Analytical Formulae for Calculating SRM Modal Frequencies for Reduced Vibration and Acoustic Noise Design

Transcription

1 Analtical Formulae for Calculating SM Modal Frequencies for educed Vibration and Acoustic Noise Design W. Cai Delco em America, Inc. 9 Enterprise Drive, Anderson, IN 4613, caiw@delcorem.com P.Pilla Department of ECE Clarkson Universit Potsdam, NY pillap@clarkson.edu A.Omekanda Delpi esearc Labs 1786 Selb Parkwa Selb Townsip, MI avoki.omekanda@delpiauto. ABSTACT- Electronic tecniques for controlling acoustic noise and vibration depend on an accurate knowledge of te stator resonant frequenc. Analtical models are developed, wic allows te calculation of te first several modal frequencies. Te impact of te stator stack lengt on te accurac of te formulae is also examined. Experimental validation is included. I. INTODUCTION sapes. Te analtical formulae are compared wit a more accurate calculation using te finite element metod and experimental results. Te applicable range of te formulae is discussed, wic act as a guide to designers in te use of tese formulae. II. DEIVATION OF MODAL FEQUENCY FOMULAE One of te disadvantages of te Switced eluctance Motor (SM) is its vibration and corresponding acoustic noise. Man of te metods to reduce acoustic noise of te SM, like andom Pulse Widt Modulation (PWM) or active noise control using two or tree stage commutation, assume tat te resonant frequenc of te stator is known or measurable. It is essential to be able to predict te resonant frequencies and te response caracteristics of te stator in order to design a quiet SM or to avoid operating te motor near te resonant frequenc during drive operation. Te mode sape provides valuable information as to te excitation pattern to produce a particular resonance. Te resonant frequencies can be obtained b analtical calculation, numerical computation (finite element metod etc.) and/or experimental tecniques. Te experimental metod can be performed onl after a prototpe as been built. Terefore, deriving analtical formulae and building numerical computation models, wic ave been validated experimentall, are essential for motor designers. Because of te complexit of numerical computation, analtical metods are more popular in te earl stages of motor design. Te tradeoffs between different design parameters are easier to analze wit analtical models. Tere are advantages in combining te analtical metod wit numerical computation, i.e., to use te numerical metods to correct te deviation from analtical predictions due to simplifications necessar in te analtical model. Tis paper firstl derives a formula to predict te second order resonant frequenc of stator laminations. Tis is usuall te resonant frequenc of most interest. Te formula can also be used in cases were smoot frame structures are used to support te stator lamination stack. A more general formula is also developed to calculate te integer order resonant frequencies of te first few mode Formula for te frequenc of te nd order mode Te metod of computing approximate natural frequencies of a vibration sstem, using an energ approac, as become known as "aleig's metod. Te resonant frequenc of te fundamental mode (te nd order mode) is obtained b equating te peak kinetic energ T max wit te potential energ of te maximum deformation U max wen te stator lamination stack is modeled as a slender clindrical sell. Te derived formula for te nd order modal frequenc is given as follows f ω kc E = = (1) π π ρ ( were ω is te angular frequenc; is te tickness of stator oke; ( is te average radius of oke; E is Young's modulus of te stator lamination material; ρ Fe is te mass densit of te lamination materials; and te coefficient k c can be expressed as follows: k c Fe 1 4 k1 k = π π 1 3(1 ν )( k 1 + k1 k 4 were ν is te Poisson's ratio and te coefficients k 1, k can be expressed b Poisson's ratio and Hoop-stress deformation factors α.: 1 k1 = 1+ (7 + 1ν ) α k = 1 α ) 1 () (a)

2 1 ( ln[( α = 1 ( 1(1 ν ) ( ( + 1) ), ( ( 1)],.6 ( ( < > 8 8 (b) α is not equal to zero for te common dimensions of SM stator. Wen ( (a straigt beam) for given oke tickness, α and k 1 k 1wic was used in [4]. An error in te deformation calculation can occur if k 1 k 1 is used witout cecking te ratio ( /. Equation (1) is also called te "tick pipe model". Wen te average radius of te stator oke is muc larger tan te oke tickness, i.e., ( >>, te coefficients k 1 k 1. Furtermore taking Poisson's ratio ν =.3, te coefficient k c =.66, wic makes equation (1) identical to te formula in reference [4]. If (1-ν ) in Eq.() & Hook deformation coefficient expression is replaced b 1, te corresponding model is called te "tin ring model". III. FEQUENCY ESTIMATION OF MULTI-ODE MODES Equation (1) can be used to approximatel estimate te frequenc of te nd order vibration mode of SM stator. Te metod for accurate determination of te resonant frequenc of a complex stator structure in an induction motor as been described [8]. An alternative formula for modal frequencies of te rt order mode is derived [1] and given below 1 kr fr = (3) π m Were m is te equivalent mass per square meter (kg/m ) on te clindrical surface at average oke radius, wic can be expressed b te following formula: M m = (4) π L ( ( eff) and te equivalent spring stiffness coefficient per square meter k r (N/m 3 ) in Eq.(3) on te clindrical surface at average oke radius can be written as[1,9,1] E for r = ( k r = kd kd = witout sock absorber ) for r = 1 () r ( r 1) E for r 4 (1 + r ) 1( It sould be noticed tat te formulas (3)~() can be applied to predict onl te natural frequencies of te first several low order vibration modes. Te ma lead to unacceptable errors if te formulas are used to calculate te mode greater tan te t order. Tis is sown in te comparison wit te numerical metod and te measured values. IV. PACTICAL CALCULATION AND ESULT COMPAISON In order to verif te validit of te above formulas, extensive numerical calculations and an experimental investigation were carried out on a 4kW, 8/6 pole, 4 pase switced reluctance motor. Te resonant frequenc results of te nd order mode from te analtical models are compared wit te corresponding results from te finite element metod, as sown in Table 1. Te finite element result (817.3Hz) is based on te 3D free vibration model. Te errors of te analtical models are witin te allowable range of engineering error. Effect of Lamination Lengt: To explore te effects of stator lamination lengt on te resonant frequenc of te nd order in-plane vibration, te nd order frequencies are calculated b FEM troug te lamination lengt range of 89.8 ~19mm wile te cross section dimensions of te stator laminations are kept uncanged. Te finite element results are given in Fig.1. Te nd order resonant frequencies in Fig. sow onl a sligt variation (.4%) wit cange of stator lamination lengt wen te ratio between lengt/radius varies from 1 to. Furtermore, te resulting errors of te analtical TABLE 1. COMPAISON OF FEQUENCY ESULTS BETWEEN MODELS Models Tick Pipe Tin ing Multi-order Frequenc(Hz) Error *(%) <*> Te base value is FE result Hz nd order modal frequenc (Hz) Lamination lengt (mm) Fig.1 Te nd order modal frequenc vs. lamination lengt b FE

3 Percentage error of frequenc 1 tick pipe model tin ring model Percentage error of frequenc tick pipe model tin ring model Lamination lengt (mm) Fig. Percentage error of frequenc vs. lamination lengt formulas wit lamination lengt canges, compared wit tat b numerical calculations, are demonstrated in Fig.. All te D analtical formulas for analzing in-plane flexural modes in te paper are accurate enoug to satisf industrial applications. Effect of Stator Yoke adius/tickness: To searc for te applicable range of te analtical formulas presented in te paper, te nd order resonant frequencies of te stator lamination are calculated troug a range of oke radius of 8.8~99.8mm. In te calculation, te lengt of te laminations is kept at te actual lengt of te stator stack (11mm) of te 4kW SM. Te curves of te relationsip between te nd mode frequenc and oke outer radius are sown in Fig.3. Compared wit te frequenc values from te finite element metod, te errors of te tree analtical nd order modal frequenc (Hz) tick pipe model tin ring model FE free vibration Outer radius of stator oke (mm) Fig.3 nd order modal frequenc vs. oke radius wen te lamination lengt is kept uncanged atio of average oke radius to tickness Fig.4 Percentage error of frequenc vs. ratio of oke radius/tickness wen lamination lengt = const models are drawn in Fig.4. In tis figure, te ratio of average oke radius to tickness is taken as te abscissa in order to generalize te applicable range of te analtical formulas. Te ratio from 4.16 to 11.1 corresponds to te outer radius range of stator laminations from 99.8 to 8.8mm in te real 4kW SM wen its oke inner radius fixed at 78.4mm. From Fig.4, te frequenc errors are witin % wen te radius/tickness of te stator oke falls in te range of 4~7.3 in te tick pipe model,.4~9. in te tin ring model, 4~1 in te multi order model. Witin tese ratio ranges, te corresponding formulas meet te needs of engineering requirements for estimation of te nd resonant frequenc. Te applicable range for frequenc estimation b te multi order model seems muc wider. Low Order Frequenc Estimation: Te formulas of te multi order model can teoreticall be applied to predict te frequencies of integer order vibration modes, but te real calculation sows onl limited order frequencies ma be estimated b te formulas (3)~(). Te frequenc errors produced b te multi order model are sown in Fig., compared wit te values wit finite element results. Te mode sapes of te vibrations include te t, nd, 3 rd and 4 t order, wose deformations are sown as in Fig.6. Te 1st order mode reflecting free vibration of te stator oke can be estimated b reference [1] if te feet of te motor are fixed on an infinite base, wic is similar to te rotating vibration under mass unbalance of rotor. From Fig., te percentage errors of te resonant frequencies from formulas (3)~(), can be restricted to witin % if te radius/tickness of te stator oke falls into te following ranges: 4~8.8range for te t order, 4.~11 for nd order, more tan 8. for te 3 rd order, and 4.4~7. for te 4 t order. Obviousl tese formulas are accurate enoug to predict te frequencies of te t, nd and some 4 t order vibration modes. But it sould be noticed tat

4 1 1 -order mode nd mode 3rd mode 4t mode Percentage error of frequenc atio of average oke radius to tickness Fig. Percentage error of frequenc vs. oke radius/tickness under multi order model (order,,3,4) (a) t order mode (b) nd order mode Fig.7 Frequenc spectrum under single pase excitation (epetition=4hz) Dominant frequenc=134hz (c) 3 rd order mode (d) 4 t order mode Fig.6 Deformation of 4 order mode sapes te multi order model brings unacceptable errors if it is used to calculate te frequenc beond te t order vibration mode. V. COMPAISON WITH MEASUEMENTS To verif te validit of te formulas, te calculated values of te nd order modal frequenc is compared wit te measurement results of a 4kW SM. Te contact between laminations and case is assumed to be perfect due to te encased structure. Te frequenc measurements are performed wit an accelerometer beind te stator pole wile te pase windings on tese two poles are conducting pulse current. Te frequenc of te exciting current is muc lower tan te nd mode vibration frequenc so tat te acceleration signals from te accelerometer are allowed to deca to a low enoug value during te interval between te adjacent current pulses. Fig.7 sows te waveform of te suppl voltage, current pulse and te acceleration in te time domain as well as te power spectrum in te frequenc domain. Te acceleration waveform clearl demonstrates tat te vibrations occur at switc-off of te pase current, instead of during switced-on. Tis result confirms conclusions drawn in previous papers [~4]. Te calculated results wit te analtical formulas and corresponding percentage errors relative to te measured values are given in Table. It sould be pointed out tat te measured SM as radiator ribs, wic ma reduce te resonant frequenc of te nd order mode sape [1]. Te measurement frequenc (138Hz) includes te effects of tis factor, and almost exactl matces te numerical results

5 TABLE. COMPAISON TO MEASUEMENT ESULTS Models Tick Pipe Tin ing Multi-order Frequenc(Hz) Error *(%).6.1. <*> Te base value is test result 134 Hz (1366Hz) b finite element model including ribs and terminal box of te macine. Te ribs are considered as extra masses attaced to te stator oke in te "multi order model". VI. CONCLUSION Te prediction of resonant frequencies is ver important for te design of quiet switced reluctance macines and for drive control of low noise SMs. Two simplified formulas are derived to calculate te nd order modal frequenc of te SM, and teir applicable ranges are discussed in te paper. A more sopisticated formula can also be used to predict te t, 3 rd, and 4 t order modal frequencies witin a certain range of oke radius/tickness. Troug practical measurements and a comparison wit numerical results, several useful conclusions are drawn as follows: Te effect of te lengt of te stator laminations on inplane natural vibration frequencies can be neglected, so tat D analtical models for a SM wit smoot frame is accurate enoug to satisf te application requirements in te prediction of in-plane vibration modes. Te effects of frame ribs ave been neglected. Wit stator poles and winding space kept uncanged, te nd order modal frequenc increases linearl wit te outer radius of te stator oke. Te multi order model is accurate to a wider range of practical stator dimensions to calculate te modal frequencies. If te ratio of average oke radius to tickness falls into te.4~9. range, te tin ring model can be applied to estimate te modal frequenc and te error is less tan %. Te multi order formula can be used to predict te low order modal frequencies. Te accurac of te formulas is ig enoug for te calculation of te t, nd and 4 t order (witin ( /H = 4.~7.3) modal frequencies. Te FE results are closer to te experimental results because te complexit of te stator structure must be reduced before developing te analtical models. Additional natural frequencies are found in te numerical results besides tose corresponding to te integer order mode sapes. Te measurement results sowed tat te frame ribs lowered te nd order frequenc. Te measured waveform demonstrated tat te nd order modal vibration occurred at te turn-off moment of te current pulse, not switced-on, tus validating man former papers [1~4]. Acknowledgements: Te autors acknowledge te support of Delpi esearc Labs, Selb Townsip, Micigan, and te US Nav, troug te Office of Naval esearc for an equipment researc grant. EFEENCES [1] P.Pilla, W. Cai, An investigation into vibration in switced reluctance motor, Conference ecord of te 1998 IEEE Industrial Application Societ, 33 t IAS Annual Meeting, Vol.1, St. Louis, USA, Oct. 1-16, [] D.E. Cameron, J.H. Lang, S.D.Umans, Te origin and reduction of acoustic noise in doubl salient variable-reluctance motors, IEEE Trans. on Industr Applications, Vol.8, No.6, November/December, 199, pp.1~1. [3] C. Pollock, C.Y.Wu, Acoustic noise cancellation tecniques for switced reluctance drives, Conference ecord of te 199 IEEE Industrial Application Societ, 3 t IAS Annual Meeting, Vol.1, Orlando, USA, Oct. 8-1, 199, pp [4].S.Colb, F.Mottier, T.J.E.Miller, Vibration modes and acoustic noise in a 4-pase switced reluctance motor, Conference ecord of te 199 IEEE Industrial Application Societ, 3 t IAS Annual Meeting, Vol.1, Orlando, USA, Oct. 8-1, 199, pp.441~447. [] P.Pilla,.M.Samudio, M.Amed, P.T.Patel, A Copper-controlled SM drive for reduced acoustic noise and improved ride-troug capabilit using super-capacitor, IEEE Trans. on Industr Applications, Vol.31, No., September/October, 199, pp.19~138. [6] Singiresu S. ao, Mecanical Vibrations (3 rd edition), ISBN: , Addison Wesle, 199. [7] Warren C. Young, oark's Formulas for Stress & Strain (sixt edition), McGraw-Hill, ISBN , 1989, pp.67, 63. [8] S.P.Verma,.S.Girgis, Metod for accurate determination resonant frequencies and vibration beaviour of stators of electrical macines, Proceedings of IEE, Part B, Vol.18, No.1, Januar 1981, pp [9] obert D. Blevins, Formulas for Natural Frequenc and Mode Sape, Van Nostrand einold, New York, ISBN , 1979, pp.. [1] Sikun Cen, Electrical Maciner Design (1 st edition), Press of Maciner Industr, Beijing, Cina, 198 (in Cinese).