Static And Dynamic Vibration Analyses Of Switched Reluctance

IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

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Static and Dynamic Vibration Analyses of Switched Reluctance Motors Including Bearings, Housing, Rotor Dynamics, and Applied Loads K. N. Srinivas, Member, IEEE, and R. Arumugam, Member, IEEE

Abstract—This paper presents a thorough numerical study of vibration analysis in electric motors, with particular application to switched reluctance motors (SRMs), using three-dimensional finite-element analysis (3-D FEA) methodology. It covers the major vibration sources: mechanical, magnetic, and applied load. The following analyses are presented. 1) A 3-D modal analysis including stator and rotor structures, shaft, end shields, bearings, and housing. 2) An unbalanced rotor dynamics analysis of the rotor, which is important for deciding on the eccentricity of the rotor mass to ensure that the vibration of rotor and housing is within acceptable limits. 3) A harmonic analysis to identify the range of speeds producing high vibration and noise that should be skipped over quickly during acceleration. 4) A stress analysis under different loading conditions (a simulation of load testing) to predict the deformation of the shaft and rotor. Apart from frequently reported modal analysis on the stator of SRMs in two dimensions, these 3-D vibration analyses are essential to examine the vibration in SRMs as a whole. Index Terms—End shields, loads and housing, pulleys, switched reluctance motor, three-dimensional finite-element analysis, vibration analysis including bearings.

I. HISTORY OF VIBRATION AND ACOUSTIC NOISE IN SRM AND ORGANIZATION OF CHAPTERS

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IBRATION is the to and fro oscillation of the part or structure about its mean aligned position. In any electric machine, tiny amplitudes in vibration may be ignored because their contribution toward oscillation will be negligibly small when compared to the mass of the machine. But large amplitude of vibration is harmful to machine and human due to the following reasons. • It can result in premature fatigue failure due to large dynamic stresses. • The high inertia forces can damage components such as bearings, gears, etc. • It can cause damage to internal organs of humans. All of the above reasons are to be necessarily analyzed. The cross section of switched reluctance motors (SRMs) under investigation is shown in Fig. 1. The SRM is a “doubly salient and singly excited” machine. The stator and rotor have projecting poles in unequal numbers. Each stator pole carries Manuscript received April 7, 2003; revised March 8, 2004. K. N. Srinivas is with the Department of Electrical and Electronics Engineering, Crescent Engineering College, Chennai 600 048, India (e-mail: [email protected]). R. Arumugam is with Anna University, Chennai 25, India (e-mail: arumugam @annauniv.edu). Digital Object Identifier 10.1109/TMAG.2004.828034

Fig. 1. SRM under study.

a concentrated winding. The windings in the diametrically opposite stator poles are joined together to form “a phase” of the motor. When a phase winding is excited the rotor moves so as to align itself with the magnetic axis of the excited phase, producing the basic torque of the motor. In SRMs, there exists a strong radial magnetic force in the total force developed, apart from the torque producing tangential magnetic field (as detailed understanding of principles of SRMs is well documented [1]–[5], it is avoided here). When a phase is excited, the magnetic flux from the excited stator pole crosses the air gap in radial direction producing large radial forces on the excited stator poles, which deform the stator into an oval shape, called ovalization. The ovalization or modal deformation of the stator back iron as a ring, and the lateral rocking of the stator poles, both together, produces radial vibration of the stator leading to acoustic noise in SRMs. This is the way vibration is produced due to magnetic reasons. It is imperative to know the frequencies (called the modal frequencies) at which the radial forces are induced, as, the coincidence of the natural frequency of the stator with any of the modal frequencies will cause

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

Fig. 2. 3-D model of SRM. (a) 3-D model of SRM under study. (b) 3-D model of SRM with shaft, bearings, end-shield housing, and foundation.

resonance resulting in vibration and noise. A modal study will yield the possible frequencies (and hence the respective speeds) to be skipped for a quiet operation of the machine. The modal analysis of stator was investigated and reported earlier [6]–[9], and hence, even though the same exercise is carried out for the SRM under study and modes are identified, the procedure is not discussed in this paper. The modal analysis can be performed in frequency domain or in time domain. Analysis of acoustic noise in frequency domain could provide the spectrum and dominant components of the noise and vibration [6], whereas a time-domain analysis can provide clear links between the vibration and the timing of the applied excitation current in windings of the stator. A study by Wu and Pollack [8] shows that greater noise suppression could be achieved when a proper excitation current profiling is carried out. Half-cycle counterexcitation at the turn-off angle rather than at the turn-on angle was suggested. Pillay made notable contributions on the vibration analysis based on the controller side research in the past decade [9], [10]. Tang and Radan [12] demonstrated the design aspects and experiments for reducing vibration. These are the major papers that investigated vibrations in SRMs and concentrated on the two-dimensional (2-D) modal analysis on SRMs considering the stator frame alone. Although the stator is the major portion of noise production, there are contributions from other parts like rotor core, end shield, etc., which were believed to be negligible in the above earlier at-

tempts. A three-dimensional finite-element analysis (3-D FEA) is capable of accounting for all these parts. In this paper, a 3-D finite-element vibration analysis is performed considering the rotor core, end rings, bearings, shaft, and applied loads. Also, the mechanical structure of the stator is a major mechanical source of vibration. The coincidence of the natural frequency of the stator with any of the modal frequencies will cause resonance and acoustic noise. The rotor does not react to most radial waves as it is stiffer than the stator. So it cannot be deformed easily and does not suffer any notable vibrations, but, however, a modal analysis on the full model of SRM including rotor, housing, and end shields can enlighten vibration modes of the machine as a whole. As there is going to be series of modes in which the machine as a whole will vibrate, there is a necessity to perform modal analysis for the whole SRM assembly. Section II describes the modal analysis on SRM in three dimensions considering the rotor, shaft, end shields, bearings, and housing. In a vibration study on machines, study on rotor dynamics called the unbalanced response analysis is essential to verify whether the vibration of rotor including housing is within the acceptable limits. Rotor unbalance gives rise to dynamic rotor vibration and eccentricity, which will produce nonsupply frequency-based vibrations. This vibration will finally converge to the housing and its ovalization. In fact, the eccentricity of rotor mass including shaft will help in deciding on the mass of the

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rotor. Section III records the simulation procedure for unbalanced rotor dynamic analysis on SRM. This section also reports on the harmonic analysis to identify the frequencies at which the vibration is maximum due to the rotor eccentricity. The load induced source of noise, which forms another major category of source of noise, includes noise due to coupling with a load, pulley, and belt tension, and mounting of SRM on foundation. Auxiliaries, such as brushes and commutators (of course, which are absent in SRM) and shaft-keys, also contribute to the noise. Section IV is a documentation of vibration analysis on SRM due to the applied loads. Modeling of different auxiliaries and consideration of different loads are the highlights of the section. Section V concludes this paper. ANSYS version 6 is used for all the four analyses. Vibration analysis due to electronic sources is not dealt with in this paper. II. MODAL ANALYSIS INCLUDING HOUSING The SRM model under consideration is shown in Fig. 2 and the meshes formed during FEA are shown in Fig. 3. The length of the stator stack is 90 mm. The end shield has thickness of 10 mm. The shaft has a diameter of 25 mm. The main values set during simulation are: the Young’s modulus, N/m ; specific mass of winding kg/m ; total mass kg/m ; Poisson’s ratio, . density In the SRM, it is found that resonance occurs if the phase frequency or add harmonics coincides with the stator natural frequency, resulting in a peaking of the stator frequency. The is given by [13] phase frequency (1) where is the speed in radians per second and is the number of rotor poles. Vibration is maximum if any of the frequencies (2) are coincident with the natural frequency of the machine given by [14] (3) where is the stator iron thickness in meters, is the mass is the density of the material in kilograms per cubic meter, , mean radius of the stator shell in meters given by where is the outer diameter of the stator. The governing Laplace equation that is solved iteratively to find the modal frequencies is (4) where is the modal vector and is the frequency of vibration. The solution is the th mode shape and is the corresponding natural frequency. The vibration modes of housing with foundation are shown in Fig. 4. There are modes at which the housing as a whole vibrates, as depicted in Fig. 4(a) and (c), at mode frequencies

Fig. 3. Meshed model of SRM. (a) Rotor. (b) Stator with stand. (c) Whole SRM model with end shield and housing.

1017 Hz (approximately 10 000 rpm) and 1217 Hz, respectively. Although the vibration spreads fully over the housing, the noise will be of submerged humming in nature and will not be severe, as the housing is well mounted onto the foundation. At a frequency of 520 Hz [Fig. 4(e)] and at a frequency of 402 Hz [Fig. 4(f)], there exists vibration modes twisting the shaft with the rotor and arresting the housing, which is a severe phenomenon. Other frequencies of higher order are neglected because they occur at several thousand revolutions per minute, which are considered impractical. Frequencies at which the rotor and shaft undergoes bending leading to higher acoustic noise is depicted in Fig. 5. There is a severe shaft deformation extending to rotor core at 333 Hz, whereas there is shaft end deformation which may lead to a twist of connected pulley, at frequencies 2259, 1910, and 3089, which are respectively shown in Fig. 5(b), (c), and (d). It can be observed that the shaft end vibration does not spread to rotor or housing, but, as the pulley and connected loads will be put into vibration, the noise will be high. It can also be noted that a 2-D analysis will not be demonstrating in detail such involved vibrations of the machine which are practically possible.

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

Fig. 4 (a) Mode frequency: 1017 Hz, housing vibration. (b) Mode frequency: 2419, severe shaft bending. (c) Mode frequency: 1217 Hz, housing vibration. (d) Mode frequency: 333 Hz. (e) Mode frequency: 520 Hz. (f) Mode frequency: 402 Hz.

III. UNBALANCED ROTOR DYNAMICS A harmonic frequency analysis has been performed to identify whether the vibration of rotor and housing is within safe range. It is a technique used to determine the steady-state response of SRM to loads that vary harmonically with time. The aim is to obtain the SRM structure response at several frequencies with respect to displacement. Peak responses are identified and plotted as a graph and stresses are reviewed at these frequencies.

The weight of the rotor (w) is 3.75 kg. The balancing quantity ( ) and the damping ratio ( ) were assumed to be 2.5 and 0.02, respectively, which are the usual standard values prescribed by Indian Standard specifications ISO 1940, for high-speed machines. The rated speed is 3000 rpm. The centrifugal force is calculated using the formula w g

(5)

As all the units are in millimeters, the g used is 9810, which gives the centrifugal force ( ) as 0.3. This is applied to the

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Fig. 5 (a) Modal shape at 1910 Hz, rotor and shaft bending. (b) Modal shape at 2259 Hz, shaft bending. (c) Modal shape at 1910 Hz. (d) Modal shape at 3089 Hz, having back ovalization and severe shaft bending. (e) Modal shape at 2520 Hz, having back ovalization and severe shaft bending.

center of the rotor. The center node on the rotor is selected to is applied as a lateral load. The arrow mark at the which . middle of the rotor in Fig. 6 indicates the application of Arresting the nodes at the foundation, which forms one of the boundary conditions to model that the SRM is bolted to the foundation, can also be noted. These boundary conditions set up the condition of a simply supported beam on two bearings. In case of SRM, the whole housing also sit on the bearings. So, is set throughout the outer housing surface and to the the front and rear bearings. This is a constant force (forms the boundary condition) applied over a frequency range. The frequency range was assumed to be 400, which is on the upper side. The “harmonic analysis” is run on this model using an FE package.

Actually, the unbalance force

, at a frequency

, is (6)

But, the force which the FE package finds will be at . It has to be converted to satisfy the relation, as given in (7), using a FORTRAN program (7) The result of simulation is shown in Fig. 7. It can be observed that the rotor eccentricity reaches a maximum of 6 m, whereas that of outer frame is 1 m. As this eccentricity is of negligible micrometers, it is conclusive that the rotor dynamics of the considered SRM is in acceptable limits.

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

Fig. 6. Applied lateral force at the central node (or surface) of the rotor and bearings, for unbalanced rotor dynamic analysis. Fig. 8.

Bearing model.

Fig. 9. Upward arrow marks indicate the location of applied loads and pulley weight at the shaft end for the conduct of stress analysis (the load test).

Fig. 7. Rotor dynamic-harmonic analysis results showing displacement versus speed: (a) for rotor and front bearing and (b) for outer frame (housing) and rear bearing.

IV. STATIC STRESS ANALYSIS Static analysis is simulating a load test on SRM for observations in limit violations of stress and deformation at different

places. Additional models required are bearings and pulley. Bearing is the element in SRM to hold the housing and the rotor mass at the shaft. So, bearing is simulated as four springs attached to the housing from the bearing locations. Two such models are made at the locations of front and rear bearings. The node at the front bearing location and four nodes at the housing each displaced by 90 in and axes are selected and joined. A similar procedure is repeated at the rear side of the shaft. The spring stiffness (21 000) is assigned and bearing is thus modeled (Fig. 8). The spring holding the housing with the shaft forms respectively the front and rear bearings. The pulley is modeled by assigning its weight at the end of the front shaft. The procedure adopted in the conduct of static stress analysis (or load test) on SRM through a 3-D FEA is described below. The required geometrical data are as follows: radius of the mm; outer diameter of pulley mm; thickness shaft

SRINIVAS AND ARUMUGAM: STATIC AND DYNAMIC VIBRATION ANALYSES OF SRMs

Fig. 10. ISO view of stress plot (kgf/mm ), showing the distribution of stress at the shaft, at full load. Maximum stress is 21.99 kgf.

Fig. 11. Zoomed view of stress plot (kgf/mm ), showing the distribution of stress at the shaft, at full load.

Fig. 12. ISO view of deformation (in mm) that takes place in rotor due to the application of full load. Maximum deformation is 1.205 mm.

of pulley mm; weight of pulley 2.85 kg; belt prestress and kg. The full load of 3.63 N m is considered. As the housing assignment of load, etc., are in force units, the equivalent kgf m kgf m. Using the radius of the shaft and is

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+

Fig. 13. ISO view of torsional rotation (in rad) occurring in shaft rotor apart from the rotating torque developed at the application full load.

Fig. 14. ISO view of stress plot (kgf/mm ), showing the distribution of stress at the shaft, at half load. Maximum stress is 10.9 kgf.

converting all quantities to a single unit, the force is kg. This force is spread over all the finite-element nodes at the shaft end. It is observed that there are 12 nodes. Thus, the equivalent force applied to each of the 12 nodes for a load direction, as of 3.63 N m is 5.1. This force applied in the this is the load tensioning the belt-pulley mechanism and acts upwards. Further, the sum of the weight of pulley-belt loading mechanism, belt’s prestress and the housing with stator, which amounts to 7.85 kg, is divided by all the 12 nodes in a similar direction, as they act downwards. way and applied at the The zoomed view of the above applied loads is shown in Fig. 9. At this full-load model, a “stress analysis” is run whose output is shown in Figs. 10–13. Winding weight is considered as net mass along with the weight of the stator. A similar “load test” needed to be conducted on the SRM at different working loads to ascertain whether the deformation and torsional rotor rotation are within limits. The simulated experiment results at half load and twice full loads are presented in Figs. 14–16 and Figs. 17–19, respectively. The half-load torque is 1.815 N m and twice full-load toque is 7.26 N m. The FEA

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Fig. 15. Zoomed view of stress plot (kgf/mm ), showing the distribution of stress at the shaft, at half load.

Fig. 16. Front view of deformation (in mm) that takes place in rotor due to the application of half-load. Maximum deformation is 1.522 mm.

IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

Fig. 18. Zoomed view of stress plot (kgf/mm ), showing the distribution of stress at the shaft, at twice full load.

Fig. 19. ISO view of deformation (in mm) that takes place in rotor due to the application of twice full load. Maximum deformation is 4.685 mm. TABLE I SUMMARY OF LOAD TEST RESULTS OF STRESS AND DEFORMATION

The stress and the deformation in the three cases considered viz., full, half, and twice full loads, are tabulated in Table I. The material considered has a maximum tensile stress of 45 kg/mm . From this table, it can readily be observed that the factor of safety is 2.05 for the full-load case, 4.35 for the half-load case, and is 1.4 for the twice full-load case. Thus, the SRM can be operated up to twice full load without any mechanical threat. V. CONCLUSION Fig. 17. ISO view of stress plot (kgf/mm ), showing the distribution of stress at the shaft, at twice full load. Maximum stress is 31.869 kgf.

procedure described for full-load case has to be repeated appropriately for these two cases.

The vibration that is triggered in SRMs due to magnetic, mechanical, and load-induced reasons has been addressed in this paper [15]. The stator, stator frame, rotor, end rings, bearings, shaft, pulleys, and applied loads have been modeled in three dimensions to study the vibration in SRMs as a whole.

SRINIVAS AND ARUMUGAM: STATIC AND DYNAMIC VIBRATION ANALYSES OF SRMs

A step-by-step 3-D vibration analysis procedures for modal, dynamic, harmonic, and static stress analyses are presented for a 6/4 SRM. The eccentricity in the SRM rotor including housing is found to be much less from the rotor dynamic study. From the static stress analysis, the reviewed stresses reveal that, up to twice the full load, the stresses are under safe limit. The procedure can be adopted for any pole combination and dimension of SRM to thoroughly perform vibration study in three dimensions. This will help to fine tune the design to be declared for end product. REFERENCES [1] P. J. Lawrenson, “Development of application of reluctance motor,” Proc. Electron. Power, pp. 195–198, June 1965. [2] P. J. Lawrenson, J. M. Stephenson, P. T. Blenkinsop, J. Corda, and N. N. Fulton, “Variable-speed switched reluctance motors,” Proc. Inst. Elect. Eng. B, vol. 127, pp. 253–265, July 1980. [3] M. R. Harris, H. R. Bolton, P. A. Ward, J. V. Byrne, G. B. Smith, J. Merrett, F. Devitt, R. J. A. Paul, K. K. Schwartz, M. F. Mangan, A. F. Anderson, R. Bourne, P. J. Lawrenson, J. M. Stephenson, and N. N. Fulton, “Discussions on variable speed switched reluctance motors,” Proc. Inst. Elect. Eng. B, vol. 128, pp. 260–276, Sept. 1981. [4] T. J. E. Miller and M. McGilip, “PC CAD for switched reluctance drives,” in Proc. IEE Electric Machines and Drives Conf., London, U.K., Dec. 1987, pp. 360–366. [5] N. M. Fulton, “The application of CAD to switched reluctance drives,” in Proc. IEE Electric Machines and Drives Conf., London, U.K., Dec. 1987, pp. 275–279. [6] C. G. C. Neves, R. Carlson, N. Sadowski, J. P. A. Bastos, N. S. Soeiro, and S. N. Y. Gerges, “Vibrational behavior of switched reluctance motors by simulation and experimental procedures,” IEEE Trans. Magn., vol. 34, pp. 3158–3161, Sept. 1998. [7] C. Yongxiao and W. Jianhua, “Analytical calculations of natural frequencies of stator of switched reluctance motor,” in Proc. 8th Int. Conf. Electric Machines and Drives, 1997, pp. 81–85. [8] C.-Y. Wu and C. Pollock, “Analysis and reduction of vibration and acoustic noise in the switched reluctance drive,” IEEE Trans. Ind. Applicat., vol. 31, pp. 91–98, Jan./Feb. 1995.

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[9] P. Pillay and W. Cai, “An investigation into vibrations in switched reluctance motors,” IEEE Trans. Ind. Applicat., vol. 35, pp. 589–596, Mar./Apr. 1999. [10] P. Pillay, R. M. Samudio, M. Ahmed, and P. T. Patel, “A chopper-controlled SRM drive for reduced acoustic noise and improved ride through capability using super capacitors,” IEEE Trans. Ind. Applicat., vol. 31, pp. 1029–1038, Sept./Oct. 1995. [11] A. Michaelides and C. Pollock, “Reduction of noise and vibration in switched reluctance motors,” in IEEE-IAS Annu. Meeting, 1996, pp. 771–778. [12] Y. Tang, “Characterization, numerical analysis and design of switched reluctance motors,” IEEE Trans. Ind. Applicat., vol. 33, pp. 1542–1552, Nov./Dec. 1997. [13] J. F. Lindsay, R. Arumugam, and R. Krishnan, “Finite-element analysis characterization of a switched reluctance motor with multi-tooth per stator pole,” Proc. Inst. Elect. Eng. B, vol. 133, pp. 347–353, Nov. 1986. [14] P. Vijayraghavan and R. Krishnan, “Noise in electric machines,” IEEE Trans. Ind. Applicat., vol. 35, pp. 1007–1014, Sept. 1999. [15] K. N. Srinivas, “Analysis and characterization of switched reluctance motors,” Ph.D. dissertation, Anna Univ., Chennai, Tamilnadu, India, Jan. 2003.

K. N. Srinivas (M’03) is an Assistant Professor in the Department of Electrical and Electronics Engineering, Crescent Engineering College, Chennai, India. His research mainly includes performance simulation of electric drives. Dr. Srinivas is the recipient of IECON 2000 and IECON 2003 awards and fellowships from the IEEE Industrial Electronics Society.

R. Arumugam (M’03) is Professor and the Director of the Electrical and Electronics Engineering Department, Anna University, Chennai, India. His main research area is electric drives. He is an industrial consultant with major Indian industries such as Lucas TVS, BPL Telcom Ltd., and Tamilnadu Electricity Board.