocheov model ne upošteva eliminacije odmika histerezne zanke. Za še natančnejšo določitev histereznih krivulj bi bilo treba upoštevati dodatne kinematične parametre, kar pa zahteva dodatno delo in preračune.

Rezultati kažejo, da vsi trije numerični modeli zelo dobro popišejo izmerjene rezultate. Prednost numeričnega modela s Prandtlovimi operatorji je v tem, da model ne upošteva vpliva odmika histerezne zanke, kar je prednost v primerjavi s Chabocheovim modelom. Primerjava rezultatov je izvedena tudi za kompleksnejše obremenitvene primere. Pri tem je opazno, da oblika histereznih zank numeričnih modelov odstopa od eksperimentalnih rezultatov, kar je lahko tudi posledica dejstva, da temperature pri eksperimentalnem obremenjevanju ni možno spreminjati tako hitro in linearno kot pri numerični obravnavi temperaturnega vpliva.

V članku je predstavljena primerjava rezultatov treh različnih numeričnih modelov z eksperimentalnimi rezultati. Uporabljen je nelinearni kinematični model z upoštevanim vplivom temperature. Opazno je dobro ujemanje vseh treh numeričnih modelov. ©2011 Strojniški vestnik. Vse pravice pridržane.

Ključne besede: ciklično obremenjevanje, elastoplastična analiza, kinematično utrjevanje, napetostno-deformacijsko stanje, termomehansko utrujanje

*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za strojništvo, Aškerčeva 6, 1000 Ljubljana, Slovenija, henrik.zaletelj@fs.uni-lj.si

SI 91

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 92

Prejeto: 13.04.2010 Sprejeto: 12.01.2011

Analiza kritičnosti elementov zgloba jarmovega droga lahkega gospodarskega vozila Ćatić, D. – Jeremić, B. – Djordjević, Z. – Miloradović, N. Dobrivoje Ćatić* – Branislav Jeremić – Zorica Djordjević – Nenad Miloradović Univerza v Kragujevcu, Fakulteta za strojništvo, Srbija

Članek prikazuje rezultate teoretične študije in praktične uporabe kvantitativne analize načinov, učinkov in kritičnosti odpovedi (FMECA) na elementih strojnih sistemov. Analiza kritičnosti elementov je še zlasti pomembna pri sistemih, kot je na primer krmilni sistem motornega vozila, kjer bi odpovedi lahko ogrozile varnost ljudi. Odvisno od zahtev in možnosti pridobitve ustreznih podatkov je analizo FMECA mogoče izvesti kvantitativno ali kvalitativno.

Razvit je postopek za kvantitativno analizo FMECA za ocenjevanje kritičnosti elementov elektronskih sistemov. Pri uporabi te analize je uporabljena osnovna predpostavka, da je intenzivnost odpovedi elementov sistema konstantna, kar močno poenostavlja postopek ocenjevanja kritičnosti elementa. Uporaba te metode v primerih, ko je intenzivnost odpovedi funkcija časa, pa lahko privede do velikih napak in popačenja realne predstave o kritičnosti elementov. Lastnosti elementov mehanskih sistemov, povezane z intenzivnostjo napak, in objektivna nezmožnost določitve intenzivnosti odpovedi, kot funkcije časa za vsako možno odpoved elementa zahtevajo poseben pristop k tem problemom. Namen raziskave, je prilagoditi obstoječe metode kvantitativne analize FMECA za uporabo na elementih strojnih sistemov.

Namesto intenzivnosti odpovedi kot funkcijskega indeksa zanesljivosti je za vrednotenje kritičnosti elementa strojnega sistema uporabljen srednji čas med odpovedmi, določen na osnovi statističnih podatkov o času obratovanja do odpovedi. Razlike med strojnimi elementi z vidika možnosti popravil, deleža pojava odpovedi v celotnem obratovalnem času sistema, ki pripada določenemu konstrukcijskemu sestavu itd., zahtevajo poseben pristop k določitvi srednjega časa obratovanja do odpovedi. Le v tem primeru je možna primerjava kritičnosti elementov, izračunanih s tem parametrom.

Rezultat sta postopka za kvantitativno analizo FMECA elementov strojnih sistemov in računalniški program, ki omogoča učinkovito uporabo metode. Z uporabo predlagane metode in programa je bila na osnovi podatkov iz eksploatacije opravljena analiza kritičnosti načinov odpovedi elementov zgloba jarmovega droga lahkih gospodarskih vozil. Določitev elementov sistema, ki so kritični z vidika zanesljivosti in varnosti delovanja, ter uveljavitev ustreznih ukrepov za zmanjšanje njihove kritičnosti, je v splošnem optimalen način za povečanje zanesljivosti celotnega sistema.

V prihodnjih raziskavah je treba preučiti možnosti integracije metode FMECA z drugimi orodji in tehnikami sistema zagotavljanja kakovosti. Še posebno pomembno je, da je možno uporabiti tudi obstoječe zbirke podatkov o zgradbi sistemov, potencialnih načinih odpovedi elementov itd. Pri analizi zgloba jarmovega droga so potrebne nadaljnje raziskave za ugotavljanje dejavnikov, ki vplivajo na obrabo kritičnih sestavnih elementov, in ukrepi za izboljšanje kakovosti izdelka na podlagi rezultatov tehnoekonomske analize.

Z uporabo predlaganega postopka za kvantitativno analizo FMECA je mogoče dobiti realen vpogled v vpliv elementov na zanesljivo in varno obratovanje strojnega sistema. Uvedba korakov za določanje absolutne kritičnosti elementov omogoča razvrščanje elementov po kritičnosti brez dodatnih zahtevnih izračunov. Izračun absolutne kritičnosti je v celoti možen v primeru, ko je relativna kritičnost načinov odpovedi elementov enakomerno razporejena po kategorijah posledic. ©2011 Strojniški vestnik. Vse pravice pridržane.

Ključne besede: zanesljivost, FMECA, krmilni sistem lahkega gospodarskega vozila, zglob jarmovega droga

SI 92

*Naslov avtorja za dopisovanje: Univerza v Kragujevcu, Fakulteta za strojništvo, 6 Sestre Janjic, 34000 Kragujevac, Srbija, caticd@kg.ac.rs

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 93

Prejeto: 04.08.2010 Sprejeto: 05.01.2011

Matematični model avtoklava

Preglej, A. ‒ Karba, R. ‒ Steiner, I. ‒ Škrjanc, I. Aleksander Preglej1,* ‒ Rihard Karba2 ‒ Igor Steiner1 ‒ Igor Škrjanc2 1 INEA d.o.o., Slovenija 2 Univerza v Ljubljani, Fakulteta za elektrotehniko, Slovenija V prispevku je predstavljeno matematično modeliranje procesov segrevanja, hlajenja in spreminjanja tlaka v avtoklavu. Avtoklav je tlačna posoda cilindrične oblike in prostornine 5600 l, v katero se na kovinsko ploščo nad električne grelce postavijo polizdelki (kot so deli čolnov, kioskov, kopalnih kadi, letal, helikopterjev, avtomobilov, otroških igral itd.) iz kompozitnih materialov (kot so smola, kovina, keramika, steklo, karbon itd.), katere nato pod povišanim tlakom in pri določenih temperaturah segrevajo, da postanejo trši in kvalitetnejši.

Namen modeliranja je bil izgradnja podrobnega matematičnega modela, s katerim bomo lahko v okolju Matlab simulirali delovanje omenjenih procesov in izboljšali že izvedeno vodenje temperature (PFC-regulacija) in tlaka (ON-OFF regulacija) v avtoklavu. Nadalje bomo lahko na pridobljenem matematičnem modelu načrtovali in preizkušali različne napredne uni- in multivariabilne algoritme vodenja.

Matematični model avtoklava je zgrajen na podlagi teorij o prehodu toplote in spreminjanju tlaka.

Medtem ko spreminjanje tlaka ni kompliciran proces, saj obsega le masno ravnotežno enačbo, parcialne odvode gostote zraka in masni pretok, pa procesa segrevanja in hlajenja obsegata kompleksne fenomene toplotne prevodnosti, prestopnosti in sevanja, kot so energijska ravnotežna enačba s toplotnimi tokovi, koeficienti toplotne prehodnosti, prevodnosti in prestopnosti ter brezdimenzijska Nusseltova, Prandtlova in Reynoldsova števila. V matematičnem modelu so bile upoštevane nekatere poenostavitve, zato so bile uporabljene korelacije prehoda toplote ob ravnih ploščah. Večina podatkov je realnih in pridobljenih od proizvajalca avtoklava, kjer pa realnih podatkov ni bilo mogoče dobiti, je bila uporabljena metoda prilagajanja odziva modela izmerjenim podatkom z uporabo kriterijske funkcije vsote kvadratov pogreška.

Na ta način smo pridobili simulirane odzive, ki so zelo podobni posnetim realnim odzivom procesa, med njimi pa so opazna le manjša odstopanja. Tako lahko sklenemo, da je pridobljeni matematični model uporaben za načrtovanje različnih aplikacij procesnega vodenja.

Matematični model avtoklava je bil izdelan na podlagi enega sklopa meritev z realnega procesa pri določenih pogojih, nato pa validiran na podlagi drugega sklopa meritev pri drugih pogojih. Izkazalo se je, da dobljeni matematični model pri drugih pogojih nekoliko odstopa od posnetih realnih meritev, kar pomeni, da matematični model dobro deluje v okolici pogojev, na podlagi katerih je bil izdelan, za dobro pokrivanje celotnega območja delovanja procesa pa bi bilo potrebno matematični model še nekoliko preizkusiti in dodelati.

Glede na pregled najnovejše literature je v prispevku nov koncept celotnega matematičnega modela delovanja procesov znotraj avtoklava, saj so se številni avtorji v svojih prispevkih že ukvarjali s procesi toplotne prehodnosti, nekateri tudi s procesi toplotne prehodnosti znotraj avtoklava, vendar so se le-ti osredotočili na prehod in distribucijo toplote znotraj kompozitnega materiala in določitev optimalnega temperaturnega profila. Drugi so toplotno prestopnost določili eksperimentalno in posebej upoštevali toplotno sevanje, ki v našem matematičnem modelu ni zanemarjeno, ampak upoštevano v koeficientih Nusseltovih števil. Že sam postopek izgradnje matematičnega modela je lahko uporaben za modeliranje podobnih procesov, dobljeni matematični model avtoklava in ugotovitve pa so uporabni za vse, ki se ukvarjajo z različnimi simulacijami in načrtovanjem najrazličnejših oblik vodenja enakih ali podobnih procesov. ©2011 Strojniški vestnik. Vse pravice pridržane.

Ključne besede: avtoklav, matematični model, prehod toplote, toplotna prevodnost, toplotna prestopnost, temperatura, tlak

*Naslov avtorja za dopisovanje: INEA d.o.o., Stegne 11, 1000 Ljubljana, Slovenija, aleksander.preglej@inea.si

SI 93

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 94

Prejeto: 03.11.2009 Sprejeto: 23.03.2011

Teoretična analiza vzmetne konstante in efektivne mase za polkrožno upognjeni nosilec v pospeškomeru MEMS Wong, W.C. ‒ Azid, I.A. ‒ Majlis, B.Y. Wai Chi Wong1,2,* ‒ Ishak Abdul Azid1 ‒ Burhanuddin Yeop Majlis3 1Univerza Sains, Šola za strojništvo, Malezija 2Univerza Tenaga Nasional, Tehnična fakulteta, Oddelek za strojništvo, Malezija 3Univerza Kebangsaan, Inštitut za mikroinženiring in makroelektroniko, Malezija

Namen tega članka je teoretična izpeljava vzmetne konstante in efektivne mase za polkrožno upognjeni nosilec, ki bosta uporabljeni za napovedovanje delovanja tovrstne konstrukcije.

Vzmetna konstanta in resonančna frekvenca, kot predmeta te raziskave, sta dve najpomembnejši funkcionalnosti, ki ju zahteva konstrukcija vsake naprave MEMS s premičnimi deli. Zlasti senzorji in aktuatorji pogosto zahtevajo posebno vzmetno konstanto in resonančno frekvenco za zagotavljanje zmogljivosti in ponovljivosti delovanja. Vzmetna konstanta in efektivna masa, ki upoštevata tako materialne lastnosti kot fizično geometrijo, določata občutljivost in resonančno frekvenco pospeškomera MEMS. Občutljivost pospeškomera je merilo za odmik v odvisnosti od pospeška. Resonančna frekvenca pa po drugi strani določa pasovno širino pospeškomera. Obešalni nosilec je lahko izveden na različne načine, odvisno od namena uporabe pospeškomera MEMS. Strukturna analiza obešalnega nosilca pri pospeškomerih s prsti, nanizanimi kot pri glavniku, doslej še ni bila opravljena. Analitična izpeljava teh dveh parametrov je na voljo samo za enostavne zasnove, kot so ravni nosilci. Avtorjem ni znano, da bi bila formula za polkrožno upognjene nosilce že izpeljana.

Polkrožno upognjeni nosilec je za analizo mogoče razstaviti na tri komponente: dva enostransko vpeta nosilca in polovični obroč. Vzmetna konstanta se določi s pomočjo deformacijske energije in Castiglianovega teorema o odmikih, medtem ko se efektivna masa določi po Rayleighovem principu.

Vzmetna konstanta in efektivna masa se rešujeta po posameznih komponentah in nato kombinirata po metodi superpozicije. Polkrožno upognjeni nosilec v pospeškomeru MEMS je bil analiziran tudi s paketom za analize po metodi končnih elementov ANSYS® 8.1. Rezultat simulacije bo uporabljen za primerjavo z analitičnim rezultatom.

Izpeljana je bila efektivna vzmetna konstanta za polkrožno upognjeni nosilec 1 1  L3 3 (1 + µ ) L 24r 3  π 1 1 = + + 3  − +  5w ke Et  2 w3 wr  16 π 2π

2  3π L r  + .  3   4 wr 

Vzmetna konstanta in efektivna masa, ki ju dajejo analitično izpeljane enačbe, se dobro ujemata z rezultati simulacije MKE; razlika je manjša od 5%. Zato je mogoče zaključiti, da je z izpeljanimi analitičnimi formulami mogoče pridobiti efektivno maso in vzmetno konstanto polkrožno upognjenega nosilca v pospeškomeru izvedbe z obešenimi prsti, nanizanimi kot pri glavniku.

Glavni prispevek tega članka je možnost napovedovanja vedenja polkrožno upognjenega nosilca pri pospeškomeru MEMS z izpeljanimi enačbami za vzmetno konstanto in efektivno maso. Rezultati, izračunani z uporabo teh izpeljanih formul, se dobro ujemajo z rezultati programske opreme za MKE ANSYS. ©2011 Strojniški vestnik. Vse pravice pridržane.

Ključne besede: efektivna masa, upognjen nosilec, pospeškomer MEMS, vzmetna konstanta, deformacijska energija

SI 94

*Naslov avtorja za dopisovanje: Univerza Sains, Šola za strojništvo 14300 Nibong Tebal, Seberang Perai Selatan, Pulau Pinang, Malezija, WaiChi@uniten.edu.my

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 95-96 Navodila avtorjem

Navodila avtorjem Članke pošljite na naslov: Strojniški vestnik Journal of Mechanical Engineering Aškerčeva 6, 1000 Ljubljana, Slovenija Tel.: 00386 1 4771 137 Faks: 00386 1 2518 567 E-mail: info@sv-jme.eu strojniski.vestnik@fs.uni-lj.si Članki morajo biti napisani v angleškem jeziku. Strani morajo biti zaporedno označene.

Prispevki so lahko dolgi največ 10 strani. Daljši članki so lahko v objavo sprejeti iz posebnih razlogov, katere morate navesti v spremnem dopisu.

Kratki članki naj ne bodo daljši od štirih strani.

Navodila so v celoti na voljo v rubriki “Informacija za avtorje” na spletni strani revije: http://en.sv-jme.eu/ Prosimo vas, da članku priložite spremno pismo, ki naj vsebuje: 1. naslov članka, seznam avtorjev ter podatke avtorjev; 2. opredelitev članka v eno izmed tipologij; izvirni znanstveni (1.01), pregledni znanstveni (1.02) ali kratki znanstveni članek (1.03); 3. opredelitev, da članek ni objavljen oziroma poslan v presojo za objavo drugam; 4. zaželeno je, da avtorji v spremnem pismu opredelijo ključni doprinos članka; 5. predlog dveh potencialnih recenzentov, ter kontaktne podatke recenzentov. Navedete lahko tudi razloge, zaradi katerih ne želite, da bi določen recenzent recenziral vaš članek.

OBLIKA ČLANKA Članek naj bo napisan v naslednji obliki: Naslov, ki primerno opisuje vsebino članka.

Povzetek, ki naj bo skrajšana oblika članka in naj ne presega 250 besed. Povzetek mora vsebovati osnove, jedro in cilje raziskave, uporabljeno metodologijo dela, povzetek rezultatov in osnovne sklepe.

- Uvod, v katerem naj bo pregled novejšega stanja in zadostne informacije za razumevanje ter pregled rezultatov dela, predstavljenih v članku.

- Teorija.

- -

-

Eksperimentalni del, ki naj vsebuje podatke o postavitvi preskusa in metode, uporabljene pri pridobitvi rezultatov.

- Rezultati, ki naj bodo jasno prikazani, po potrebi v obliki slik in preglednic.

- Razprava, v kateri naj bodo prikazane povezave in posplošitve, uporabljene za pridobitev rezultatov. Prikazana naj bo tudi pomembnost rezultatov in primerjava s poprej objavljenimi deli. (Zaradi narave posameznih raziskav so lahko rezultati in razprava, za jasnost in preprostejše bralčevo razumevanje, združeni v eno poglavje.)

- Sklepi, v katerih naj bo prikazan en ali več sklepov, ki izhajajo iz rezultatov in razprave.

- Literatura, ki mora biti v besedilu oštevilčena zaporedno in označena z oglatimi oklepaji [1] ter na koncu članka zbrana v seznamu literature.

Enote - uporabljajte standardne SI simbole in okrajšave. Simboli za fizične veličine naj bodo v ležečem tisku (npr. v, T, n itd.). Simboli za enote, ki vsebujejo črke, naj bodo v navadnem tisku (npr. ms1, K, min, mm itd.)

Okrajšave naj bodo, ko se prvič pojavijo v besedilu, izpisane v celoti, npr. časovno spremenljiva geometrija (ČSG).

Pomen simbolov in pripadajočih enot mora biti vedno razložen ali naveden v posebni tabeli na koncu članka pred referencami.

Slike morajo biti zaporedno oštevilčene in označene, v besedilu in podnaslovu, kot sl. 1, sl. 2 itn. Posnete naj bodo v ločljivosti, primerni za tisk, v kateremkoli od razširjenih formatov, npr. BMP, JPG, GIF. Diagrami in risbe morajo biti pripravljeni v vektorskem formatu, npr. CDR, AI. Vse slike morajo biti pripravljene v črnobeli tehniki, brez obrob okoli slik in na beli podlagi.

Ločeno pošljite vse slike v izvirni obliki Pri označevanju osi v diagramih, kadar je le mogoče, uporabite označbe veličin (npr. t, v, m itn.). V diagramih z več krivuljami, mora biti vsaka krivulja označena. Pomen oznake mora biti pojasnjen v podnapisu slike.

Tabele naj imajo svoj naslov in naj bodo zaporedno oštevilčene in tudi v besedilu poimenovane kot Tabela 1, Tabela 2 itd.. Poleg fizikalne veličine, npr t (v ležečem tisku), mora biti v oglatih oklepajih navedena tudi enota. V tabelah naj se ne podvajajo podatki, ki se nahajajo v besedilu.

SI 95

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 95-96

Potrditev sodelovanja ali pomoči pri pripravi članka je lahko navedena pred referencami. Navedite vir finančne podpore za raziskavo.

REFERENCE Seznam referenc MORA biti vključen v članek, oblikovan pa mora biti v skladu s sledečimi navodili. Navedene reference morajo biti citirane v besedilu. Vsaka navedena referenca je v besedilu oštevilčena s številko v oglatem oklepaju (npr. [3] ali [2] do [6] za več referenc). Sklicevanje na avtorja ni potrebno.

Reference morajo biti oštevilčene in razvrščene glede na to, kdaj se prvič pojavijo v članku in ne po abecednem vrstnem redu. Reference morajo biti popolne in točne. Vse neangleške oz. nenemške naslove je potrebno prevesti v angleški jezik z dodano opombo (in Slovene) na koncu Navajamo primere: Članki iz revij: Priimek 1, začetnica imena, priimek 2, začetnica imena (leto). Naslov. Ime revije, letnik, številka, strani.

[1] Zadnik, Ž., Karakašič, M., Kljajin, M., Duhovnik, J. (2009). Function and Functionality in the Conceptual Design Process. Strojniški vestnik – Journal of Mechanical Engineering, vol. 55, no. 7-8, p. 455-471.

Ime revije ne sme biti okrajšano. Ime revije je zapisano v ležečem tisku.

Knjige: Priimek 1, začetnica imena, priimek 2, začetnica imena (leto). Naslov. Izdajatelj, kraj izdaje [2] Groover, M. P. (2007). Fundamentals of Modern Manufacturing. John Wiley & Sons, Hoboken.

Ime knjige je zapisano v ležečem tisku.

Poglavja iz knjig: Priimek 1, začetnica imena, priimek 2, začetnica imena (leto). Naslov poglavja. Urednik(i) knjige, naslov knjige. Izdajatelj, kraj izdaje, strani.

[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. Članki s konferenc: Priimek 1, začetnica imena, priimek 2, začetnica imena (leto). Naslov. Naziv konference, strani.

[4] Štefanić, N., Martinčević-Mikić, S., Tošanović, N. (2009). Applied Lean System in Process Industry. MOTSP 2009 Conference Proceedings, p. 422-427.

SI 96

Standardi: Standard (leto). Naslov. Ustanova. Kraj.

[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.

Spletne strani: Priimek, Začetnice imena podjetja. Naslov, z naslova http://naslov, datum dostopa.

[6] Rockwell Automation. Arena, from http://www. arenasimulation.com, accessed on 2009-09-27.

RAZŠIRJENI POVZETEK Ko je članek sprejet v objavo, avtorji pošljejo razširjeni povzetek na eni strani A4 (približno 3.000 - 3.500 znakov). Navodila za pripravo razširjenega povzetka so objavljeni na spletni strani http://sl.svjme.eu/informacije-za-avtorje/.

AVTORSKE PRAVICE Avtorji v uredništvo predložijo članek ob predpostavki, da članek prej ni bil nikjer objavljen, ni v postopku sprejema v objavo drugje in je bil prebran in potrjen s strani vseh avtorjev. Predložitev članka pomeni, da se avtorji avtomatično strinjajo s prenosom avtorskih pravic SV-JME, ko je članek sprejet v objavo. Vsem sprejetim člankom mora biti priloženo soglasje za prenos avtorskih pravic, katerega avtorji pošljejo uredniku. Članek mora biti izvirno delo avtorjev in brez pisnega dovoljenja izdajatelja ne sme biti v katerem koli jeziku objavljeno drugje.

Avtorju bo v potrditev poslana zadnja verzija članka. Morebitni popravki morajo biti minimalni in poslani v kratkem času. Zato je pomembno, da so članki že ob predložitvi napisani natančno.

Avtorji lahko stanje svojih sprejetih člankov spremljajo na http://en.sv-jme.eu/.

PLAČILO OBJAVE Domači avtorji vseh sprejetih prispevkov morajo za objavo plačati prispevek, le v primeru, da članek presega dovoljenih 10 strani oziroma za objavo barvnih strani v članku, in sicer za vsako dodatno stran 20 EUR ter dodatni strošek za barvni tisk, ki znaša 90,00 EUR na stran.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 97-99 Osebne objave

Doktorati, magisteriji, specialistična dela in diplome

DOKTORAT ZNANOSTI

MAGISTERIJ ZNANOSTI

Na Fakulteti za strojništvo Univerze v Ljubljani je z uspehom obranil svojo doktorsko disertacijo: dne 19. maja 2011 Uroš ZUPANC z naslovom: »Integriteta površine po mehanskem utrjevanju aluminijevih zlitin« (mentor: prof. dr.

Janez Grum); V doktorskem delu so predstavljene raziskave učinkov hladnega mehanskega utrjevanja površine visokotrdnostne aluminijeve zlitine AlZn5,5MgCu (ENAW 7075). Mikroplastično utrjevanje materiala ob trkih kroglic na površino poviša trdoto površinskega sloja in v materialu inducira zaostale tlačne napetosti. Enakomerno utrjen površinski sloj zavira nastanek in širjenje utrujenostnih razpok. Zaostale tlačne napetosti namreč znižujejo koncentracije nateznih napetosti na površini pri utrujanju materiala in tako povišajo njegovo dinamično trdnost. V delu so obravnavane korelacije med pogoji hladnega utrjevanja površine, integriteto površine materiala po utrjevanju ter uporabniškimi lastnostmi površinsko utrjenih vzorcev, ki so izpostavljeni dinamičnim obremenitvam in degradaciji materiala pri korozijski izpostavljenosti. V raziskavi integritete površine so zajete meritve hrapavosti površine po hladnem utrjevanju in po izpostavljenosti koroziji, mikrostrukturna analiza, meritve zaostalih napetosti, testi utrujanja materiala ter elektrokemijske obstojnosti pri različnih korozijskih pogojih. Slabša korozijska obstojnost izbrane aluminijeve zlitine v vlažnem okolju s klorovimi ioni vpliva na drastičen padec njene odpornosti na utrujanje. Pri dinamično obremenjenem materialu predstavljajo lokalne korozijske poškodbe mesto nastanka in širjenja utrujenostne razpoke. Eksperimentalni rezultati dokazujejo, da mikroplastično utrjevanje površinskega sloja ob trkih kroglic poveča odpornost materiala na utrujanje tudi v korozivnem okolju, obenem pa ne poslabša njegove elektrokemijske obstojnosti.

Na Fakulteti za strojništvo Univerze v Ljubljani sta z uspehom zagovarjala svoje magistrsko delo: dne 13. maja 2011 Dejan NOŽAK z naslovom: »Odcepitev toka v nadzvočni konični šobi« (mentor: doc.dr. Tadej Kosel); dne 17. maja 2011 Suvad BAJRIĆ z naslovom: »Optimizacija termo-hidravličnega distribucijskega sistema« (mentor: prof. dr. Iztok Žun). * Na Fakulteti za strojništvo Univerze v Mariboru sta z uspehom zagovarjala svoje magistrsko delo: dne 17. maja 2011 Boštjan OGRIZEK z naslovom: »Napredno razvijanje in načrtovanje hladilnih aparatov« (mentor: prof. dr. Andrej Polajnar); dne 17. maja 2011 Gregor ŽALEC z naslovom: »Razvoj in načrtovanje proizvodnje ohišja gonila« (mentor: prof. dr. Andrej Polajnar).

SPECIALISTIČNO DELO Na Fakulteti za strojništvo Univerze v Mariboru je z uspehom zagovarjal svoje specialistično delo: dne 20. maja 2011 Klemen PETRIČ z naslovom: »Optimizacija komutatorja za elektromotor pralnega stroja« (mentor: prof. dr.

Ivan Anžel).

DIPLOMIRALI SO Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv univerzitetni diplomirani inženir strojništva: dne 25. maja 2011: David BEZEK z naslovom: »Nadzor sistema za brizganje in lakiranje leč avtomobilskih žarometov v realnem času« (mentor: izr. prof. dr.

Peter Butala); SI 97

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 97-99

Blaž PALADIN z naslovom: »Vrednotenje konstrukcijskih rešitev tečaja pokrova motorja« (mentor: prof. dr. Boris Štok); Jure SEDEJ z naslovom: »Mehansko vrednotenje konstrukcijske rešitve grelnega elementa z vidika njegove tehnološke izdelave« (mentor: prof. dr. Boris Štok, somentor: doc. dr.

Nikolaj Mole); Blaž ŠTRANCAR z naslovom: »Robotizirano zarezovanje testenih štruc na osnovi merilnega sistema s kamero« (mentor: prof. dr. Alojzij Sluga); dne 27. maja 2011: Matija KRAJNC z naslovom: »Hidravlično preizkuševališče za zvezno delujoče potne ventile« (mentor: doc. dr. Jožef Pezdirnik); Mohor KUNŠIČ z naslovom: »Zagotavljanje notranjega okolja, potrebnega za kakovostno konzervacijo kulturne dediščine« (mentor: prof. dr. Vincenc Butala); Urban PAVLOVČIČ z naslovom: »Laserski triangulacijski sistem za določanje volumna in barve ran« (mentor: prof. dr. Janez Možina, somentor: doc. dr. Matija Jezeršek).

Teo REBERŠEK z naslovom: »Možnost nadgraditve funkcij centralno nadzornega sistema klimatskih sistemov v stavbi« (mentor: prof. dr.

Vincenc Butala, somentor: doc. dr. Matjaž Prek); Miha POLAJNAR z naslovom: »Geometrijska optimizacija vitke konzole pri problemih bočne stabilnosti« (mentor: prof. dr.

Franc Kosel, somentor: doc. dr. Tomaž Videnič); Gregor POVŠE z naslovom: »Uvajanje metod LCA v pedagoško in raziskovalno delo« (mentor: prof. dr. Sašo Medved); Matjaž ZAPUŠEK z naslovom: »Kozičasta prenosnica za skladišče širine 75 m« (mentor: izr. prof. dr. Janez Kramar). * Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv univerzitetni diplomirani inženir strojništva: dne 4. maja 2011: Sebastijan VRABL z naslovom: »Parametrična analiza sušenja v razpršilnem postroju s poudarkom na porabi toplote« (mentor: prof. dr. Matjaž Hriberšek); dne 20. maja 2011: Matej GRACEJ z naslovom: »Določanje optimalne oblike odrezka pri struženju Al-zlitin« SI 98

(mentor: prof. dr. Franci Čuš, somentor: dr. Peter Cvahte); dne 26. maja 2011: Matic DRAŽNIK z naslovom: »Ocena varnosti robotske celice z vgrajenim robotom Acma XR701 na podlagi varnostnih standardov ISO 10218-1:2006 in ISO 10218-2« (mentor: izr. prof. dr. Karl Gotlih, mentor FERI: prof. dr. Riko Šafarič); Ivan IGNATOVSKI z naslovom: »Optimizacija izdelave dvokrakih vzmeti« (mentor: izr. prof. dr. Miran Brezočnik, somentor: prof. dr. Anton Hauc); Gregor KAUČIČ z naslovom: »Uporaba alternativnih goriv iz trdnih odpadkov« (mentor: prof. dr. Niko Samec, somentor: dr. Filip Kokalj); Jože SENICA z naslovom: »Računalniško podprto konstruiranje in simulacija hidravlične vpenjalne naprave« (mentor: izr. prof. dr. Miran Ulbin). * Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva: dne 9. maja 2011: Uroš BODLAJ z naslovom: »Pregled fenskih lopatic turbofenskega motorja V2500« (mentor: doc. dr. Tadej Kosel); Jožef POLLAK z naslovom: »Analiza instrumentalnega vodenja letal s pomočjo mobilne radio-navigacijske oddajne enote « (mentor: pred.

Miha Šorn, somentor: doc. dr. Tadej Kosel); Marko ŠINK z naslovom: »Ekonomsko vrednotenje energetske oskrbe stanovanjskega naselja« (mentor: prof. dr. Sašo Medved); dne 12. maja 2011: Matjaž BAJEC z naslovom: »Razvoj orodja za rezanje cevi« (mentor: izr. prof. dr.

Zlatko Kampuš); Sebastjan RAVNIKAR z naslovom: »Uporaba uporovnih merilnih lističev v merilni tehtnici za tehtanje letal« (mentor: izr. prof. dr.

Ivan Bajsić); dne 15. maja 2011: Stojan ČRV z naslovom: »Uporaba metodologije Šest sigma v procesu proizvodnje hidrostata« (mentor: prof. dr. Mirko Soković); Aron MRMOLJA z naslovom: »Varjenje aluminija in njegovih zlitin ter zagotavljanje kakovosti varilskih del« (mentor: prof. dr. Janez Tušek);

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)6, SI 97-99

Igor VEBER z naslovom: »Izdelava orodja za brizganje leče stranske luči pri avtomobilu« (mentor: prof. dr. Janez Kopač, somentor: doc. dr.

Franci Pušavec). * Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv diplomirani inženir strojništva: dne 4. maja 2011: Marko MEŽNAR z naslovom: »Numerična analiza tokovno-toplotnih razmer v avtodomu« (mentor: prof. dr. Matjaž Hriberšek); dne 10. maja 2011: Zoran KOVAČIČ z naslovom: »3D-modeliranje orodja za litje ohišja diferenciala« (mentor: izr. prof. dr. Miran Brezočnik, somentor: prof. dr. Jože Balič); dne 26. maja 2011: Matjaž BAJGOT z naslovom: »Uvajanje sistema vodenja kakovosti v skladu s standardom SIST EN ISO/IEC 17020« (mentor: izr. prof. dr. Bojan Ačko, somentor: izr. prof. dr. Borut Buchmeister);

Robert DOLER z naslovom: »Integrirani sistemi vodenja kakovosti v podjetju HTZ Velenje« (mentor: izr. prof. dr. Bojan Ačko, somentor: izr. prof. dr. Borut Buchmeister); Andreja HANŽEKOVIČ z naslovom: »Konstrukcija nosilca antene NAPS« (mentor: doc. dr. Aleš Belšak, somentor: izr. prof. dr.

Stanislav Pehan); Jože OČKO z naslovom: »Uvedba vitke linije za sestavo komolčnika Opel Astra Delta II« (mentor: prof. dr. Andrej Polajnar, somentorica: doc. dr. Nataša Vujica Herzog); Aleš PETROVIČ z naslovom: »Načrtovanje in vodenje proizvodnje v Orodjarni Koban d.o.o.« (mentor: izr. prof. dr. Borut Buchmeister, somentor: doc. dr. Iztok Palčič); Denis RAZBORŠEK z naslovom: »Analiza TVO jekla P92 namenjenega za delo na povišanih temperaturah v kotlogradnji« (mentor: izr. prof. dr. Vladimir Gliha, somentor: dr. Tomaž Vuherer); Aleš ZEVNIK z naslovom: »Preiskava TVO jekla P91 za delo na povišanih temperaturah po toplotni obdelavi« (mentor: izr. prof. dr.

Vladimir Gliha, somentor: dr. Tomaž Vuherer).

SI 99

Platnica SV-JME 57(2011)6_kor4.pdf 2 20.6.2011 14:45:31

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 Co-Editor Borut Buchmeister University of Maribor Faculty of Mechanical Engineering, Slovenia Technical Editor Pika Škraba University of Ljubljana Faculty of Mechanical Engineering, Slovenia

C

M

Y

CM

MY

CY

CMY

K

Editorial Office University of Ljubljana (UL) Faculty of Mechanical Engineering SV-JME Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386-(0)1-4771 137 Fax: 386-(0)1-2518 567 E-mail: info@sv-jme.eu http://www.sv-jme.eu Founders and Publishers University of Ljubljana (UL) Faculty of Mechanical Engineering, Slovenia University of Maribor (UM) Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia Chamber of Commerce and Industry of Slovenia Metal Processing Industry Association Cover: Top: Test rig for measurement of energetic, cavitational and dynamic characteristics of Francis turbines. For better flow observation the conical part of the draft tube was made of plexiglass. At part load cavitating vortex rope can be observed and pressure pulsations caused by its rotation were measured. Bottom: The same shape of the rotating vortex rope was obtained by numerical simulation. Also pressure distribution on the runner and streamlines in the inlet part of the draft tube are presented.

Image courtesy: Turboinštitut d.d., Slovenia

ISSN 0039-2480 © 2011 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 Book Agency.

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 Mech. Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mech. Engineering, Slovenia Imre Felde, Bay Zoltan Inst. for Mater. Sci. and Techn., Hungary Jože Flašker, UM, Faculty of Mech. Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mech. 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 Mech. Engineering, Slovenia Franc Kosel, UL, Faculty of Mech. Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mech. 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 Mech. Engineering, Slovenia Leopold Škerget, UM, Faculty of Mech. 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 President of Publishing Council Jože Duhovnik UL, Faculty of Mechanical Engineering, Slovenia Print Tiskarna Present d.o.o., Ižanska cesta 383, Ljubljana, Slovenia General information Strojniški vestnik – The 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/.

You can advertise on the inner and outer side of the back cover of the magazine. The authors of the published papers are invited to send photos or pictures with short explanation for cover content.

We would like to thank the reviewers who have taken part in the peer-review process.

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.

57 (2011) 1 6

Platnica SV-JME 57(2011)6_kor4.pdf 1 20.6.2011 14:45:31

Since 1955

C

M

Y

CM

MY

CY

CMY

Journal of Mechanical Engineering - Strojniški vestnik

K

6 year 2011 volume 57 no.