Journal of Mechanical Engineering 2014 11 by Darko Svetak

http://www.sv-jme.eu

60 (2014) 11

Since 1955

Papers

685

Boštjan Novak, Aleš Babnik, Janez Možina, Matija Jezeršek: Three-Dimensional Foot Scanning System with a Rotational Laser-Based Measuring Head

694

Emiliano Pipitone, Stefano Beccari, Marco Cammalleri, Giuseppe Genchi: Experimental Model-Based Linearization of a S.I. Engine Gas Injector Flow Chart

709

Andrzej Zbrowski, Krzysztof Matecki: The Use of Computed Tomography to Analyse Grinding Smudges and Subsurface Defects in Roller Bearing Rings

716

Changyi Liu, Gui Wang, Matthew S Dargusch: Mechanics and Dynamics of Helical Milling Operations

725

Daniel Kozelj, Zoran Kapelan, Gorazd Novak, Franci Steinman: Investigating Prior Parameter Distributions in the Inverse Modelling of Water Distribution Hydraulic Models

747

Nima Jafarzadeh Aghdam, Soran Hassanifard, Mir Mohammad Ettefagh, Arvin Nanvayesavojblaghi: Investigating Fatigue Life Effects on the Vibration Properties in Friction Stir Spot Welding Using Experimental and Finite Element Modal Analysis

742

Uroš Karadžić, Vladimir Bulatović, Anton Bergant: Valve-Induced Water Hammer and Column Separation in a Pipeline Apparatus

Journal of Mechanical Engineering - Strojniški vestnik

Contents

11 year 2014 volume 60 no.

Strojniški vestnik Journal of Mechanical Engineering

Strojniški vestnik – Journal of Mechanical Engineering (SV-JME) Aim and Scope The international journal publishes original and (mini)review articles covering the concepts of materials science, mechanics, kinematics, thermodynamics, energy and environment, mechatronics and robotics, fluid mechanics, tribology, cybernetics, industrial engineering and structural analysis. The journal follows new trends and progress proven practice in the mechanical engineering and also in the closely related sciences as are electrical, civil and process engineering, medicine, microbiology, ecology, agriculture, transport systems, aviation, and others, thus creating a unique forum for interdisciplinary or multidisciplinary dialogue. The international conferences selected papers are welcome for publishing as a special issue of SV-JME with invited co-editor(s). Editor in Chief Vincenc Butala University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

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

Founding Editor Bojan Kraut University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

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

University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Vice-President of Publishing Council Jože Balič University of Maribor, Faculty of Mechanical Engineering, Slovenia

Cover: Badly-fitting shoes are one of the major causes of pain, foot related diseases and injuries of the feet. Therefore three-dimensional measurements of the feet is crucial for the correct design and selection of shoes. Upper figure presents a new system for 3D footshape measurements which is based on the laser-multiple-line-triangulation principle. Example of unprocessed raw measurement and the extracted feet is shown below. Courtesy: Studio Miklavc (upper image) and University of Ljubljana, Faculty of Mechanical Engineering (lower images)

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11 Contents

Contents Strojniški vestnik - Journal of Mechanical Engineering volume 60, (2014), number 11 Ljubljana, November 2014 ISSN 0039-2480 Published monthly

Papers Boštjan Novak, Aleš Babnik, Janez Možina, Matija Jezeršek: Three-Dimensional Foot Scanning System with a Rotational Laser-Based Measuring Head Emiliano Pipitone, Stefano Beccari, Marco Cammalleri, Giuseppe Genchi: Experimental Model-Based Linearization of a S.I. Engine Gas Injector Flow Chart Andrzej Zbrowski, Krzysztof Matecki: The Use of Computed Tomography to Analyse Grinding Smudges and Subsurface Defects in Roller Bearing Rings Changyi Liu, Gui Wang, Matthew S Dargusch: Mechanics and Dynamics of Helical Milling Operations Daniel Kozelj, Zoran Kapelan, Gorazd Novak, Franci Steinman: Investigating Prior Parameter Distributions in the Inverse Modelling of Water Distribution Hydraulic Models Nima Jafarzadeh Aghdam, Soran Hassanifard, Mir Mohammad Ettefagh, Arvin Nanvayesavojblaghi: Investigating Fatigue Life Effects on the Vibration Properties in Friction Stir Spot Welding Using Experimental and Finite Element Modal Analysis Uroš Karadžić, Vladimir Bulatović, Anton Bergant: Valve-Induced Water Hammer and Column Separation in a Pipeline Apparatus

685 694 709 716 725 735 742

Received for review: 2014-05-09 Received revised form: 2014-07-09 Accepted for publication: 2014-07-23

Three-Dimensional Foot Scanning System with a Rotational Laser-Based Measuring Head Novak, B. – Babnik, A. – Možina, J. – Jezeršek, M. Boštjan Novak1 – Aleš Babnik2 – Janez Možina2 – Matija Jezeršek2,* 2University

1Alpina d.o.o., Slovenia of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Three-dimensional (3D) measurements of the feet is crucial for the correct design and selection of shoes. Badly-fitting shoes are one of the major causes of pain, foot related diseases and injuries of the feet. This article presents a new system for 3D foot-shape measurements which is based on the laser-multiple-line-triangulation principle. The main part of a system is the measuring head comprising a three laser lines projection unit and two cameras, which rotate around the centre of the platform that the customer stands on, and measures both feet simultaneously. The developed software analyzes the different foot dimensions and suggests the most suitable model and size of a shoe from a database to the customer. Validation experiments have been presented to demonstrate the measuring precision of the system. The results show that the standard deviation for all feet dimensions is better than 0.6mm in case of test objects. Keywords: 3D foot measurement, laser-multiple-line-triangulation, foot dimensions, footwear fit

0 INTRODUCTION Knowledge of the exact three-dimensional (3D) shape of the feet is extremely important for the footwear industry, since the correct fit between the shoe and the foot is an important comfort factor. Badly fitting shoes are the major cause of pain, foot related diseases and injuries [1] to [4]. 3D feet measurement therefore provides state of the art data for: (i) producing an adaptive design of a standard shoe; (ii) personalizing the shoe to individual foot dimensions; (iii) creating a better fitting for standard mass produced shoes and (iv) determining the best-fitting shoe for customers in a shop selling standard shoes. Traditionally, foot measuring techniques used callipers and measuring tapes. Simple mechanical devices were developed later, such as the Brannock device [5] and [6] or the Ritz stick length measuring device [7], which only measures a few of the most important foot dimensions, such as foot length, arch length and foot width. The next step in the foot measurement evolution are two-dimensional foot scanners, which measure the shape and dimensions of the footwear using a photo capturing in a flat plane [8]. The major drawback of these systems is the lack of foot height measurement. So-called two-and-half-dimension scanners, which measure foot contours from the top and side view [9] to [11] enable extraction of foot’s length, width and height in any cross-section. But the girths of the crosssections, their curvature and local foot deformation are still not known. The complete 3D scanning systems are the logical progression to the mentioned limitations of the earlier systems. These systems are mainly based on the laser triangulation principle. One of the first

such systems was the measuring system made by the VORUM research corporation [12]. It has four laserline projectors and eight cameras, which measure the lateral cross-section of the foot, defined by the projected light plane. The entire 3D foot shape is further scanned by moving the complete projectorscameras assembly along the longitudinal foot axis (from the toes to the heel). Since the complexity of this system is relatively high due to the many cameras and lasers, which should be actuated during the measurement procedure, the price of the system is rather high. Similar system, but low cost, was later developed by Kouchi and Mochimaru [13]. Such systems are therefore mainly used in research and medical applications [14] to [17]. There are many attempts to overcome the economical drawback of above solution, which is still a gold standard in terms of measurement precision. One direction of development was to eliminate the scanning procedure. A representative of this technique is described in [18] where the foot is measured by four stationary measuring modules based on laser multiple line triangulation. The methods based on the fringe projection technique are also presented in [19] to [21]. The major improvement regarding the scanning techniques is the shorter measuring time, which in principle enables the study of the foot shape during walking [18] and [22] to [25]. The second line of development went into the reduction of the number of cameras and lasers. Such an example is the method described by Boer and Dulio [26]. The person being scanned steps on the platform, which has photogrammetric landmarks for the detection of the current position of the measuring head, consisting of a camera and lights.

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

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The measuring head rotates around the foot making a full circle and measures the 3D foot shape based on the stereo principle. The weak point of this solution is that prior to the measurement the subject needs to wear specially textured socks. Foot scanning system presented in [27] is based on laser line triangulation where the measuring head moves around single foot in oval shaped trajectory. Again the measurement procedure consist of consecutive right and left foot scanning, where each takes approximately 13 seconds. The new 3D foot measuring system represents an innovation in terms of simplification and speedup measurement. This is based on simultaneous measurement of both feet without using special socks. The apparatus is intended to be used mainly in shoe shops and in outpatient clinics; therefore its design is robust and simple to operate. In the first part of this article an optomechatronic design based on the laser triangulation principle is described. Then the algorithms for a 3D shape reconstruction and the extraction of the dimensions of the feet are presented. The presentation is concluded with measurement examples which demonstrate the measuring precision of the system. 1 OPTOMECHATRONIC DESIGN The 3D feet scanning system with a rotational measuring head is schematically presented in Fig. 1. The measuring head (MH) is attached to the rotating arm (RA), which rotates around the centre of the

standing platform (SP). The basic 3D measuring principle is laser-multiple-line triangulation [18] which is incorporated into a MH. Since the MH measures only three profiles of the measured surface at a time, a rotational scanning technique is implemented to measure the entire surface of both feet simultaneously. The rotation of the RA and its marginal positions are also pointed out in Fig. 1a. The RA is driven by a stepper motor (SM) by using a synchro-belt transmission. The SM and the MH are controlled by the control unit (CU), which is connected to a personal computer (PC). The main part of the system is the MH which consists of a laser projection unit (LPU) and two cameras (C1 and C2) which are symmetrically positioned on each side of the LPU (see Fig. 2). The LPU projects three vertically directed and equally inclined laser light planes on the measured surface. The boundary laser light planes, LP-1 and LP1, are displaced for the interbeam angles δ1 and δ2 according to the central plane LP0, which is directed toward the centre of the SP. Since the cameras acquire images of the illuminated measured surface from different viewpoints, the positions of the laser contour points in the image space are directly related to the distance between the MH and the measured points. The MH is positioned approximately 200 mm above the SP and is oriented towards the centre of the SP (see Fig. 1b). This position ensures that almost all the foot’s surface area which is not in contact with the SP can be measured. It should be stressed that the sole part of

Fig. 1. Schematics of the 3D foot scanning system with a rotational measuring head, a) top view, b) front view: MH - measuring head , CU – control unit, SM – stepper motor drive, LF and RF – left and right measured foot, RA – rotating arm, SP – standing platform

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 685-693

the foot, which is in contact with the SP has a planar shape and therefore its geometry is known. The measuring head configuration with two cameras and three laser light planes was chosen to reduce the shadowing effects which appear on the inner sides of both feet. Figs. 3a and b show the situations where the MH is in a position where one camera (C2 in Fig. 3a or C1 in Fig. 3b) cannot see the illuminated surface of the opposite foot due to an occlusion with one of the ankles. So, two symmetrically positioned cameras are used to avoid these occlusions.

dashed region in Fig. 3c). Therefore it is important that the regions where the light is blocked do not interfere. By using this rule, we can determine the position of the foot on the SP and the interbeam angles δ1 and δ2 so that the maximum length of the measured foot can be up to 325 mm.

Fig. 2. Components of the measuring head – top view: C1 and C2 - two cameras, LPU - laser projection unit, LP-1 to LP1 – laser light planes

Fig. 4. Isometric view of the 3D foot scanning system with the rotational measuring head

The ankle also blocks the laser light planes from reaching the opposite foot (see Fig. 3c). Each light plane is blocked in a certain region (dashed areas on Fig. 3c) which is governed by the position and size of the ankle section and by the interbeam angles δ1 and δ2. We can see that by using only a central light plane LP0, the inner side of the toes are hidden (see the red

Fig. 4 shows the final design of the measuring system, which consists of two main subassemblies: the measuring part and the operating console. Both can be easily detached for long-distance transportation, but in case of room-to-room movement, the system has two wheels under the operating console. The cameras (producer: Unibrain; model: Fire-I; CCD sensor size:

Fig. 3. Schematic presentation of the regions; a) and b) where the camera’s views are obstructed and c) where the laser light planes are obstructed Three-Dimensional Foot Scanning System with a Rotational Laser-Based Measuring Head

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 685-693

¼″, 640×480 pix), are attached symmetrically to the steel plate of the MH. The LPU consists of three laser line projectors (producer: World Star Tech.; model: FLL 5-3.5P-635-75). It is attached on the same plate in the middle position as is shown in Fig. 4c. The stepper motor drives the RA by using a synchro belt transmission with a reduction ratio of 20. The CU is implemented with an 8 bit μP PIC 16f876 which drives the 4-poles unipolar SM and sends the data of the exact RA angular position for a single acquired camera-image to the PC. Communication between the CU and the PC is implemented by using the RS232. The PC (Asus Pundit P1-AH2) is the central unit for controlling the cameras, lasers and the rotation of the MH. On the other hand, the PC stores the acquired images, extracts the 3D shapes and dimensions of the feet, calculates the fit between the feet and various footwear models and finally shows the results to the customer. A touch screen monitor (ELO 1715L 17″) is used for a user-friendly interaction and measurement display.

3.1 Three-Dimensional Reconstruction The reconstruction of the 3D shape from the sequence of detected contours is performed in the next step. The algorithm is divided into partial transformations, where the first four are the same as described in [18]. Afterwards we get the coordinates of the measured point T in the MH coordinate system (XMH, YMH, ZMH; see Fig. 6.), where the Z axis is collinear with the optical axis of the middle laser projector (LP0), the origin is at the light planes’ cross-section and the X axis is parallel with the camera’s sensor plane as is shown in Fig. 6.

2 MEASURING PROCEDURE The measuring procedure starts when the customer steps on the SP and enters their gender and age. The measurement takes 10 seconds while the MH encircles both feet. During that time, the person must not move. After that the measured feet are reconstructed (see the next chapter) and shown in 3D together with the main dimensions, such as the length and maximum width, height and girth in the forefoot region. If the customer or operator is satisfied with the results, the complete 3D measurement is saved to the PC, otherwise the measurement is repeated. If the system is used in a shoe shop then it follows the so called matching algorithm, where the geometrical difference between the foot and the shoe’s internal volume is calculated for an entire assortment of selected shoe models [14], [26] and [28]. The models with the best fit are then suggested to the customer.

3 DATA PROCESSING

b) Fig. 5. Images of the illuminated feet and the standing platform acquired by cameras; a) C1, and b) C2

The data processing begins based on the images of the illuminated feet acquired by both cameras (see Fig. 5). The first processing step is a sub-pixel line detection [29], where the laser-line contours locations are determined along the vertical direction in each column of the image. Afterward, the correlation between the detected contours and the laser light planes is determined, which is done by sorting the segments along the vertical.

The transformation into the World Coordinate System (WCS) is the last step, which considers the rotation and translation of the MH. The WCS’s XY plane lies on the SP, its origin is coincident with the centre of the SP and the XW axis is directed toward to the operating console (see Fig. 6). By using such a convention, we can calculate the point radii vector (rW) by using the following equations:

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 685-693

rW = ( ( R MH rMH − TMH ) Rφ R SP ) − TW , (1)

where rMH is the point radii vector described in the MH coordinate system and the rotation matrix RMH describes the rotation of the points around the ZMH and YMH axis considering the β and γ angles respectively:

R MH

 sin β cos γ  =  − sin β sin γ   − cos β

sin γ cos γ 0

cos β cos γ   − cos β sin γ  . (2)  sin β 

modules, where both use the same LPU. In this way we get six partially overlapping surfaces, where each surface belongs to one light plane measured by both cameras. An example of such measurement is shown in Fig. 7a, where the colours represent different surfaces. This kind of measurement is also called raw data, because post-processing steps must be done prior to extracting the dimensions of the feet.

Vector TMH describes the translation of the points for the distances H and R as shown in Fig. 7:

R   TMH =  0  . (3) H  

The rotation matrix Rϕ considers the rotation of the MH around the rotation axis:

 cos φ  Rφ =  − sin φ  0 

sin φ cos φ 0

0  0  , (4) 1 

where ϕ is the angle between the XW axis and the rotation arm. The exact value of this angle is obtained from the CU during the scanning procedure for each acquired image. The last rotation matrix describes the deviation of the rotation axis from the perpendicular orientation relative to the SP (see Fig. 6):

 sin κ  R SP =  − sin κ  sin ε 

0 cos ε sin ε

cos κ   − cos κ  , (5) cos κ 

where ε and κ are the angles of inclination of the SP around the axes XW and YW respectively. Finally, the points are translated for vector TW, which describes the small misalignment between the calibration etalon and the origin of the rotation axis:

 ∆X    TW =  ∆ Y  . (6)  0   

The misalignment in the X and Y direction is denoted by ΔX and ΔY, respectively. The above-described calculation is performed for all the detected points of all three line segments and for all the acquired images from both cameras (typically 300 images from each camera). Since the basic laser triangulation arrangement consists of one camera, we treat our MH as an assembly of two measuring

Fig. 6. Schematic representation of the measured point (T) transformation from the MH coordinate system to the World Coordinate System (XWYWZW)

The first step is to create two groups, where the first includes all the points lying on the left hand side of the XWZW plane (YW > 0) and the second all other points. Cross-sections perpendicular to the XW axis are extracted at every millimetre in the second step. Each cross-section is further divided into the foot and supporting plate (SP) part. The points which describe the SP are then deleted, but the section where the foot is in contact with the SP is linearly interpolated. Fig. 7b shows only point cloud of measured SP, where the sole contact area is clearly visible. Finally, the points between the consecutive cross-sections are triangulated, which results in two surfaces describing the left and right foot. These surfaces are then filtered by using a 3D filter where the radius of the averaging sphere is 3 mm. An example of the post-processed measured feet is shown in Fig. 7c.

Three-Dimensional Foot Scanning System with a Rotational Laser-Based Measuring Head

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3.2 Extraction of Global Foot Dimensions Linear foot dimensions such as length, width, height and girth are also called global dimensions since they describe the foot as a whole. These dimensions are dependent on the measuring direction; therefore the first step in the extraction procedure is the alignment of the foot into a standard position, which is described in [18]. In such a position, the foot length is defined as a maximum X coordinate; the width is defined as the difference between the maximum and minimum Y coordinate of the cross-section at 66% of the foot length; and the height and girth are determined at the same cross-section. Besides the above-mentioned dimensions, the cross-sectional dimensions (width, height and girth) at other positions are also extracted to calculate the fit between the foot and shoe geometry. 4 MEASUREMENT EXAMPLES To demonstrate the measuring precision of the system, three experiments were made: (i) measurements of test object to show the pure system dependent precision; (ii) measurements of living human to show the influence of foot deformability; and (iii) comparison between the traditional and 3D measurements according to ISO 20685 standard [30]. In the second case, none of the participants had before or at the time of measurement, injured or in any other way harmed their feet. The participants were informed of the methods used and their rights as participants. Written informed consent was obtained from all participants.

positioned ten times randomly within the marked area of measurement (see Fig. 4) and measured. The measuring precision of the feet length (L), width (W) and girth (G) were analysed by calculating the standard deviation (SD) of each dimension separately for the left and right foot. The results are shown in Table 1. 4.2 Validation by a Living Human In the second step, the measurement repeatability of one male subject (age 48 years, height 1.70 m, and weight 65 kg) was studied. The subject wore white cotton socks on both feet. After each measurement, the subject had to step down from the SP and wait for 1 minute. An analysis of the measurements was made for the same dimensions as in the first step. The results for ten measurements are presented in Table 2. 4.3 Comparison between Traditional and 3D Measurements Twenty subjects (10 males and 10 females) of age 46.5±9 years were measured using the presented system and traditionally using calliper and tape. Dimensions were then compared. Means, standard deviations and 95% confidence intervals for 3D minus traditional measurements of length, width and girth are presented in Table 3. The 95% confidence intervals were calculated as mean ±1.96∙SD. Table 1. Results of the test object repeatability measurement. Plastic feet were measured 10 times

4.1 Validation by a Test Object The plastic feet were copies of natural feet made from high density polyethylene. The length was 272.2 and 278.7 mm for the left and right foot respectively. Both feet were dressed in white cotton socks. They were

MIN MAX MEAN SD

Length [mm] 271.3 273.1 272.2 0.6

Left feet Girth [mm] 248.2 249.7 248.9 0.5

Width [mm] 100.0 101.2 100.4 0.4

Length [mm] 277.5 279.4 278.7 0.6

Right feet Girth [mm] 273.0 275.1 274.4 0.6

Fig. 7. Example of a) entire raw measured data, b) only raw data of measured supporting platform and c) the extracted feet

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Novak, B. – Babnik, A. – Možina, J. – Jezeršek, M.

Width [mm] 110.7 112.3 111.8 0.5

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Table 2. Results of the living human repeatability measurement. The subject was measured 10 times Length [mm]

Left feet Girth [mm]

Width [mm]

Length [mm]

MIN

274.1

252.8

104.9

270.6

253.2

104.7

MAX

276.9

256.0

106.9

272.0

255.8

106.1

MEAN

275.7

254.3

105.5

271.5

254.6

105.2

1.0

0.7

0.4

0.8

0.4

Right feet Girth Width [mm] [mm]

Table 3. Comparison between traditionally and 3D measured feet dimensions; number of feet is 40; units are millimeters

Length Girth Width

Mean difference

Standard deviation

1.3 1.8 1.0

0.7 1.0 0.6

95% confidence limits Lower Upper 0.1 2.7 0.1 3.7 0.4 1.6

5 DISCUSSION The main objective of this paper is to show the operation principle and precision of the presented foot measuring system. Repeatability test of the plastic feet (presented in Table 1) shows that the SD for all feet dimensions is better than 0.6 mm. The difference between the maximum and minimum foot length is 1.8 and 1.9 mm for the left and right foot respectively. The maximum difference for the width is 1.5 and 1.6 mm, meanwhile the maximum difference of the girth is 1.5 and 2.1 mm for the left and right foot respectively. The comparison of the results between the left and right plastic feet does not show any essential difference. The results of repeatability measurements of living human show (see Table 2) that the SD and maximum differences are approximately 60% higher compared with the plastic feet. The maximum SD corresponds to the length and girth of the left feet, which is 1.0 mm. The maximum difference is in the girth measurement, where it is 3.2 mm. The smallest difference and the SD correspond to the width, where the SD is 0.7 mm and the maximum difference is 2 mm. We assume that the lower repeatability is mainly related to the transient fluctuations in feet dimensions, especially with a deformable foot, whose shape is mostly determined by the current pose of the subject [25]. Comparison between traditional and 3D measurement (see Table 3) shows that dimensions obtained by 3D measurement are systematically larger for 1.3, 1.8 and 1.0 mm in case of length, girth and width respectively. Further, the SD of differences

are within the same range as SD obtained from repeatability measurements of living human. According to ISO 20685 [30], the 3D foot scanning system can be said to give results sufficiently comparable to traditional methods if the 95% confidence interval for the 3D minus traditional measurements is within ±2 mm. Table 3 shows that only the width is measured with the sufficient accuracy, while the length and the girth measurements do not meet the requirement. The main reasons are large systematic offsets, which are more than three times greater than in case of test object (plastic feet). According to this we assume that the offsets originate mainly from the slight deformation of the live foot in the vicinity of the contact area during traditional measurement. In future software version this offset can be minimised with a simple calibration in order to provide compatible anthropometric data. The presented results show that the measuring precision of the key foot-dimensions are better than the recommended precisions by Goonetilleke [31], therefore it can be concluded that the system is suitable for use in the footwear industry. The collected data are mainly used for: (i) automatic shoe size selection according to the customers feet and (ii) for development of new lasts. On the other hand, the system is not suitable for insole customization since it does not measure the entire sole area. 6 CONCLUSION The presented system for 3D measuring of the foot is based on the laser-multiple-line triangulation principle. The main part of the system is the measuring head which rotates around the centre of the standing platform where the treated person stands and measures both feet simultaneously in 10 seconds. The analysis of the foot shape is performed for the length, width, height and girth. Three experiments were done to validate the measuring precision. Validation by plastic feet shows that the system measures the length, girth and width with a precision better than 0.6 mm. Validation by living human shows that the repeatability is better than 1.0mm. Comparing with traditional measurements, dimensions are systematically larger for approximately 1.5 mm in case of 3D system. This is a consequence of foot deformation during traditional measurement. The presented results show that the new system is versatile and yet affordable and easy to use tool for accurate feet measurement in footwear development [32], selling and research applications. At the moment a pilot production of 14 systems was already

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conducted, and the design is ready for industrial production. 7 ACKNOWLEDGEMENT The research has been partly supported and financed by the European Union, European Social Fund, 2009. 8 REFERENCES [1] Xiong, S., Zhao, J., Jiang, Z., Dong, M. (2010). A computer-aided design for foot-feature-based shoe last customization. International Journal of Advanced Manufacturing Technology, vol. 46, no. 1-4, p. 11-19, DOI:10.1007/s00170-009-2087-7. [2] Mickle, K.J., Munro, B.J., Lord, S.R., Menz, H.B., Steele, J.R. (2010). Foot shape of older people: implications for shoe design. Footwear Science, vol. 2, no. 3, p. 131-139, DOI:10.1080/19424280.2010. 487053. [3] Coughlin, M.J., Thompson, F.M. (1994). The high price of high-fashion footwear. The Journal of Bone and Joint Surgery, vol 76, no. 10, p. 1586-1593. [4] Menz, H.B., Morris, M.E. (2005). Footwear characteristics and foot problems in older people. Gerontology, vol. 51, no. 5, p. 346-351, DOI:10.1159/000086373. [5] The Brannock Device Co., Inc. (2013). The Brannock Device, from http://www.brannock.com/, accessed on 2013-07-01. [6] Goonetilleke, R.S., Ho, E.E.H., So, R.H.Y. (1997). Foot sizing beyond the 2-D Brannock method. Annual Journal of the Institute of Industrial Engineers, Hong Kong, p. 28-31. [7] Foot Measuring Devices (2013). Ritz Stick, from http:// www.footmeasure.com/ritz-stick, accessed on: 2013-0701. [8] NVOS-Orhobanda (2002). Orthopedische shoentechniek. Boek 3 – maatnemen en Leesten (Orthopedic shoe technology. Book 3 - the Measurements and Lasts). NVOS-Orthobanda. [9] Kolšek, T., Jurca, A., Vidič, T. (2011). Survey on parents’ selection of children’s footwear. Footwear Science, vol. 3, supplement no. 1, p. S88-D90, DOI:10. 1080/19424280.2011.575848. [10] Lu, X., Jain, A.K., Colbry, D. (2006). Matching 2.5D face scans to 3D models. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 28, no. 1, p. 31-43, DOI:10.1109/TPAMI.2006.15. [11] Hajati, F., Raie, A.A., Gao, Y. (2012). 2.5D face recognition using patch geodesic moments. Pattern Recognition, vol. 45, no. 3, p. 969-982, DOI:10.1016/j. patcog.2011.08.025. [12] Yeti TM. (2013). VORUM research coorporation, from http://www.vorum.com/english/footware/measurementcarving-yeti-3d-scanner.php, accessed on 2013-07-04.

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[13] Kouchi, M., Mochimaru, M. (2001). Development of a low cost foot-scanner for a custom shoe making system. Proceedings of 5th Symposium on Footwear Biomechanics, Zürich, p. 58-59. [14] Witana, C.P., Feng, J., Goonetilleke, R.S. (2004). Dimensional differences for evaluating the quality of footwear fit. Ergonomics, vol. 47, no. 12, p. 1301-1317, DOI:10.1080/00140130410001712645. [15] Xiong, S., Goonetilleke, R.S., Zhao, J., Li, W., Witana C.P. (2009). Foot deformations under different loadbearing conditions and their relationship to stature and body weight. Anthropological Science, vol. 117, no. 2, p 77-88, DOI:10.1537/ase.070915. [16] Luximon, A., Goonetilleke, S.R., Zhang, M. (2005). 3D foot shape generation from 2D information. Ergonomics, vol. 48, no. 6, p. 625-641, DOI:10.1080/0014013050070970. [17] Grimmer, R., Eskofier, B., Schlarb, H., Hornegger, J. (2011). Comparison and classification of 3D objects surface point clouds on the example of feet. Machine Vision and Applications, vol. 22, no. 2, p. 235-243, DOI:10.1007/s00138-009-0230-y. [18] Jezeršek, M., Možina, J. (2009). High speed measurement of foot shape based on multiple-laserplane triangulation. Optical Engineering, vol. 48, no. 11, DOI:10.1117/1.3265522. [19] Pan, J., Huang, P.S., Chiang, F.P. (2005). Color-coded binary fringe projection technique for 3-D shape measurement. Optical Engineering, vol. 44, no. 2, DOI:10.1117/1.1840973. [20] Huang, P.S.S., Zhang, C.P., Chiang, F.P. (2003). Highspeed 3-D shape measurement based on digital fringe projection. Optical Engineering, vol. 42, no. 1, p. 163168, DOI:10.1117/1.1525272. [21] Zhang, S., Huang, P.S. (2006). High-resolution, realtime three-dimensional shape measurement. Optical Engineering, vol. 45, no. 12, DOI:10.1117/1.2402128. [22] Kouchi, M., Kimura, M., Mochimaru, M. (2009). Deformation of foot cross-section during walking. Gait & Posture, vol. 30, no. 4. p. 482-486, DOI:10.1016/j. gaitpost.2009.07.113. [23] Schmelzpfenning, T., Plank, C., Krauss, I., Aswendt, P., Grau, S. (2009). Dynamic foot scanning - A new approach for measurement of the human foot shape while walking. Footwear Science, vol. 1, suppl. 1, p. 28-30, DOI:10.1080/19424280902977111. [24] Leardini, A., Benedetti, M.G., Berti, L., Bettinelli, D., Nativo, R., Giannini, S. (2007) Rear-foot, mid-foot and fore-foot motion during the stance phase of gait. Gait & Posture, vol. 25, no. 3, p. 453-462, DOI:10.1016/j. gaitpost.2006.05.017. [25] Novak, B., Možina, J., Jezeršek, M. (2014). 3D laser measurements of bare and shod feet during walking. Gait & Posture, vol. 40, no. 1, p. 87,93, DOI:10.1016/j. gaitpost.2014.02.015. [26] Boer, C.R., Dulio, S. (2007). Mass customization and footwear: Myth, Salvation or Reality, Springer Verlag London.

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[27] Fujita, H., Fukumoto, S., Yoshida, H., Wakasugi, Y., Kano, H. (2004). A 3D Foot Scanning System with a Sensor Head Guided around the Foot, from https:// www.jstage.jst.go.jp/article/iscie1988/17/8/17_8_330/_ pdf, accessed on 2014-07-07. [28] Nácher, B., Alemany, S., González, J.C., Alcántara, E., Garcia-Hernandes, J., Heras S., Juan A. (2006). A footwear fit classification model based on anthropometric data. SAE International Conference and Exposition of Digital Human Modeling for Design and Engineering, Lyon, SAE paper 2006-01-2356.

[29] Jezeršek, M., Možina, J. (2003). A laser anamorph profilometer. Strojniski vestnik - Journal of Mechanical Engineering, vol. 49, no. 2, p. 76-89. [30] ISO 20685:2010. 3-D Scanning Methodologies for Internationally Compatible Anthropometric Databases, International Organization for Standardization, Geneva. [31] Goonetilleke, R.S. (2013). The Science of Footwear. CRC Press - Taylor and Francis Group, Boca Raton. [32] Mandić, V., Ćosić, P. (2011). Integrated product and process development in collaborative virtual engineering environment. Tehnički vjestnik – Technical Gazette, vol. 18, no. 3, p. 369-378.

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Received for review: 2013-07-18 Received revised form: 2014-03-25 Accepted for publication: 2014-07-08

Experimental Model-Based Linearization of a S.I. Engine Gas Injector Flow Chart

Pipitone, E. – Beccari, S. – Cammalleri, M. – Genchi, G. Emiliano Pipitone* – Stefano Beccari – Marco Cammalleri – Giuseppe Genchi University of Palermo, Italy Experimental tests previously executed by the authors on the simultaneous combustion of gasoline and gaseous fuel in a spark ignition engine revealed the presence of strong nonlinearities in the lower part of the gas injector flow chart. These nonlinearities arise via the injector outflow area variation caused by the needle impacts and bounces during the transient phenomena that take place in the opening and closing phases of the injector and may seriously compromise the air-fuel mixture quality control for the lower injection times, thus increasing both fuel consumption and pollutant emissions. Despite the extensive literature about the operation and modelling of fuel injectors, there are no known studies focused on the nonlinearities of the gas injector flow chart and on the way they can be reduced or eliminated. The authors thus developed a mathematical model for the prediction of mass injected by a spark ignition (S.I.) engine gas injector, validated through experimental data. The gas injector has been studied with particular reference to the complex needle motion during the opening and closing phases, which may strongly affect the amount of fuel injected. In this work, the mathematical model previously developed has been employed to study and determine an appropriate injection strategy in order to linearize the injector flow chart to the greatest degree possible. The injection strategy proposed by the authors is based on minimum injection energy considerations and may be easily implemented in current engine control units (ECU) without any hardware modification or additional costs. Once calibrated by means of simulation, this strategy has been validated by experimental data acquired on an appropriately equipped injector test bench. As a result, the real injector flow chart has been substantially improved, reducing its deviation from linearity to one third of the original flow chart, which is an excellent result, especially if the typical measurement dispersion of the injected mass is taken into account. The injection strategy proposed by the authors could extend the linear behaviour of gas injectors and improve the fuel supply by means of a simple software update of the ECU, thus obtaining higher engine efficiency and lower pollutant emissions. Keywords: fuel injector, injection strategy, spark ignition engine, modelling and optimization

0 INTRODUCTION Since the introduction of polluting emission regulations for passenger cars, fuel injection systems have become an indispensable part of modern spark ignition engines, due to their capability to accurately meter the mass of fuel employed at each engine cycle. A typical gaseous fuel (liquefied petroleum gas, LPG, or compressed natural gas, CNG) multi-point injection system for spark ignition (S.I.) engines is composed of the elements reported in Fig. 1: the regulator reduces the gas pressure from the high level in the tank (where LPG is stored at around 10 bar and CNG at around 200 bar) to the low level in the fuel rail (about 2 bar for LPG and 10 bar for CNG); hence, when the injector is activated, the fuel arrives to the inlet duct (port injection) of the engine; the flow through the gas injector can be assumed to be equivalent to the flow through a convergent nozzle: in a chocked flow condition (i.e. supposing that, as usual, the ratio between fuel rail pressure and manifold pressure is ≥2), the gas flow depends only on pressure and temperature upstream from the injector; this makes the injected mass directly proportional to the “injection time” (i.e. the duration of the time interval during which the injector is activated), regardless of 694

the pressure level in the intake manifold of the engine. This proportionality makes the injector characteristic almost linear on the injector flow chart, which is the diagram used to represent the amount of fuel injected for any injection time. An electronic control unit (ECU) adjusts the injected fuel mass, and then the airfuel ratio, acting on the injection time, whose values are stored in memory, by means of proper tables, as a function of engine speed and load. In some operating conditions, typically lower engine loads, this openloop control is integrated by means of a more accurate closed-loop control, which, using the lambda sensor output signal, performs continuous adjustment on the amount of fuel injected to maintain air-fuel ratio at the stoichiometric value, thus minimizing pollutant emissions. This simple closed-loop control, based on a single feedback parameter (e.g. lambda sensor output signal), can be performed due to the linearity of the injector characteristic. Previous experimental tests carried out by the authors of this paper showed, however, the existence of strong nonlinearities in the lower part of the gas injector flow chart. These nonlinearities may compromise the air-fuel ratio control performed by the engine ECU, causing unstable corrections of the

*Corr. Author’s Address: University of Palermo, Viale delle Scienze, Palermo, Italy, emiliano.pipitone@unipa.it

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injected fuel mass, thus leading to both poor fuel economy and high pollutant emissions (spark ignition engine catalytic converters has a very low efficiency for non-stoichiometric mixtures).

Fig. 1. Gaseous fuel multi point injection setup: 1) ECU, 2) pressure regulator, 3) filter valve, 4) gas injectors, 5) lambda sensor, 6) gas tank, and 7) fuel rail

With the aim of studying and reproducing the nonlinear behaviour of the gas injector, the authors created a mathematical model [1] for the evaluation of the complex needle motion during the entire injection event, with maximum accuracy on the opening and closing phases, which have been recognized to be determinant in the generation of the nonlinearities. In the present work, the authors employed the developed model to study and propose a suitable injection strategy with the aim of suppressing the nonlinear behaviour of an actual gas injector, thus linearizing its flow chart and extending its range of utilization towards the lower injection times. The main advantage of the determined injection strategy is its ease of implementation in current production engines, since a simple ECU software update is required. 1 LITERATURE OVERVIEW An extensive literature is currently available on the simulation and modelling of internal combustion engine injection systems. Compression ignition (C.I.) engines typically have high pressure (1600 to 2000 bar) common rail injectors which, activated by a solenoid or by a piezoelectric element, use the high fuel pressure to move the needle and open the nozzle. Spark ignition engines may be port injected or direct injected: in the first case, low pressure (3 to 10 bar, depending on fuel type) injectors are usually employed, while in the second case significantly higher injection pressure may be involved (100 to

500 bar). Despite the extensive available literature on injection system simulation, very few works cover the dynamic modelling of the injector needle motion, which is the focus of this paper. With regard to common rail injection systems, the needle motion has been dealt with extensively in literature: for example, the fluid-dynamic model presented in [2] allows predicting the injection pressure variations and deriving control laws for the rail pressure controller, while in [3] the model developed using a commercial code also predicts needle lift and injection rate for different injection pressures; the common rail piezoelectric injector model realized in [4] takes into account both the hydraulic part (fluid flow, discharge coefficients) and the mechanical part (needle movement, seats elastic deformation) with the aim of predicting different flow rate profiles. With regard to gasoline direct injection, the model developed in [5] refers to a piezoelectric injector and compares the capability of lumped parameters and distributed parameters to describe the needle motion and the behaviour of piezoelectric elements. On account of the dumping effect of liquid fuels, which completely suppress any needle bounces, none of the abovementioned works [2] to [5], however, report any injector flow chart nonlinearities. The problem is a feature of gas injection systems. The modelling of gas injection in S.I. engines is handled in [6], where the details of fuel spray formation and mixture with air are explored, while the dynamic behaviour of the injector needle is discussed in [7], where different model predictive control (MPC) schemes are presented for the control of an electromagnetically actuated mass-spring-damper system for automotive applications: in this work, however, the solenoid power voltage is assumed to vary between zero and 350 V while, in the present paper, the authors considered a constant power voltage of 13 V, which complies with the actual automotive electric system specifications. The natural gas injection system modelled in [8] presents control strategies for the optimization of the injection system operation focusing on the fluid-dynamic behaviour of the whole injection system (fuel rail, pressure control valve, injectors). Even if focusing on gas injections, however, none of the abovementioned works [6] to [8] deals with the nonlinearities produced by the needle bounces during the opening and closing phases of the injector. Only the work presented in [9] focuses on the suppression of gas injectors needle bounces, even if with an entirely different purpose, i.e. the prevention of fatigue stress

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damages. Moreover, contrary to the approach followed in the present paper, the implementation of the method proposed in [9] requires a substantial modification of the injector power supply system. The presence of a nonlinear zone in the injector flow chart, however, has never been studied in detail, least of all its correlation with the needle motion. This consideration led the authors to develop a proper mathematical dynamic model of the gas injector and to study a proper injection strategy with the aim to linearize the injector flow chart: this would allow improving the air-fuel mixture quality control while minimizing both fuel consumption and polluting emissions.

needle is moved by the electromagnetic force from the closed towards the open position, thus knocking against the stopping surface at the end of the lift. Here, the needle bounces and moves towards the closed position, where another impact may occur. Under the action of the electromagnetic field, the needle will, however, be pushed toward the open position, thus producing other bounces. If the injection time is long enough, the needle will conclude all the bounces and then, compelled by the electromagnetic field, remain in the open position.

2 SOLENOID INJECTOR DYNAMICS Fig. 2 reports a cutaway of the solenoid gas injector [10] used in the test, while Fig. 3 shows a typical electrical circuit used to energize the injector solenoid; this circuit is composed of the power supply, the injector solenoid and the power transistor activated by TTL pulses, which may be generated by the engine ECU or by a personal computer. The injector is mainly composed of a mechanical part (the needle) and an electric part (the solenoid), and these two parts interact, influencing each other through the electromagnetic field. The needle movement influences the solenoid current, which in turn, acts on the needle by the electromagnetic force.

Fig. 3. Schematic representation of the usual electrical circuit involved in injector operation

Once finished the injection time, the ECU deactivates the transistor that opens the circuit, producing an instantaneous drop of the solenoid current; the needle is then forced to return to the closed position by the fuel pressure and the spring load, thus knocking against the closed position seat and producing other bounces. Figs. 4 and 5 show the output signal from an accelerometer mounted on the armature of the injector used for test, during the injector opening and closing phases.

Fig. 2. Cutaway of the fuel injector used in the test: 1) pintle, 2) needle, 3) armature, 4) spring, 5) solenoid winding, 6) electrical terminals, 7) fuel strainer

When the solenoid is not energized (i.e. the electrical circuit is open), the needle is kept in closing position by both the fuel pressure and the spring load. When the ECU activates the transistor (which can be considered a “digital switch”), this closes the electrical circuit, and the current rises in the solenoid windings (see Fig. 3), according to the R-L circuit law; the 696

Fig. 4. Waveforms of solenoid current and armature acceleration during injector opening phase

As can be observed, the substantial impacts that occur both in the opening and closing phases cause prominent spikes on the accelerometer output signal.

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The same diagrams also show that the measured solenoid current, which, during the opening phase, is characterized by the presence of several cusps: due to the reciprocal interaction between needle movements and coil-winding current [1] and [9], the abrupt velocity variation during an impact causes a rapid change in the current first derivative. These current cusps are not present in the closing transient since, after the end of the injection, the electric circuit is open, and the solenoid current is null. Fig. 6. Experimental injector flow chart obtained with air at 10 bar

Fig. 5. Waveforms of solenoid current and armature acceleration during injector closing phase

Fig. 4 also shows that, for the tested injector fed with air at 10 bar, in the opening phase, the bounces continue for about 4 ms, while in the closing phase (see Fig. 5) their duration is shorter, i.e. about 3 ms. The importance of these bounces relies on the significant variations they produce on the injected mass, since the instantaneous flow section depends on the needle position; therefore, assuming a linear correlation between the flow section area and needle position, it results that the injected mass depends on the value reached by the integral of the needle position over time. Moreover, when the injection time is below the opening phase transient duration (≈4 ms for the injector tested fed with air at 10 bar), not only is the needle transient not completed, but it is also influenced by the duration of the injection itself; the impact energy of the needle on the opening stop surface, in effect, depends on its kinetic energy, which, in turn, is related to the duration of the electromagnetic force applied, and hence to the injection time. It results hence that, for injection times shorter than 4 ms, changing the injection duration modifies the needle movement and hence the integral of its position, which causes a variation in the injected mass. This introduces a non-linear dependence between the injected mass and the injection time, as evident in the injector flow chart shown in Fig. 6.

This diagram reports the measured injected mass for each of the injection time imposed to the injector, fed with air at 10 bar. It is noteworthy that this diagram does not represent the integral of the gas mass flow as a function of time, but rather the measured total injected mass at the end of each single injection, whose duration is the injection time ∆t. As can be noted, for injection durations shorter than the bounces duration (≈4 ms), the needle bounces have a considerable influence on the total injected mass, causing the presence of strong nonlinearities, which may seriously compromise the injected mass control; the ideal flow chart is represented by a straight line, suitable for a pure linear control of air-fuel ratio; therefore, the deviation from the ordinary least square (OLS) line can be considered a measure of the injected mass control quality: the higher the deviation is, the worse the control is. As can be seen in Fig. 6, for the injector flow chart obtained with air at 10 bar, this deviation amounts to almost ±1 mg, which means that using the OLS line instead of the experimental data, the fuel mass control would be subjected to a maximum error of 1 mg, which, depending on the amount of gas to inject, can cause a very large error. The strong nonlinearities of such a flow chart arise from the flow section variations caused by the needle bounces [1], whose intensities, as already pointed out, are related to the needle’s kinetic energy, which depends on the duration of the electromagnetic force applied. During the opening phase, due to the bounces on the two stopping surfaces, the needle frequently reverses its motion, while the electromagnetic force always acts in the same direction: this implies that, depending on the needle velocity, the electromagnetic thrust may accelerate or slow down the needle, thus changing its effect in terms of the needle’s kinetic energy and, in turn, in terms of integral of the needle position, which is proportional to the injected mass. On account of this, it can be understood that, during the opening phase, increasing the injection time may

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have opposite effects on the injected mass, depending on the needle position and velocity. This conclusion has been confirmed by experimental observation, performed using a 100 MHz oscilloscope, of the solenoid current and armature acceleration waveforms together with mass flow data acquisition carried out for air at 9 bar and injection times between 1.8 and 2.5 ms [1]. When the injection time is long enough to let the needle complete all the opening bounces (i.e. ≥4 ms for 10 bar air pressure), the complete opening and closing transient phenomena repeat at each single injection and thus have no effect on the total injected mass, which then becomes a linear function of the injection time, as observable in Fig. 6. The nonlinearities of the injector flow chart can cause inaccurate control over the engine’s air-fuel ratio; this can lead to both higher fuel consumption and higher pollutant emissions, also due to the low efficiency of the catalytic converter for nonstoichiometric air-fuel mixtures. These nonlinearities have not been observed using gasoline: therefore, this study focuses on gaseous fuel injector dynamics. It could be argued that the problem could be overcome by properly selecting the gas injector, so as to always let it operate on its linear range, thus making the nonlinearities an unused part of the flow chart; unfortunately, this is not always possible, since maximum injection time must respect limits imposed by the available time at the maximum engine speed; moreover, the “ideal” injector may not be available from the manufacturer, or may not be economically favourable; this may lead to the installation of a gas injector that, for the particular engine, operates in the nonlinear part of its diagram. For example, the injector used in this study is part of the CNG injection system of a series production bifuel engine from FIAT; data acquired on the engine test bench revealed a maximum injection time of 8.8 ms, which corresponds to the injection of the full load fuel mass; in contrast, the injector flow chart revealed the nonlinear range to lie between injection times of 1.8 and 3.5 ms: experimental data show that when the injector exits the nonlinear range (i.e. with injection time of 3.5 ms), the injected mass is between 18 and 36% of the full load mass (depending on engine speed), and the engine torque is between 10 and 23% of the maximum, which, as example, could be a typical condition in urban areas. In this case the characteristics of the selected injector do not allow operating exclusively in the linear range. A very accurate (and time-consuming) calibration of the ECU injection map, however, may attenuate the effects of 698

the nonlinearities, thus allowing the engine to comply with current pollutant emission limits and regulations. Other particular engine applications may however involve injector operations in the nonlinear range, such as supercharged engines or Double-Fuel combustion. In a supercharged engine, basically, a compressor is used to increase the air pressure in the manifold, thus letting the engine to draw a greater amount of air-fuel mixture and produce higher power; this obviously implies higher fuel flows, and thus longer injection times, which, may compel the adoption of larger injectors; for the lower engine loads (e.g. urban operative condition), the same injectors could thus be operated with injection times so short that they enter the nonlinear range. The second case refers instead to the simultaneous combustion of gasoline and gaseous fuels, such as LPG or CNG, which has been successfully tested by the authors [11], and [12] and by other research groups [13] to [15]; this kind of combustion can be easily implemented in bi-fuel engines as a third operative mode realized by the injections of both gasoline and gas within the same engine cycle, requiring the adoption of shorter gas injection times (even 20% of nominal values): this may induce the gas injector to operate in the nonlinear range, thus causing poor air-fuel ratio control, with consequent increases in pollutant emissions and decreases in engine efficiency. The injection strategy proposed could be effectively employed to linearize the injector flow chart and allow a better control of the engine air-fuel ratio on a wider part of the injector flow chart. The approach followed in the present paper relies on the use of numerical simulations for the definition of a proper injection strategy capable of avoiding needle bounces; the optimal injection strategy determined has been implemented on a real injector test bench and experimentally optimized, thus removing most of the unwanted nonlinearities from the real injector flow chart. 3 SIMULATIONS As already mentioned, in previous works [1] and [16] the authors realized a mathematical model for the simulation of the complex needle motion during the opening and closing phases of a gas injector, in order to predict the amount of fuel injected for each injection time; the model, whose main equations and structure are briefly resumed in Appendix B, has been calibrated by means of the experimental data obtained on a proper test bench using a natural gas injector fed with air at 9 bar, and successfully validated by means

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of the experimental data obtained injecting air at 8 and 10 bar. Fig. 7 shows a comparison between measured and simulated injector flow charts relative to the injection of air at 10 bar. The results explored in this paper, both from simulations and from experimental tests, all refer to the injection of air at 10 bar absolute pressure.

injection time of the linear part in the simulated flow chart of Fig. 7.

Fig. 8. Measured and simulated solenoid current Fig. 7. Comparison between measured and computed injected mass

As shown, a very good fit has been obtained between experimental data and model prediction, since the nonlinearities of the experimental diagram are accurately replicated by the model. Fig. 7 also shows the error (i.e. difference between calculated and measured injected mass) distribution, whose mean and maximum values resulted in 0.22 and 0.74 mg, respectively. In all the cases, the model evaluation accuracy resulted in being comparable to the test measurement uncertainties [1], which are not related to mass flow measurement errors (always less than 1%) but rather to the typical injected mass measurement dispersion around the mean values (as reported in Fig. 19). A further confirmation of the model’s predictive capacity has been obtained by the comparison between the measured and the evaluated solenoid current in the same condition of air pressure and injection time; for example, the diagram of Fig. 8 shows the good agreement between the experimental and the numerical current during the injector opening phase; the first cusp is due to variation of the steel magnetic permeability, while the other cusps are connected to the sudden speed change of the needle due to the impacts on the seat surface. Fig. 9 shows a typical model output, i.e. the solenoid current and the needle displacement as a function of time: specifically, the diagram refers to a 5 ms injection of air at 10 bar. The opening phase bounces are evident both in the needle displacement and in the solenoid current, and their duration is about 3.6 ms; consequently, this is also the minimum

Fig. 9. Simulated solenoid current and needle displacement (∆t = 5 ms)

The closing phase bounces have a duration of about 2.5 ms and are evident only in the displacement waveform, since, at the end of the injection, the transistor (see Fig. 3) is deactivated and this opens the electric circuit causing the current to immediately fall down to zero.

Fig. 10. Simulated needle displacement for two injections with ∆t = 1.7 and 2.0 ms

Fig. 10 shows the needle displacement evaluated by the model for two different injection durations, i.e.

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1.7 and 2.0 ms, with air at 10 bar. In the case of the 2 ms injection, the electromagnetic force still acts after the first impact, thus slowing down the needle, whose successive impacts have lower energy and thus cause smaller bounces. As a result, the 2.0 ms injection is characterized by a lower value of the integral of the needle position, which, as already mentioned, means a lower value of the integral of the mass flow, and hence, a lower injected mass; the simulated injector flow chart in Fig. 7 confirms that the 1.7 ms injection gives a higher injected mass than the 2.0 ms injection.

that, in a conventional way, the injection time has been always considered the time interval between the first rising front and the last falling front of the injection pulse, as shown in Fig. 11.

4 OPTIMAL INJECTION STRATEGY The needle bounces on the stopping surfaces originate from the excess of kinetic energy acquired by the needle during the opening lift, which in turn is due to the excess energy transferred through the electromagnetic field, and hence by the solenoid. It can be easily understood that in order to avoid any bounce, the needle should arrive at the opening stop surface with no kinetic energy, and be maintained in this position by the electromagnetic thrust. This could be pursued by a proper modulation of the solenoid current in order to progressively reduce the electromagnetic thrust on the needle during the lift, thus involving the minimum energy necessary to shift the needle from the closed to the open position. The entire excess of energy transferred to the needle, with respect to the minimum required, is completely dissipated during the bounces by the mechanical friction between the needle and guides, by the gas viscous forces and by the energy loss at each impact: the more the needle moves or impacts, the more energy it dissipates. The modulation of the solenoid current would however require the modulation of the voltage supplied to the injector solenoid, which instead, as typical in automotive engines, is constant and equal to the battery voltage. Given the difficulty of operating with a variable solenoid voltage supply, the energy transferred to the needle during the opening lift can be elsewhere modulated by acting on the duration of the injection pulse: this can be then divided in half, with the first part dedicated to shifting the needle from the closed to the open position without bounces, and the second part dedicated to maintaining the needle in the open position and let the fuel flow. The authors thus focused on this division, which can be realized by the simple interruption of the injection pulse, characterized by two parameters: the interruption delay δ with respect to the start of injection and the duration τ, both indicated in Fig. 11. It is noteworthy 700

Fig. 11. Injection pulse with and without interruption

The authors used the mathematical model previously developed to determine the two parameters values, which allow avoiding needle bounces, so as to linearize as much as possible the injector flow chart. Before the research for the optimal interruption parameters was started, the model was further improved in order to adequately take into account a phenomenon not revealed by the first experimental campaign. As reported in Fig. 12, the solenoid current measured during an interrupted injection shows a sort of extra-current that substantially modifies the current waveform (and hence the needle motion) and is due to the partial discharge of the energy accumulated by the solenoid that occurs during the injection interruption. Details on this phenomenon and on the model modifications introduced by the authors are given in Appendix A.

Fig. 12. Measured solenoid current for an interrupted injection pulse (injection time = 5 ms)

Once refined, the model was employed to perform several simulations with the aim of determining the optimal values to assign to the interruption parameters δ and τ in order to avoid any needle bounce. According

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to what was already explained above, this condition should also minimize the energy E employed for the needle shift from the closed to the open position; the authors thus adopted the energy transferred to the needle in the opening phase as objective function ϕ of the search algorithm:

has a very short duration and should be placed before the first impact to occur. The effects of these optimal interruption parameters on the 5 ms injection of air at 10 bar are reported in Fig. 15 in terms of both the solenoid current and needle displacement.

ϕ = E = V ⋅ ∫ i ⋅ dt , (1) 0

being t* the time necessary to let the needle complete all the opening phase bounces and stops in the open position (i.e. the opening phase duration), V the constant supply voltage and i the solenoid current. As a first step, the authors considered the injection time of 5 ms, which, as reported in Fig. 9, gives rise to several bounces in the opening phase transient. A quite simple search algorithm has been employed, since an entire matrix of interruption delay δ (ranging from 1.55 to 1.76 ms with steps of 0.003 ms) and duration τ (ranging from 0.01 to 0.15 ms with steps of 0.002 ms) has been tested using the model, evaluating the objective function ϕ on the basis of the resulting simulation output. This procedure allowed tracing the ϕ surface, shown in Fig. 13 as a function of the two variables delay δ and duration τ; as can be noted, the absolute minimum region is visible, whose coordinates represent hence the best (minimum energy) values of interruption delay and duration. This is also shown in Fig. 14, which shows a contour plot of the ϕ surface; the best interruption parameters are δ = 1.64 ms, τ = 0.038 ms.

Fig. 14. Contour plot of the opening phase energy [mJ] as function of the two interruption parameters

Fig. 15. Model-predicted solenoid current and needle displacement with optimal interruption parameters

Fig. 13. Surface of the opening phase energy as a function of the two interruption parameters

In conclusion, for the injector tested and fed with air at 10 bar, the optimal injection pulse interruption

As can be seen, the modulation of the injection energy actuated by means of the pulse interruption has the effect of letting the needle reach the open stop surface without impacts and hence without producing bounces; obviously, the best result is attained simply by preventing the first impact from occurring: once the needle rests in the open position, the mass flow remains constant and the injected mass becomes a linear function of the injection time. As a result, Fig. 16 shows the injector flow chart obtained, adopting the optimal interruption parameters for each injection; as is evident, a satisfying linearization has been achieved, since most of the nonlinearities have been suppressed, and the injector characteristic is now a monotone function of the injection time whose deviation from linearity is ±0.22 mg, which is significantly lower than the ±1 mg of the original injector flow chart.

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According to these results, a very good linearization of the injector flow chart can be obtained by means of a simple injection pulse modulation; this kind of power supply strategy can be easily implemented in the current production engines by means of a simple ECU software update and without any hardware change. As already mentioned, the suppression of gas injectors’ needle bounces have also been studied [9] in order to prevent fatigue stress damages; in this case, however, the proposed method relies on a substantial modification of the injector power supply system.

after metering, air pressure was reduced to 10 bar before reaching the gas injector, whose activation power was generated by a 13 volt DC power supply. In place of the engine ECU, a National Instruments PCI 6602 counter board programmed with LabVIEW has been employed for the generation of the 0-5 volt TTL pulses necessary for injector actuation.

Fig. 17. Injector test bench layout: 1) air cylinder, 2) mass flow meter, 3) pressure regulator, 4) pressure sensor, 5) injector, 6) accelerometer, 7) ammeter, 8) transistor, 9) power supply, 10) signal acquisition and generation system Fig. 16. Simulated optimal injector flow chart (δ = 1.64 ms, τ = 0.038 ms)

It is worth mentioning, however, that for a fixed injection time, the injected mass mainly depends on the solenoid current (and hence on the power supply voltage) and on the gas pressure; these two parameters are therefore of crucial importance for the optimization of the injection interruption. With regard to the application in passenger vehicles, the gas pressure can be considered constant, while battery voltage may change during engine operation; the determination of the optimal interruption parameters should therefore be carried out for different supply voltage levels, so as to always adopt the best couple of delay δ and duration τ. 5 EXPERIMENTAL VALIDATION Once determined through simulations, the optimal injection strategy has been put to the test to experimentally prove its capability to linearize a real injector flow chart. An experimental campaign has been carried out on a suitably equipped test bench, whose main elements are shown in Fig. 17. The air flow from the cylinder was measured using a Bronkhorst mini CORI-FLOW M13, a Coriolis-type mass flow meter that features a measuring range of 100 to 2000 g/h with an accuracy of ±0.2% of the measured value; 702

As also shown in Fig. 3, a transistor was used to transform the low power digital pulses into the high current square waveforms necessary for injector solenoid excitation. The injector was activated with frequencies ranging from 10 to 70 Hz so as to obtain mass flows in the measurable range; for each injection time, the experimental injected mass mexp was derived from the measured mass flow m and injection frequency finj:

m exp =

m . (2) f inj

The needle impacts were detected via the output signal from a Bruel & Kjaer Cubic DeltaTron 4502 accelerometer placed on the injector armature, while a clamp-on ammeter LEM PR20 (with 20 kHz frequency response) was used to acquire the solenoid current. All the necessary quantities have been acquired by means of a National Instruments DAQ board PCI-6133, employing a sample frequency of 400 kHz and using the generated TTL pulse as trigger for data acquisition. For each injection time, the complete waveforms of power supply voltage, solenoid current and accelerometer output were recorded for 100 consecutive injections, while mass flow, gas pressure and temperature were recorded as mean values over the 100 injections. In this way, a complete injector chart could be obtained, e.g. as

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shown in Fig. 6: here the total injected gas mass is reported for each injection time between 1 and 5 ms. The experimental validation of the optimal injection strategy obviously started from the best pulse interruption parameters determined in the simulation (δ = 1.64 ms, τ = 0.038 ms), even if a certain shift from these values was expected. In effect, although the model replicated with unexpected accuracy the nonlinearities of the injector flow chart, the imperfect correspondence between the experimental results and the model output may, however, cause substantial differences and must be taken into account; for example, the difference in terms of injected mass between model prediction and real measure is shown in Fig. 7, while Fig. 8 shows the phase differences between simulated and real impacts. These differences, even though minuscule, may have a nonnegligible effect on the needle motion and hence on the total injected mass, thus compromising the success of the linearization process. As a consequence, the interruption parameters determined by the model may not represent, as a general rule, the optimal choice even for experimental test. The determination of the best interruption parameters was thus carried out monitoring the output signal from the injector armature accelerometer, searching for the pulse interruption delay δ and duration τ, which allowed to minimize the needle impacts energy. The best solution was found for δ = 1.70 ms and τ = 0.1 ms, which are not so far from the model optimal values. In particular, the real best interruption delay δ was revealed to be very close to the one determined by simulation, while the interruption duration τ instead showed a greater difference. This can be easily explained taking into account the delay sequence in the injection pulse actuation: the insulated gate bipolar transistor (IGBT) employed in the test, in effect, is characterized by typical current falling and rising times on the order of some tens of microsecond, which indeed delays the needle actuation and contributes to extending the total interruption duration. The result of the experimental linearization achieved is shown in Fig. 18: as can be seen, the linearity of the optimized injector flow chart is not as good as the one obtained by simulation, since the real optimal flow chart revealed a ±0.35 mg deviation from the ordinary least square line, which is higher than the ±0.22 mg of the simulated optimal flow chart of Fig. 16. The reason for this higher deviation from linearity can be found in the natural dispersion of the measured injected mass around the mean value: as shown in Fig. 19, the experimental measurement dispersion, evaluated by means of the standard deviation recorded

for each of the 100 consecutive mass flow samples acquired during the test, can be as high as 0.59 mg. This means that, for example, for the fixed injection time of 1.85 ms, even if the mean injected mass is 2.73 mg, as reported in the diagrams in Fig. 6 and Fig. 7, the measured 100 consecutive values used to compute this mean are scattered in a 0.59 mg wide range. It should be mentioned that this data dispersion is not caused by mass flow measurement errors, which (as already stated) are less than 1% of the measured value. The cause of these high dispersions is instead related to the real needle movement, which, even at fixed injection times, does not repeat identically at each injection, thus causing significant variations on the total injected mass.

Fig. 18. Experimental optimal injector flow chart (δ = 1.70 ms, τ = 0.10 ms)

Fig. 19. Dispersion range of the measured injected mass

In conclusion, the result obtained by the simple pulse interruption strategy allowed a substantial improvement of the injector flow chart, whose deviation from linearity has been reduced to about one third of the original chart of Fig. 6. With the aim of evaluating the benefit introduced by the proposed injection strategy on the control of the engine airfuel ratio, the deviation from the OLS lines has been determined for the original injector flow map and for both the simulated and experimental optimized charts; as shown in Fig. 20, the use of an OLS regression

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line in place of the original injector flow chart would cause an air-fuel ratio error up to 37% in the lower injection time zone; once optimized by means of pulse interruption strategy, the improved linearity of the injector flow chart allows using the OLS line with a maximum error of 10% in the same lower injection time region: the optimization performed hence may noticeably improve the engine air-fuel ratio control for the lower injection time.

substantially improve the injector flow chart, whose deviation from linearity was reduced to one third of the original flow chart: the authors consider this result to be undeniably good, especially if the measurement dispersion of the injected mass is considered. As outlined, power supply voltage and gas pressure are of crucial importance for the optimization of the injection interruption. For application in passenger vehicles, gas injection pressure can be considered constant, while battery voltage variations may occur; to take this into account, the determination of the optimal interruption parameters should be carried out for different supply voltage levels. 7 NOMENCLATURE

Fig. 20. Air-fuel ratio error caused by the use of the OLS line in place of injector flow map

6 CONCLUSIONS A mathematical model previously realized by the authors for the simulation of the needle motion of a natural gas injector and for the evaluation of the injected mass has been now employed to determine an optimal injection strategy with the aim of linearizing to the greatest extent the nonlinear part of a real S.I. engine gas injector. The analysis of the needle motion, together with some considerations on energy conservation, led to the definition of a proper objective function, which guided the authors toward one possible solution, whose main advantage is to be easily implementable in current engine ECU without any hardware modification or additional costs: it consists of the modulation of the energy transferred to the needle by means of an injection pulse interruption in order to avoid any bounces. The pulse interruption strategy was implemented in the mathematical model, thus allowing the effective linearization of the simulated injector flow chart. On the basis of the good results obtained with simulations, the authors proceeded to the validation of the solution found by means of experimental tests carried out on a suitably equipped test bench. A real injector was controlled with the pulse interruption strategy, adopting as initial parameters the values suggested by simulation. The best interruption parameters were experimentally fixed, and allowed to 704

C.I. Compression Ignition CNG Compressed Natural Gas ECU Electronic Control Unit IGBT Insulated Gate Bipolar Transistor LPG Liquefied Petroleum Gas MPC Model Predictive Control OLS Ordinary Least Square S.I. Spark Ignition TTL Transistor to Transistor Logic E energy transferred to the injector needle during the opening phase [mJ] Ecoil energy stored in the solenoid coil [mJ] finj injection frequency [Hz] i solenoid current [A] L solenoid inductance [mH] mexp experimental injected mass [mg] mass flow [g/s] m R equivalent resistance [Ω] t time [ms] t0 time at the end of the rapid discharge phase [ms] V voltage [V] V0 voltage at the end of the rapid discharge phase [V] V1 asymptotic voltage of the discharge phase [V] δ time delay of the injection interruption [ms] ϕ objective function of the optimal condition search algorithm [mJ] τ duration of the injection interruption [ms] Δt injection time [ms] t* opening phase duration [ms] A solenoid cross-section area [mm2] B magnetic induction in the steel [T] B0 magnetic induction in the air [T] c viscous damping coefficient [N/(m/s)] ambient pressure force [N] Famb

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Fem Ffr Fgas Fv H H0 i k M N R V x xai xai xbi  x δS ϕB μ0 μr σ ξ

electromagnetic force [N] Coulomb frictional force [N] gas pressure force [N] gas viscous force [N] magnetic field in the steel [A/m] magnetic field in the air [A/m] solenoid coils current [A] spring constant [N/mm] needle mass [g] number of coils enclosed by the loop solenoid electric resistance [Ω] injector power supply voltage [V] needle position [mm] needle velocity [m/s] needle velocity after impact [m/s] needle velocity before impact [m/s] needle acceleration [mm/ms2] spring preload deformation [mm] magnetic induction flux [Wb] space magnetic permeability [H/m] steel relative magnetic permeability [H/m] path along the loop [mm] coefficient of restitution [-] 8 REFERENCES

[1] Cammalleri, M., Pipitone, E., Beccari, S., Genchi, G. (2013). A mathematical model for the prediction of the injected mass diagram of a S.I. engine gas injector. Journal of Mechanical Science and Technology, vol. 27, no. 11, p. 3253-3265, DOI:10.1007/s12206-0130848-6. [2] Lino, P., Maione, B., Rizzo, A. (2007). Nonlinear modelling and control of a common rail injection system for diesel engines. Applied Mathematical Modelling, vol. 31, no. 9, p. 1770-1784, DOI:10.1016/j. apm.2006.06.001. [3] Seykens, X.L.J., Somers, L.M.T., Baert, R.S.G. (2005). Detailed modelling of common rail fuel injection process. Journal of Middle European Construction and Design of Cars, vol. 3, no. 2-3, p. 30. [4] Le, D., Shen, J., Ruikar, N., Shaver, G.M. (2013). Dynamic modeling of piezoelectric fuel injector during rate shaping. International Journal of Engine Research, vol. 15, no. 4, p. 471-487, DOI:10.1177/1468087413492737. [5] Mehlfeldt, D., Weckenmann,H., Stohr, G. (2008). Modeling of piezoelectrically actuated fuel injectors. Mechatronics, vol. 18, no. 5-6, p. 264-272, DOI:10.1016/j.mechatronics.2008.03.001. [6] Baratta, M., Catania, A.E., Spessa, E., Herrmann, L., Roessler, K. (2009). Multi-dimensional modeling of direct natural-gas injection and mixture formation in a stratified-charge SI engine with centrally mounted injector. SAE International Journal of Engines, vol. 1, no. 1, p. 607-626, DOI:10.4271/2008-01-0975.

[7] Di Cairano, S., Bemporad, A., Kolmanovsky, I.V., Hrovat, D. (2007). Model predictive control of magnetically actuated mass spring dampers for automotive applications. International Journal of Control, vol. 80, no. 11, p. 1701-1716, DOI:10.1080/00207170701379804. [8] Lino, P., Maione, B., Amorese, C., De Matthaeis, S. (2008). Modeling and predictive control of a new injection system for compressed natural gas engines. Control Engineering Practice, vol. 16, no. 10, p. 12161230, DOI:10.1016/j.conengprac.2008.01.008. [9] Dyntar, D., Guzzella, L. (2004). Optimal control for bouncing suppression of CNG injectors. Journal of Dynamic Systems, Measurement and Control (ASME), vol. 126, no. 1, p. 47-53, DOI:10.1115/1.1648311. [10] Bosch Automotive Handbook (1996). Robert Bosch GmbH, Bentley Publishers, Cambridge, from: http:// www.bentleypublishers.com/c/H016, accessed on 201403-25. [11] Pipitone, E., Beccari, S. (2009). Performances improvement of a S.I. CNG bi-fuel engine by means of double-fuel injection. SAE International, Technical Paper 2009-24-0058, DOI:10.4271/2009-24-0058. [12] Pipitone, E., Beccari, S. (2010). Performance and emission improvement of a S.I. engine fuelled by LPG/ gasoline mixtures. SAE International, Technical Paper 2010-01-0615, DOI:10.4271/2010-01-0615. [13] Obiols, J., Soleri, D., Dioc, N., Moreau, M. (2011). Potential of concomitant injection of CNG and gasoline on a 1.6L Gasoline Direct Injection Turbocharged Engine. SAE International, Technical Paper 2011-011995, DOI:10.4271/2011-01-1995. [14] Momeni Movahed, M., Basirat Tabrizi, H., Mirsalim, M. (2014). Experimental investigation of the concomitant injection of gasoline and CNG in a turbocharged spark ignition engine. Energy Conversion and Management, vol. 80, p. 126-136, DOI:10.1016/j. enconman.2014.01.017. [15] Veiga, M., Mansano, R., Silva, R., Gomes, C. (2010). Injection system for tri-fuel engines with control of power by simultaneous use of CNG and ethanol or gasoline, SAE International, Technical Paper 2010-360195, DOI:10.4271/2010-36-0195. [16] Beccari, S., Pipitone, E., Cammalleri, M., Genchi, G. (2014). Model-based optimization of injection strategies for SI engine gas injectors. Journal of Mechanical Science and Technology, vol. 28, no. 8, p. 1-13, DOI:10.1007/s12206-014-0742-x. [17] Fairchild Semiconductor Corporation, “HGTP14N40F3VL/HGT1S14N40F3VLS Data sheet Rev. B1”, February 2002, http://www.datasheetlib. com/datasheet/184055/hgtp14n40f3vl_fairchildsemiconductor.html, accessed on 2014-03-25.

9 APPENDIX A As already mentioned, the first tests carried out interrupting the injection pulse with the aim of

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modulating the energy transferred to the needle showed a phenomenon not observed in the previous work: the experimental data in effect revealed an extra-current whose duration and magnitude depends on the duration of the pulse interruption, as shown in Fig. 21. Here, different solenoid current waveforms, obtained by varying the pulse interruption duration, are represented as functions of time: as can be seen, the extra-current amplitude decreases when the interruption duration increases.

As can be seen, the voltage induced in the solenoid circuit exceeds 390 V (which is also the oscilloscope maximum visible value): as a consequence, the IGBT intrinsic protection system, endowed of Zener diodes and internal resistances, permits this high voltage to discharge through itself toward the ground of the counter board used to generate the digital pulses (current i1 in Fig. 22). This first part of the solenoid energy discharge is very rapid, as can be noted in Fig. 23.

Fig. 21. Measured solenoid current for different duration of the pulse interruption

Fig. 23. Measured solenoid voltage during the injection pulse interruption

The analysis of the injection electric circuit (here reported in Fig. 22) and of the IGBT characteristics [17], together with some voltage measurements (carried out between points A and B of the electric circuit) led the authors to believe that the phenomenon is related to the dissipative discharge of the energy accumulated in the solenoid coil.

Once below 130 V (approximately 0.02 ms after the start of pulse interruption), due to the IGBT intrinsic protection system properties [17], the path to the counter board ground through the IGBT closes; therefore, the energy continues to discharge through the internal structure of the injector itself; this second stage of the discharge process, as can be observed in Fig. 23, is slower, and the voltage exhibits a gradual decrease. Considering a simple R-L circuit, this voltage waveform has been fitted by an exponential function of time t:

Fig. 22. Electrical circuit involved in the injector operation

When the injection is interrupted, the IGBT is deactivated, and this abruptly opens the electric circuit (between points A and B); the solenoid current immediately falls down to zero, and this causes an abrupt decrease of the solenoid magnetic flux, which, according to the Faraday-Lenz law, induces a very high voltage in the solenoid. A waveform of this high voltage has been recorded by means of a 100 MHz oscilloscope and is reported in Fig. 23. 706

V = V1 + (V0 − V1 ) ⋅ e

−( t −t0 )⋅

R L

, (3)

where t0 and V0 represent the time and the voltage at the end of the rapid discharge phase (0.02 ms and 130 V respectively), V1 is the asymptotic voltage (fixed by the power supply system) and L/R is the time constant of the circuit: in particular L is the known solenoid inductance while R is the equivalent resistance, which has been determined fitting the data of Fig. 23 with Eq. (3). The instantaneous energy stored in the solenoid can be always expressed as:

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ECoil =

1 2 ⋅ L ⋅ i ( t ) , (4) 2

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where, according to Eq. (3), the current i(t) during the slow discharge phase can be evaluated as:

(V0 − V1 ) −(t −t0 )⋅ RL . (5) ⋅e R

If, during this slow discharge phase, after a sufficiently short time (less than 0.2 ms from the start of pulse interruption) the electric circuit is closed again (i.e. the IGBT is reactivated), the residual energy still stored in the solenoid suddenly discharges through the power supply cathode, producing the extra current (i2 in Fig. 22), whose value depends on the energy still available in the solenoid, and can be evaluated by Eq. (5); the value of this extra current is therefore related to the duration of the interruption itself: this explains the current waveforms represented in Fig. 21.

closed loop to the electric current flowing through the loop.

∫ H ⋅ dσ = N ⋅ i, (6)

where N is the number of coils enclosed by the loop.

Fig. 25. Schematic representation of the injector magnetic circuit

While the magnetic H-field changes passing from the steel to the air gap, the magnetic induction B does not change, hence: B0 = B , (7)

Fig. 24. Injection pulse interruption: measured and simulated solenoid current for air at 10 bar

Eqs. (3) and (5) have been implemented in the model with the aim to take account of the extra current phenomenon: Fig. 24 shows the good agreement between experimental measure and simulation output. 10 APPENDIX B The mathematical model realized by the authors [1] is able to predict the needle motion during an entire injection event and to evaluate the total injected mass for any fixed operative condition. A brief description of the main physical principles taken into consideration to realize the mathematical model follows. From the electrical point of view, the injector has been modelled as an electromagnet, schematically represented in Fig. 25. When the current i flows through the injector solenoid coils, a magnetic field H appears in the steel core according to the Ampère’s circuital law, which correlates the integrated magnetic field around a

where subscript 0 refers to the air gap. B and H fields are connected by the constitutive equation:

B = μ0·μr·H ,

(8)

being μ0 the space permeability and μr the steel relative permeability (function of H). Combining the above equations allows definition of the relation B = B(i, x) between the magnetic B-field, the solenoid current i and the needle position x. Furthermore, Ohm’s law together with FaradayLenz’s law yields:

R ⋅i = V −

dϕ B , (9) dt

where R represents the solenoid electric resistance, ϕB = N · B · A = ϕB(i, x) the flux of the magnetic B-field and A is the solenoid cross-section area; in this way, the solenoid current is connected to its time derivative and to the needle motion. The relation between the electromagnetic force Fem and the solenoid current i can be easily determined considering that for any virtual displacement dx of

Experimental Model-Based Linearization of a S.I. Engine Gas Injector Flow Chart

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the needle, the work produced by Fem must equal the electromagnetic energy variation in the gap:

[ B(i, x)] ⋅ S ⋅ dx, (10) B02 ⋅ S ⋅ dx = 2 ⋅ µ0 2 ⋅ µ0 2

Fem ⋅ dx =

being

∫

H ⋅ dB =

B02 , (11) 2 ⋅ µ0

the magnetic energy per unit volume in the gap. From the mechanical point of view, the injector has been modelled as a mass-spring system, damped by Coulomb and viscous frictional forces, and subject to variable electromagnetic and constant gas pressure forces. Based on the free body diagram of Fig. 26, the needle dynamic equilibrium equation can be written as:

where xai and xbi are the needle velocities after and before impact respectively, and ξ is the coefficient of restitution (i.e. the ratio between the kinetic energy after and before impact). In summary, in the mathematical model realized, both solenoid current and needle motion are fully described by the two coupled differential equations (Eqs. (9) and (12)), together with the Eq. (13) applied at each impact. Assuming the injector equivalent to a choked flow convergent nozzle, the total injected mass results proportional to the value of the integral of the needle displacement over time; this allows determination of the total amount of fuel supplied at the end of each injection event. The above physical equations were then expressed in a dimensionless form [1], thus strongly reducing the number of parameters required for model calibration.

M ⋅  x + c ⋅ x + k ⋅ x = x = Fem − k ⋅ δ S − Ffr − Fgas − Famb , (12) x

(

)

where M represents the needle mass, c the viscous damping coefficient, k and δS the spring constant and preload deformation, respectively; Ffr is the coulomb frictional force, Fgas is the force exerted by the gas pressure while Famb is the ambient pressure force. This equation correlates the needle position x to its first and second time derivative and to the solenoid current i. Finally, to take in to account the needle impacts and bounces on the seat surfaces, the following equation has been introduced:

708

xai = − ξ ⋅ xbi , (13)

Fig. 26. Injector needle free body diagram

Pipitone, E. – Beccari, S. – Cammalleri, M. – Genchi, G.

Received for review: 2014-03-20 Received revised form: 2014-06-27 Accepted for publication: 2014-07-24

The Use of Computed Tomography to Analyse Grinding Smudges and Subsurface Defects in Roller Bearing Rings Zbrowski, A. – Matecki, K. Andrzej Zbrowski* – Krzysztof Matecki

National Research Institute, Institute for Sustainable Technologies, Poland This article shows that changes occurring on the surface of roller bearings in the form of grinding smudges stem from the subsurface material defects of these elements. The authors discuss how the smudges are created and show the results of computed tomography tests conducted for roller bearing rings with the above-mentioned defects. The ring reconstruction images are presented, and the defects are located and described with the use of reverse engineering. The defects identified are presented in radiographs. The topography tests confirmed the existence of subsurface defects that emerge on the surface in the form of smudges, once the grinding of an element starts. Keywords: roller bearing rings, material defects, non-destructive tests, computed tomography

0 INTRODUCTION Defects in roller bearing rings stem from incorrectly conducted technological processes, e.g. metallurgical processes, forging, thermo-mechanical processing, or machining. Each such mistake made can be the cause of a different kind of defect. The most dangerous are internal material defects as cracks, micro-shrinkage, overlapping, etc., which appear just under the surface. The high load in the defect area results in excessive stress travelling deep below the surface. As a result, there is a sudden and unexpected damage to the roller bearing, shortening the expected time of its operation. Detection of these defects during the manufacturing process is possible only when they appear on the surface of the element and become visible to the naked eye, or can be spotted using specialised defectoscopy instrumentation dedicated to mass production systems. The defects usually “make themselves visible” during the final stage of processing, i.e. during the grinding of the surface of the element. The incorrect parameters of this final process cause surface defects in the form of different surface hardnesses, tension, micro-cracks, grinding burns and smudges. While the first of them are the results of thermal loads, the smudges can result from the “open (exposed)” subsurface defect, which cannot be observed with the naked eye. The defect can be confirmed only when the element is subjected to defectoscopy tests by means of computed tomography [1] to [3]. The tests were intended to prove that there is a direct connection between the inner material defects and the occurrence of grinding smudges on the surface of the element studied. The purpose of the test was to detect defects in the internal structure of the ring under the ground surface on which the smudges were formed. The authors used computed tomography methods to look for defects

including voids, blisters, cracks, and porosity, which can be found during the grinding process. 1 APPLICATION OF COMPUTED TOMOGRAPHY FOR NON-INVASIVE TESTS The X-ray computed tomography (CT) is commonly used in industrial non-destructive tests for technical objects [4] and [5]. Based on the X-rays in different sections, spherical images are generated, which are then used for dimensional analyses, and constitute an important element of reverse engineering. CT enables material defectoscopy tests in which cracks, discontinuity, inclusions, or structural defects are detected. The fundamental advantage of the CT method is the possibility of conducting non-contact, spatial analysis of the internal structure of the tested element with a resolution of up to 1 µm [6]. The precision of the method is so high that it enables the determination of the spatial arrangement of crystallites of the tested material [7]. Therefore, the CT method was used for the observation of the microstructure and the propagation of the defect in a model composite material in elements manufactured using the sintering method [8] and [9]. In order to determine the influence of the material deformation on the internal structure of the foamed metal using the CT method, an analysis of the structure of these metals was conducted [10] and [11], followed by an analysis of the foamed polypropylene [12] and [13]. The results obtained enabled the experimental verification of the computer simulation of the deformation. Due to its high resolution, the CT method was used for the inspection of the quality of the metal foaming process [14]. The examination of the microstructure of the walls of inner cells of the foamed metals was also conducted; as a result, blisters were detected [15]. In the case of structure deformation tests, the CT

*Corr. Author’s Address: National Research Institute, Institute for Sustainable Technologies, 26-600 Radom, Pułaskiego 6/10 str, Poland, andrzej.zbrowski@itee.radom.pl

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method enables the recording of the sintering process [16]. The observation of the microstructure of the sintered material, in turn, allowed the experimental verification of computer simulations. The CT method is common in tests on the internal structure of metal elements [17], as well as in the study of the microstructure of the casts of non-ferrous metals [18]. The tomography tests were used to determine the Young modulus of metal-ceramic composites [19]. Based on the spatial model, a computer simulation was carried out for the deformation of the sample. The Young modulus, determined using computer simulation, was then compared to the results of experimental tests. The CT method allows the spatial mapping of the structure of metallic and non-metallic materials [20]. The use of the CT method also enables the determination of the distribution of the density of the sintered metal [21], which in turn allows the simulation of geological processes [22] and/or the simulation of the impact of the pile foundation on the ground [23].

During the grinding of the inner rings of tapered roller bearings, grinding wheels with 508 mm in diameter, 7 to 40 mm in width, hardness J and K, and structure 8 are used. The abrasive used in them is alumina or sintered alumina with a grain size of 140 µm. The cutting speed is 60 m/s. During grinding, the wheel encounters discontinuities located just under the surface of the material ground. Such material defects resulting from the rolling, forging or casting processes are usually filled with impurities from metallurgical or forging processes. The grinding wheel opens up the material discontinuities, extracts the impurities, and smudges them over the ground surface, which creates smudges of different colours and lustre. The size of the smudge depends on the kind, shape, and location of the discontinuity. The greater the size of the discontinuity, the wider and clearer the smudge. Even when the opening in the discontinuity of the material is small, the smudge can be visible to the naked eye, not to mention specialised optical inspection systems. A roller bearing ring with the detected defect in form of a grinding smudge should be excluded from any further manufacturing process.

2 CREATION OF GRINDING SMUDGES

3 TEST STAND

The grinding is the final stage of surface generation. It has an influence on the accuracy of the shape, size, and smoothness of working surfaces and the condition of the surface layer. The grinding process affects the utility properties of the roller bearings and the safety of their operation. During the grinding process, certain subsurface defects concerning the inner structure of the element may become visible. Any detected defect automatically discredits an element and necessitates its exclusion from further manufacturing processes. Visible symptoms of the existence of inner defects in the structure of the investigated material are surface smudges stemming from the grinding process (Fig. 1).

The rings were tested using the Phoenix v|tome|x s240 X-ray computed scanner, which enabled the projection of objects from different directions and helped to obtain the reconstruction images of the layered roller bearing rings, which in turn facilitated the spatial imaging of the defects under the surface of the tested objects. A CT device is an automatic device employing VGStudio MAX 2.1 software. The data are collected and processed using Phoenix datos/x 2 acq software, but for the reconstruction, Phoenix datos/x 2 rec is used. The time of reconstruction is approximately 30 min for a resolution of ~ 80 μm. Table 1. Scanner setting parameters Parameter

210 kV

Current

210 μA

Voxel size Number of images

Fig. 1. Grinding smudges on the inner ring of the roller bearing

A smudge is created on the surface using a grinding wheel. 710

Value

Accelerating voltage

≈ 98.5 μm 1600

Exposure time

200 ms

Radiation filter

0.2 mm Cu

The steel that the rings are made of is characterised by high density, which significantly

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reduces the possibility of transmittance, which makes it difficult to obtain a desired X-ray contrast and the reconstruction image. Some of the defects, particularly the small ones, can be unclear, and their imaging with the use of reconstruction images requires proper accuracy of tests, which depends, inter alia, on the number of projections. The scanner setting parameters are presented in Table 1. The detection of internal defects is also possible when magnetic, ultrasonic methods and eddy current methods are applied. Computed tomography, however, is the only method providing a quantitative and qualitative assessment of the ring structure and enabling the three-dimensional visualization of the defect. 4 TEST OBJECT

5 TEST RESULTS 3D images were obtained for both A and B rings. A method of computed tomography was used (Fig. 4) to detect the location of the defect (marked with numbers 1 and 2).

The A and B roller bearing rings with visible grinding smudges made of the 100Cr6 steel were tested (Fig. 2). The hardness of the surface after the heat treatment was 58 to 62 HRC. The geometry of the test object is presented in Fig. 3.

a) b) Fig. 2. Test objects for rings: a) A and b) B

Fig. 4. Tomography reconstruction image for rings a) Ring A and b) Ring B

In the case of Ring A, two discontinuities were detected; whereas only one defect was detected in the case of the other ring. In order to enable a more detailed description of the defects, three perpendicular sections of the reconstructed objects were created. They allowed the determination of the character, orientation and the topography of the discontinuity. The examples of the images of Defect 1 in Ring A in the defined axes are depicted in Fig. 5. In order to make the images of the defect more readable, the geometry of the object was reconstructed (Fig. 6). The imaging of the geometry of the defect was conducted using the results of the measurements taken for the reconstructed 3D model. Discontinuity 1 in Ring A is similar to the crack or the flattened cavity with extended topography in the direction perpendicular to the inner surface of the ring (Fig. 6). The examples of the images of the area in which Defect 2 was observed for Ring A are shown in Fig. 7.

Fig. 3. Geometry of the test object; dimensions in mm The Use of Computed Tomography to Analyse Grinding Smudges and Subsurface Defects in Roller Bearing Rings

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a) a)

d) Fig. 5. Selected images of sections in xyz axes of Defect 1 in the reconstruction of a) ring A, b) xy intersection, c) yz intersection, d) xz intersection

d) Fig. 7. Selected images of Defect 2 cross-section in planes XY, YZ, XZ in reconstruction of Ring A: a) ring A, b) xy intersection, c) yz intersection, d) xz intersection

a) b) c) Fig. 6. Cross-sections and a draft of the defect propagation – Defect 1, Ring A – dimensions in mm: a) intersection, b) part section A c) part section B

a) b) c) Fig. 8. Cross-sections and a draft of the defect propagation – Defect 2, Ring A – dimensions in mm: a) intersection, A b) part section B c) part section C

Defect 2, similar to Discontinuity 1, has the form of a crack or a flattened cavity. It is smaller than Defect 1, and its shape is more irregular (Fig. 8). Fig. 9 depicts the radiographs for Ring A with a clearly visible Defect 1 and a tiny, even at greater magnification, Defect 2.

In the central part of Ring B, a discontinuity of extended structure was observed. The middle of the defect has the form of a cavity in which the propagating discontinuities (crack-like) are rooted (Fig. 10). The defect is also visible in Fig. 11 presenting the radiograph of Ring B.

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Fig. 11. Radiograph for Ring B

6 CONCLUSIONS

b) Fig. 9. Radiograph for Ring A: a) defect 1, b) defect 2

The computed tomography tests conducted for roller bearing rings with grinding smudges confirmed the existence of structural discontinuities underneath the smudges. These defects become visible after the grinding process but are not visible to the naked eye. The possibility of multidirectional irradiation, and with it, the multifaceted reconstruction of the tested rings, allows speculations concerning the character, size, shape, and orientation of the defect with reference to the working surfaces in roller bearing rings. The reconstructions indicate that the discontinuity in Ring A has the form of a crack or cavity. It has an irregular shape oriented perpendicularly to the inner area of the ring. Defect 2, with irregular shapes, also has the form of a crack or a flattened cavity from which the crack propagates towards the inner area of the ring. This defect is smaller and less visible than Defect 1. The defect located in Ring B has the form of a cavity with discontinuities directed towards the inner area of the ring as well. Based on the tests conducted, the relation between the grinding defects and subsurface material defects, such as cavities or discontinuities, were determined. These defects are the results of metallurgical processes, and they disqualify the rings from operation. Since the method is rather time-consuming, it cannot be used in mass production systems for the inspection of the quality of roller bearing rings. However, it is crucial as far as off-line verification of elements qualified as defects is concerned. It can be particularly useful in comparison tests of different defectoscopy techniques applied in mass production. 7 ACKNOWLEDGEMENT

d) Fig. 10. Selected images of Defect 1 cross-section in planes XY, YZ, XZ in reconstruction of Ring B: a) ring B, b) xy intersection, c) yz intersection, d) xz intersection

This scientific work was executed within the Strategic Programme “Innovative Systems of Technical Support for Sustainable Development of Economy”

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within 2010 to 2014 Innovative Economy Operational Programme. 8 REFERENCES [1] Schillinger, B., Lehmann, E., Vontobel, P. (2000). 3D neutron computed tomography: requirements and applications. Physica B: Condensed Matter, vol. 276278, p. 59-62, DOI:10.1016/S0921-4526(99)01254-5. [2] Auditore, L., Barna, R.C., Emanuele, U., Loria, D., Trifiro, A., Trimarchi, M. (2008). X-ray tomography system for industrial applications. Nuclear Instruments and Methods in Physics Research B, vol. 266, no. 10, p. 2138-2141, DOI:10.1016/j.nimb.2008.02.082. [3] Bartscher, M., Hilpert, U., Goebbels, J., Weidemann, G. (2007). Enhancement and proof of accuracy of industrial computed tomography (CT) measurements. Annals of the CIRP - Manufacturing Technology, vol. 56, no. 1, p. 495-498, DOI:10.1016/j.cirp.2007.05.118. [4] Krimmel, S., Stephan, J., Baumann, J. (2005). 3D computed tomography using a microfocus X-ray source: Analysis of artifact formation in the reconstructed images using simulated as well as experimental projection data. Nuclear Instruments and Methods in Physics Research A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol. 542, no. 1-3, p. 399-407, DOI:10.1016/j.nima.2005.01.171. [5] Buffiere, J.-Y., Maire, E., Adrien, J., Masse, J.-P., Boller, E. (2010). In situ experiments with X ray tomography: An attractive tool for experimental mechanics. Experimental Mechanics, vol. 50, no. 3, p. 289-305, DOI:10.1007/s11340-010-9333-7. [6] Salvo, L., Cloetens, P., Maire, E., Zabler, S., Blandin, J.J., Buffière, J.Y., Ludwig, W., Boller, E., Bellet, D., Josserond, C. (2003). X-ray micro-tomography an attractive characterisation technique in materials science. Nuclear Instruments and Methods in Physics Research B: Beam Interactions with Materials and Atoms, vol. 200, p. 273-286, DOI:10.1016/S0168583X(02)01689-0. [7] Ludwig, W., King, A., Reischig, P., Herbig, M., Lauridsen, E.M., Schmidt, S., Proudhon, H., Forest, S., Cloetens, P., Rolland du Roscoat, S., Buffière, J.Y., Marrow, T.J., Poulsen, H.F. (2009). New opportunities for 3D materials science of polycrystalline materials at the micrometre length scale by combined use of X-ray diffraction and X-ray imaging. Materials Science and Engineering A, vol. 524, no. 1-2, p. 69-76, DOI:10.1016/j.msea.2009.04.009. [8] Babout, L., Maire, E., Fougères, R. (2004). Damage initiation in model metallic materials: X-ray tomography and modelling. Acta Materialia, vol. 52, no. 8, p. 2475.2487, DOI:10.1016/j.actamat.2004.02.001. [9] Babout, L., Maire, E., Buffière, J.Y., Fougères, R. (2001). Characterization by x-ray computed tomography of decohesion, porosity growth and coalescence in model metal matrix composites. Acta

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mechanical and non-destructive tests and microCT based computational modeling. Computational Materials Science, vol. 77, p. 19-30, DOI:10.1016/j. commatsci.2013.04.007. [20] Maire, E., Fazekasb, A., Salvob, L., Dendievelb, R., Youssefa, S., Cloetensc, P., Letangd, J.M. (2003). X-ray tomography applied to the characterization of cellular materials. Related finite element modeling problems. Composites Science and Technology, vol. 63, no. 16. p. 2431-2443, DOI:10.1016/S0266-3538(03)00276-8. [21] Burch, S. (2002). Measurement of density variations in compacted parts using X-ray computerized

tomography. Metal Powder Report, vol. 57, no. 2, p. 24-28, DOI:10.1016/S0026-0657(02)85009-3. [22] Ueta, K., Tani, K., Kato, T. (2000). Computerized X-ray tomography analysis of three-dimensional fault geometries in basement-induced wrench faulting. Engineering Geology, vol. 56, no. 1-2, p. 197-210, DOI:10.1016/S0013-7952(99)00143-X. [23] Eskisar, T., Otani, J., Hironaka, J. (2012). Visualization of soil arching on reinforced embankment with rigid pile foundation using X-ray CT. Geotextiles and Geomembranes, vol. 32, p. 44-54, DOI:10.1016/j. geotexmem.2011.12.002.

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Received for review: 2013-12-04 Received revised form: 2014-05-09 Accepted for publication: 2014-06-16

Mechanics and Dynamics of Helical Milling Operations Liu, C. – Wang, G. – Dargusch, M.S. Changyi Liu1,* – Gui Wang2 – Matthew S. Dargusch2

1 Nanjing

University of Aeronautics & Astronautics, College of Mechanical and Electrical Engineering, China 2 The University of Queensland, School of Mechanical and Mining Engineering, Australia

This paper presents analytical models dealing with time domain cutting forces and the time and frequency domain dynamics of helical milling operations. The cutting forces both on the side-cutting edges and on the end-cutting edges along the helical feed path are modelled by considering the tangential and the axial motions of the tool. The dual periodicity, which is caused by the spindle rotation, as well as the period of the helical feed of the cutting tool has been included. The models of the machining process dynamics and of the chatter stability problems are decomposed into two parts: the critical axial depth of cut, and the critical radial depth of cut, both of which are solved. The dynamics model allows the selection of processing parameters, including axial cutting depth, radial cutting depth, and spindle rotation velocity in the chatter-free zone and the prediction of chatter vibrations during helical milling. Experiments have also been performed for real machining observations in order to verify the chatter-free zone obtained from the analytical model. Keywords: cutting, process modelling and planning, helical milling, cutting dynamics, chatter

0 INTRODUCTION Helical milling has been applied to generate or enlarge boreholes by means of a milling tool being fed on a helical path into the work piece. It is an advanced hole-making technology, used mainly for typically difficult-to-cut materials. Helical milling is shown to be able to machine H7 quality holes with a surface finish of 0.3 μm Ra in hardened AISI D2 tool steel, and to be able to enhance the tool life [1]. Helical milling was utilized in the hole-making in carbon fibre-reinforced plastic, and the cutting force coefficients were identified and the mechanistic modelling technique was used to predict cutting forces [2]. The method was employed on CFRP-titanium layer compounds, to model the non-deformed chip geometries for the process, and to explain the effect of the helical feed on the cutting forces [3]. Accuracy and precision are important considerations for machining products along with efficiency and cost. The investigation of machining stability and chatter vibration is significant for process planning. Milling can be one of the most complicated cutting process. A cutting forces prediction model considering the tool’s geometrical features and the tool-work piece interaction is the foundation for the chatter modelling. Most research analyses the cutting forces on discrete milling tools elements, and then integrates them to result in the cutting forces on cutting tools. The cutter is divided into a number of discrete disk elements along its axis, and each elements’ surface is considered as the plane of cut [4] and [5]. The factors that influence milling forces during the multi-axis milling process, such as the varying lead 716

and tilt angles, the cutting tool deflections and form errors, have been included [6] and [7]. During the helical milling processing, both side-cutting edges and end-cutting edges involved simultaneously is an important feature. The cutting forces of the end-milling process using a flat end milling tool considering both the side edge and the side edge cuttings simultaneously were modelled [8]. For helical milling operations, as well as plunge milling operations, axial feed is another feature that should not be neglected. By considering the rigid body motion of the cutter and three translational and torsional vibrations of the structure, models to predict cutting forces, vibrations, and the chatter stability in the frequency domain for plunge milling process have been presented [9] and [10]. The finite element method (FEM) and finite difference method have been applied as a novel approach to simulate the dynamics of the machining system. A coupled finite element (FE) model was created to simulate the dynamic properties at the tool end point of the whole mechanical system, including the tool, spindle, and machine tool frame [11]. FEM has also been used to model the dynamics and vibration. This approach simulates the process, simultaneously including both vibration and the chip formation [12]. With regard to machining dynamic effects, machining geometric defects have been modelled, and the dynamic displacements due to clamping and machining forces have been defined using FEM [13]. The resonance frequencies of the axial vibration mode of the milling tool were predicted using FEM model of the vibration milling tool and demonstrated that high-frequency vibrations superimposed onto the tool lead to stabilization of the

*Corr. Author’s Address: College of Mech. and Electrical Engineering, Nanjing University of Aeronautics & Astronautics, 210016, Nanjing, China, liuchangyi@nuaa.edu.cn

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milling process with superior surface finishes [14]. These FEM models reflected the physical processes in the machining system accurately and visually, but such approaches are time-consuming and require intensive computing resources. Machining chatter was detected using on-process methods, such as acquiring and processing cutting forces, vibration signals and acoustics signals, or by using off-process methods [15]. A new method for the detection of chatter in the end-milling operation based on the wavelet transform has been suggested, which provides various ways to determine chatter characteristics real-time or post process [16]. Sound pressure, machining force and tool displacements are measured during the process to evaluate the stability of high-speed milling, and tool displacements are used as input data to simulate the chip thickness variation during the process [17]. Helical milling has been presented as an alternative enabling technology for drilling operations [1] to [3]. Recently, a few analyses of the cutting forces of the helical milling process have emerged [18] and [19]. Although the mechanisms and dynamics of special operations with axial feed characteristics, such as plunge milling and drilling, or with the tangential feed properties, such as multi-axis milling and circular milling, were investigated, the dynamics and the chatter stability model for helical milling or milling with axial feed have not been found. The characteristics of the helical milling operation are discussed in this paper, including modelling the cutting force influenced by the axial feed; to build the dynamics model in the four degrees of freedom (DOF) cutting tool – machine tool vibration system,

and to simulate the chatter limitation for both the axial cutting depth and the radial cutting depth. This research based on the analytical cutting forces model, combining the interactions both on the sidecutting edges and on the end-cutting edges of helical milling operations, focusing on the machining process dynamics model and chatter stability problems, which will dealt with into two parts: the critical axial depth of cut, and the critical radial depth of cut. Both the chatter stability limitations of the critical axial depth of cut and the critical radial depth of cut during helical milling process will be solved. 1 CUTTING FORCE MODEL FOR HELICAL MILLING The motion curve of a certain point on the cutting edge of the milling tool performing helical milling is composed of the helical motion of the tool axis and the circular motion of the point relative to the axis. The diameter of the bore ΦB and the endmill diameter Dm, the rotating angular velocity Ωh of the helix feed, the axial feed speed fva and the tangential feed speed fvt, the tangential feed rate per tooth fzt, the axial feed rate per tooth fza, the pitch of the helix curve of the reference frame P, and the spindle rotational velocity Ω, are depicted in Fig. 1. To define the movement of the cutting tool, and the cutting force, an X, Y, Z global coordinate system (GCS) is attached the work piece, and an x, y, z local coordinate system (LCS) is attached to the cutter. While modelling the cutting forces of the helical milling process, there have been two considerations that might also affect the dynamics of the helical milling process. One is the cutting force

Fig. 1. Schematics of helical milling Mechanics and Dynamics of Helical Milling Operations

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fluctuating periodically as a result of the cutting tool circumferential feed; the other is the cutting force component due to the cutting tool axial feed. The axial feed force mostly occurs at the end-cutting edge of the milling tools. The detailed interpretation and the development of the relative equations in this section refer to the authors’ previously published literature [18]. The cutting forces loaded on the tool are a combination of side-edge cutting forces and end-edge cutting forces; these two parts are modelled and then summarized:    (1) F = F i + F *,   where, F i and F * are cutting force components on the side-cutting edge and on the end-cutting edge, respectively. The cutting force components on the jth endcutting edge to the LCS coordinate can be represented as:  Fx , j *   − cos ϕ j 0 0 0  *   F 0 sin ϕ 0 0    y, j   j *  f za   [ Θ ]  K     ,(2)  * =  0 0 1 0   1   Fz , j    T j *   0 0 0 1  

 FX , j   cos Ω ht sin Ω ht F    Y , j   − sin Ω ht cos Ω ht  = 0  FZ , j   0  TZ , j   0 0  

0 0   Fx , j i + Fx , j *    0 0   Fy , j i + Fy , j *   . (3)  1 0   Fz , j i + Fz , j *    T j* 0 1   

Finally, the superposition of all the cutting forces on the Nm cutting teeth is the cutting forces on the cutting tool. The established time domain analytical cutting forces model for helical milling operations reflects both the cutting process on the side-cutting edge and on the end-cutting edge, incorporating the influence of the spindle motion and the tool helical feed. 2 DYNAMICS OF HELICAL MILLING Helical milling is a typical interrupted cutting operation with a tangential feed and an axial feed. The machine tool-spindle-milling tool system was considered to be flexible in four degrees of measurement (DOM), i.e. in the global X, Y, Z and C directions, as shown in Fig. 2. This flexibility could lead to chatter instability if the axial depth of cut, the radial depth of cut and the spindle rotation speed were not selected properly.

where K*vc, K*ve, K*nc, K*ne are the cutting force constants on the end-cutting edges along the circumferential and normal directions respectively,

 K vc*   K *  =  K nc*  K rc* 

K ve*   K ne*  , K re* 

 cos θ  sin θ Θ = dr [ ] ∫D2m −ae*  0   r cos θ Dm 2

N m f zt cos ϕ j 2π

− sin θ cosθ 0 −r sin θ

, θ = arg tan

0 0  N f , A = m za , 0 2π  0

A , ϕj is the relative r+B

rotational angle of the cutting tooth j, r is the distance from centre point to a certain point on the end-cutting edge. The side-cutting edge force components are added to the end-cutting force components on the jth tooth, then converted to GCS coordinates. 718

Fig. 2. Four DOM dynamics system of helical milling

The proposed cutting force model was used to build the dynamics model of the helical milling system and to predict the limitation of the chatter stability. Based on the regenerative chatter mechanism, the time domain dynamics relationship between the cutting force and dynamic displacement can be expressed as:

Liu, C. – Wang, G. – Dargusch, M.S.

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For helical milling operations, transform to the X-Y coordinates,

[ M ]{R ( t )} + [C ]{R ( t )} + [ K ]{R ( t )} = {F ( t )} , (4) where R is the dynamic displacement or amplitude vector with four components X, Y, Z and C. From the proposed cutting forces model, the cutting forces have a functional relationship with ap, ae, fzt, fat, Ω, Ωh etc. The objective of chatter prediction during the helical milling operations should be to find the proper relationship between ap, ae, and Ω in order to avoid chatter vibration. To find the chatter lobe including the ap, ae, Ω directly is difficult. Therefore, a certain level of idealization and assumption has to be introduced. According to the cutting forces model that resulted from cutting forces on the side-cutting edges and cutting forces on the end-cutting edges, the dynamics model and chatter limitation problem was decomposed into two parts: the X and Y direction, and Z and C direction.

FX  1 t  ∆x    = K tc a p [ AXY ]   , ∆y   FY  2

(7)

 cos ( Ω ht ) sin ( Ω ht )  where, [ AXY ] =    Axy  are the  − sin ( Ω ht ) cos ( Ω ht )  time-varying directional dynamic helical milling force coefficients. If the influence of the Ωh was neglected, the procedure to obtain the approximate solution could be identical with that in literature [5].

2.1 The Chatter Limitation of Axial Depth of Cut The X and Y components of the cutting forces on the cutting tools mainly result from the interaction of the side-cutting edges, because the magnitude of the interaction on the end-cutting edges is smaller, and their sum is almost zero. Hence, the axial depth of cut is affected by the X and Y components of the cutting forces. Regenerative chatter is the result of variation in the dynamic chip thickness as depicted in Fig. 3. The variation of dynamic chip thickness can be described as:

( )

Fig. 3. Dynamic chip thickness variation in x and y direction

Considering the X and Y force components, and the Fourier transform, Eq. (4) becomes Eq. (8).

∆h ϕ j = hst − h ϕ j =

(

)

= f zt sin ϕ j − ∆x sin ϕ j + ∆y cos ϕ j . (5)

(

)

(

)

(8)

The critical chatter frequency of the spindle is ωc,

{R (ω )} = G (ω ) {F (ω )} ,

If the edge effect is neglected, and the dynamic chip thickness into the side-cutting edge forces model is substituted, the sum of cutting forces on the Nm flutes can be given as, (Eq. (6)):

i - K i tc sin 2ϕ j − K i rc 1 − cos 2ϕ j  Fx  Nm 1  i  = ∑ ap g j  i  K tc 1 − cos 2ϕ j − K i rc sin 2ϕ j  Fy  j =1 2 

{[ K ] − ω [ M ] + iω [C ]}{R (ω )} = {F (ω )}. c

(9)

where G (ω )  is the transfer function, or frequency response function. From Eq. (5), a frequency-

(

)

- K i tc 1 + cos 2ϕ j − K i rc sin 2ϕ j   ∆x  1 ∆x    = K i tc a p  Axy    , (6)   i i  ∆ y ∆ K tc sin 2ϕ j − K rc 1 + cos 2ϕ j   2  y 

(

)

1, ϕ j ∈ [ Φ st , Φ ex ] , and K rci = kr K tci . where g j =  0, ϕ j ∉ [ Φ st , Φ ex ] Mechanics and Dynamics of Helical Milling Operations

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dependent dynamic chip thickness vector as in Fig. 3 shown, ∆X (ω )  {∆R (ωc )} =  ∆Y (ω c )  = {R (ωc )} − {R0 (ωc )} =  c  

(

= 1 − e-iωcT

){R (ω )} = (1 − e ) G (ω ) {F (ω )}. (10) -iωcT

Substitute Eq. (10) to Eq. (9), we obtain: {F (ωc )} = 12 K itc a p 1 − e-iωcT ⋅ ⋅  AXY (ωc )  G (ωc )  {F (ωc )}. (11)

(

)

The characteristic function of Eq. (11) is: 1

[ I ] − 2 K i tc a p (1 − e-iω T )  AXY (ωc ) G (ωc ) c

= 0. (12)

The response function has a nontrivial solution only if its determinant is zero, therefore, det [ I ] + Λ  AXY (ωc )  G (ωc )  = 0 , and the nontrivial solution of Eq. (11) exists. Where Λ is the eigenvalue of Eq. (12), and the vibration system has the critical chatter frequency. Experimental modal analysis was performed to extract the modal parameters of the helical milling tool-machine tool system. The frequency response function (FRF) G (ω )  of the milling tool attached to a Mikron 710 machining centre was identified through hammer impact modal tests, as described in the literature [20]. In order to describe the directional dynamic helical milling force coefficients  AXY (ω )  , the zero order approximation of its Fourier series expansion was applied, and a single frequency solution derived. The predicted stability lobe identifying the critical axial depth of cut (aplim) and spindle speed to construct the lobes can be determined, as depicted in Fig. 4.

2.2 The Chatter Limitation of Radial Depth of Cut The Z and C components of cutting force (FZ, T) on the cutting tools mainly result from the interaction of the end-cutting edges, because the end-cutting edges engage predominantly in the axial feed direction and create continuous cutting. Hence, the radial depth of cut is affected by the Z and C components of cutting forces. Considering the influence of the dynamic chip thickness on the axial vibration and torsional vibration, as presented in the literature, the regenerative chatter in the Z and C direction is the result of the variation in the dynamic chip thickness with period τ0 as depicted in Fig. 5. The variation of dynamic chip thickness can be described as:

( )

( ( )

h ϕ j = hZ − ∆hϕ = ∆Z ϕ j − f za ∆ϕ j / Φ p

)

cosθ .(13)

b) Fig. 5. Dynamic chip thickness variation in a) Z and b) C direction

Neglecting the edge effect on the end-cutting edges, and substituting dynamic chip thickness into the end-cutting edge force model results in the following relation: dFa , j *   ∆Z  * =  K vc dr ⋅  *   dTj  ∆ϕ j 

Fig. 4. The stability lobes prediction of axial depth of cut

720

Liu, C. – Wang, G. – Dargusch, M.S.

 tan θ + kn* ⋅  r − k *r tan θ n 

(

)

− tan θ + kn* f za Φ p  , (14) − r − kn*r tan θ f za Φ p  

(

)

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 716-724

where, K nc* = kn* K vc* . Integrating Eq. (15), the dynamic cutting forces on an end-cutting edge can be described as,  A ln r + B + kn*r  F  *   = K vc  1 2 *  r - kn A ( r − B ln r + B ) −  T  2 * a, j * j

(

1 2

 FZ  Nm  FZ , j   ∆Z  *   = ∑  = K vc  AZ Φ ( t )    , T  T  j =1  j  ∆ϕ 

(16)

where the directional dynamic helical milling force coefficients are:

(

)

 1 − f za Φ p ⋅  Dm  +B Nm  + k *n ae A ln Dm2  + B − ae  AZ Φ ( t )  = ∑   2 j =1     Dm 2 * * 1   2 ae − 2 ae − k n Aae + k n AB ln 

Dm +B 2 Dm + B − ae 2

     . (17)     

The Z and C components of the displacement are:  Z ( t )  ,  ϕ ( t ) 

{R ( t )} =

{R ( t )} = 0

 Z ( t − T )  ,  ϕ ( t − T ) 

{R (ω )} =

 Z (ω )  ,  ϕ (ω ) 

{∆ (ω )} =

 ∆Z (ω )   = {R (ω )} − {R0 (ω )} =  ∆ϕ (ω ) 

(

= 1 − e-iωT

{R (ω )} = 0

)

r - k A ( r − B ln r + B ) f za 2

Sum up all the flutes and dynamic cutting force components on the cutting tool (FZ, T) as follows:

)

− A ln r + B + kn*r f za Φ p * n

Dm 2

  ∆Z      Φ p  ∆ϕ j  D 

m 2

(15)

. − ae

trivial solution of Eq. (19) exists, Λ is the eigenvalue of Eq. (20). However, for the directional dynamic helical milling force coefficients  AZΦ ( t )  the situation is more complicated, and the variable ae is not linear. In the range from 0 to the end-edge length (Dm/2, half the tool diameter),  AZΦ ( t )  can be simplified to: * Nm   k a  AZ Φ ( t )  = ∑  D n e 2  1 − f za Φ p . m 1 j =1   2 ae − 2 ae 

{

}

(21)

If the influence of the fza was neglected, the procedure to obtain the approximate solution would be identical with that in literature [9]. It is then necessary to solve the eigenvalue and obtain the predicted stability lobe that identifies the critical radial depth of cut (aelim) and the spindle speed, as shown in Fig. 6.

e-iωcT {R (ω )} ,

){R (ω )} = (1 − e ) G (ω ) {F (ω )}. (18) -iωT

Substitute Eq. (18) to Eq. (16), for the chatter frequency ωc, then:

{F (ω )} = K (1 − e )  A (ω ) ⋅ ⋅ G (ω )  {F (ω )}. *

-iωcT

ZΦ

(19)

The characteristic function of Eq. (19) is:

[ I ] − K (1 − e * vc

-iωcT

)  A (ω ) G (ω ) ZΦ

Fig. 6. The s tability lobes prediction of radial depth of cut

= 0. (20)

Theoretically, similar to the aforementioned resolution approach to the chatter limitation of ap, when det [ I ] + Λ  AZ Φ (ωc )  G (ωc )  = 0 , the non

3 EXPERIMENTAL METHODS The chatter-free machining experiments of helical milling process were performed under dry conditions. The radial cutting depth of ae, cutting velocity of vc,

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axial feed rate of fza, and tangential feed rate of fzt are independent processing parameters, on which spindle frequency n and axial cutting depth ap depend. The experiments were designed to orthogonal trails with three variable factors (ae, vc, fza) where each factor had three levels. The range of n, ap and ae based on the solved stability lobe diagrams shown in Figs. 4 and 6, along with the recommendations of the milling tool manual and the consideration that the processing parameters would be suitable for the finishing operation. All the processing parameters were depicted in Table 1. The rough wrought Ti6Al4V alloy blank has the geometry of a cuboid (160 mm width, 160 mm length, and 20 mm height) with a hole in a diameter of 60 mm in the centre as depicted in Fig. 1. M.A. Ford 20 mm 5-flute flat end cylindrical carbide milling tools (17878703A) were employed. The occurrence of chatter could be detected by measuring the amplitude of the cutting forces, as well as the machining accuracy (surface roughness, roundness, and cylindricity). Once chatter occurs, the cutting force amplitude increases drastically, the surface roughness escalates, and the geometric accuracy of the hole is lost. Experiments were performed on a five-axis Mikron UCP-710 machining centre. A Kistler 9265B piezo-electric dynamometer, a Mahr S3P 2D surface profile tester, and a Hexagon 3D coordinate measuring machine were setup to measure the cutting forces, machined surface roughness, roundness and cylindricity, respectively. Table 1. Helical milling experiment parameters Test vc No. [m/min] 1 100 2 100 3 100 4 120 5 120 6 120 7 135 8 135 9 135

[rpm] 1591 1591 1591 1910 1910 1910 2149 2149 2149

[mm] 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75

[mm] 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5

fza

fzt

[mm/tooth] [mm/tooth] 0.48 0.060 0.60 0.075 0.72 0.090 0.72 0.090 0.48 0.060 0.60 0.075 0.60 0.075 0.72 0.090 0.48 0.060

4 EXPERIMENTAL RESULTS AND DISCUSSION The inspection of the operation accuracy, including the surface roughness of the inner surface of the hole, the roundness of the hole cross section, and the 722

cylindricity of the hole were measured, and shown in Table 2. Table 2. Helical milling experiment results Test No. 1 2 3 4 5 6 7 8 9

Surface roughness [µm] < 1.6 < 1.6 < 1.6 < 1.6 < 1.6 < 1.6 < 1.6 < 1.6 < 1.6

Roundness [µm] 3.6, 4.7, 5.4 4.8, 5.2, 6.3 6.2, 6.9, 7.5 3.8, 5.1, 7.3 4.5, 5.4, 7.9 4.3, 5.6, 8.1 8.3, 9.5, 10.3 8.0, 8.0, 10.9 8.4, 8.7, 12.0

Cylindricity [µm] 7.01 8.66 8.95 9.72 11.2 11.9 12.3 15.2 15.3

During the experimental process, chatter vibration between the cutter and work piece did not occur. In order to assess the chatter-free conditions during the experiment, the cutting forces signal was analysed to confirm that there was no abnormal forces impact and fluctuation. The experimental and simulated cutting force results of helical milling with typical cutting conditions, cutting speed vc 100 m/min, axial feed rate fza 0.5 mm/tooth, tangential feed rate fzt 0.1 mm/tooth, were compared in Fig. 7. The amplitude of the experimental cutting forces in comparison to the simulated cutting forces, which were typical stable force signals in Fig. 7a and b, has a margin of error within 10%. No impact was detected. The amplitude spectrum of the experimental result shown in Fig. 7c properly matches the simulated cutting forces shown in Fig. 7d. The unmatched spectrum in the experimental cutting forces might be due to the serrate chip formation and other environmental influences. From the observation of the experiment processes and the measurement of the machining accuracy and the cutting forces, it could be concluded that the given helical milling processing parameters can deliver the needed machining accuracy, and the operation can be chatter free within the stability limitation predicted by the presented analysis and model. 5 CONCLUSION Based on the modelling of cutting forces, the machining dynamics and the chatter stability of the helical milling process have been modelled in both the time and frequency domains. The cutting forces model predicts both the interaction of the side-cutting

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Fig. 7. Experiment and simulation of the helical milling cutting forces results; a) experiment; b) simulation; c) amplitude spectrum by Fourier transform of experiment; and d) amplitude spectrum by Fourier transform of simulation

edges with the work piece and the end-cutting edges with the work piece along the helical feed path. The helical milling dynamics model represents the special tangential feed and axial feed. This model includes the dual periodicity which is caused by spindle rotation, as well as the period of the helical feed of the cutting tool. The frequency domain solutions of the critical axial cutting depth and critical radial cutting depth with spindle velocity have also been identified. The 4 DOM vibration system has been decomposed to two 2 DOM systems and the critical axial cutting depth, and critical radial cutting depth have been solved separately. On the given machining condition of the helix milling operation, the experimental results indicate that the simulation from the presented models effectively forecasts the cutting forces and chatterfree process parameters. Chatter did not occur during machining, and the helical milling operation satisfied the required surface roughness, roundness and cylindricity.

6 ACKNOWLEDGEMENTS This work was supported by the Nanjing University of Aeronautics & Astronautics Fundamental Research Funds [grant number NS2013048]. 7 REFERENCES [1] Iyer, R., Koshy, P., Ng, E. (2007). Helical milling: An enabling technology for hard machining precision holes in AISI D2 tool steel. International Journal of Machine Tools & Manufacture, vol. 47, no. 2, p. 205-210, DOI: 10.1016/j.ijmachtools.2006.04.006. [2] Wang, H., Qin, X., Li, H., Ren, C. (2013). Analysis of cutting forces in helical milling of carbon fiber– reinforced plastics. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, vol. 227, no. 1, p. 62-74, DOI:10.1177/0954405412464328. [3] Denkena, B., Boehnke, D., Dege, J.H. (2008). Helical milling of CFRP–titanium layer compounds. CIRP

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Journal of Manufacturing Science and Technology, vol. 1, no. 2, p. 64-69, DOI:10.1016/j.cirpj.2008.09.009. [4] Larue, A., Altintas, Y. (2005). Simulation of flank milling processes. International Journal of Machine Tools and Manufacture, vol. 45, no. 4-5, p. 549-559, DOI:10.1016/j.ijmachtools.2004.08.020. [5] Altintas, Y., Engin, S. (2001). Generalized Modeling of Mechanics and Dynamics of Milling Cutters. CIRP Annals - Manufacturing Technology, vol. 50, no. 1, p. 25-30, DOI:10.1016/s0007-8506(07)62063-0. [6] Ozturk, E., Budak, E. (2007). Modeling of 5-axis milling processes. Machining Science and Technology: An International Journal, vol. 11, no. 3, p. 287 - 311, DOI:10.1080/10910340701554808. [7] Lazoglu, I., Liang, S.Y. (2000). Modeling of ballend milling forces with cutter axis inclination. Journal of Manufacturing Science and EngineeringTransactions of the ASME, vol. 122, no. 1, p. 3-11, DOI:10.1115/1.538885. [8] Dang, J.-W., Zhang, W.-H., Yang, Y., Wan, M. (2010). Cutting force modeling for flat end milling including bottom edge cutting effect. International Journal of Machine Tools and Manufacture, vol. 50, no. 11, p. 986-997, DOI:10.1016/j.ijmachtools.2010.07.004. [9] Ko, J.H., Altintas, Y. (2007). Dynamics and stability of plunge milling operations. Journal of Manufacturing Science and Engineering-Transactions of the ASME, vol. 129, no. 1, p. 32-40, DOI:10.1115/1.2383070. [10] Altintas, Y., Ko, J.H. (2006). Chatter stability of plunge milling. CIRP Annals-Manufacturing Technology, vol. 55, no. 1, p. 361-364, DOI:10.1016/S00078506(07)60435-1. [11] Kolar, P., Sulitka, M., Janota, M. (2011). Simulation of dynamic properties of a spindle and tool system coupled with a machine tool frame. International Journal of Advanced Manufacturing Technology, vol. 54, no. 1-4, p. 11-20, DOI:10.1007/s00170-010-29177. [12] Mahnama, M., Movahhedy, M.R. (2010). Prediction of machining chatter based on FEM simulation of chip formation under dynamic conditions. International Journal of Machine Tools & Manufacture, vol. 50, no. 7, p. 611-620, DOI:10.1016/j.ijmachtools.2010.03.009. [13] Chaari, R., Abdennadher, M., Louati, J., Haddar, M. (2011). Modelling of the 3D machining geometric

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defects accounting for workpiece vibratory behaviour. International Journal of Simulation Modelling, vol. 10, no. 2, p. 66-77, DOI:10.2507/ijsimm10(2)2.173. [14] Ostasevicius, V., Gaidys, R., Dauksevicius, R., Mikuckyte, S. (2013). Study of vibration milling for improving surface finish of difficult-to-cut materials. Strojniski vestnik-Journal of Mechanical Engineering, vol. 59, no. 6, p. 351-357, DOI:10.5545/ sv-jme.2012.856. [15] Bajić, D., Celent, L., Jozić, S. (2012). Modeling of the influence of cutting parameters on the surface roughness, tool wear and the cutting force in face milling in off-line process control. Strojniški vestnik Journal of Mechanical Engineering, vol. 58, no. 11, p. 673-682, DOI:10.5545/sv-jme.2012.456. [16] Yoon, M.C., Chin, D.H. (2005). Cutting force monitoring in the endmilling operation for chatter detection. Proceedings of the Institution of Mechanical Engineers Part B-Journal of Engineering Manufacture, vol. 219, no. 6, p. 455-465, DOI:10.1243/095440505x32292. [17] Polli, M.L., Weingaertner, W.L., Schroeter, R.B., Gomes, J.d.O. (2012). Analysis of high-speed milling dynamic stability through sound pressure, machining force and tool displacement measurements. Proceedings of the Institution of Mechanical Engineers Part B-Journal of Engineering Manufacture, vol. 226, no. A11, p. 1774-1783, DOI:10.1177/0954405412460128. [18] Liu, C., Wang, G., Dargusch, M. (2012). Modelling, simulation and experimental investigation of cutting forces during helical milling operations. The International Journal of Advanced Manufacturing Technology, vol. 63, no. 9-12, p. 839-850, DOI:10.1007/ s00170-012-3951-4. [19] Wang, H., Qin, X., Ren, C., Wang, Q. (2012). Prediction of cutting forces in helical milling process. The International Journal of Advanced Manufacturing Technology, vol. 58, no. 9-12, p. 849-859, DOI:10.1007/ s00170-011-3435-y. [20] Damir, A., Ng, E.G., Elbestawi, M. (2011). Force prediction and stability analysis of plunge milling of systems with rigid and flexible workpiece. International Journal of Advanced Manufacturing Technology, vol. 54, no. 9-12, p. 853-877, DOI:10.1007/s00170-0102982-y.

Liu, C. – Wang, G. – Dargusch, M.S.

Received for review: 2014-02-08 Received revised form: 2014-04-02 Accepted for publication: 2014-04-04

Investigating Prior Parameter Distributions in the Inverse Modelling of Water Distribution Hydraulic Models Kozelj, D. – Kapelan, Z. – Novak, G. – Steinman, F. Daniel Kozelj1,* – Zoran Kapelan2 – Gorazd Novak3 – Franci Steinman1 1University

of Ljubljana, Faculty of Civil and Geodetic Engineering, Slovenia of Exeter, School of Engineering and Computer Science, U.K. 3Institute for Hydraulics Research, Slovenia

2University

Inverse modelling concentrates on estimating water distribution system (WDS) model parameters that are not directly measurable, e.g. pipe roughness coefficients, which can, therefore, only be estimated by indirect approaches, i.e. inverse modelling. Estimation of the parameter and predictive uncertainty of WDS models is an essential part of the inverse modelling process. Recently, Markov Chain Monte Carlo (MCMC) simulations have gained in popularity in uncertainty analyses due to their effective and efficient exploration of posterior parameter probability density functions (pdf). A Bayesian framework is used to infer prior parameter information via a likelihood function to plausible ranges of posterior parameter pdf. Improved parameter and predictive uncertainty are achieved through the incorporation of prior pdf of parameter values and the use of a generalized likelihood function. We used three prior information sampling schemes to infer the pipe roughness coefficients of WDS models. A hypothetical case study and a real-world WDS case study were used to illustrate the strengths and weaknesses of a particular selection of a prior information pdf. The results obtained show that the level of parameter identifiability (i.e. sensitivity) is an important property for prior pdf selection. Keywords: Bayesian inference, calibration, generalized likelihood, Markov Chain Monte Carlo, differential evolution adaptive metropolis, pipe networks, hydraulics, water distribution systems

0 INTRODUCTION Inverse modelling is the reciprocal process of the forward modelling problem in which a physical theory is used to predict the behaviour of a real system. Data from the indirect observations of unknown model parameters can be inferred to adequately represent the observed system behaviour. Inverse modelling of water distribution system (WDS) models, commonly referred to as calibration or parameter estimation, has been investigated extensively since the 1980s, providing valuable insight for modellers when tackling the nonlinear and highly combinational calibration process. Throughout this period, different types of model parameters have been estimated, e.g. pipe friction coefficients, pipe diameters, nodal demands, etc. Approaches of WDS model calibration can be divided into three categories: iterative trail-and-error approaches, explicit models, and implicit models, e.g. optimization approaches. The development of implicit models has proved to be the most effective in the exploration of the non-linear parameter space. A wide variety of global optimization methods has been studied for parameter estimation problems. Those methods can be divided into non-evolutionary and evolutionary methods. Among the evolutionary methods, genetic algorithms in particular have proved their applicability to large and complex realworld calibration problems with multimodal search (parameter) spaces. For a comprehensive review of

calibration methods, we refer the reader to Savic et al. [1]. The assessment of parameter and predictive uncertainty is an essential part of the modelling process in order to perform model comparison and selection [2]. One shortcoming of the summarized optimization methods is their ability to only identify near optimal parameter values, while they lack the ability to estimate the parameter and predictive uncertainty. However, formulating the inverse modelling problem as a probabilistic Bayesian approach, and solving it with a Markov Chain Monte Carlo (MCMC) method exhibits the capability of estimating parameter values and their associated parameter and predictive uncertainties in a single optimization run [3]. Alternatively, in a recent study, the uncertainty analysis of pipe roughness coefficients by using grey numbers was proposed, which led to uncertainty intervals without defining any probability distribution [4]. Bayesian inference is a concept of probability theory whereby model parameters are represented as probabilistic variables having a joint posterior probability density function (pdf). The joint posterior pdf is derived from combining information on the prior distribution of model parameters and data likelihood. Bayesian-type approaches have some distinct advantages in comparison to existing WDS calibration methods: probabilistic definition of prior pdf of parameters, retrieving joint and marginal

*Corr. Author’s Address: University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova cesta 2, 1000 Ljubljana, Slovenia, daniel.kozelj@fgg.uni-lj.si

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posterior pdf, and no requirement of derivative calculation [3]. Recently, developments have led to significant improvements in the efficiency of MCMC simulations and extended their feasibility to complex, multimodal search problems [5], [6]. The differential evolution adaptive metropolis (DREAM) scheme is a new MCMC sampler, which runs multiple chains simultaneously for global exploration and automatically tunes the scale and orientation of the proposal distribution during the search process. We use a recent variant of DREAM, called DREAM(ZS), which uses sampling from past states and a mix of parallel direction and snooker updates to generate proposals in each chain [5]. The aim of this paper is to demonstrate the benefits of including prior information to improve the identifiability of estimated parameters. We investigate the effect of different sampling strategies of pipe roughness coefficients in the inverse modelling of WDS hydraulic models. The paper is organized as follows: following this introduction, we provide the governing equations that constitute the WDS forward modelling approach. Afterwards, a formulation of the Bayesian inference approach is given. The presented approach is applied in Sections 3 (artificial case study) and 4 (real-world case study) to estimate the parameter and predictive uncertainty of WDS model parameters. The results of each case study are discussed in their corresponding sections. Section 5 summarizes our findings and relevant conclusions are drawn. 1 WATER DISTRIBUTION SYSTEM MODELLING The main purpose of WDS is to supply its users with the required quantities of water under adequate pressure for various loading conditions. Common constituents of WDS are water sources (i.e. reservoirs, pumping stations), distribution storage (water tanks), and distribution pipe networks. To appropriately perform operational tasks, as well as development and rehabilitations measures, the utility operator is assisted by WDS models. Hydraulic simulations of WDS models provide insight into the flow and pressure conditions of even the most complex WDS. The interconnection of the WDS components is governed by the conservation of energy and the conservation of mass. The conservation of energy means that the difference in energy between nodes is equal to the pipe friction and minor losses and the energy added to the flow in components between the observed nodes: 726

∑h

L ,i

+ ∑ hP , j = ∆E , (1)

where hL,i is the energy loss in pipe network component i, hP,j the added energy by pump j, and ΔE the difference in energy between observed nodes [7]. A commonly used fictional energy loss model is the Darcy-Weisbach equation:

hL , frict = λ

L v2 ε  , λ = λ  Re,  , (2) d 2g d 

where L is pipe length, d pipe diameter, v fluid flow velocity, g gravitational constant, λ the DarcyWeisbach friction factor dependent on the Reynolds number (Re), and relative pipe roughness (ε/d), ε equivalent roughness. The minor (i.e. local) energy losses of valves and fittings are typically expressed as: hL ,local = ζ

v2 , (3) 2g

where ζ is an empirical coefficient. The pump energy gain is given by:

(

)

hP = −ω 2 h0 − r ( Q / ω ) , (4) n

where h0 is pump shutoff head, ω variable pump speed, r and n pump curve coefficients. The conservation of mass of each junction node is:

∑Q − ∑Q in

out

= qext , (5)

where Qin and Qout are pipe flow into and out of a junction node, and qext is the external demand at junction node [7]. When steady-state simulations are extended to extended-period simulations, which mimic a quasi-dynamic WDS behaviour, the conservation of mass Eq. (5) is extended to account for storage in tanks:

∑Q − ∑Q in

out

−

dVT = qext , (6) dt

where dV is a change in storage volume, and dt the time period between steady-state simulations. Changing tank water levels are updated by:

dH T =

dVT , (7) AT

where dHT is a change in tank level and AT tank cross-section. The set of mass continuity and energy equations for a WDS model are most efficiently solved by the gradient method, and its implementation can be found in the widely known EPANET2 network solver [8].

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2 INVERSE MODELLING The inverse modelling problem is usually based on a nonlinear regression model [3]. First, let us consider a model, f, that simulates a vector of model predictions. In a general form, the model can be written as:

Y = f ( X | θ ) µ s , (8)

where Y is a vector of model predictions, X a vector of known model inputs, θ a vector of unknown model parameters, μs a bias factor to account for model input error which is defined as:

µ s = exp ( µhY ) , (9)

where μh is a bias parameter to be inferred from the observations [2]. In order to provide a measure of model adequacy, it is common to compare the model f simulated response Y with measurements of the observed system behaviour Y͂ . The nonlinear regression model describes the random component of residuals as the difference between the deterministic components of model predictions of a WDS model, Y, and observations, Y͂ :

e (θ ) = Y ( X | θ ) − Y , (10)

where e(θ) is a vector of residuals {e1, …, eN}, N the number of observations Y͂ . Residuals, e(θ), are defined as a statistical model describing a priori expected behaviour. Frequently, residuals are assumed to be independent and identically distributed (i.i.d.) according to a normal distribution with zero mean and a constant variance, i.e. homoscedasticity, and are not showing any autocorrelation. Occasionally, these assumptions are violated and an alternative description of the residual is needed. In this study, we adopt the generalized likelihood function of Schoups and Vrugt [2] that can account for residual errors that are correlated, heteroscedastic, and non-Gaussian. First, we describe the statistical model of residuals, while the generalized likelihood function is provided in Section 2.1. To account for correlation and non-normality residuals, e(θ) are described by:

Φ p ( B ) es (θ ) = σ e ,s as , (11)

where Φp(B) is an autoregressive polynomial with parameters ϕp, B a backshift operator, σe,s a standard deviation of residuals, as i.i.d. random error described by a skew exponential power distribution as ~ SEP(0,1,ξ,β) with zero mean, unit variance, and

with the parameters ξ and β accounting for skewness and kurtosis. The heteroscedasticity of residuals is accounted for by assuming that the standard deviation σe,s linearly increases with model predictions:

σ e ,s = σ 0 + σ 1Ys , (12)

where σ0 and σ1 are parameters to be inferred from the observations. Details of this approach can be found in Schoups and Vrugt [2]. 2.1 Likelihood Function If the inverse problem is stated as a probabilistic framework the criterion (i.e. measure) to estimate the residuals of a model, the response variables vs. observations is called the likelihood. The likelihood L(θ|Ŷ) quantifies the “probability” that observed data were simulated by a particular set of parameters [9]. A general likelihood function presented in Schoups and Vrugt [2] is adopted to account for conditions of correlation, non-constant variance, i.e. heteroscedasticity, and non-normality of residuals. Their formulation of a log likelihood ℓ(θ|Ŷ) functions is: N 2σ ω (θ h ,e | Y ) = N log ξ −β1 − ∑ log (σ e ,s ) ξ +ξ s =1

−cβ ∑ aξ ,s

2 (1+ β )

(13)

s =1

and residual errors aξ,s are given as: − sign( µξ +σ ξ as ) aξ ,s = ξ µξ + σ ξ as , (14)

(

)

where μξ, σξ, cβ and ωβ are variables defined as functions of ξ and β, which are provided in Appendix A of Schoups and Vrugt [2]. 2.2 Parameter Uncertainty By considering model parameters as the only source of uncertainty, the posterior parameter pdf p(θ|Ŷ) can be estimated from the Bayes theorem: p (θ ) p (Y | θ ) p (θ | Y ) = , (15) p (Y ) where p(θ) is a prior parameter pdf, p(Ŷ) a normalization constant or “model evidence”, p(Ŷ|θ) ≡ L(θ|Ŷ) likelihood function. Since only parameters are of interest, we can ignore the normalization constant p(Ŷ) and infer parameter samples from the posterior parameter pdf p(θ|Ŷ) that is proportional to the prior

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parameter pdf p(θ) multiplied by the likelihood function L(θ|Ŷ):

p (θ | Y ) ∝ p (θ ) L(θ | Y ). (16)

Parameter uncertainty after observing data is directly derived from the posterior parameter pdf p(θ|Ŷ) [9]. The term p(θ) denotes prior knowledge of the parameter vector θ before inferring it to the observational data Ŷ. In the present case, pipe roughness coefficients are under investigation. Since prior information on the parameter pdf is limited and vague, we will consider three cases to sample the parameter sets θ from the prior parameter space. Initial parameter sampling will be from a continuous uniform pdf (also known as rectangular distribution) of the parameter space. The continuous uniform distribution p(θ)~U(μθ,σθ) is a bounded domain distribution and samples values between given lower μθ and upper σθ bounds, respectively. The second type of prior pdf is the normal (Gaussian) distribution with a given mean value and standard deviation. The prior p(θ)~U(μθ,σθ) is given by the mean parameter value μθ and its standard deviation σθ. Finally, we provide the gamma distribution for describing the prior information on parameters. The gamma distribution p(θ)~Г(α,β) is defined by a shape parameter α and a scale parameter β. The gamma distribution closely approximates a normal distribution with the advantage that the gamma distribution has density only for positive real numbers, which is compliant to the physical nature of our parameters. All samples (parameter values) generated from prior pdf are trunked at 0 (only positive values are allowed). Prior information about parameters can significantly improve parameter identifiability and provides an effective and robust approach of parameter value estimation [6]. Additional information on the selected prior pdf parameter values is provided in Section 3. The assembled Bayesian framework, i.e. the prior pdf of model parameters, the likelihood function in Eq. (13), and the joint posterior parameter pdf can be calculated using Eq. (16). MCMC simulations are used to efficiently derive the joint posterior parameter pdf by repeated sampling of parameter sets [2] and [3]. 2.3 Predictive Uncertainty In addition to the evaluation of parameter uncertainty, the predictive uncertainty is also of significant interest. The predictive uncertainty derives from predictive 728

percentiles Yα, which correspond to the exceedance probability P(Ŷ≤ Ŷα|X), and can be calculated as: P Y ( X | θ ) + e (θ )  ≤ Y α | X = α , (17) 1... N

(

)

where Ŷα is exceedance probability 1–α, α significance levels, Nθ number of MCMC sampled parameter sets θ. The prediction percentiles Ŷα are obtained from the set of J predictions of the sampled parameter set θ and its corresponding response Y(X|θ) and residuals e(θ). Evaluating the 95% predictive uncertainty bands requires the selection of α = 0.025 and α = 0.975, i.e. the 97.5% and 2.5% prediction percentiles, respectively [10]. 2.4 Sensitivity Analysis and Parameter Identifiability A complex real-world WDS comprises numerous uncertain model parameters (e.g. pipe roughness coefficients, nodal demands, pipe diameters, etc.) that could be investigated. To reduce the number of calibrated parameters, a sensitivity analysis of the pipe roughness coefficient was performed by applying the forward finite difference approximation of the first derivative of model response against all investigated model parameters [11] and [12]. This approximate approach is warranted since it serves only as a measure of model parameter identifiability for the given measurement layout (the model structure is assumed to be given and not addressed here). The low sensitivity of the model response to a parameter can lead to the reduced identifiability of the investigated parameters [12]. Sensitivity and uncertainty are closely related, e.g. greater parameter sensitivity results in greater uncertainty propagation from that parameter. The sensitivity analysis facilitates the selection and differentiation between more and less identifiable (i.e. sensitive) model parameters. The classification between the cases is applied in conjunction with model parameter’s prior information. If prior information on model parameters is vague, sensitive parameters could still be identifiable by applying a uniform prior pdf, and less sensitive ones by applying an informative (e.g. normally distributed) prior pdf. 2.5 DREAM(ZS) Algorithm MCMC simulations are an increasingly popular method in a wide range of engineering problems [3], [6], [13] and [14]. In inverse modelling, Bayesian frameworks proved their ability to effectively estimate the posterior pdf of parameters. In our study, we used the DREAM(ZS) algorithm [5] provided by J. A. Vrugt.

Kozelj, D. – Kapelan, Z. – Novak, G. – Steinman, F.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 725-734

The DREAM MCMC scheme runs multiple Markov chains simultaneously for effective global exploration of the parameter space and provides efficient evolution of the proposal distribution to its target distribution, especially for complex, highly non-linear and multimodal target distributions [5]. The DREAM(ZS) algorithm differs from its predecessor by using sampling from past states and a mix of parallel direction and snooker updates to generate proposals in each chain. Some of the distinct advantages of DREAM(ZS) are that sampling from the past reduces the need to use a large number of chains; outliers can be redirected to the region of exploration; the independence of the current state of chains enables integration in multi-processor environments [5] and [6]. These improvements lead to the acceleration of convergence to the target distribution, especially for high-dimensional problems (d > 20, i.e. number of parameters). DREAM(ZS) can work with d up to 50 to 100 with far fewer chains, e.g. NZS = 3, while still accurately assessing the target distribution once convergence has been achieved [5]. Other DREAM(ZS) algorithm parameters are DEpair the number of chain pairs to generate candidate points; NCR the crossover value, pup the fraction of parallel direction updates, k the thinning parameter for appending position of chains and corresponding posterior density values to sample history, Zm0 the initial size of thinned sample history (past states), pjump the probability of selecting a jump rate of 1, Neval the number of function evaluation. 3 HYPOTHETICAL WDS CASE STUDY This study aims to demonstrate the performance of the suggested approach of parameter and predictive uncertainty analysis by applying the Bayesian framework on the “Anytown” WDS model and has been used in various calibration studies [3], [11] and [15]. In a previous study of the Anytown model, a Bayesian-type procedure was applied to investigate the uncertainties of HW C-factor pipe roughness estimations [3]. The present study aims to investigate pipe roughness coefficients for the equivalent roughness ε of the Darcy-Weisbach (DW) friction model. The Anytown model consists of 34 pipes and their roughness coefficients are grouped into six pipe roughness groups (PG) (Nθ = 6). Their true DW ε values are provided in Table 1. Observational data sets are generated by simulating the model response via the Epanet2 hydraulic solver [8]. Pressure measurements collected at four junction nodes (i.e. 40, 90, 120 and 140) and five independent LC represent the

observational data (N = 20) for the presented case. The imperfect observational data was generated through altering the perfect observational data by introducing random normally distributed noise with zero mean and a standard deviation of 0.10 m. Incorporation of prior knowledge on calibration parameters (DW equivalent roughness ε) is performed by using three prior information pdfs. A continuous uniform prior pdf p(θ)~U(0.001, 15) is first used for all PGs. Then, the pdf parameters of the normal and gamma distribution are estimated on the basis of the approximate equivalent roughness ε values by consulting literature sources relating the original HW C-factors [3] to the DW ε values used in this study [16]. The distribution parameters of the normal and gamma priors are provided in Table 1. Table 1. Anytown: True D-W ε values for PGs and parameters of normal and gamma prior pdf DW ε PG1 PG2 PG3 PG4 PG5 PG6

0.525 11.75 2.5 0.3 1.2 1.2

Normal σθ μθ 0.75 0.5 11.0 1.0 2.5 1.0 0.5 0.5 1.25 1.0 1.25 1.0

Gamma α

1.0 10.0 3.0 0.5 2.0 2.0

1.0 1.0 1.0 1.0 1.0 1.0

The generalized likelihood (GL) function given by Eq. (13) is used with fixed values of residual model parameters ϕ1 = 0 and μh = 0, while parameters σ0, σ1, β, and ξ, are inferred additionally to the model parameters. Uniform prior pdfs are assumed for the GL parameters and their upper and lower bounds are as follows: σ0 [0, 1], σ1 [0, 1], β [–1, 1] and ξ [0.1, 10]. This results in a total number of Nθ = 10. The DREAM(ZS) algorithm was set up with the following parameters: Nθ = 10, NZS = 3, DEpair = 1, NCR = 3, pup = 0.9, Zm0 = 10×Ndim = 60, pjump = 0.2, Neval = 50,000. The DREAM(ZS) algorithm converged in approximately 35,000 function calls with a total simulation time of 315 s on a 403 MFLOPS PC. The inferred residual model parameters θe of the GL function were evaluated at σ0 = 0, σ1 = 0.0011, β = 1, and ξ = 0.583 for the proposal prior pdf p(θ) scheme. Very similar values were also observed at other simulations. The GL function parameters indicate that residuals are non-normally distributed and heteroscedastic. The SEP parameters β and ξ indicate that the residual distributions are peaked (β = 1) and negatively skewed (ξ = 0.583).

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The model parameters uncertainty simulation results can be observed in Fig. 1. The presented box plots of PG1 to PG6 provide information on the following statistical values: median (middle line), lower (first) and upper (third) quartiles (i.e. the interquartile range (IQR)) of posterior parameter pdf samples, and the 95% confidence interval (vertical lines). The actual parameter values are given in Table 1. The obtained parameter statistics show that even “uninformative” prior distributions (e.g. uniform pdf) can adequately identify parameters values. This can be observed for parameter groups PG1 to PG4. However, PG5 and PG6 show greater deviations of the median parameter estimates as well as their IQR and the 95% confidence intervals. This is caused by their small parameter sensitivity for the given observational layout. Therefore, the incorporation of prior information is narrowing the IQR ranges by deriving independent information on pipe roughness states. Identification of parameters with small parameter sensitivity can be very difficult, since the given observational data do not provide sufficient information to provide reasonable parameter estimates and narrow posterior pdf [17]. PG5 and PG6 are not identifiable by the uniform pdf, while normal and gamma prior pdfs slightly deviate in their marginal posterior pdf. Applying a normal or gamma prior distribution narrows the parameter uncertainty. The differences in shift and broadness of IQR and 95% confidence intervals arise from the prior pdf used and the observational information available. By examining Fig. 1, the shape and position of both normal and

gamma pdfs are identifiable from the marginal posterior parameter statistics. This indicates that the posterior parameter pdf, and their estimated values of insensitive parameters benefit or suffer from the applied prior distribution. This is evident since the likelihood function does not force the joint posterior parameter pdf towards their “true” values. Here lies the true added value of prior information of calibration parameter estimates. Based on the information given in Sections 2.2 and 2.4 and the findings from the previous paragraph, we used a fourth prior information scheme by combining the synergies of prior parameter information and parameter sensitivity. A continuous uniform prior pdf p(θ)~U(0.001, 15) is used for PGs with higher parameter sensitivity (PG1 to PG4), while PGs with lower sensitivity are estimated by their associated gamma prior distribution given in Table 1 (PG5 and PG6). When compared to the normal and gamma prior pdf results, a close resemblance in terms of parameter mean, IQR and 95% confidence intervals can be observed. Table 2. Anytown: Model fit statistics for the four prior pdf schemes RMSE R2 Bias

Uniform 0.068 0.998 –0.035

Normal 0.071 0.998 –0.021

Gamma 0.062 0.998 –0.016

The predictive uncertainty results in terms of root-mean-squared error (RMSE), coefficient of determination R2 and bias are presented in Table 2. All different prior pdfs generated an excellent model fit. Fig. 2 presents the histograms of marginal parameter

Fig. 1. Anytown: DW ε roughness statistics for the marginal posterior pdf of parameters PG1 to PG6a

730

Proposal 0.065 0.999 –0.011

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 725-734

Fig. 2. Anytown; histograms of marginal distributions and two-dimensional correlation plots of posterior parameter samples

distributions and a two-dimensional correlation plot of posterior parameter samples for the fourth (i.e. mixed prior) approach. Correlation values of any pair of parameter groups are low, while only PG3 and PG4 share a higher correlation coefficient of −0.882. These features were also observed in [3]. 4 REAL-WORLD WDS CASE STUDY The aim of this section is to demonstrate the Bayesian framework of parameter estimation on a real-world WDS network and to show the effect of assumed prior pdfs on calibrated parameter values. The prior information approach presented in this paper is applied to exhibit its applicability to real-world WDS networks. The selected model parameters are the equivalent roughness ε of the DW pipe friction model. The analysed system is part of a bigger WDS, but hydraulically independent of the rest of the WDS. The WDS of Šentvid serves a population of approximately 34,000 inhabitants, and its estimated average demand is 93.87 l/s. From the available WDS data, an Epanet2 hydraulic model was assembled consisting of three reservoirs, two tanks, three pumps,

one pressure reducing valve, 812 junction nodes and 1072 pipes. The complete measurement campaign consists of 11 fire flow tests were performed throughout the WDS network. Sixteen pressure loggers (PL) (Memmy NT, measurement range: 0 to 20 bar, measurement error: ±0.05% max. measurement range)), four ultrasonic flow metering devices (Krohne UFM 610P, measurement range: 0.006 to 14.89 m/s, measurement error: ±2.0% (v ≥ 1 m/s) and ±0.02 m/s (v < 1 m/s)) and SCADA measurements (five flow meters and two tank level gauges) were recording measurements. In this study, 11 steady-state hydraulic simulations were performed to represent the 11 fire flow loading conditions (LC) during the measurement campaign. A total of 176 observations (16 PL × 11 LC) are considered in the observational data set. Flow and SCADA measurements were used to define the boundary conditions of the hydraulic simulations. The PGs were established based on the criterion of pipe diameter, material and age, resulting in a total of 93 PGs. A second grouping criterion involved only pipe material and age, resulting in 25 PGs. Only the last criterion was investigated, since the quantity of observational data would not support the higher

Fig. 3. Marginal posterior densities of the individual PGs (1 to 8) for the real-world WDS network and GL residual model parameters (9 to 12) (× indicates the maximum a posterior (MAP) values) Investigating Prior Parameter Distributions in the Inverse Modelling of Water Distribution Hydraulic Models

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exponentially distributed. Standard deviations σ0 and σ1 show small heteroscedasticity. The first four PGs (Fig. 3, 1 to 4) show high parameter sensitivity; therefore, the uniform prior pdf was p(θ)~U(0.001, 15). All other PGs had gamma prior pdf applied. Parameter uncertainty can be expressed in terms of the spread of the posterior marginal parameter pdf. A greater spread indicates higher uncertainty. The asbestos-cement (AC) and ductile iron (NL) PGs have a narrower posterior pdf in combination with a uniform prior pdf, indicating smaller parameter uncertainty for those two PGs. In contrast, some PGs (e.g. cast iron (LZ)) show higher parameter uncertainties due to their greater spread. The next four PGs (Fig. 3, 5 to 8) have smaller parameter sensitivity values and were inferred using a gamma prior pdf p(θ)~ Г(α,β). As can be observed by the posterior pdf, a general shape of the gamma prior is recognizable, while the likelihood functions provided a drift towards the observational information content. Fig. 4 illustrates how the marginal posterior pdf (i.e. parameter uncertainty) translates into a 95% pressure head predictive uncertainty. The

dimensionality of the parameter estimation problem [18]. For the GL residual model, we used fixed values of residual model parameters ϕ1 = 0 and μh = 0, while parameters σ0, σ1, β, and ξ, were inferred with uniform prior distributions as described in Section 3. This resulted in a total of Nθ = 29. The PGs prior pdf were estimated by literature given pipe roughness values ε for different pipe materials [19]. The parameter values of the gamma pdf p(θ)~Г(α,β) were kept close to the higher estimates of roughness values for new pipes (i.e. α parameter), while their righttailed shape provided a possible drift towards higher roughness values if some pipe aging was present (i.e. β parameter). The DREAM(ZS) algorithm was set up with the same parameters as in Section 3, except Ndim = Nθ = 29 and Neval = 75,000. Approximate posterior parameter pdfs and of equivalent roughness ε in [mm] maximum a posterior (MAP) values are given in Fig. 3. Additionally, posterior densities for the GL parameters θe are provided in Fig. 3 (numbers 9 to 12). The inferred GL parameters β = 1 and ξ = 1 indicate that SEP distribution of residuals is symmetrically double

65 60 55 Pressure head [m]

50 45 40 35 30 25 20 15 0

80 100 Number of observations

120

140

160

180

Fig. 4. Real-world WDS: 95% posterior parameter (dark grey) and prediction (light grey) uncertainty ranges and corresponding pressure observations (solid circles) 70

Quantiles of posterior sample

Model predictions [m]

RMSE = 0.96408 R2 = 0.9903 Bias = -0.028863

10 15

Observational data [m]

1 0 -1 -2 -3 -4 -3

-2

-1

Standard normal quantiles

Fig. 5. Real-world WDS: a) scatter plot of observational data against model predictions, b) residuals as a function of model predictions

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light grey region depicts the predictive uncertainty, while the dark grey region corresponds to parameter uncertainty. These and other results from both case studies are shown for the calibration data set only, i.e. no results are shown for the validation data set. Since observational data is very limited, it was all used for calibration only. Ideally, validation on an independent data set should be done. The WDS model fits very well with the observational data with an associated RMSE of 0.458 m. Additionally, Fig. 4 shows that all observations fall inside the 95% predictive uncertainty bounds. In a post-processing analysis, assessment of the underlining assumptions made in Section 2.1 is required, i.e. the likelihood function. Two diagnostic tests were conducted to verify the assumptions on the statistical model of residuals. Fig. 5a plots the model predictions against the observational data. In addition to the RMSE, the coefficient of determination R2 = 0.997 and bias = –0.052 indicate a very good model fit. Fig. 5b presents residuals as a function of model predictions. It can be observed that residual show some heteroscedastic behaviour. We can, therefore, conclude that the model residual distribution, the posterior parameter pdf and predictive uncertainties are adequately represented. 5 CONCLUSIONS This paper presents a study of uncertainty analyses of pipe roughness parameter estimates, their corresponding parameter and predictive uncertainties. The analyses were conducted on a hypothetical and a real-world WDS model. Identifiability of pipe roughness parameters is difficult, especially in a realworld WDS model due to the limited information content of the observational data. Mapping samples from prior distributions of the parameter space to the likelihood space results in the identification of plausible ranges of parameter sets through given observational data and allows estimation of both types of uncertainties. The generalized likelihood function was used to adequately represent the residual distribution. Using this formal Bayesian approach, the inference should lead to unbiased parameter estimates [2]. Incorporation of the prior distribution has proved to be an efficient and effective approach to estimate the posterior parameter pdf. We used three different prior pdfs. The results of this study demonstrate that prior information on pipe roughness parameters and correct representation of residual distributions significantly improves identifiability and reduces parameter and

predictive uncertainties. Since definition of prior pdf is difficult, we suggested an approach that resembles the parameter identifiability. It proves to be important to provide accurate prior information in order to narrow the ranges of uncertainties of posterior parameter pdfs and to obtain confidence in the optimised/expected parameter values [17]. Using this approach, we successfully inferred the posterior parameter pdf and derived parameter and predictive uncertainties for a real-world WDS model. 6 ACKNOWLEDGEMENTS We are obliged to Jasper A. Vrugt and Cajo ter Braak for providing the code of the DREAM(ZS) algorithm and graphical post-processing software. 7 REFERENCES [1] Savic, D.A., Kapelan, Z.S., Jonkergouw, P.M.R. (2009). Quo vadis water distribution model calibration? Urban Water Journal, vol. 6, no. 1, p. 3-22, DOI:10.1080/15730620802613380. [2] Schoups, G., Vrugt, J.A. (2010). A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and non-Gaussian errors. Water Resources Research, vol. 46, no. 10, p. 1-17, DOI:10.1029/2009WR008933. [3] Kapelan, Z.S., Savic, D.A., Walters, G.A. (2007). Calibration of water distribution hydraulic models using a Bayesian-type procedure. Journal of Hydraulic Engineering, vol. 133, no. 8, p. 927-936, DOI:10.1061/ (ASCE)0733-9429(2007)133:8(927)). [4] Alvisi, S., Franchini, M. (2010). Pipe roughness calibration in water distribution systems using grey numbers. Journal of Hydroinformatics, vol. 12, no. 4, p. 424-445, DOI:10.2166/hydro.2010.089. [5] Ter Braak, C.J., Vrugt, J.A. (2008). Differential evolution Markov chain with snooker updater and fewer chains. Statistics and Computing, vol. 18, no. 4, p. 435-446, DOI:10.1007/s11222-008-9104-9. [6] Scharnagl, B., Vrugt, J.A., Vereecken, H., Herbst, M. (2011). Inverse modelling of in situ soil water dynamics: Investigating the effect of different prior distributions of the soil hydraulic parameters. Hydrology and Earth System Sciences, vol. 15, no. 10, p. 3043-3059, DOI:10.5194/hess-15-3043-2011. [7] Mays, L.W. (2000). Water Distribution System Handbook. McGraw-Hill Professional, New York. [8] Rossman, L.A. (2000). EPANET 2 – User manual. United States Enviromental Protection Agency, Washington, D.C. [9] Box, G.E.P., Tiao, G.C. (1992). Bayesian inference in statistical analysis. Wiley Classics Library. John Wiley & Sons, New York, DOI:10.1002/9781118033197. [10] Branisavljević, N., Prodanović, D., Ivetić, M. (2009). Uncertainty reduction in water distribution

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network modelling using system inflow data. Urban Water Journal, vol. 6, no. 1, p. 69-79, DOI:10.1080/15730620802600916. [11] Banovec, P., Kozelj, D., Šantl, S., Steinman, F. (2006). Sampling design for water distribution system models by genetic algorithms. Strojniški vestnik--Journal of Mechanical Engineering, vol. 52, no. 12, p. 817-834. [12] Kang, D.S., Pasha, M.F.K., Lansey, K. (2009). Approximate methods for uncertainty analysis of water distribution systems. Urban Water Journal, vol. 6, no. 3, p. 233-249, DOI:10.1080/15730620802566844. [13] Seifollahi-Aghmiuni, S., Haddad, O.B., Omid, M.H., Marino, M.A. (2013). Effects of pipe roughness uncertainty on water distribution network performance during its operational period. Water Resources Management, vol. 27, no. 5, p. 1571-1599, DOI:10.1007/s11269-013-0259-6. [14] Kuščer, L., Diaci, J. (2013). Measurement Uncertainty assessment in remote object geolocation. Strojniški vestnik – Journal of Mechanical Engineering, vol. 59, no. 1, p. 32-40, DOI:10.5545/sv-jme.2012.642.

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[15] Ormsbee, L.E. (1989). Implicit network calibration. Journal of Water Resources Planning and Management, vol. 115, no. 2, p. 243-257, DOI:10.1061/(ASCE)07339496(1989)115:2(243). [16] Travis, Q.B., Mays, L.W. (2007). Relationship between Hazen-William and Colebrook-White roughness values. Journal of Hydraulic Engineering-ASCE, vol. 133, no. 11, p. 1270-1273, DOI:10.1061/(ASCE)07339429(2007)133:11(1270). [17] Kapelan, Z., Savic, D.A., Walters, G.A. (2004). Incorporation of prior information on parameters in inverse transient analysis for leak detection and roughness calibration. Urban Water Journal, vol. 1, no. 2, p. 129-143, DOI:10.1080/15730620412331290029. [18] Giustolisi, O., Berardi, L. (2011). Water distribution network calibration using enhanced GGA and topological analysis. Journal of Hydroinformatics, vol. 13, no. 4, p. 621-641, DOI:10.2166/hydro.2010.088. [19] Lamont, P.A. (1981). Common pipe-flow formulas compared with the theory of roughness. Journal American Water Works Association, vol. 73, no. 5, p. 274-280.

Kozelj, D. – Kapelan, Z. – Novak, G. – Steinman, F.

Received for review: 2013-07-21 Received revised form: 2013-12-04 Accepted for publication: 2014-01-17

Investigating Fatigue Life Effects on the Vibration Properties in Friction Stir Spot Welding Using Experimental and Finite Element Modal Analysis

Aghdam, N.J. – Hassanifard, S.– Ettefagh, M.M. – Nanvayesavojblaghi, A. Nima Jafarzadeh Aghdam* – Soran Hassanifard – Mir Mohammad Ettefagh – Arvin Nanvayesavojblaghi University of Tabriz, Mechanical Engineering Department, Iran Frequency response and vibration study of friction-welded specimens with different fatigue life properties is very important. Friction stir spot welding (FSSW) is one of the welding methods applied in different types of manufacturing processes such as in the vehicle and aerospace industries. Therefore the main purpose of this paper is better understanding of the correlation between natural frequencies and fatigue crack initiation of the FSSW in four different welding processes. For this purpose fatigue and experiment modal analysis tests were carried out at different fatigue strength levels. The experimental modal analysis was carried out on four different types of friction stir spot welded specimens after different fatigue life tests. Finite element (FE) modelling was performed in ABAQUS software using the Lanczos method for comparing the obtained results with the experimental tests. Some important findings were determined from the experimental and FE modelling results. One of the interesting findings was similar behaviour between the FE model and the experimental test for defined frequency ranges. Keywords: friction spot welding, fatigue damage, frequency response, experimental modal analysis, finite element modelling, vibration characteristics

0 INTRODUCTION Joining metal sheets by welding different materials is absolutely necessary in the automotive and other industries. However, different welding methods may be used depending on the applications and materials. Spot welding is one of the applicable methods, which includes resistance spot welding (RSW) and friction stir spot welding (FSSW). Each method has its individual characteristics and advantages or disadvantages. RSW is one of the most popular welding methods, but in joining light-metals such as aluminium alloys, the application of this method would lead to some major difficulties [1]. Recently, automotive, aerospace and other industries have begun to pay serious attention to the FSSW method for joining metal sheets [2] to [3]. This method could be considered as an upgraded variant of the friction stir welding (FSW) process. Both FSW and FSSW processes can be termed as standard methods of welding aluminium alloys. FSW is a solid-state joining process which means that the metal is not fully melted and instead it is just softened [4]. One of the most important features of the FSW process is that the main characteristics of the materials remain unchanged as much as possible. In order to apply the FSSW method correctly, it is necessary to provide a proper model, which would allow for a more exact study of the fatigue and dynamic properties of the welding. There are some studies that deal with modelling and prediction of fatigue life of metals

welded by FSSW methods in various applications. Some of these studies will be described below. The fatigue life of FSSW in aluminium 6061-T6 sheets were investigated by Wang and Chen [5]. In this work, tool geometry, the rotational speed, the holding time and the downward force were considered the most important processing parameters. They modelled the three dimensional FSSW’s crack as a twodimensional crack problem. The fatigue life for FSSW has also been investigated by Lin et al. [6] to [8] and by Ericsson et al. [9]. They used fracture mechanics to solve the modelling problem. In all of the abovementioned papers the weld nugget is modelled as a cone. For example, in [6] to [8], a fatigue crack growth model was adopted based on the Paris law and the local equivalent stress intensity factors. Kinked cracks were applied to predict the fatigue life of FSSW and the fatigue crack propagation; the modelling results were then compared with experimental ones. In [9] the global stress intensity factor was used to predict the crack growth rate of the FSSW. These stress intensity factors were calculated from FE models. The main purpose of the above-mentioned works was to predict fatigue life using proper modelling, but additional goal of other studies has been to model welding in order to study the dynamic and vibrational characteristics of the model, which are very important for designing proper welding techniques, especially in the automotive and aerospace industries. Shang [10], Wang et al. [11] and Shang et al. [12] and [13] investigated the effects of fatigue cracks on the frequency responses of RSW joints and concluded that the natural frequencies of the

*Corr. Author’s Address: University of Tabriz, 29th Bahman Blvd., Iran, Tabriz, n.j.aghdam91@ms.tabrizu.ac.ir

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Table 1. Chemical properties of aluminium 7075-T6 (Maximum values if range not presented) a) Chemical composition of aluminium 7075-T6 [% weight] Si Fe Cu Mn Mg 0.4 0.5 1.2 to 2 0.3 2.1 to 2.9 b) Mechanical properties of aluminium 7075-T6 Density ×1000 [kg/m3] Elastic modulus [GPa] 2.81 71

Cr 0.18 to 0.28

Zn 5.1 to 6.1

Ultimate tensile strength [MPa] 578

Ti 0.2

Others, each 0.5

Yield strength [MPa] 469

Others, total 0.15

Al Balance

Elongation [%] 33

Table 2. Main parameters of the four types of FSSW

Type1 Type2 Type3 Type4

Penetration depth of the shoulder [mm] 1.3 1.3 0.9 1.3

Rotational speed of the rotating tool [rpm] 2500 1600 2000 2000

Vertical pin advance speed [mm/min] 8 8 8 8

Force duration or holding time [ms] 5 to 6 5 to 6 5 to 6 5 to 6

1 FRICTION STIR SPOT WELDING SETUP AND PROCESSING

To start the welding process, two sheets of aluminium 7075-T6 are joined in a lap configuration as shown in Fig. 1. The lower sheet is put on a supporting plate while a rotating tool, as shown in Fig. 2, presses the upper sheet. The back plate supports the downward force of the rotating pin while the rotational speed of the pin results in softening the aluminium but not melting it. The downward force of the pin causes penetration of the pin into the softened material. By the time the shoulder of the tool reaches the surface of the upper sheet, more heat is generated due to a larger contact surface and additional friction between the tool shoulder and the aluminium sheet, which causes a larger region of the material to be softened. So the softened regions of the sheets are pressed and at the same time stirred, which leads to a metallurgical bond around the rotating pin. At the end of the welding process, the tool was retracted and a characteristic hole in the middle of the weld was left. The main parameters of the welding process for different sets of welded joints are summarized and presented in Table 2. Hereafter, these four different sets of friction stir spot welded joints will be called Type1, Type2, Type3 and Type4, respectively. The experimental fatigue test setup, provided in the fatigue test laboratory, is illustrated in Fig. 3. Using this setup the fatigue test was carried out to generate different types of fatigue life.

In this investigation, distinct Aluminium 7075T6 sheets were welded using four different FSSW process parameters. The chemical composition of the aluminium 7075-T6 sheet is presented in Table 1a and the mechanical properties of the sheets are presented in Table 1b.

Fig. 1. The two sheets of aluminium 7075-T6, joined by FSSW

welds change a lot when the specimens are put under cyclic loading based on counting the cycles or stress reversals until destruction. Wang and Barky [14] and [15] also studied fatigue cracks and their effects on the specifications of dynamic responses of RSW joints and used their results to detect and diagnose fatigue cracks. In this paper, fatigue and experiment modal analysis test were used to come to a better understanding of the correlation between the natural frequencies and fatigue crack initiation of the FSSW in four different welding processes. For this purpose, four types of specimens, each of them using different FSSW process to weld two aluminium 7075-T6 sheets together, were used and were then exposed to cyclic loading to generate different levels of fatigue. At each level, the welded sheets were tested by experimental modal analysis method in order to extract the natural frequencies of the FSSW specimens in each of the four types. An FE model of the four types of FSSW specimens were provided in ABAQUS and the natural frequencies of the FE model were extracted at different fatigue levels, which were modelled as a proper crack. Finally the obtained results in the experimental and modelling sections of this work will be illustrated and discussed.

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and it is first initiated near tip of the circumferential notch around the nugget as shown in Figs. 5 and 6. It then grows perpendicular to the applied load and finally the crack reaches through the thickness of the plate in the direction of the specimen width [12] to [13]. It should be added that the fatigue cracks are assumed to initiate at a distances of 1 mm from the nugget edge. The site of the simulated crack initiation in the 3D FE model is shown in Fig. 6 by a cut-away view of the FE model as illustrated in Fig. 5.

Fig. 2. The rotating tool

Fig. 4. FE mesh and simulation of FSSW specimen

Fig. 3. The experimental fatigue test setup

2 FRICTION STIR SPOT WELDS (FSSW) FINITE ELEMENT MODELING Three-dimensional FE analysis was used to determine the relationship between the length of fatigue cracks and the frequency response characteristics (natural frequencies and mode shapes). The severity of the fatigue damage can be modelled using the fatigue crack length or depth [12]. In this study, the finite element mesh contained approximately 34000 elements with a maximum element length of 3 mm. The element type was 10-node quadratic tetrahedron (Fig. 4). To simulate the experimental conditions, clamped-free condition were applied. The mechanical properties, considered for the FSSW specimens were a density of 2810 kg/m3, a Young’s modulus of 71 GPa, and a Poisson’s ratio of 0.33. In this paper, the steps of fatigue crack growth modelling are as follows: The crack was modelled in the form of an elliptical surface

Fig. 5. FE model of the fatigue crack

Fig. 6. Fatigue crack growth steps in the model

It should be noted that according to the aforementioned experimental observations, the shape of the propagating surface cracks in a plate is approximately semi-elliptical. Therefore, the models built in this study assumed that the growing crack

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was of a semi-elliptical shape [12] that maintained a constant aspect ratio (the ratio between the surface crack length and the crack depth) until the surface crack extended through the thickness of plate [13]. The aspect ratios were taken as 0.5. The steps of the fatigue crack development process that were assumed from initiation to propagation are shown in the FE model in Fig. 6. A microscopic image of a fatigue crack in one of the FSSW types is shown in Fig. 7.

the fatigue crack initiation of the four FSSW type’s specimens and, for this reason, all the specimens are loaded into an HCF regime before the modal analysis test.

Fig. 8. Fatigue test results of friction stir spot welded specimens; ∆τ =∆P/(nugget area), and (nugget area) = π(ro2 – ri2) where ro is shoulder radius and ri is pin radius

4 EXPERIMENTAL MODAL ANALYSIS TESTS In order to identify the natural and mode shape of the four different FSSW types, experimental modal analysis [16] was implemented as shown in Fig. 8. Every FSSW types was excited by a shaker (type 4809 B&K) at the free end by applying white Gaussian noise in the frequency range of 10 kHz.

Fig. 7. Cross-section of a frictional stir spot weld after creation of a fatigue crack under cyclic loadings

3 EXPERIMENTAL FATIGUE TESTS Fatigue tests of four different types of FSSW specimens were conducted at different load levels ranging from 1.5 to 5 kN using a load controlling technique at a frequency of 10 Hz and a load ratio of 0.1 by applying a 250 kN Zwick/Amsler fatigue testing machine. Fatigue test results of the four FSSW types are shown in Fig. 8. As it can be seen in Fig. 8, the fatigue life of the specimens is reduced with decreasing the tool rotational speed, especially in a low cycle fatigue regime. Increasing the tool penetration depth from 0.9 to 1.3 mm has no major impact on increasing the fatigue life. Although the four types of FSSW specimens have the same total fatigue lifetime in a high cycle fatigue regime, they may have a different crack initiation and propagation life. Therefore, modal analysis should be used primarily for assessing 738

Fig. 9. Experimental modal analysis set-up

The dynamic response is measured by a piezoelectric acceleration sensor (type 4507 B&K) attached to the middle of the specimen. The analogue data gathered from each sensor was processed using a Pulse system [17] and converted to digital data and saved in a personal computer. The frequency response function (FRF) was then extracted and analysed by MeScope software [18] using the polynomial modal analysis method to find out the natural frequencies of the

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specimens. The above mentioned modal analysis test was applied to the specimen after applying the fatigue test with different fatigue lives. In order to validate the modal analysis test, the FRF and corresponding coherence of one of the FSSW types is shown in Fig. 10 as an example.

Fig. 10. The a) FRF and b) corresponding coherence of Type1

5 RESULTS AND DISCUSSION As described in previous sections, four different FSSW types were provided for experimental fatigue and modal analysis testing. Figs. 11 to 14 present experimental modal analysis test results for every FSSW type by depicting the percentage of frequency reduction versus the percentage of fatigue damage. It should be noted that to obtain the values of the fatigue damage percentage, the fatigue test of the specimens were terminated at some specific fatigue life value, which was smaller than the total fatigue life. Therefore the fatigue damage percentage is calculated in percent by dividing the terminated life by the total life. As shown in these figures, the major trend noted is a lowering of all the 1st to 5th natural frequencies as the fatigue crack grows in each of the types. However, the first natural frequencyâ&#x20AC;&#x2122;s decrease is more visible than that of the other frequencies. Some disorders are seen in the overall process of frequency lowering while fatigue damage increases which is a result of a possible variation in the specimensâ&#x20AC;&#x2122; dimensions due to the fatigue test. For example, a small length increase during modal analysis test may cause the frequency to increase. This is normal in experimental tests, because noise, especially measurement noises, can have very serious effects on modal analysis tests. As another

example, it is not possible to provide the boundary condition as an ideal cantilever beam and preserve this condition in different tests of the specimens when opening and closing them on a fixture during modal analysis. In addition, positioning the measurement sensors and exciting shaker in exactly the same position for different tests of the specimens is another reason for having some errors in extracting the natural frequencies. Another issue that contributes to errors is the length positioning of the specimens when installing them in the cantilever fixture for the modal analysis test, which strongly contributes to change the natural frequencies. The results of the numerical modal analysis of the FE model, introduced in Section 3 of this paper, are presented in Fig. 15. In this figure, the percentage of frequency reduction versus fatigue crack length is illustrated. In order to better illustrate this, the FEM and experimental results have been depicted in one figure after normalization of the percent of fatigue damage and frequency reduction, which correspond to the horizontal and vertical axis, respectively (Fig. 16). By considering the above-mentioned figures, it maybe deduced that firstly, the same decreasing behaviour in the natural frequencies seen in the experimental results is obvious for different frequency domains. Secondly, despite increases in the frequency domain, frequency reduction is not reduced and this phenomenon is also obvious in the experimental results. In other words, fatigue/crack damage effects on frequency reduction are not directly related to the frequency domain (frequency band). Although at some frequencies, the reduction may be more than at other frequencies. For example, in Fig. 15, the frequency reduction in the 2nd frequency (a low band frequency) is more than in other frequencies. As another example, a frequency reduction in the 4th frequency (a high band frequency) is less than other frequencies. This kind of trend can be found in the experimental results as well. For example, considering Figs. 10 to 13, the frequency reduction in the 1st frequency is more than the other frequencies. As another example, Figures 11 to 14 show that frequency reduction in the 5th frequency is less than in other frequencies in the above-mentioned experimental results. The reason for the different frequency numbers and presence of a frequency reduction when comparing both experimental and numerical results may be due to the fact that that the FE modelling frequency domain is different compared to the corresponding domain in the experimental test due to some unknown modelling errors. This error may be minimized by a more exact selection of the mechanical property such as module

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of elasticity in the FE model. This task may be carried out by FE model updating in future.

Fig. 15. Results of numerical modal analysis of the FE model Fig. 11. Results of the experimental modal analysis test on the Type1 FSSW by illustrating the percentage of frequency reduction against fatigue damage percent

6 CONCLUSIONS

Fig. 13. Results of experimental modal analysis test on Type3 FSSW by illustrating percentage of frequency reduction against fatigue damage percent

In this paper experimental and numerical modal analysis test was designed to study and analyze the effects of fatigue damage on frequency response function (vibration characteristics) of the four types of FSSW with different welding processes. The experimental modal analyses were carried out on four types of FSSW after different fatigue life tests. FE modelling was also carried out in ABAQUS and by using the Lancsoz method the obtained results were compared with experimental test. One of the important findings of this research is that the fatigue damage effect on the frequency response function of the four types of FSSW has no direct relationship with the frequency domain level. In other words it was shown in experimental tests’ results that at some low frequency domains, the fatigue damage effect is more than the high frequency domain and result were also validated by FE model results. Therefore,as the frequency domain increases, the fatigue effect may decrease under some conditions or asthe frequency domain decreases the fatigue effect may increases. Another important finding is that for each individual frequency band with increasing fatigue damage level, the frequency reduction is obvious in the experimental test except in a few cases and this result is also illustrated in the FE model results.

Fig. 14. Results of experimental modal analysis test on Type4 FSSW by illustrating percentage of frequency reduction against fatigue damage percent

Fig. 16. Comparing the numerical modal analysis of the FE model and the experimental test (Type1 FSSW)

Fig. 12. Results of the experimental modal analysis test on Type2 FSSW by illustrating percentage of frequency reduction against fatigue damage percent

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7 REFERENCES [1] Spinella, D.J., Brockenbrough, J.R., Fridy, J.M. (2005). Trends in aluminium resistance spot welding for the auto industry. Welding Journal, vol. 84, p. 34-40. [2] Baek, S.W., Choi, D.H., Lee, C.Y., Ahn, B.W., Yeon, Y.M., Song, K., Jung, S.B. (2010). Microstructure and mechanical properties of friction stir spot welded galvanized steel. Materials Transactions, vol. 51, no. 5, p. 1044-1050, DOI:10.2320/matertrans.M2009337. [3] Hancock, R. (2004). Friction welding of Aluminum cuts energy cost by 99%. Welding Journal, vol. 83, no. 2, p. 40. [4] Bilici, M.K. (2012). Effect of tool geometry on friction stir spot welding of polypropylene sheets. Express Polymer Letters, vol. 6, no. 10, p. 805-813, DOI:10.3144/expresspolymlett.2012.86. [5] Wang, D.A., Chen, C.H. (2009). Fatigue lives of friction stir spot welds in aluminum 6061-T6 sheets. Journal of Materials Processing Technology, vol. 209, no. 1, p. 367-375, DOI:10.1016/j.jmatprotec.2008.02.008. [6] Lin, P.C., Pan, J., Pan, T. (2005). Investigation of fatigue lives of spot friction welds in lap-shear specimens of aluminum 6111-T4 sheets based on fracture mechanics. SAE Technical Paper, no. 2005-01-1250. [7] Lin, P.C., Pan, J., Pan, T.(2006). Fatigue failures of spot friction welds in aluminum 6111-T4 sheets under cyclic loading conditions. SAE Technical Paper, no. 2006-011207. [8] Lin, S.H., Pan, J., Wung, P., Chiang, J. (2006). A fatigue crack growth model for spot welds under cyclic loading conditions. International Journal of Fatigue, vol. 28, no. 7, p. 792-803, DOI:10.1016/j.ijfatigue.2005.08.003. [9] Ericsson, M., Jin, L.Z., Sandström, R. (2007). Fatigue properties of friction stir overlap welds. International Journal of Fatigue, vol. 29, no. 1, p. 57-68, DOI:10.1016/j.ijfatigue.2006.02.052.

[10] Shang, D.G. (2009). Measurement of fatigue damage based on the natural frequency for spot-welded joints. Materials and Design, vol. 30, no. 4, p. 1008-1013, DOI:10.1016/j.matdes.2008.06.048. [11] Wang, R.J., Shang, D.G., Li, L.S., Li, C.S. (2008). Fatigue damage model based on the natural frequency changes for spot-welded joints. International Journal of Fatigue, vol. 30, no. 6, p. 1047-1055, DOI:10.1016/j. ijfatigue.2007.08.008. [12] Shang, D.G., Barkey, M.E., Wang, Y., Lim, T.C. (2003). Fatigue damage and dynamic natural frequency response of spot-welded joints. SAE Technical Paper, no. 2006-01-0695. [13] Shang, D.G., Barkey, M.E. (2006). Analysis of fatigue crack behavior based on dynamic response simulations and experiments for tensile-shear spot-welded joints. Fatigue & Fracture of Engineering Materials & Structures, vol. 29, no. 1, p. 23-30, DOI:10.1111/ j.1460-2695.2006.00955.x. [14] Wang, G., Barkey, M.E. (2005). Fatigue crack identification in tensile-shear spot welded joints by dynamic response characteristics. Transactions of the ASME, vol. 127, no. 3, p. 310-317, DOI:10.1115/1.1925286. [15] Wang, G., Barkey, M.E. (2004). Fatigue cracking and its influence on dynamic response characteristics of spot welded specimens. Experimental Mechanics, vol. 44, no. 5, p. 512-521, DOI:10.1007/BF02427963. [16] Ewins, D.J. (2000). Modal Testing: Theory, Practice and Application. Research Studies Press, Baldock. [17] PULSE (2006). Analyzers and Solutions, Release 11.2, Bruel&Kjaer, Sound and Vibration Measurement A/S, Nærum. [18] ME’scopeTM (2001). Version: 2.0.0.21. Vibrant Technology, Scotts Valley.

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Received for review: 2014-04-16 Received revised form: 2014-08-14 Accepted for publication: 2014-09-04

Valve-Induced Water Hammer and Column Separation in a Pipeline Apparatus Karadžić, U. – Bulatović, V. – Bergant, A. Uroš Karadžić1,* – Vladimir Bulatović1 – Anton Bergant2

1 University

of Montenegro, Faculty of Mechanical Engineering, Montenegro 2 Litostroj Power, Slovenia

A flexible experimental apparatus for investigating water hammer and column separation in an unsteady friction-dominated pipelines has been developed and designed. The apparatus has been tested for steady and unsteady flow conditions. Transient cavitation and column separation phenomena have been observed in a number of experimental runs. Water hammer has been triggered by both the closing and opening of electro-pneumatic (EV) and hand-operated (HV) valves. The experimental data have been compared with results given from in-house numerical code written in Visual Fortran based on the method of the characteristics (MOC) with a convolution-based unsteady friction model (CBM) included. The column separation and transient cavitation phenomena are modelled via a discrete gas cavity model (DGCM). There is good agreement between the experimental and numerical results; the model is robust and, therefore, it is recommended for engineering practice. In addition, the influence of variations of the pressure wave speed and the uncertainty in flow rate measured by the electromagnetic flow meter are also investigated. Keywords: water hammer, experimental setup, column separation, unsteady friction, piping systems

0 INTRODUCTION The phrase “water hammer” describes the generation, propagation and reflection of pressure waves along pipelines of pressurized liquid systems that are associated with changes in flow conditions. Uncontrolled water hammer can disturb the operation of hydraulic systems and, in the worst case, damage and destroy system components. Rises or drops in water hammer pressure may be controlled by installing protecting devices, the appropriate control of operating regimes or the redesigning of an originally developed pipeline layout [1] and [2]. The classic form of water hammer may be affected by transient cavitation and column separation, unsteady friction effects, visco-elastic behaviour of the pipe wall and fluid-structure interaction (FSI) [3] and [4]. Transient vaporous cavitating pipe flow occurs when the pressure drops to the liquid vapour pressure. The fluid also contains a small amount of free and released gas. The gas and vapour bubbles form pockets (cavities) [1] and [5], which can break the fluid column at the system boundaries or at the high points, i.e. a phenomenon known as “column separation” [6] and [7]. The collapse of a vapour cavity may induce short-duration pressure pulses with values higher than the pressure initially given by the Joukowsky equation [8]. The value of the friction factor during the water hammer event is different than its value during the steady flow. The friction factor can be expressed as a sum of two parts: quasi-steady (fq) and unsteady (fu) [9]. The unsteady part attempts to represent transientinduced changes in the velocity profile [10] to [12], 742

and it is important for fast transients [13] and [14]. For pipelines that are not completely fixed, FSI effects have to be taken into consideration [15] to [17]. The viscoelastic behaviour of the pipe wall is significant in cases in which the pipe is made from plastic materials, such as polyethylene PE, high-density polyethylene HDPE, polyvinyl chloride PVC and acrylonitrile butadiene styrene ABS [18] to [21]. The experimental test rig has been developed primarily for an investigation of the transient cavitation, column separation and unsteady friction during the water hammer events. In the first part of the paper, mathematical tools for modelling water hammer, unsteady friction and transient vaporous cavitation (liquid column separation) are presented. Water hammer is fully described by two hyperbolic partial differential equations: the continuity and the momentum equation that are traditionally numerically solved by the method of the characteristics [1] and [2]. The improved convolution-based unsteady friction model [22] is explicitly incorporated into the staggered grid of the method of characteristics. The developed numerical code is further improved by including a discrete gas cavity model for modelling transient cavitation and column separation [1] and [23]. The Ghidaoui et al. coefficient P is calculated, and it is proved that developed experimental setup is unsteady friction dominated [24]. The paper continues with detailed description of the experimental apparatus for measurement of water hammer pressure waves. In the second part of the paper, the influence of variations of pressure wave speed [25] and uncertainty in flow rate measured by the electromagnetic flow meter are

*Corr. Author’s Address: University of Montenegro, Faculty of Mechanical Engineering, Dzordza Vasingtona nn, 81000 Podgorica, Montenegro, uros.karadzic@ac.me

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investigated. The paper concludes with a number of comparisons between the experimental and the numerical results given from the fast closing and opening of the downstream end valve. The numerical scheme that includes discrete gas cavities and unsteady friction effects yields accurate and robust results. 1 WATER HAMMER WITH COLUMN SEPARATION Water hammer is manifested by a pressure rise or drop along the liquid-filled pipeline due to a change of flow conditions. The simplified form of the equation of continuity and the momentum equation is [1] and [2]:

∂H a ∂Q + = 0, (1) ∂t gA ∂x 1 ∂Q f Q | Q | ∂H + + = 0. (2) ∂x gA ∂t 2gDA2 2

The unknown variables in Eqs. (1) and (2) are the piezometric head H and the discharge Q. The method of the characteristics (MOC) transformation of Eqs. (1) and (2) gives the following set of the compatibility equations that are solved algebraically [1] and [2]:

dH a dQ af Q | Q | ± ± = 0, (3) dt gA dt 2gDA2

that are valid along the characteristic lines dx/dt=±a. At a boundary (reservoir, valve), a device-specific equation replaces one of the MOC water hammer compatibility equations. The cavitating pipe flow usually occurs as a result of very low pressures during water hammer events. Cavitation may occur as localized cavitation with a large void fraction (column separation) or as distributed cavitation with a small void fraction. To date, numerous numerical models have been developed for simulating transient vaporous cavitation, one of which is a discrete gas cavity model (DGCM) that performs accurately over a broad range of input parameters [8]. The DGCM allows gas cavities to form at computational sections within the MOC numerical grid. A liquid phase with a constant wave speed is assumed to occupy the computational reach. The DGCM is fully described by the two water hammer compatibility equations and two additional equations; the continuity equation for the gas volume and the ideal gas equation with assumption of isothermal behaviour of the free gas, respectively, [1] and [23],

dVg = Qout − Qin , (4) dt

 p*  Vg = α 0V  *0  . (5) p   g

The numerical solution of Eqs. (3) to (5) can be found elsewhere [1] and [26]. The investigations of Ghidaoui et al. [24] indicate that accurate, physically based, unsteady friction models are required if the ratio of radial diffusion time scale to the pressure wave time scale is in order of one or less. The ratio is defined as [24]:

2D/

(

)

f Q/ A . (6) L/a

When the parameter P is close to one, then the vorticity generated at the pipe wall by the water hammer pressure wave diffuses through the complete pipe core and alters the pre-existing turbulent state [27]. This is the situation in the cases investigated in this paper; it implies that the developed experimental apparatus is an unsteady friction dominated one, i.e. the unsteady friction model is needed for proper estimation of skin friction losses during rapid transient events. For this purpose, an improved convolution based unsteady friction model is used [22]. Typical industrial examples of this situation are oil-hydraulic systems [28]. Such systems are part of a number of control systems in power industry, and their accurate modelling is essential for control simulations [29]. The convolutionbased model (CBM) has been analytically developed by Zielke for transient laminar flow [30]. This model produces correct results for a number of flow types using analytical expressions [31]. Column separation is a relatively short duration event with a wide range of flow event types [8]. Simple instantaneous acceleration-based models need to be calibrated (empirical coefficients) [12] and fail for certain types of flow [31]. In the improved Vítkovský et al. CBM model [22] the unsteady friction factor is expressed as a finite sum of Nk functions yk(t),

fu =

32ν A Nk ∑ yk (t ), (7) DQ Q k =1

with,

t yk (t ) = ∫ ∂Q* mk exp − nk K  t −t*   dt* , (8)    0 ∂t

where the constant K = 4ν/D2 converts the time t into the dimensionless time τ = 4νt/D2. The maximum

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number of exponential terms is Nk,max = 10. The coefficients of the exponential sum mk and nk have been developed for Zielke’s [30] and Vardy-Brown’s weighting functions [32] and [33] for laminar and turbulent transient flow, respectively. 2 EXPERIMENTAL APPARATUS A flexible experimental apparatus for investigating water hammer and column separation in unsteady friction dominated pipeline has been designed and constructed [34]. The apparatus shown in Fig. 1 is composed of a high pressure upstream end reservoir (HPR) (design pressure: 22 bar), a 54.23 m long steel pipeline (design pressure: 250 bar) with an inner diameter of 18 and 2 mm wall thickness, a fast closing electro-pneumatic ball valve (EV) (maximum working pressure: 63 bar; operating pressure: from 2 to 4 bar; closing and opening time: from 10 to 20 ms) that induces a transient event and a low pressure downstream end reservoir (LPR). The EV is operated by a solenoid valve (Burkert 5/2) and pneumatic actuator (Prisma). In addition, water hammer can be induced by a hand-operated valve (HV1) (maximum working pressure: 63 bar), which enables closures with different closing time. Both valves (EV and HV1) are equipped with a displacement sensor (VAS) (Positek P500.90BL, measurement range: 0 to 90°, frequency response: > 10 kHz) which measures the change of the valve angle during its closing or opening. Four dynamic pressure transducers (DPT) (Dytran 2300V4, pressure range: from 0 to 69 bar, sensitivity: 5 mV/0.069 bar, uncertainty [35]: ±0.1%) have been installed equidistantly along the pipeline for capturing high frequency pressure changes. At the upstream and the downstream end of each DPT, a hand-operated valve is installed. Dynamic pressure transducers are marked as D1 (next to the EV), D2 (18.4 m upstream from the EV), D3 (36.1 m upstream from the EV), and D4 (next to the HPR; see Fig. 1). For the evaluation of the initial conditions in the system, two pressure transducers (SPT) (Endress+Hauser PMP131, pressure range: from 0 to 10 bar, uncertainty: ±0.5%) are installed, one at the HPR and one at the downstream end of the pipeline just in front of the needle valve (NV) (Swagelok, maximum pressure: 344 bar). The needle valve is used for adjustment of the initial pipe flow rate (discharge). The initial discharge and consequently the initial average flow velocity is measured by the electromagnetic flow meter (EF) (Krohne OPTIFLUX 4000F IFC 300C, uncertainty: ±0.2%) and by the redundant ultrasonic one (UF) (Krohne UFM 610P, uncertainty: ±2%). 744

Pressure in the HPR is kept constant during the transient event by compressed air that is supplied from the compressor (CP) and the air reservoir (AR). The high precision air pressure regulator (HPPR) (SMC AF40-F04D, pressure range: from 0 to 1 MPa) is used for control of the initial pressure in the system as well as for control of the EV closing and opening pressure. All measured data are collected by the data acquisition system (DAS) (Measurement Computing USB1608FS, sample rate: up to 100 kHz) that is connected to PC. HPR is supplied with water from the tap water supply system. The lime-scale neutralizer (LN) and the check valve (CV) are installed in the water supply line. The water temperature is continuously monitored by the thermometer (TM) installed in LPR. Five sets of measurements have been performed. For all sets the initial pressure in HPR was adjusted to 4 bar. In the first set of measurements, the water hammer event has been initiated by the fast closing or opening of the downstream end valve using either the electro-pneumatic valve or hand-operated valve at different pipe velocities (from 0.26 to 2.34 m/s; 192 measurements). In the second set of measurements, the water hammer event has been initiated only with the hand-operated valve. Closing and opening of the valve has been done with different closing/opening times at different flow velocities (from 1.20 to 2.12 m/s; 18 measurements). The water hammer event in the third set of measurements has been triggered by closing the hand-operated valves along the pipeline. Measurements have been carried out for four positions of the hand valves (valves HV1 to HV4; see Fig. 1; 106 measurements). In the fourth set of measurements, the HV4d at the HPR and the HV1 at the downstream end of the pipeline have been closed at the same time (26 measurements). In the fifth set of the measurements three types of experiments have been performed: (1) rapid opening of the hand-operated valve HV4d at the HPR (other HVs are open) for different openings of the needle valve (filling of the pipeline); (2) filling the last third of the pipeline – rapid opening of HV2d and closed valve HV1 at the downstream end of the pipeline: (3) emptying the pipeline; rapid opening of the HV1 at the downstream end of pipeline with HV4u at the HPR closed (24 measurements). 3 COMPARISON OF EXPERIMENTAL AND NUMERICAL RESULTS The experimental apparatus has been tested for a number of steady and unsteady flow conditions. The experiments have been performed for the different initial flow velocities (from 0.26 to 2.34 m/s) at the

Karadžić, U. – Bulatović, V. – Bergant, A.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 742-754

b) Fig. 1. Water hammer and column separation pipeline apparatus; a) schematic layout of the apparatus; b) longitudinal profile of the pipeline Valve-Induced Water Hammer and Column Separation in a Pipeline Apparatus

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constant initial pressure in the HPR (pr = 4 bar). All experiments have been carried out as follows: the initial pressure in the HPR was adjusted and maintained at a constant level during the transient tests, using a high precision air pressure regulator (HPPR). After that, the initial velocity in the pipeline was adjusted by an appropriate opening of the needle valve. The water hammer event was triggered by fast closing or opening of the downstream end valve, using either the EV or the HV1. In addition, slow closing and opening of the HV1 were also investigated. In this paper, the convergence and stability of the used numerical model were checked first. Then, the influence of variations of the pressure wave speed were investigated. This is done because the wave speed is usually not known with accuracy better than 5% [25]. Furthermore, the uncertainty of the flow rate measured by the electromagnetic flow meter was investigated because it is well known that this device could not measure the flow rate (average pipe flow velocity) very accurately. The measured value of the initial pipe flow velocity varied ±2%. Then, the

examples of fast closing and opening of the EV at the downstream end of the pipeline were examined. This paper ends with the examination of the impact of the pipeline length on the water hammer head rise by the fast closing of hand-operated valves along the pipeline. All investigated cases with their description are summarized in the Table 1. Table 1. Summary of investigated cases Description Test A Test B Test C Test D Test E

Fast closing of EV Fast closing of EV Fast closing of EV Fast opening of EV Fast closing of HVs along pipeline

Initial/final pipe velocity [m/s] 2.07 2.01 2.05 2.01 2.00

3.1 Convergence and Stability of Numerical Model The numerical solution of the developed numerical code should satisfy the convergence and stability criteria. Convergence relates to the behaviour of the

Fig. 2. Numerical analysis for Test A: pr = 4 bar, v0 = 2.07 m/s, a = 1349 m/s, P = 2.74; a) N = {54, 108, 216} at D1, b) N = {216, 432, 864} at D1, c) N = {54, 108, 216} at D3, d) N = {216, 432, 864} at D3

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solution as Δx and Δt tends to zero while the stability is concerned with round-off error growth [2]. The influence of different numbers of computational reaches N = {54, 108, 216, 432, 864} is investigated. Fig. 2 shows the numerical results for the fast closing of EV (Test A) with initial flow velocity v0 = 2.07 m/s with severe cavitation. The water hammer wave speed used in simulations is a = 1349 m/s and the weighting factor used in numerical solution of Eq. (4) [1] and [26] is ψ = 1. The numerical results are consistent for higher number of reaches (Figs. 2b and d). For a smaller number of reaches, the numerical results are practically the same for the first three pressure pulses (Figs. 2a and c). After that, the discrepancies are obvious. Along the pipeline, a number of discrete vapour cavities occur as do distributed cavitation zones. The collapses of small cavities along the pipe produce high-frequency pressure peaks that are not repeatable in experiments nor in computations [26]. Some pressure spikes along the pipe occurred in different times as number of computational reaches increase. However, high frequency pressure peaks along the pipe do not significantly affect the main pressure pulses. Generally, the magnitude and timing of the main pressure pulses predicted by the developed numerical model converge as the number of reaches is increased.

values, numerical calculations are performed and compared with the results of measurements.

3.2 Sensitivity Analysis to Input Parameters An important feature of the numerical analysis is the sensitivity of numerical model results to input parameters. The influence of variations the wave speed and the flow rate will be investigated. The calculated pressure wave speed in the pipeline is:

K/ρ = 1402.7 m/s, (9) 1 + c1 ( KD / Ee )

in which, the dimensionless parameter that describes the effect of pipe constraint condition on the wave speed is c1 = 1.12 [1], the water bulk modulus of elasticity K = 2.18 GPa, Young´s modulus of elasticity of pipe material E = 200 GPa, and the water density ρ = 998.2 kg/m³. The measured value of the pressure wave speed is obtained from the measured time for a water hammer wave to travel between the closed valve (position of dynamic pressure transducer D1) and the position of the first nearest transducer D2; its value is a = L/t = 18.4/0.012957 = 1420 m/s (uncertainty: ±0.1%). The measured value of the pressure wave speed is then varied between ±5%, i.e. a–5% = 1349 m/s and a+5% = 1491 m/s. With these three wave speed

Fig. 3. Comparisons of heads at D1 for Test B; a) a = 1420 m/s, b) a-5% = 1349 m/s, and c) a+5% = 1491 m/s at v0 = 2.01 m/s

The experimental run with rapid EV closure has been selected for the analysis (initial pressure in HPR: pr = 4 bar; initial flow velocity in the system: v0 = 2.01 m/s; Test B). The initial flow for this run is turbulent with Reynolds number Re = 36,100. For

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the measured value of the wave speed a = 1420 m/s and the initial pipe flow velocity v0 = 2.01 m/s, the Ghidaoui et al. parameter is P = 2.97. An even number of pipeline reaches of N = 108 has been selected and the corresponding numerical time step is Δt = 3.59×10-4 s. Fig. 3 shows comparisons of heads at the downstream end valve (position D1). The EV measured closing time is tc = 0.016 s, which is much shorter than the water hammer wave reflection time 2L/a = 0.0764 s. The fast closing of the EV produces water hammer with liquid column separation [1] and [5]. The existence of a large vapour cavity at the valve is represented by a vapour pressure line. The maximum measured head at the EV (DPT D1) is HmaxD1 = 333.6 m. The maximum calculated head occurs when the first reflected wave arrives back to the EV and its value is HmaxD1 = {338.6; 351.5; 368.1} m for a = {1349, 1420, 1491} m/s, respectively. The absolute difference between the measured and the calculated values is {5.0; 17.9; 34.5} m for a = {1349, 1420, 1491} m/s, respectively. The results with the wave speed a–5% = 1349 m/s yield the best fit with the results of measurement for the first pressure pulse; therefore, the lowest value of the wave speed is a–5% = 1349 m/s is used this paper. The next step is to verify the influence of flow rate measured by the electromagnetic flow meter. The value of the initial pipeline flow velocity is varied ±2% i.e. v0–2% = 2.01 – 2% = 1.97 m/s and v0+2% = 2.01 + 2% = 2.05 m/s. The value of the pressure wave speed is a–5% = 1349 m/s and the numerical time step is Δt = 3.77×10-4 s. Fig. 4 presents comparisons of heads at the dynamic pressure positions D1 for different values of the initial velocity. The maximum calculated heads are HmaxD1 = {333.4; 338.6; 345.1} m for v0 = {1.97; 2.01; 2.05} m/s, respectively. The absolute differences between the measured and the calculated values are {0.2; 5.0; 11.5} m for v0 = {1.97; 2.01; 2.05} m/s, respectively. The maximum value of the calculated head with initial flow velocity reduced by 2% compared to the measured value coincides the best (Fig. 4a). Furthermore, the numerical results with v0–2% have less phase difference than the results with v0 and v0+2% when they are compared with the results of measurements. Thus, the measured initial flow velocity is decreased by 2%.

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Fig. 4. Comparisons of heads at D1 for Test B: a–5% = 1349 m/s; at a) v0–2% = 1.97 m/s, b) v0 = 2.01 m/s, and c) v0+2% = 2.05 m/s

3.3 Fast Closing and Opening of the Electro-Pneumatic Valve The fast closing and opening of the EV is used to validate the developed numerical model. In this section, two different experimental tests results are presented. The first experimental run represents fast closure of the EV at the initial pressure in the

Karadžić, U. – Bulatović, V. – Bergant, A.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 742-754

Fig. 5. Comparisons of heads at a) D1, b) D2, c) D3 and d) D4 for Test C: pr = 4 bar, v0 = 2.05 m/s, Δt = 3.77x10-4 s, P = 2.76

HPR of pr = 4 bar and the initial flow velocity in the system of v0 = 2.05 m/s (Test C). The flow for Test C is turbulent flow with Reynolds number Re = 36,900. The EV measured closing time is tc = 0.018 s. The second run represents the fast opening of the EV with the adjusted pressure in the HPR of pr = 4 bar and the final pipe velocity of vf = 2.01 m/s (Test D). Fig. 5 presents comparisons of heads at the dynamic pressure transducer positions D1, D2, D3 and D4 for Test C. The maximum measured heads at all dynamic pressure transducer positions occur during the first pressure pulse except D4 next to the HPR (Fig. 5d). At this position, the maximum head occurs at time t = 0.46 s as a short duration pressure pulse. The maximum measured values are Hmax = {341.7; 336.4; 357.7; 181.9} m for D1, D2, D3 and D4 positions, respectively. The corresponding head rise is ΔH = {314.6; 302.6; 316.8; 134.2} m. The maximum computed heads with the corresponding head rise are as follows: Hmax = {343.9; 338.7; 330.9;

236.4} m and ΔH = {319.2; 305.5; 290.1; 188.6} m for D1, D2, D3 and D4 transducer positions, respectively. The corresponding relative differences between the measured and the computed maximum head values are, Hmeas – Hcomp = {0.6; 0.7; 8; 23}%, respectively. The developed numerical model effectively determines the values of maximum head at positions of D1 and D2 (Figs. 5a and b); this is not the case at positions D3 and D4 (Figs. 5c and d). The maximum measured head occurs at position D3 (Fig. 5c). In the numerical model, the maximum head is at the valve (Fig. 5a). At the valve, there is an alternating growth and collapse of cavitation bubbles, which occur at a constant vapour pressure head (Fig. 5a). Measured and calculated duration of the first large cavity at the EV is tcav ={0.344; 0.335} s, respectively. Good agreement between the experimental and the computed results may also be observed along the pipeline at the transducer positions D2, D3 and D4. In cavitation regions along the pipeline, the collapse of a number of vapour bubbles cause small pressure

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Fig. 6. Calculated void fraction volume at a) D1, b) D2, c) D3 and d) D4 for Test C

fluctuations that are registered in the experimental run and simulated in numerical calculations. The calculated results of the void fraction volume for Test C are shown on Fig. 6. As expected, the maximum void fraction volume occurs at the dynamic pressure transducer position D1. Fig. 7 shows comparisons of heads at the dynamic pressure transducer positions D1, D2, D3 and D4 for Test D. The EV-measured opening time is to = 0.011 s. The results of measurements show a characteristic appearance of a high frequency pressure peak at the beginning of the valve opening that is not simulated by the numerical model (Fig. 7a). The peak may be attributed to FSI effects of the EV. Otherwise, the numerical model shows a good match with the results of measurement for the case of the valve opening. After the valve is opened, the head at the D1 drops to the minimum value (Hmin = {3.1 (measured); 1.5 (computed)} m) and reaches a new steady state without significant oscillations (Fig. 7a). In contrast, head fluctuations at the D2 and D3 positions, after the valve is opened, are much larger (Figs. 7b and c). The maximum measured head of Hmax = 53.4 m occurred 750

at the position D3 at the time t = 0.145 s and has a higher value than the initial system’s head. This is not the case in the numerical model, where the maximum head is lower than the initial one. The minimum measured head of Hmin = –2.7 m also occurred at position D3 at the time of t = 0.047 s. The head oscillations at position D4 next to the HPR is of minor importance (Fig. 7d). The cavitation does not occur in the considered case of the valve opening. The maximum measured head rise and drop have been observed at dynamic pressure transducer position D3 for both investigated cases of EV closing and opening. This may be attributed to FSI effects. These effects will be investigated by the authors in the near future. 3.4 Impact of the Pipeline Length on the Water Hammer Head Rise It is commonly known that the water hammer head rise, after rapid valve closure, depends on the initial flow velocity and pressure wave speed according the Joukowsky formulae. In this section, the impact of

Karadžić, U. – Bulatović, V. – Bergant, A.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 742-754

Fig. 7. Comparisons of heads at a) D1, b) D2, c) D3 and d) D4 for Test D: pr = 4 bar, vf = 2.01 m/s, Δt = 3.77×10–4 s, P = 2.82

the pipeline length on the water hammer head rise is investigated. Fast closures of the HVs along the pipeline are performed and with appropriate numerical calculations water hammer in the different system’s length is simulated. The hand valves downstream of the dynamic pressure transducers at the positions D1, D2, D3 and D4 have been fast closed in the separate experimental runs and noted as Test E. The initial flow velocity and the pressure in the HPR are the same for all tests investigated, v0 = 2.0 m/s and pr = 4 bar. The corresponding pipe length is L = {54.23; 35.83; 18.13; 1.74} m, and the hand valve closing time is tc = {0.13; 0.11; 0.1; 0.06} s. The HV closing time is nearly the same for D1, D2 and D3 positions. The pressure wave speed in all numerical calculation is adopted as a = 1349 m/s and the numerical time step is Δt = {3.77×10-4; 2.49×10-4; 1.2×10-4; 0.15×10-4} s, respectively. The water hammer wave reflection time is 2L/a = {0.080; 0.053; 0.027; 0.002} s and it is shorter than the HV closing time in all cases investigated. It means that when the water hammer pressure wave arrives back at the valve, the valve is still open and incomplete water hammer occurs. Fig. 8

shows comparisons of heads at the dynamic pressure transducer positions D1, D2, D3 and D4 for Test E. The maximum measured head in all cases occurs after the valve is closed, and its values are Hmax = {287.1; 293.2; 272.0; 71.0} m, respectively. The corresponding measured head rises are ΔH = {261.6; 260.2; 231.8; 25.0} m. The computed maximum head and the head rise are Hmax = {291.2; 289.8; 269.6; 71.0} m and ΔH = {261.4; 254.7; 228.6; 25.0} m. The largest increase in head occurs in the longest pipeline. However, based on the results presented here, a general conclusion cannot be drawn, because the closing time of the valves is not accurately measured but rather is read from the diagram of the pressure changes. To investigate whether the length of the pipeline has an impact on the head rise due to water hammer and how important this impact is, it is necessary that the valve closing time to be much shorter than the pressure wave reflection time. In this way, the full water hammer and head rise could be calculated using the Joukowsky equation and compared with results given by the measurements. In the considered case, using the Joukowsky equation, the calculated head rise is

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 742-754

Fig. 8. Comparisons of heads at a) D1, b) D2, c) D3 and d) D4 for Test E: pr = 4 bar, v0 = 2.0 m/s, a = 1349 m/s, P = 2.83

ΔH = 275 m, which is much higher than the measured and calculated values obtained for the valve closing time longer than the pressure wave reflection time. The numerical model shows good agreement with the results of measurements (Figs. 8a, b and c) except to some extent for the case of the hand valve closure at the position D4 (Fig. 8d), where the maximum head is calculated well but the shape differs. The fast closure of the HV close to the HPR causes a head increase and then oscillations that quickly damped out. However, further improvement of the numerical model is necessary in order to successfully simulate water hammer in very short pipelines. 4 CONCLUSIONS In this paper, an experimental apparatus for investigating water hammer and column separation in pipelines has been described in detail. Based on the value of the Ghidaoui et al. parameter P, it is concluded that the developed experimental setup is an unsteady friction dominated one. In-house 752

numerical code using a discrete gas cavity model and a convolution-based unsteady friction model has been developed. The numerical results have been compared with the results of measurements for the cases of fast closing and opening both the electro-pneumatic and the hand-operated downstream end valve. The impact of different numbers of computational reaches was first investigated, and examination of the computed results reveals numerically robust behaviour of the developed numerical model as the number of reaches increases. The influence of variations of pressure wave speed and the uncertainty of electromagnetic flow meter have been investigated, and it is concluded that pressure wave speed should be decreased by 5%, and the initial pipe velocity should be reduced by 2% compared with initially measured values. The numerical results show very good matches with the results of measurements for the case of fast closing and opening of the EV. The maximum measured head rise and drop have been observed at dynamic pressure transducer position D3 for both investigated cases. In the case of the valve opening,

Karadžić, U. – Bulatović, V. – Bergant, A.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, 742-754

the maximum measured head is higher than the initial one, which is not simulated by the numerical model. This may be attributed to FSI effects in the system. Future research may be seen in investigations of FSI impact of the EV as well as possible influence of the pipeline bends on the occurrence of the maximum head in the system. Due to the very good agreement between the computed and measured results and the robust numerical algorithm, the discrete gas cavity model using the convolution based unsteady friction term is recommended for engineering practice. 5 ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the Ministry of Science of Montenegro (MSM) and the Slovenian Research Agency (ARRS) conducted through the projects “Analysis of transient phenomena in hydraulic and aeromechanical systems” (MSM), “Investigation of water hammer effects in a test facility” (MSM, ARRS) and “Unsteady skin friction modelling in hydraulic piping systems” (ARRS). 6 NOMENCLATURE A a D E f g H K L mk , nk N P p Q Qin Qout Re t, t* tc to V v W x yk α Δt Δx

pipe area [m2], water hammer wave speed [m/s], pipe diameter [m], Young’s modulus of elasticity [Pa], friction factor [-], gravitational acceleration [m/s2], piezometric head (head) [m], water bulk modulus of elasticity [Pa], pipe length, length [m], exponential sum coefficients [-], number of computational reaches [-], Ghidaoui et al. parameter [-], pressure [Pa], discharge [m3/s], node upstream end discharge [m3/s], node downstream end discharge [m3/s], Reynolds number [-], time [s], valve closing time [s], valve opening time [s], volume [m3], velocity [m/s], weighting function [-], distance [m], component of the weighting function [-], gas void fraction [-], time step [s], reach length [m],

ν kinematic viscosity [m2/s], τ dimensionless time [-], ψ weighting factor [-]. Subscripts: f final g gas max maximum q quasi-steady u unsteady 0 steady state conditions Superscripts: * absolute pressure 7 REFERENCES [1] Wylie, E.B., Streeter, V.L. (1993). Fluid Transients in Systems. Prentice-Hall, Englewood Cliffs. [2] Chaudhry, M.H. (2014). Applied Hydraulic Transients. 3rd ed., Springer, New York, DOI:10.1007/978-1-46148538-4. [3] Bergant, A., Tijsseling, A.S., Vítkovský, J.P., Covas, D.I.C., Simpson, A.R., Lambert, M.F. (2008). Parameters affecting water hammer wave attenuation, shape and timing. Part 1: Mathematical tools. IAHR Journal of Hydraulic Research, vol. 46, no. 3, p. 373381, DOI:10.3826/jhr.2008.2848. [4] Bergant, A., Tijsseling, A.S., Vítkovský, J.P., Covas, D.I.C., Simpson, A.R., Lambert, M.F. (2008). Parameters affecting water hammer wave attenuation, shape and timing. Part 2: Case studies. IAHR Journal of Hydraulic Research, vol. 46, no. 3, p. 382-391, DOI:10.3826/jhr.2008.2847. [5] Bergant, A., Simpson, A.R., Tijsseling, A.S. (2006). Waterhammer with column separation: A historical review. Journal of Fluids and Structures, vol. 22, no. 2, p. 135-171, DOI:10.1016/j.jfluidstructs.2005.08.008. [6] Adamkowski, A., Lewandowski, M. (2012). Investigation of hydraulic transients in a pipeline with column separation. ASCE Journal of Hydraulic Engineering, vol. 138, no. 11, p. 935-944, DOI:10.1061/ (ASCE)HY.1943-7900.0000596. [7] Bergant, A., Kruisbrink, A., Arregui, F. (2012). Dynamic behavior of air valves in a large-scale pipeline apparatus. Strojniški vestnik - Journal of Mechanical Engineering, vol. 58, no. 4, p. 225-237, DOI:10.5545/ sv-jme.2011.032. [8] Bergant, A., Simpson, A.R. (1999). Pipeline column separation flow regimes. ASCE Journal of Hydraulic Engineering, vol. 125, no. 8, p. 835-848, DOI:10.1061/ (ASCE)0733-9429(1999)125:8(835). [9] Vardy, A.E. (1980). Unsteady flow: fact and friction. Proceedings of the 3rd International Conference on Pressure Surges, BHRA, Cantenbury, p. 15-26. [10] Brunone, B., Karney, B.W., Mecarelli, M., Ferrante, M. (2000). Velocity profiles and unsteady pipe friction in transient flow. Journal of Water Resources

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11 Vsebina

Vsebina Strojniški vestnik - Journal of Mechanical Engineering letnik 60, (2014), številka 11 Ljubljana, november 2014 ISSN 0039-2480 Izhaja mesečno

Razširjeni povzetki Boštjan Novak, Aleš Babnik, Janez Možina, Matija Jezeršek: Tri-dimenzionalni merilnik stopal z lasersko rotacijsko merilno glavo Emiliano Pipitone, Stefano Beccari, Marco Cammalleri, Giuseppe Genchi: Linearizacija diagrama pretoka šobe za vbrizgavanje plinastega goriva pri motorju z vžigalnimi svečkami na podlagi eksperimentalnega modela Andrzej Zbrowski, Krzysztof Matecki: Uporaba računalniške tomografije za analizo lis od brušenja in napak pod površino na obročih kotalnih ležajev Changyi Liu, Gui Wang, Matthew S Dargusch: Mehanika in dinamika procesov rezkanja po vijačnici Daniel Kozelj, Zoran Kapelan, Gorazd Novak, Franci Steinman: Preiskovanje apriornih porazdelitev parametrov v inverznem modeliranju hidravličnih modelov vodovodnih sistemov Nima Jafarzadeh Aghdam, Soran Hassanifard, Mir Mohammad Ettefagh, Arvin Nanvayesavojblaghi: Raziskava vpliva utrujanja na vibracijske lastnosti točkovnih tornih zvarov z eksperimentalno modalno analizo in MKE Uroš Karadžić, Vladimir Bulatović, Anton Bergant: Vodni udar in pretrganje kapljevinskega stebra inducirana z ventilom v preizkusni postaji Osebne vesti Doktorske disertacije, diplomske naloge

SI 133 SI 134 SI 135 SI 136 SI 137 SI 138 SI 139

SI 140

Prejeto v recenzijo: 2014-05-09 Prejeto popravljeno: 2014-07-09 Odobreno za objavo: 2014-07-23

Tri-dimenzionalni merilnik stopal z lasersko rotacijsko merilno glavo Boštjan Novak1 – Aleš Babnik2 – Janez Možina2 – Matija Jezeršek2,* 2Univerza

1Alpina d.o.o., Slovenija v Ljubljani, Fakulteta za strojništvo, Slovenija

Slabo prileganje obutve k stopalu je eden glavnih razlogov za bolečine v stopalih ter bolezni in poškodbe stopal. Ustrezne 3-dimezionalne meritve stopala zato omogočajo pridobivanje podatkov za izdelavo ustreznih oblik standardne obutve, prilagajanje obutve obliki stopala individualnega uporabnika, oblikovanje boljšega prileganja obutve pri masovni proizvodnji in določanje kupcu najbolj prilegajočega modela v posamezni trgovini. Slabosti obstoječih sistemov so predvsem v visoki ceni in njihovi kompleksnosti. Zato se večinoma uporabljajo le v raziskovalne in medicinske namene. V članku predstavljamo inovativni merilnik stopal, posebej razvit za merjenje stopal v prodajalnah z obutvijo in specializiranih zdravstvenih ambulantah. Merilnik odlikuje nizka cena izdelave, konstrukcija omogoča visoko stopnjo mobilnosti, poenostavljen in pospešen je postopek merjenja. Razviti merilnik stopal deluje na principu laserske večlinijske triangulacije. Najpomembnejši sestavni del merilnika je merilna glava, ki sestoji iz treh laserskih linijskih projektorjev in dveh kamer, ki sta simetrično postavljeni na vsako stran laserskih projektorjev. Merilna glava, ki je nameščena na merilno roko, se med izvedbo meritve zavrti okrog centra merilne plošče, na kateri stoji merjena oseba in s tem izmeri celotno obliko obeh stopal. Z uporabo dveh kamer in treh laserskih ravnin smo preprečili nastanek senčenja oz. zastiranja, ki se pojavi na notranjih straneh posameznega stopala, ko nasprotno stopalo bodisi zastre pogled posamezne kamere, bodisi prepreči popolno lasersko osvetlitev. Čas merjenja obeh stopal je 10 sekund. Po končani meritvi se izvede 3-D rekonstrukcija stopal in izmera osnovnih dimenzij, kot so dolžina in maksimalna širina, višina in obseg v sprednjem predelu stopala. Posebej razvit algoritem za izračun prileganja obutve stopalu omogoča individualno svetovanje kupcu, kateri model se najbolje prilega izmerjenemu stopalu. Evaluacijo merilnika smo opravili z izvedbo treh vrst meritev. V prvem primeru smo izvedli 10 meritev levega in desnega plastičnega stopala, v drugem primeru smo izvedli 10 meritev stopal ene osebe, medtem ko smo v zadnjem primeru izvedli primerjavo rezultatov meritev 40 različnih stopal (20 različnih oseb) po klasični metodi in z merilnikom glede na standard ISO 20685. Rezultati analiz testnih meritev so pokazali, da je standardni odklon meritev dolžine, širine in obsega plastičnih stopal manjši od 0,6 mm, medtem ko je standardni odklon meritev živih stopal do 60% večji. Razlog je povezan z deformacijo stopala, ki se pojavlja tekom različnih obremenitev med posamezno meritvijo. Primerjava meritev po klasični metodi in s 3-D merilnikom pokaže, da so dimenzije po klasični metodi sistematično manjše za približno 1,5 mm zaradi dotičnega načina merjenja. Možnost dodatne nadgradnje sistema vidimo predvsem v uvedbi meritve podplatne površine skupaj z merjenjem pritiskov stopala na podlago. Na osnovi tako pridobljenih podatkov bi lahko izdelovali tudi prilagojene notranjike oz. steljke. Ključne besede: 3D meritve stopala, večlinijska laserska triangulacija, dimenzije stopala, prileganje obutve

*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za strojništvo, Aškerčeva 6, 1000 Ljubljana, Slovenija, matija.jezersek@fs.uni-lj.si

SI 133

Prejeto v recenzijo: 2013-07-18 Prejeto popravljeno: 2014-03-25 Odobreno za objavo: 2014-07-08

Linearizacija diagrama pretoka šobe za vbrizgavanje plinastega goriva pri motorju z vžigalnimi svečkami na podlagi eksperimentalnega modela Emiliano Pipitone* – Stefano Beccari – Marco Cammalleri – Giuseppe Genchi Univerza v Palermu, Italija

Eksperimenti avtorjev na področju vbrizgavanja plina pri avtomobilskih motorjih so pokazali močno nelinearnost diagramov pretoka nekaterih šob za vbrizgavanje plina. Te nelinearnosti lahko vplivajo na nadzor razmernika goriva, ki se izvaja v elektronski krmilni napravi motorja, ter povzročijo nestabilne korekture vbrizgane mase goriva. Posledično se poveča poraba goriva, zaradi nizke učinkovitosti katalizatorja pri nestehiometričnem mešalnem razmerju pa se pojavijo tudi visoke emisije onesnaževalcev. Avtorji so v predhodnih raziskavah že preučili in reproducirali nelinearno vedenje šobe za vbrizgavanje plina. Postavili so matematični model za vrednotenje kompleksnega gibanja igle med postopkom vbrizgavanja, pri tem pa so se osredotočili predvsem na fazi odpiranja in zapiranja, ki sta ključni pri uvajanju nelinearnosti. Kljub temu, da znanstvena literatura dobro pokriva modeliranje vbrizgavanja goriva, pa ni bilo mogoče najti prispevkov, ki bi se ukvarjali z nelinearnostmi zaradi odbojev igle med fazo odpiranja in zapiranja šobe za vbrizgavanje plina. Postavljeni model je bil validiran z eksperimentalnimi podatki in dosežena je bila visoka stopnja ujemanja, saj model točno replicira nelinearnosti iz eksperimentalnega diagrama. Točnost vrednotenja modela je bila v vseh primerih primerljiva z negotovostjo testnih meritev, ki je povezana z značilnim raztrosom meritev vbrizgane mase okrog srednje vrednosti. Avtorji so se na podlagi svojih izkušenj v tem delu odločili uporabiti model za preučitev predloga primerne strategije vbrizgavanja, ki odpravlja nelinearno vedenje realne šobe za vbrizgavanje plina, linearizira diagram pretoka ter razširja uporabnost tudi na območje krajših časov vbrizgavanja. Analiza gibanja igle ob upoštevanju ohranitve energije je privedla do opredelitve primerne ciljne funkcije, ki je avtorje pripeljala do možne rešitve problema. Glavna prednost rešitve je v enostavni implementaciji v današnjih elektronskih krmilnih napravah motorja, ki ni povezana z nobenimi spremembami strojne opreme ali z dodatnimi stroški: s prekinitvijo vbrizgalnega impulza se energija, ki se dovaja igli, modulira tako, da ne prihaja do odbojev. Strategija prekinitve impulzov je bila uporabljena v matematičnem modelu, ki je z opredelitvijo minimuma ustrezno opredeljene ciljne funkcije omogočil določitev optimalnih vrednosti parametrov prekinitve in s tem učinkovito linearizacijo diagrama pretoka simulirane šobe za vbrizgavanje. Optimalna strategija vbrizgavanja, določena s simulacijami, je bila nato še eksperimentalno preizkušena. Dokazano je bilo, da je z njo mogoče linearizirati diagram pretoka realne šobe za vbrizgavanje. Avtorji so v naslednjem koraku opravili vrsto eksperimentov na ustrezno opremljenem preizkuševališču. Strategija prekinitve impulzov je bila udejanjena z zrakom na realni šobi za vbrizgavanje goriva, začetni parametri pa so bili nastavljeni na optimalne vrednosti, ki izhajajo iz simulacij. Diagram pretoka vbrizgalne šobe je bil občutno izboljšan z eksperimentalno optimizacijo: odstopanje od linearne karakteristike se je v primerjavi z izhodiščno karakteristiko zmanjšalo za dve tretjini. Rezultat je po mnenju avtorjev nedvomno dober, še posebej ob upoštevanju merilnega raztrosa vbrizgane mase. Predlagano strategijo vbrizgavanja bi bilo torej mogoče učinkovito uporabiti za linearizacijo vedenja šobe za vbrizgavanje goriva in s tem boljši nadzor nad razmernikom goriva v širšem delu diagrama pretoka, s tem pa bi se zmanjšala emisija onesnaževalcev in izboljšala poraba goriva. Ključne besede: modeliranje vbrizgavanja goriva, motor z notranjim zgorevanjem, motor z vžigalnimi svečkami, plinasta goriva, optimalna strategija vbrizgavanja

SI 134

*Naslov avtorja za dopisovanje: Univerza v Palermu, Viale delle Scienze, Palermo, Italija, emiliano.pipitone@unipa.it

Prejeto v recenzijo: 2014-03-20 Prejeto popravljeno: 2014-06-27 Odobreno za objavo: 2014-07-24

Uporaba računalniške tomografije za analizo brusilnih lis in napak pod površino na obročih kotalnih ležajev Andrzej Zbrowski* – Krzysztof Matecki Nacionalni raziskovalni institut, Institut za trajnostne tehnologije, Poljska

Članek predstavlja rezultate preskusov obročev kotalnih ležajev, pri katerih je bila med kontrolo na proizvodni liniji ugotovljena prisotnost nesprejemljivih površinskih napak v obliki brusilnih lis. Napake so bile ugotovljene po brušenju površine testnih predmetov. Zato je bila postavljena hipoteza, da imajo obroči kotalnih ležajev površinske diskontinuitete ali napake pod brušeno površino, spremembe v obliki lis pa se pojavijo zaradi odpiranja teh napak in kontaminacije v procesu površinske obdelave. Obroči, ki so bili zaradi ugotovljenih napak med kontrolo odstranjeni iz proizvodnega procesa, so bili pregledani s pomočjo računalniške tomografije (CT). Pri testih je bila uporabljena programska oprema Phoenix datos/x in VG studio max. Preiskave so bile usmerjene v potrjevanje neposredne povezave med prisotnostjo lis od brušenja na obdelanih površinah ter obstojem napak pod površino materiala, kot so votline, razpoke, diskontinuitete itd. Avtorji so dokazali, da je prisotnost površinskih napak v obliki brusilnih lis tesno povezana s prisotnostjo napak v materialu pod površino, ki so bile razkrite po metodi CT. Obsevanje z več strani in s tem večkratna rekonstrukcija preskušanih obročev omogoča preučitev vrste, velikosti, oblike ter orientacije napak na delovni površini obročev kotalnih ležajev. Preskusi so bili opravljeni na dveh notranjih obročih kotalnih ležajev z lisasto površino. Preskusi s CT in rekonstrukcija po metodi vzvratnega inženirstva so pokazali, da so na obroču A napake v obliki razpok ali votlin. So nepravilnih oblik in orientirane pravokotno glede na notranjo površino obroča. Napaka 2 nepravilne oblike je prav tako v obliki razpoke ali sploščene votline, iz katere se izteza razpoka proti notranji površini obroča. Ta napaka je manjša in manj vidna od Napake 1. Napaka v obroču B je v obliki votline z diskontinuitetami, ki so prav tako usmerjene proti notranji površini obroča. Ugotovljene napake so vidne tudi na rentgenogramih preskusnih obročev. Preskusi CT obročev z vidnimi površinskimi lisami so potrdili prisotnost diskontinuitet v materialu pod njimi. Napake se razkrijejo po metalurških procesih, ni pa jih mogoče opaziti z golim očesom. Obroči s takšnimi napakami niso primerni za uporabo. Metoda CT je precej zamudna, zato ni primerna za kontrolo kakovosti obročev kotalnih ležajev v sistemih množične proizvodnje. Kljub temu pa je nepogrešljiva za karakterizacijo ugotovljenih napak zunaj proizvodne linije. Še posebej uporabna je lahko v primerjalnih preskusih različnih tehnik defektoskopije pri masovni proizvodnji. Testi CT so pokazali, da so brusilne lise kot očitne površinske spremembe nedvoumen kriterij za klasifikacijo obročev kotalnih ležajev, ki se kontrolirajo s strojnim vidom. Opažena lisa na površini torej pomeni ugotovljeno napako pod površino. To je zelo pomembno, saj so sistemi strojnega vida za razliko od metod CT zaradi svoje hitrosti primerni za 100-odstotno kontrolo kakovosti. Avtorji predlagajo povezavo hitrih tehnik strojnega vida na liniji s tomografsko preiskavo zunaj linije za tiste obroče, na katerih se ugotovijo lise od brušenja. Organizacije bodo tako uveljavile 100-odstotno kontrolo in nadzor nad proizvodnim procesom s popolno kvalitativno in kvantitativno klasifikacijo notranjih napak. Ključne besede: obroči kotalnih ležajev, napake v materialu, neporušne preiskave, računalniška tomografija

*Naslov avtorja za dopisovanje: Nacionalni raziskovalni institut, Institut za trajnostne tehnologije, 26-600 Radom, Pułaskiego 6/10 cesta, Poljska, andrzej.zbrowski@itee.radom.pl

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Prejeto v recenzijo: 2013-12-04 Prejeto popravljeno: 2014-05-09 Odobreno za objavo: 2014-06-16

Mehanika in dinamika procesov rezkanja po vijačnici Liu, C. – Wang, G. – Dargusch, M.S. Changyi Liu1,* – Gui Wang2 – Matthew S. Dargusch2

1 Univerza

za aeronavtiko in astronavtiko v Nanjingu, Oddelek za strojništvo, Kitajska v Queenslandu, Oddelek za strojništvo in rudarstvo, Avstralija

2 Univerza

Postopki rezkanja po vijačnici se uporabljajo za ustvarjanje ali širjenje izvrtin s pomočjo rezkalnega orodja, ki se podaja proti obdelovancu po poti v obliki vijačnice. Pomembni dejavniki pri izbiri obdelovalnih postopkov so predvsem natančnost, učinkovitost in stroški. Preučevanje stabilnosti obdelave in drdranja je pomembno za načrtovanje procesa. V zadnjem času je bilo objavljenih nekaj analiz rezalnih sil pri procesu rezkanja po vijačnici. Čeprav so bili raziskani mehanizmi in dinamika posebnih operacij z aksialnim podajanjem, kot sta npr. potopno rezkanje in vrtanje, oz. s tangencialnim podajanjem, kot je npr. krožno rezkanje, pa še ni bil opredeljen model dinamike in drdranja pri rezkanju po vijačnici. Cilj tega članka je torej preučitev mehanike in dinamike procesov rezkanja po vijačnici. Raziskava na podlagi modela rezalnih sil, ki vključuje interakcije na stranskih in čelnih rezalnih robovih med rezkanjem po vijačnici, je bila osredotočena na model dinamike obdelovalnega procesa in probleme stabilnosti oz. drdranja. Rezalne sile na stranskih in čelnih rezalnih robovih vzdolž vijačne poti podajanja so bile analitično modelirane ob upoštevanju tangencialnega in aksialnega gibanja orodja. Vključena je tudi dvojna periodičnost zaradi vrtenja vretena in periode podajanja rezkalnega orodja po vijačnici. Model dinamike obdelovalnega procesa in stabilnostni problem drdranja sta bila razdeljena na dva dela: kritično aksialno globino reza in kritično radialno globino reza. Kritični aksialna in radialna globina reza se rešujeta posebej. Za razrešitev kritične aksialne in radialne globine reza je bila uporabljena regenerativna teorija drdranja in kriteriji stabilnosti na podlagi karakteristične funkcije dinamike v frekvenčni domeni. Modelirana je bila tudi dinamična debelina odrezkov, ki jih ustvarjajo stranski in čelni rezalni robovi, ob upoštevanju podajanja po vijačnici. Dejanske dinamične sile predstavljajo povratne informacije za krmiljenje obdelovalnega stroja. Določeni sta kritična aksialna in radialna globina reza v odvisnosti od vrtilne frekvence vretena. Opravljeni so bili tudi eksperimenti obdelave s podajanjem po vijačnici brez drdranja, pri čemer so bili parametri obdelave znotraj določenih stabilnostnih omejitev. Pojav drdranja je bil opazovan z merjenjem rezalnih sil, hrapavosti obdelane površine, okroglosti in cilindričnosti obdelanih izvrtin. Pri eksperimentih ni prišlo do drdranja med rezkarjem in obdelovancem. Amplituda eksperimentalno določenih rezalnih sil je v primerjavi s simuliranimi stabilnimi rezalnimi silami znotraj 10-odstotne napake. Vpliva sil ni bilo mogoče zaznati. Iz rezultatov eksperimentalnega postopka ter meritev natančnosti obdelave in rezalnih sil je mogoče sklepati, da podani parametri obdelave z rezkanjem po vijačnici lahko zagotovijo zahtevano natančnost obdelave. Le-ta bo tedaj brez drdranja znotraj stabilnostnih omejitev, ki jih napovedujeta naša analiza in model. V članku je predstavljen model rezalnih sil, dinamike in stabilnostnih omejitev za drdranje pri postopkih rezkanja po vijačnici. Dinamika sistema za rezkanje po vijačnici je bila modelirana kot vibracijski sistem s štirimi prostostnimi stopnjami. Sistem s štirimi prostostnimi stopnjami je bil razstavljen v sistem z dvema prostostnima stopnjama, tako da je bilo mogoče uporabiti regenerativni pristop k napovedovanju drdranja. Kritični aksialna in radialna globina reza sta bili razrešeni ločeno. Model rezalnih sil je uporaben za napovedovanje rezalnih sil na stranskih in čelnih rezalnih robovih. Model dinamike omogoča izbiro parametrov postopka, vključno z aksialno globino reza, radialno globino reza in vrtilno frekvenco vretena, v območju brez drdranja pri rezkanju po vijačnici. Raziskava podaja tudi teoretično ogrodje za načrtovanje postopkov rezkanja po vijačnici in relativnega rezkanja, kakor tudi za praktično uporabo postopkov rezkanja po vijačnici. Ključne besede: odrezavanje, modeliranje in načrtovanje procesa, rezkanje po vijačnici, rezalne sile, dinamika rezanja, drdranje

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*Naslov avtorja za dopisovanje: Univerza za aeronavtiko in astronavtiko v Nanjingu, Oddelek za strojništvo, 210016, Nanjing, Kitajska, liuchangyi@nuaa.edu.cn

Prejeto v recenzijo: 2014-02-08 Prejeto popravljeno: 2014-04-02 Odobreno za objavo: 2014-04-04

Preiskovanje apriornih porazdelitev parametrov v inverznem modeliranju hidravličnih modelov vodovodnih sistemov Daniel Kozelj1,* – Zoran Kapelan2 – Gorazd Novak3 – Franci Steinman1 1 Univerza

2 Univerza

v Ljubljani, , Fakulteta za gradbeništvo in geodezijo, Slovenija v Exeterju, Visoka šola za inženirstvo in računalništvo, Velika Britanija 3 Inštitut za hidravlične raziskave, Slovenija

Inverzno modeliranje oziroma umerjanje je recipročni proces modeliranja naravnih procesov, ki se osredotoča na ocenjevanje modelnih parametrov, katerih vrednosti niso neposredno merljive in jih je zato treba določiti s posrednim pristopom. Modeliranje vodovodnega sistema (VS) zahteva, da se vzpostavljeni matematični opis realnega sistema in njegove rešitve (tj. modelne napovedi) z zadostno gotovostjo skladajo z dejanskim odzivom delovanja realnega sistema. Izvrednoteni hidravlični koeficienti hrapavosti cevovodov imajo razmeroma veliko negotovost, saj so njihove vrednosti odvisne od tokovnih razmer pri obratovanju in ne od tehnične hrapavosti cevi, takšne negotovosti pa lahko močno vplivajo na modelne napovedi (npr. pretoka, pritiskov ipd.). Ocena negotovosti parametrov in modelnih napovedi je bistveni del procesa modeliranja, s katerim se zagotovi zaupanje v rezultate modeliranja. Za analize je uporabljeno Bayesovo sklepanje, pri katerem so parametri podani kot verjetnostne spremenljivke z ustreznimi funkcijami gostote verjetnosti (pdf). Pristop Bayesovega sklepanja ima nekaj prednosti v primerjavi z običajnimi metodami umerjanja VS: uporablja verjetnostno opredelitev predhodnih informacij o vrednostih parametrov, omogoča pridobitev skupne in robnih posteriornih pdf ter ne zahteva numeričnega izračuna odvodov funkcij. Pri inverznem modeliranju z Bayesovim pristopom se uporabljajo metode Monte Carlo z Markovskimi verigami (MCMC), ki omogočajo neposredno določitev tako najboljših vrednosti parametrov θ, kakor tudi njihove negotovosti in negotovosti modelnih napovedi Y. Uporaba Bayesovega pristopa omogoča združitev predhodnih informacij o parametrih (tj. apriornih porazdelitev p(θ)) in informacij, ki so pridobljene iz opazovanih parametrov oz. stanja sistema, Ŷ (tj. verjetje L(θ|Ŷ)). Z združitvijo omenjenih verjetnostnih modelov je z uporabo MCMC algoritmov mogoče določiti robne posteriorne porazdelitve parametrov p(θ|Ŷ). Izboljšanje ocene negotovosti parametrov in modelnih napovedi je bilo doseženo z vključitvijo apriornih porazdelitev parametrov p(θ) in uporabe posplošene funkcije verjetja L(θ|Ŷ) v analize. Uporabljene so bile različne sheme vzorčenja parametrov, ki je bilo opravljeno na podlagi treh predpostavljenih apriornih porazdelitev vrednosti koeficientov hrapavosti, ki so: zvezna enakomerna, normalna in gama porazdelitev. V tem prispevku so prikazani rezultati MCMC metode, poimenovane »DREAM(ZS)«, ki je diferencirano evolucijsko prilagodljiv »Metropolis« algoritem in vključuje vzorčenje iz preteklih stanj ter metodo »Snooker update«. Za določanje prednosti in slabosti izbora posameznih apriornih porazdelitev je bila izdelana metodologija in preverjena tako na hipotetičnem VS, kakor tudi v obratovalnih pogojih stvarnega VS. Rezultati kažejo, da je stopnja določljivosti (oz. občutljivosti) parametrov ena najpomembnejših lastnosti, ki vpliva na ustreznost izbire tipa apriorne porazdelitve. Poleg navedenega je bilo z uporabo posplošene funkcije verjetja (GL) dokazano, da pogreški umerjanja niso vedno porazdeljeni po normalni porazdelitvi in da so heteroskedastični (varianca ni konstantna), kar sta ravno nasprotni ključni lastnosti, ki se običajno uporabljata pri običajnih metodah določevanja pogreškov. Ta pojav je bil opažen tako pri hipotetičnem, kot pri stvarnem modelu VS. Rezultati raziskav kažejo, da skrbno zbiranje predhodnih informacij o parametrih (tj. koeficientih hrapavosti cevovodov) in uporaba ustreznejšega statističnega modela pogreškov bistveno izboljšajo določljivost parametrov v procesu inverznega modeliranja, kar zmanjšuje tako stopnjo njihove negotovosti, kakor tudi območje negotovosti modelnih napovedi. Opredelitev ustrezne apriorne porazdelitve parametrov je težavna, zato je izdelan pristop, po katerem je izbira apriorne porazdelitve temelji na analizi določljivosti (tj. občutljivosti) parametrov. Izkazalo se je, da natančnejša ponazoritev apriornih porazdelitev parametrov ustrezno zmanjša razpone negotovosti njihovih robnih posteriornih porazdelitev. Pomembno je, da se s tem zagotovi večje zaupanje v vrednosti optimiziranih / umerjenih parametrov, z uporabo predstavljenega pristopa pa so uspešno določene robne posteriorne porazdelitve parametrov, kar zmanjša tako negotovost umerjenih parametrov, kakor tudi modelnih napovedi. Ključne besede: Bayesovo sklepanje, umerjanje, posplošena funkcija verjetja, metoda Monte Carlo z Markovskimi verigami, diferencirano evolucijsko prilagodljiv Metropolis algoritem, cevovodni sistemi, hidravlika, vodovodni sistemi *Naslov avtorja za dopisovanje: Univerza v Ljubljani, , Fakulteta za gradbeništvo in geodezijo, Jamova cesta 2, 1000 Ljubljana, Slovenija, daniel.kozelj@fgg.uni-lj.si

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Prejeto v recenzijo: 2013-07-21 Prejeto popravljeno: 2013-12-04 Odobreno za objavo: 2014-01-17

Raziskava vpliva utrujanja na vibracijske lastnosti točkovnih tornih zvarov z eksperimentalno modalno analizo in MKE

Točkovno torno varjenje z gnetenjem (FSSW) je vsestranski in sodoben varilski postopek, ki se uporablja v različnih proizvodnih procesih, npr. v avtomobilski, letalski in vesoljski industriji. Glavni namen tega članka je pridobiti boljše razumevanje korelacije med lastnimi frekvencami in iniciacijo utrujenostnih razpok pri zvarih FSSW, narejenih s štirimi različnimi nabori parametrov varilskega procesa. Raziskava je bila opravljena z namenom zbiranja podatkov o enem najobetavnejših postopkov varjenja – FSSW. Aluminijasta pločevina 7075-T6 je bila varjena s štirimi različnimi nabori procesnih parametrov FSSW. Na preizkuševališču so bili nato opravljeni utrujenostni preizkusi za pripravo preizkušancev v različnih fazah utrujanja. Za vsak preizkušanec z različnim deležem izpolnitve utrujenostne trajnostne dobe je bila opravljena eksperimentalna modalna analiza za ugotavljanje lastnih frekvenc. Rezultati modalne analize za vsako vrsto zvara FSSW so prikazani v obliki diagrama odvisnosti odstotnega deleža zmanjšanja frekvence od odstotka trajnostnih poškodb. Narejen je bil tudi model po metodi končnih elementov v paketu ABAQUS, rezultati pa so bili primerjani z rezultati eksperimentalnih preizkusov. Ugotovljeno je bilo dobro ujemanje. Ena glavnih ugotovitev raziskave je, da vpliv utrujenostnih poškodb na frekvenčni odziv štirih vrst zvarov FSSW ni neposredno povezan s frekvenčno domeno. Z drugimi besedami: eksperimenti so pokazali, da je vpliv utrujenostnih poškodb v nekaterih domenah nizkih frekvenc večji kot v domeni visokih frekvenc, to pa potrjujejo tudi rezultati modela po metodi končnih elementov. Pomembna je tudi ugotovitev, da se frekvenca s povečevanjem ravni utrujenostnih poškodb jasno zmanjšuje v vsakem frekvenčnem pasu. Članek bo možno dopolniti z obravnavo preizkušancev z več zvarnimi točkami v različnih konfiguracijah. Delo s preizkušanci realnih dimenzij, kot se pojavljajo v industriji, bi dalo dragocene informacije in boljši pregled nad utrujenostno trajnostno dobo zvarov FSSW. Prihodnje raziskave bi lahko obravnavale tudi druge lahke zlitine aluminija oz. ostalih kovin. Do manjših razlik med eksperimentalnimi in numeričnimi rezultati pride zato, ker ni mogoče slediti rasti razpok v vsakem koraku utrujenostnega preizkusa. Za natančnejše rezultate se lahko uporabi posebna fotografska oprema, ki omogoča natančno določanje velikosti razpok v vsakem koraku utrujenostnega preizkusa. Točkovno torno varjenje z gnetenjem je tehnologija na pohodu v avtomobilski, letalski in vesoljski industriji. Članek bo uporaben za razvojnike iz teh industrij, saj podaja boljši pregled nad utrujenostno trajnostno dobo zvarov, izdelanih s tem postopkom. Gre za enega redkih člankov, ki raziskujejo frekvenčne in vibracijske lastnosti predmetov, varjenih po postopku točkovnega tornega varjenja z gnetenjem. Ključne besede: točkovno torno varjenje, utrujenostne poškodbe, frekvenčni odziv, eksperimentalna modalna analiza, modeliranje s končnimi elementi, vibracijske lastnosti

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*Naslov avtorja za dopisovanje: Univerza v Tabrizu, Fakulteta za strojništvo, 29th Bahman Blvd., Tabriz, Iran, n.j.aghdam91@ms.tabrizu.ac.ir

Prejeto v recenzijo: 2014-04-16 Prejeto popravljeno: 2014-08-14 Odobreno za objavo: 2014-09-04

Vodni udar in pretrganje kapljevinskega stebra inducirana z ventilom v preizkusni postaji Uroš Karadžić1,* – Vladimir Bulatović1 – Anton Bergant2 1 Univerza

Črne gore, Fakulteta za strojništvo, Črna gora 2 Litostroj Power d.o.o., Slovenija

Članek obravnava vodni udar in pretrganje kapljevinskega stebra inducirana z ventilom v preizkusni postaji postavljeni v laboratoriju Univerze Črne gore, Podgorica. Glavni cilj raziskave je bil verifikacija in validacija diskretnega plinskega kavitacijskega modela z upoštevanjem nestalnega kapljevinskega trenja za različne pretočne pogoje. Osnove modela so bile postavljene v SV-JME v letu 2005. Model bazira na enčbah neustaljenega kapljevinskega toka v ceveh. Transformacija postavljenih parcialnih diferencialnih enačb hiperboličnega tipa z uporabo metode karakteristik tvori osnovo algoritma za vodni udar. V deltoidno mrežo metode karakteristik je vgrajen kovolucijski model neustaljenega stenskega trenja z uporabo zmogljivih računalniških orodij. Vgradnja plinskih kavitacij v numerična vozlišča da diskretni plinski kavitacijski model. Eksperimentalna postaja je sestavljena iz horizontalnega cevovoda, ki je vgrajen med gorvodni tlačni kotel in dolvodni ventil z iztokom v atmosfero. Dolžina jeklenega cevovoda je 54,23 m, notranji premer je 18 mm, debelina stene cevi 2 mm. Tlačne spremembe zasledujemo s pomočjo štirih hitro odzivnih zaznaval, ki so vgrajeni vzdolž cevovoda. Prehodni pojav je induciran z zapiranjem ali odpiranjem kroglastega zasuna dolvodno, vzdolž cevi in gorvodno. Na ta način lahko raziskujemo prehode v cevovodih različnih dolžin. Diskretni plinski model smo verificirali z metodo nabora cevnih odsekov v pasu med 54 in 864 odseki. Na veliki skali rezultati izračuna konvergirajo s povečanjem števila cevnih odsekov. Na mali skali nekateri visokofrekvenčni pulzi niso ponovljivi in tudi ne vplivajo na glavne tlačne utripe na veliki skali, kar je pomembno z vidika načrtovanja cevnih sistemov. Ta pojav zaznamo v področjih z blago kavitacijo vzdolž cevovoda kot posledico popisa kontinuirane kavitacije z različnim številom cevnih odsekov. Tudi meritve potrjujejo, da prehodne kavitacije vzdolž cevovoda niso homogene. Prehodna kavitacija se pojavi kot krajevna kavitacija z velikim kavitacijskim razmernikom (kavitacija ob ventilu, kavitacija pri srečanju valovnih front) ali kot nepretrgan kavitacijski tok z majhnim kavitacijskim razmernikom oziroma kombinacija obeh. Verifikaciji razvitega modela sledi validacija le tega s pomočjo primerjave rezultatov meritev in izračuna. Rezultate smo primerjali za širok nabor pretočnih pogojev. Pri tem smo upoštevali hitirost širjenja tlačnih valov in začetno pretočno hitrost dobljeni na osnovi občutljivostne analize vhodnih podatkov. Primerjali smo rezultate izračuna in meritev za hitro odpiranje in zapiranje dolvodnega ventila. Ventil smo zapirali s pomočjo elektropnevmatičnega aktuatorja in ročno. Ugotovili smo, da aktuator vzbudi blag sekundarni tlačni pulz zaradi interakcije med trdnino in kapljevino. Ta pojav bomo podrobneje raziskali v bližnji prihodnosti. Na osnovi Ghidauijevega števila in primerjave rezultatov izračuna in meritev sklepamo, da je vpliv neustaljenega trenja na prehodni proces v obravnavani preizkusni postaji znaten in ga ne smemo zanemariti. Nato smo numerično in eksperimentalno pokazali efekt dolžine cevovoda na intenziteto vodnega udara in prehodnega kavitacijskega toka kar je novost v literaturi. Raziskali smo razmere v cevovodih dolžin 54,23, 35,83, 18,13 in 1,74 m. Daljši je cevovod, bolj je razvita prehodna kavitacija in s tem tudi povečane obremenitve na stene cevovoda. To velja tako za nizkotlačne (kolaps cevi) kot za visokotlačne obremenitve (porušitev cevi). Rezultati izračuna in meritev se dobro ujemajo za širok nabor pretočnih pogojev zato postavljeni diskretni plinski kavitacijski model priporočamo za inženirsko uporabo. Ključne besede: cevni sistemi, preizkusna postaja, ventil, vodni udar, prekinitev kapljevinskega stebra, neustaljeno stensko trenje

*Naslov avtorja za dopisovanje: Univerza Črne gore, Fakulteta za strojništvo, Dzordza Vasingtona nn, 81000 Podgorica, Črna gora, uros.karadzic@ac.me

SI 139

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, SI 140-142 Osebne objave

Doktorske disertacije, diplomske naloge

DOKTORSKE DISERTACIJE Na Fakulteti za strojništvo Univerze v Ljubljani je obranil svojo doktorsko disertacijo: ● dne 22. oktobra 2014 Klemen RUPNIK z naslovom: »Razvoj termičnega merilnika masnega toka« (mentor: izr. prof. dr. Ivan Bajsić, somentor: doc. dr. Jože Kutin); Delo predstavlja rezultate raziskav in razvoja nove izvedbe potopnega termičnega merilnika masnega toka, ki ima, poleg merjenja masnega toka, tudi zmogljivost identifikacije vrste plina iz definiranega nabora plinov z znanimi elementnimi sestavami in korekcije merilne značilnice. Na osnovi enorazsežnega matematičnega modela termičnega zaznavala v potopnem termičnem merilniku so podana fizikalna izhodišča merilne metode za identifikacijo vrste plina ter možnosti in zahteve za njeno izvedbo. Za analizo prenosa toplote v termičnem zaznavalu in študij vplivov na merilno značilnico je bil izdelan ter uporabljen dvorazsežni matematični model. Z namenom eksperimentalne validacije predlagane merilne metode za identifikacijo vrste plina smo razvili inovativni potopni termični merilnik, ki vsebuje termični zaznavali z okroglim in kvadratnim prečnim prerezom, in ga umerili za pet različnih plinov z znanimi elementnimi sestavami. Če sta uporabljeni merilni značilnici za neustrezen plin, se bosta v splošnem masna tokova, izmerjena s termičnima zaznavaloma, razlikovala. Na osnovi analize izhodnih merilnih signalov, merilnih značilnic in definirane kriterijske funkcije se lahko identificira vrsta merjenega plina. * Na Fakulteti za strojništvo Univerze v Mariboru je obranil svojo doktorsko disertacijo: ● dne 3. oktobra 2014 Peter GSELMAN z naslovom: »Defekti PVD-prevlek in njihov vpliv na fizikalno-kemijske lastnosti sistema prevleka/ podlaga« (mentor: prof. dr. Franc Zupanič); Trde zaščitne prevleke, izdelane po postopku fizikalnega nanašanja iz parne faze (PVD – angl. physical vapor deposition), so se izkazale za enega izmed najučinkovitejših načinov povečanja produktivnosti odrezovalnih postopkov, vendar njihov potencial še ni popolnoma izkoriščen. Med nanašanjem v njih nastajajo defekti, ki poslabšajo SI 140

oprijemljivost prevlek na podlago, zmanjšujejo korozijsko obstojnost itd. Kakovost trdih prevlek izboljšamo, če zmanjšamo koncentracijo defektov. To lahko naredimo le, če poznamo mehanizme njihovega nastanka. V disertaciji so opisani mehanizmi nastanka defektov v posameznih stopnjah priprave prevlek in njihov vpliv na fizikalnokemijske lastnosti sistema prevleka/podlaga. Defekti so bili analizirani v nanoplastni prevleki TiAlN/CrN, ki je bila nanesena na štiri vrste jeklenih podlag (ASP30, M2, D2 in 316L) v naprševalniku s štirimi neuravnoteženimi magnetronskimi izviri. Za preučevanje izvorov, mehanizma nastanka in vpliva defektov PVD-prevlek na fizikalno-kemijske lastnosti sistema prevleka/podlaga so bile uporabljene naslednje tehnike: optični mikroskop (OM), mikroskop na atomsko silo (AFM), vrstični elektronski mikroskop (SEM), fokusiran ionski curek (FIB), presevni elektronski mikroskop (TEM), energijsko-disperzijska spektroskopija (EDS), tribometer, 3D-profilometer in 3D-rekonstrucija iz SEM-posnetkov. Ugotovljeno je bilo, da defekti v prevleki nastanejo predvsem zaradi geometrijskega senčenja, ki ga povzročajo topografske nepravilnosti na površini podlag, delci, ki so po čiščenju ostali na površini, ter delci, ki prispejo na površino podlag med pripravo prevlek (segrevanje, ionsko jedkanje, naprševanje). Kali defektov predstavljajo tudi sulfidni in oksidni nekovinski vključki. Na mestih sulfidnih vključkov v prevleki nastajajo kraterji in pore ne glede na vrsto jeklenih podlag in način jedkanja. Na intenzivno jedkanih oksidnih vključkih v prevleki nastanejo pore, medtem ko na blago jedkanih oksidnih vključkih prevleka raste nemoteno. Najpogostejše topografske nepravilnosti na površini trdih prevlek so nodularni defekti. Ti imajo tudi največji vpliv na povečanje površinske hrapavosti prevleke. Ugotovljeno je bilo tudi, da med tribološkim testom (angl. pin-on-disk) nastanejo prve poškodbe prevleke na mestih nodularnih defektov. Ti defekti so v začetni fazi tribološkega testa glavni vir abrazivnih delcev, ki povzročijo začetek obrabe prevleke. Izkazalo se je, da korozijsko obstojnost sistema prevleka/podlaga poslabšajo le tisti defekti, ki omogočajo elektrolitu prosto pot do podlage;

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, SI 140-142

DIPLOMSKE NALOGE Na Fakulteti za strojništvo Univerze v Ljubljani sta pridobila naziv univerzitetni diplomirani inženir strojništva: dne 22. oktobra 2014: Aleš TRAVNIK z naslovom: »Avtomatizacija lepljenja nosilcev na karoserijske dele« (mentor: prof. dr. Janez Diaci); dne 24. oktobra 2014: Bine MATIČIČ z naslovom: »Primerjava kritičnih parametrov lastnih nihanj vertikalnih agregatov s Kaplanovimi turbinami« (mentor: prof. dr. Miha Boltežar, somentor: izr. prof. dr. Anton Bergant). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv univerzitetni diplomirani inženir strojništva: dne 30. oktobra 2014: Mišo CVIŠIĆ z naslovom: »Primerjava investicijskih in obratovalnih stroškov med različnimi vrstami toplotnih črpalk (za novejšo nizkoenergijsko hišo in starejšo slabo izolirano hišo)« (mentor: prof. dr. Milan Marčič); * Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv magister inženir strojništva: dne 22. oktobra 2014: Domen LEVIČAR z naslovom: »Projektno vodenje izvedbe naročil trgovinskega reflektorja« (mentor: izr. prof. dr. Janez Kušar, somentor: prof. dr. Marko Starbek); Urša LOKAR z naslovom: »Določitev numeričnega modela realne fiziološke karotidne arterije« (mentor: prof. dr. Boris Štok); Andraž TRAMPUŠ z naslovom: »Obratovanje nizkotemperaturnih gorivnih celic v energetskih sistemih z velikim deležem obnovljivih virov energije« (mentor: izr. prof. dr. Mihael Sekavčnik, somentor: asist. dr. Mitja Mori); Matic ZUPANC z naslovom: »Primerjava energijske učinkovitosti ravnih sončnih kolektorjev v realnem okolju« (mentor: prof. dr. Vincenc Butala). dne 24. oktobra 2014: Anže ČAKŠ z naslovom: »Analiza dinamskega odziva pralnega stroja v okviru dinamike sistema togih teles« (mentor: prof. dr. Miha Boltežar, somentor: asist. dr. Gregor Čepon); Andrej KRAGELJ z naslovom: »Nadzor in optimizacija laserskega varjenja bakrene žice na

jekleno elektrodo grelnega telesa žarilne svečke« (mentor: doc. dr. Matija Jezeršek); Jure ZADRAVEC z naslovom: »Merjenje optodinamskih pojavov s hitro senčno fotografijo« (mentor: doc. dr. Matija Jezeršek). dne 27. oktobra 2014: Aleš MRAK z naslovom: »Tribološka analiza površinskih prevlek na orodju za stiskanje tablet« (mentor: prof. dr. Mitjan Kalin); Boris PLESAC z naslovom: »Analiza preoblikovanja fazno spremenljivih materialov pri nizkih temperaturah« (mentor: izr. prof. dr. Tomaž Pepelnjak, somentor: izr. prof. dr. Roman Šturm). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv univerzitetni diplomirani gospodarski inženir: dne 30. oktobra 2014: David KUREČIČ z naslovom: »Tehnološka in ekonomska analiza CNC proizvodnega sistema« (mentorja: prof. dr. Jože Balič, prof. dr. Duško Uršič). * Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv magister inženir strojništva: dne 29. oktobra 2014: Mitja KRIŽNIK z naslovom: »Analiza in optimizacija zajema CO2 v termoelektrarni na premogov prah« (mentor: prof. dr. Aleš Hribernik); Peter NAVOTNIK z naslovom: »Razvoj stroja za kontinuirano litje aluminijastega traku« (mentor: prof. dr. Zoran Ren, somentor: dr. Matej Borovinšek); Lio PAVLIČ z naslovom: »Numerična simulacija dvižne mize s trakom« (mentor: doc. dr. Janez Kramberger, somentor: izr. prof. dr. Tone Lerher); Nejc LAZAR z naslovom: »Sodobni kroglični rezkarji za izdelavo preoblikovalnih orodij« (mentor: prof. dr. Franci Čuš, somentor: dr. Matjaž Milfelner); Sebastjan ZADRAVEC z naslovom: »Razvoj vpenjalne priprave za varjenje stranic zaboja traktorske prikolice« (mentor: prof. dr. Franci Čuš, somentor: doc. dr. Tomaž Vuherer). * Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv magister inženir mehatronike: dne 23. oktobra 2014: Primož FIŠER z naslovom: »Integrirani pretvornik za napajanje pogona in baterij električnega vozila v režimu napajanja pogona« (mentor: izr. prof. SI 141

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)11, SI 140-142

dr. Karl Gotlih, somentorja: prof. dr. Miro Milanovič, doc. dr. Miran Rodič). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv magister inženir oblikovanja izdelkov: dne 29. oktobra 2014: Martin WERDONIG z naslovom: »Inženirsko oblikovanje raztegljivega povodca z zavoro za psa« (mentor: izr. prof. dr. Miran Ulbin). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv diplomirani inženir strojništva (UN): dne 30. oktober 2014: Aleksander PIKO z naslovom »Zasnova vrstnega primeža in optimizacija delovnega procesa za CNC frezalni stroj« (mentor: prof. dr. Franci Čuš, somentor: asist. Tomaž Irgolič). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv diplomirani gospodarki inženir (UN): dne 30. oktobra 2014: David PAJTLER z naslovom »Motiviranje pri udejanjanju idej in inovacij na področju uvajanja vitke proizvodnje v MSP« (mentorja: doc. dr. Marjan Leber, somentor: izr. prof. dr. Zdenka Ženko). * Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva: dne 15. oktobra 2014: Marko KASTELIC z naslovom: »Vpeljava elektronske pilotske torbe v uporabo, skladno z aktom EASA AMC 20-25« (mentor: izr. prof. dr. Tadej Kosel); Jan POŽAR z naslovom: » Razvoj orodja in tehnologije za injekcisjko brizganje puše« (mentor: izr. prof. dr. Tomaž Pepelnjak); dne 17. oktobra 2014: Blaž DEŽELA z naslovom: »Optimizacija parametrov rezanja cevi z elektroerozijskim postopkom« (mentor: doc. dr. Joško Valentinčič, somentor: doc. dr. Henri Orbanić); Davorin POŠEBAL z naslovom: »Analiza poliranja orodij za izdelavo steklenih izdelkov« (mentor: doc. dr. Peter Krajnik, somentor: prof. dr. Janez Kopač) SI 142

* Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv diplomirani inženir strojništva: dne 30. oktobra 2014: David naslovom: »Razvoj NOVAK z večnamenske kmetijsko-gozdarske prikolice« (mentor: doc. dr. Janez Kramberger); Boštjan RAMŠAK z naslovom: »Sprememba hrapavosti površine v odvisnosti od števila obremenitvenih ciklov« (mentor: prof. dr. Nenad Gubeljak, somentor: izr. prof. dr. Jožef Predan); Dominik ŽITEK z naslovom: »Robotsko varjenje podsklopa avtomobilske karoserije v avtomatski varilni pripravi« (mentor: doc. dr. Tomaž Vuherer, somentor: izr. prof. dr. Karl Gotlih). * Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva (VS): dne 15. oktobra 2014: Damir KNEŽEVIĆ z naslovom: »Vpliv temperature medija na agresivnost kavitacije« (mentor: izr. prof. dr. Matevž Dular, somentor: izr. prof. dr. Roman Šturm); Andrej PODLESNIK z naslovom: »Izbira zmogljivejšega transportnega letala za potrebe Slovenske vojske« (mentor: izr. prof. dr. Tadej Kosel); David PREDANIČ z naslovom: »Injekcijsko brizganje mikroleč na makroizdelku« (mentor: izr. prof. dr. Tomaž Pepelnjak); dne 17. oktobra 2014: Rok GRUBIČ z naslovom: »Analiza obratovanja zavornega diska za tirna vozila« (mentor: prof. dr. Marko Nagode). * Na Fakulteti za strojništvo Univerze v Mariboru sta pridobila naziv diplomirani inženir strojništva (VS): dne 29. septembra 2014: Sašo DOJNIK z naslovom: »Zgorevanje in merjenje emisije snovi v zrak pri uporabi lesnih peletov v mali kurilni napravi« (mentor: viš. pred. dr. Filip Kokalj); dne 30. septembra 2014: Tadej KLAVŽAR z naslovom: »Izboljšava montaže bobnov gozdarskih vitlov« (mentorja: prof. dr. Miran Brezočnik, somentor: izr. prof. dr. Karl Gotlih).

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

Founding Editor Bojan Kraut

University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Vice-President of Publishing Council Jože Balič

University of Maribor, Faculty of Mechanical Engineering, Slovenia Cover: Badly-fitting shoes are one of the major causes of pain, foot related diseases and injuries of the feet. Therefore three-dimensional measurements of the feet is crucial for the correct design and selection of shoes. Upper figure presents a new system for 3D footshape measurements which is based on the laser-multiple-line-triangulation principle. Example of unprocessed raw measurement and the extracted feet is shown below. Courtesy: Studio Miklavc (upper image) and University of Ljubljana, Faculty of Mechanical Engineering (lower images)

International Editorial Board Koshi Adachi, Graduate School of Engineering,Tohoku University, Japan Bikramjit Basu, Indian Institute of Technology, Kanpur, India Anton Bergant, Litostroj Power, Slovenia Franci Čuš, UM, Faculty of Mechanical Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mechanical Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mechanical Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mechanical Engineering, Slovenia Imre Horvath, Delft University of Technology, Netherlands Julius Kaplunov, Brunel University, West London, UK Milan Kljajin, J.J. Strossmayer University of Osijek, Croatia Janez Kopač, UL, Faculty of Mechanical Engineering, Slovenia Franc Kosel, UL, Faculty of Mechanical Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mechanical Engineering, Slovenia Miroslav Plančak, University of Novi Sad, Serbia Brian Prasad, California Institute of Technology, Pasadena, USA Bernd Sauer, University of Kaiserlautern, Germany Brane Širok, UL, Faculty of Mechanical Engineering, Slovenia Leopold Škerget, UM, Faculty of Mechanical Engineering, Slovenia George E. Totten, Portland State University, USA Nikos C. Tsourveloudis, Technical University of Crete, Greece Toma Udiljak, University of Zagreb, Croatia Arkady Voloshin, Lehigh University, Bethlehem, USA General information Strojniški vestnik – Journal of Mechanical Engineering is published in 11 issues per year (July and August is a double issue). Institutional prices include print & online access: institutional subscription price and foreign subscription €100,00 (the price of a single issue is €10,00); general public subscription and student subscription €50,00 (the price of a single issue is €5,00). Prices are exclusive of tax. Delivery is included in the price. The recipient is responsible for paying any import duties or taxes. Legal title passes to the customer on dispatch by our distributor. Single issues from current and recent volumes are available at the current single-issue price. To order the journal, please complete the form on our website. For submissions, subscriptions and all other information please visit: http://en.sv-jme.eu/. 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 peerreview process.

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Papers

685

Boštjan Novak, Aleš Babnik, Janez Možina, Matija Jezeršek: Three-Dimensional Foot Scanning System with a Rotational Laser-Based Measuring Head

694

Emiliano Pipitone, Stefano Beccari, Marco Cammalleri, Giuseppe Genchi: Experimental Model-Based Linearization of a S.I. Engine Gas Injector Flow Chart

709

Andrzej Zbrowski, Krzysztof Matecki: The Use of Computed Tomography to Analyse Grinding Smudges and Subsurface Defects in Roller Bearing Rings

716

Changyi Liu, Gui Wang, Matthew S Dargusch: Mechanics and Dynamics of Helical Milling Operations

725

Daniel Kozelj, Zoran Kapelan, Gorazd Novak, Franci Steinman: Investigating Prior Parameter Distributions in the Inverse Modelling of Water Distribution Hydraulic Models

747

742

Uroš Karadžić, Vladimir Bulatović, Anton Bergant: Valve-Induced Water Hammer and Column Separation in a Pipeline Apparatus

Journal of Mechanical Engineering - Strojniški vestnik

Contents

11 year 2014 volume 60 no.

Strojniški vestnik Journal of Mechanical Engineering