Determining a Function for the Damping Coefficient of a laminated Stack

Transcription

1 DOI: /UB.OVGU TECHNISCHE MECHANIK, 37, -5, (017), subitted: June 9, 017 Deterining a Function for the Daping Coefficient of a lainated Stack C. Zahalka, K. Ellerann The design of electrical achines is deterined by electrical as well as echanical requireents. Possible losses due to eddy currents in the stator or the rotor are coonly reduced by using stacks of lainated sheet etal. On the other hand, the design of the stator and the rotor has a significant influence on the echanical properties: Vibrations depend on the stiffness and the daping of the lainated stack. There are different ethods to deterine the stiffness coefficient of a stack, but it is uch ore difficult to obtain suitable values for the daping as there are ore influencing factors. This paper describes an experiental procedure, which deterines the influence of different paraeters on the daping of a stack. The stack used during the experients consists of 00 quadratic steel sheets with a side length of 80 and a thickness of 0.5. In accordance with the easureent data, a functional dependance based on three variables is derived. The first one is the surface pressure between the steel sheets, the second one is the frequency of the applied lateral force, and the third one is the displaceent between the steel sheets. It is the ai of this investigation to deterine the influence of variations of these paraeter values on the daping. The forces are applied onto the stack with hydraulic cylinders. The echanical deforation of the stacked etal sheets is easured by a laser-speckle-based easureent syste. This syste detects the displaceent of single steel sheets. The displaceent is easured on two steel sheets, but they are not side by side. The difference between the two easureent points is equal to the displaceent of the stack. Through the synchronization of the tie signal of the lateral force and the displaceent of the stack, a hysteresis loop can be calculated. This hysteresis depends on the lateral force and the displaceent of the stack. The area of the hysteresis corresponds to the dissipation energy between the two easureent points on the stack, 140 sheets apart fro each other. This area is calculated by nuerical integration based on the trapezoidal rule. Through the conservation of energy for this syste, it is possible to calculate an effective daping coefficient for the stack. Considering different influencing paraeters, a function for the daping coefficient can be identified by the least square ethod. This function can be used for the paraeters in a nuerical siulation of an electrical achine. Noenclature A [] Aplitude of the displaceent of the dynaic cylinder d [ Ns ] Structural daping f E [Hz] Excitation frequency F D [N] Dynaic force ˆF D [N] Aplitude of the dynaic force F Stat [N] Static force [kg] Mass p [ N ] Surface pressure t [s] Tie x [] Displaceent ˆx [] Aplitude of the displaceent W Diss [N] Dissipation energy η [ ] Loss factor ψ [rad] Phase shift ω 0 [ rad s ] Angular eigen frequency ] Angular excitation frequency ω E [ rad s 161

2 1 Introduction Inforation about the aterial properties of the lainated stack is necessary to siulate the dynaics of a rotor or a stator of an electrical achine. Due to the structure of the stack, the aterial properties vary in the different directions: in radial direction, the etal of the sheets doinates the tensile stiffness. Different layers of the stack act in parallel and the aterial with the highest value of Young s odulus contributes ost significantly to the effective stiffness of the stack. In axial direction, the contact- and laination-zone between the sheets becoes uch ore iportant as layers act in series. In this direction, the aterial with the sallest value of Young s odulus influences the tensile stiffness uch ore. These zones are also considered to be highly iportant for non-conservative effects: The transversal deforation of the stack due to a shearing otion between the etal sheets contributes significantly to the overall daping of the structure. This is ainly due to the softer aterial in the laination zone. The stiffness of a coponent can be deterined experientally or nuerically given the geoetry and tabulated aterial properties. Daping values are uch harder to deterine and are rarely available in literature. In this paper, an experiental set-up is described in order to deterine a function for the daping coefficient of a stack caused by a shearing otion in transversal direction. There are any different paraeters which have an influence on the structural daping of a coponent. In this work, three variables are considered: The surface pressure between the steel sheets, the frequency of the lateral force and the aplitude of the displaceent of the stack. All others, like the height of the stack or the size of the steel sheets, are being kept constant. In practice, the displaceent and the frequency vary with the operating states. The pressure on the other hand is ostly caused by the design and by copressing the stack in the production process: In electrical achines, the lainated sheets are ore or less loosely stacked, then copressed and finally held in place by a welded structure of tension eleents and end plates. This design and production process ake the pressure difficult to be deterined leading possibly to significant variations in echanical paraeters of the achine. There are different ethods to deterine the stiffness and the structural daping of a lainated stack. Luchscheider et al. (01) describes a set-up to easure the stiffness of a laination stack. Two plungers copress nine circular saples, cut out fro a typical laination sheet aterial. With two extensoeters, claped on the plungers, and a load cell, a force-displaceent diagra is created. With this diagra the stiffness in stacked direction of the lainated stack can be derived. Mogenier et al. (010) predicted the odal paraeters of an induction otor with an undaped finite-eleent odel. The iniation of the error between the predicted and the easured odal paraeters with the Levenberg- Marquardt algorith. This leads to the equivalents constitutive properties of the lainated stack. Clappier and Gaul (015) and Clappier et al. (015) deterine the structural daping and the stiffness in axial and in shear direction of a lainated stack. The easureent set-up for this evaluation consists of two lainated stacks and three plates. The stack is axial pretensioned with a screw connection between the plates. The excitation is effected with a shaker to one of the plates. To calculate the stiffness and the structural daping, the acceleration of the plates and the force on the excitated plate are easured. The structural daping is calculated through the deterination of the dissipated energy. The sae principle is used by Bograd et al. (008). The difference between these works is that Bograd deterines the structural daping in shear direction fro a thin layer eleent and not fro a lainated stack. Experiental set-up The coplete test stand is placed on a foundation, which is isolated fro the surrounding with an air suspension. This is necessary to be independent fro the environental influences of the building. The easureent syste is not placed on this foundation, but the offset between the easureent syste and the test bench can be reoved through an differential easureent of the displaceent. The test bench consists of two hydraulic cylinders and the claping device for the stack (see Fig. 1). The vertical cylinder (4) applies the surface pressure on the lainated stack. To avoid an inclined position of the pressure plate, there are four linear guides in axial directions around the lainated stack. These guides are not shown in Fig. 1. The second cylinder (1) on the right side applies the oscillating lateral force on the stack. 16

3 Figure 1: Experiental set-up 1. Dynaic hydraulic cylinder. Load cell 3. Flexure 4. Static hydraulic cylinder 5. Ball joint 6. Upper laination stack 7. Interediate plate 8. Lower laination stack 9. Pressure plate There are two stacks, one is above the interediate plate and the other one is below. Each of the stacks consists of 00 steel sheets with a side length of 80 and a thickness of 0.5. Between the static hydraulic cylinder and the pressure plate are a load cell and a ball joint. The load cell easures the force for the surface pressure in the lainated stack and the ball joint corrects the inaccuracies of the concentricity between the pressure plate and the cylinder. A load cell and a flexure are situated between the dynaic cylinder and the interediate plate. The load cell records the daping force and the flexure is used for the correction of the vertical position of the interediate plate. This is necessary, because the vertical position of the interediate plate depends on the surface pressure in the lainated stack. The static hydraulic cylinder is force controlled based on the load cell and the second one is stroke controlled based on the agnetostrictive easureent ethod of the cylinder. Both systes are controlled with one dual-channel controlling syste. The basic construction of the easureent syste for the displaceent of the steel sheets was described in Halder et al. (014) based on the laser-speckle principle. For the application in the test bench, the easureent syste has been adapted. The easureent construction is positioned so that the axes of the two high speed caeras are orthogonal to the front site of the lainated stack, shown on the left picture of Fig.. This caeras record the Regions of Interest (RoI) and the software calculates the displaceent of each RoI. The displaceent of the lainated stack is the difference between the two RoI in horizontal direction. The axiu sapling rate of the syste is 150 Hz by recording two easure points at each RoI. In cobination with the axiu excitation frequency of 1.5 Hz, there are at least twelve easureent points during one vibration period. In order to get different easureent points in the hysteresis loop, the sapling frequency is not an integral ultiple of the excitation frequency. The position of the two RoI is shown on the right side of Fig. and arked with a red rectangle. The distance between the RoI is liited by the iage of the caera in relation to the thickness of the sheets. For the setup under consideration the RoI are separated by 140 sheets of steel. The speckle pattern is produced through a laser light bea which is redirected with an irror to a bea expander. This expanded bea is divided into two beas with an splitter cube and ust be projected exactly on the two RoI. The detailed description of this easureent syste is in Halder et al. (014). 163

4 Figure : Principle of the easureent syste 3 Calculation of the daping coefficient Through the structural daping in the lainated stack, the force-displaceent graph is a hysteresis loop. The area inside this hysteresis corresponds to the dissipated energy WDiss. The energy can be calculated fro the daping force FD acting over a diplaceent x Z WDiss = FD dx. (1) In the considered case, WDiss deterined fro the horizontal displaceent of FD as indicated in the right hand side of Fig.. Another definition for the dissipation energy is the approach of Kelvin-Voigt, which is described in Dresig and Fidlin (014). This approach uses the daping coefficient d and replaces the integration over x by an integration over tie t Z T WDiss = d x dt = d ωe Z 0 T x sin (ωe t) dt. () 0 Here, the excitation is assued to be a sinusoidal function with frequency ωe and aplitude x. Furtherore, the response is assued to have reached steady state with constant aplitude and angular frequency. As a result of these assuptions, the daping coefficient d deterined fro Eq. () leads to d= WDiss. ωe π x (3) In addition to the structural daping, the loss factor η can be calculated. This factor is defined by the dissipation energy divided by the axiu energy of the syste. In Fig. 3, the axiu energy is shown in the dark gray triangle and the dissipation energy is the gray area of the hysteresis loop. On the other hand, the axiu energy can be calculated fro the aplitude of the dynaic force and the displaceent. Thereby, the loss factor becoes η= WDiss. 1/ FD x (4) The daping coefficient can also be evaluated fro the equation of otion of a forced oscillator: x + d FD iωe t x + ωe x= e. (5) 164

5 FD Maxiu energy Dissipation energy F D,x x Figure 3: Maxiu and dissipation energy This essentially reduces the vibrating stack to a one-degree of freedo oscillator. The ass is the effective ass of the stack. The fact, that parts of the experiental setup also ove is accounted for in the force FD. For the evaluation, only the steady state solution is relevant. A solution is given by the coplex function x = x ei(ωe t ψ), (6) where ψ is the phase shift between excitation and response. Substituting Eq. (6) and its derivatives into the equation of otion Eq. (5) leads to ω0 ωe +i FD iψ d ωe = e. x (7) In Fig. 4, the left side of Eq. (7) is plotted in the coplex plane. I Ψ ω0 - ωe Re Figure 4: Coplex plane Fro trigonoetric functions applied to the rectangular triangle in the coplex plane, the structural daping becoes d= FD sin ψ. x ωe (8) With this equation, it is possible to calculate the daping coefficient fro the phase shift ψ between the vibration exciteent and the vibration response. This function is used for the evaluation of the calculation of the area fro the hysteresis loop. 4 Experiental evaluation The ai of this experient is to derive a function for the daping coefficient depending on the excitation frequency, the displaceent and the surface pressure between the steel sheets. A total of 10 easureent series were considered: five different pressure values, four different aplitudes and six different frequencies. The pressure was varied fro N/ in steps of 0.4 N/, the aplitude fro in steps of Frequencies included were 1,.5, 5, 7.5, 10, 1.5 Hz. Each of these 10 series was repeated ten ties. 165

6 In order to exclude systeatic easureent errors, the chronological order of the easureent was randoized. All of these easureents were considered in this study, but for the illustration of the ethod, we subsequently focus on the easureent series with a pressure of 1. N/ as given in Table 1. Table 1: Measureent series with p = 1.N/ Series f E [Hz] A [] Series f E [Hz] A [] Fig. 5 shows the dynaic force and the displaceent of the lainated stack versus tie. The first three seconds of the signal include the approach of the hydraulic cylinder and the transient response. After this tie, the steady state solution is reached. For the deterination of the daping coefficient, the steady state solution is significant, see Section 3. Furtherore, only the easureent points after five seconds will be considered. After the transient response, the aplitude of the displaceent and the dynaic force is nearly constant. The reason for sall fluctuations lies in the stroke control of the hydraulic cylinder. The coparison of the plotted easureent data and the paraeter of the easureent series (see Table 1) shows a difference in the aplitude of the displaceent. This difference is caused by the elastic deflection of the flexure (see Fig. 1). The hydraulic cylinder is stroke controlled, which is easured inside the piston. So, on one side there is the displaceent of the piston and, on the other side, the displaceent of the lainated stack. Consequently, the difference between these two displaceents is the elastic deforation of the flexure. Figure 5: Measureent signal fro the easureent nuber 417 (series 38) The signal of the dynaic force and the displaceent includes a phase-shift. This results fro the structural daping in the lainated stack. To get a graph (see Fig. 6) with the dynaic force over the displaceent, the two signals ust be equal. In order to reove fluctuations fro the easureents, several points are grouped into one by averaging. With this operation, there is a inor error fro the calculation of the area. The area is calculated with a nuerical integration, based on the trapezoidal rule. A linear connection between the points is 166

7 satisfactory exact, because the error fro the easureent is higher than the error through the linearization. Such linear connections are shown in the right graph of Fig. 6. Figure 6: Hysteresis loop left: easureent data; right: linearized data Fig. 7 shows the boxplots of the easureent series with a surface pressure of 1.N/. All easureents which are in one arked rectangle have the sae excitation frequency and fro left to right an increasing aplitude (see Table 1). The daping coefficient decreases with an ascending aplitude and by an ascending frequency. Both of these connections have a siilar behavior and can be approxiated through an exponential function with a negativ exponent. Another detail is shown in Fig. 7: Lower aplitudes and frequencies lead to a larger difference between the first and the third quartile. The reason for this lies in the absolut easureent and calculation error which is in all cases roughly the sae, but through the saller easureent values the relative error is uch bigger. Figure 7: Boxplot of the easureent series with a surface pressure of 1. N/ Corresponding to the five different pressure values, five different functions for the daping coefficient are deterined. These functions depend on the frequency and the displaceent of the lainated stack and are assued to take the for d(f, A) = C 1 f + C A + C 3 f A. (9) Paraeters C 1, C and C 3 are calculated fro a least squares approxiation separately for each pressure value: For each variation of the paraeters (C 1, C, C 3 ) the su of all squared differences between each easureent point and the function value is calculated. The best approxiation of the function is found when the su reaches 167

8 a iniu. In order to control the quality of the solution, the coefficient of deterination (R ) is calculated. It is defined in Birkes and Dodge (1993) as R = (ŷi ȳ) (yi ȳ). (10) The range of R is fro 0 to 1, whereby 1 correspondents to the best approxiation of the data values. As an exaple, the function for a constant surface pressure of 1. N/ is found to be d(f, A) = f A f A [ ] Ns. (11) Figure 8: Daping coefficient for a surface pressure of 1. N/ and an excitation frequency of 7.5 Hz Fig. 8 illustrates Eq. (11) using a frequency of 7.5 Hz and the results of the corresponding easureents. Inserting the value of the frequency into eq. (11) gives d(a) = A [ ] Ns. (1) Again, as an exaple, the function for the daping coefficient at a surface pressure of 1. N/ in Fig. 9 is shown. The easureent points are arked with red crosses in this figure. The two-sided 95 % confidence interval is built fro all 10 easureent series. The calculation of the confidence interval is described in Mittag (015). The coplete function is inside ost of the confidence intervals. Coparing the results fro the different pressure values, only the paraeter C 1 is found to differ significantly. The correlation between the paraeter values and the associated pressures is nearly linear. With a linear regression, the coplete function for the daping coefficient can be derived and this function is d(f, A, p) = p + 18 f A f A [ ] Ns. (13) 168

9 Figure 9: Daping coefficient for a surface pressure of 1.N/ 5 Conclusion This paper describes a ethod to identify a paraetric odel for a daping coefficient of a stack of sheet etal. This function depends on the surface pressure, the excitation frequency and the displaceent of the stack. A test stand was developed and 10 easureent series were recorded. Fro this easureent data, the daping coefficient was deterined by eans of the dissipated energy. In order to control the obtained coefficients, a second approach was used. All the calculated daping coefficients were fitted into a global function. The best approxiating function was derived by the least square ethod. At last, the function was copared with the confidence interval of the easureent data. The function reveals a significant dependance of the daping on the different paraeters for the considered test case. The results ay be used for ultibody siulation analyses of a stack, which is loaded by an oscillated force with a constant excitation frequency. References Birkes, D.; Dodge, Y.: Other Methods, pages John Wiley & Sons, Inc. (1993). Bograd, S.; Schidt, A.; Gaul, L.: Joint daping prediction by thin layer eleents. In: Proceedings of the IMAC XXVI: Conference & Exposition on Structural Dynaics, Orlando, FL, USA (008). Clappier, C.; Gaul, L.; Westkper, E.: Experiental deterination of aterial properties in stacking direction of lainated stacks belonging to electrical achine rotors using a dilatation test. In: Proceedings of the nd International Congress on Sound and Vibration, Florence, Italy (015). Clappier, M.; Gaul, L.: Experiental Investigation of Structural Daping of Lainated Stacks of Electrical Machine Rotors, pages Springer International Publishing, Cha (015). Dresig, H.; Fidlin, A.: Schwingungen echanischer Antriebssystee: Modellbildung, Berechnung, Analyse, Synthese. Springer Berlin Heidelberg (014). Halder, C.; Thurner, T.; Mair, M.: Developent of a laser-speckle-based easureent principle for the evaluation of echanical deforation of stacked etal sheets. In: Proc. SPIE, vol. 913, pages 9131F 9131F 8 (014). Luchscheider, V.; Willner, K.; Maidorn, M.: Developent of a odel to describe the stiffness of an electric otor laination stack. In: 01 nd International Electric Drives Production Conference (EDPC), pages 1 5 (Oct 01). 169

10 Mittag, H.: Statistik: Eine Einführung it interaktiven Eleenten. Springer-Lehrbuch, Springer Berlin Heidelberg (015). Mogenier, G.; Dufour, R.; Ferraris-Besso, G.; Durantay, L.; Barras, N.: Identification of laination stack properties: Application to high-speed induction otors. IEEE Transactions on Industrial Electronics, 57, 1, (010), Address: Graz University of Technology, Institute for Mechanics, Kopernikusgasse 4/IV, Graz, A-8010 Austria eail: 170