# Nonlinear Resonance Responses of Electromechanical Integrated Magnetic Gear System

Authors: Xiu-hong Hao; Hong-qian Zhu; Deng Pan
Journal: Shock and Vibration
 (2018)

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## Abstract

Nonlinear differential equations for an electromechanical integrated magnetic gear (EIMG) system are developed by considering the nonlinearity in the magnetic force of the system components. Expressions for the main resonances and superharmonic resonances are obtained for output wave frequencies close to the natural frequency and half the natural frequency of the derived EIMG system. The response laws are discussed in detail. The magnetic coupling stiffness among the components is found to exhibit distinct nonlinearity, leading to strong main resonances and superharmonic resonances. Smaller values of the magnetic coupling stiffness and damping result in larger response amplitudes and transient responses that slowly decay to zero. When the main resonances and superharmonic resonances occur, the dominant frequency of the response is the natural frequency of the derived EIMG system, and the amplitudes of different components of the resonance display large differences.


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## Body

## 1. Introduction

Magnetic drives, and especially magnetic gear boxes, are extensively used in the medical, food, and chemical industries. These systems have the advantages of being contactless, wear-free, and lubrication-free, with low levels of vibration and noise, and built-in overload protection [1]. The field-modulated magnetic gear (FMMG) has a coaxial topology, which significantly improves the permanent magnet (PM) utilization rate and can generate much higher torque and torque density than traditional magnetic gears that use a parallel shaft topology [2]. Therefore, FMMGs are widely used in the vehicle, wind, marine, and aerospace fields [3–5]. Various high-performance FMMG-based systems have been proposed, such as axial-flux magnetic gears, linear magnetic gears, and intersecting axes magnetic gears [6–8]. Various direct current PM rotors and alternating current PM motors have been developed for use in electric vehicles and wind power systems [9, 10]. Together, these innovations have significantly promoted the development of magnetic gears.Electrical and control techniques are widely used in mechanical engineering, resulting in the development of many cutting-edge composite drives such as the electromagnetic harmonic drive and electromechanical integrated toroidal drive. One of the authors has previously presented a drive system known as the electromechanical integrated magnetic gear (EIMG) [11]. This integrates power and control systems into FMMG. Thus, EIMG possesses the advantages of FMMG alongside a compact structure with torque and speed control. Compared with FMMG, EIMG can generate higher torque at lower speeds. EIMG can transmit torque by magnetic field coupling, unlike mechanical gears, which require direct contact to do so. The magnetic coupling stiffness of EIMG is much lower than the contact stiffness of mechanical gears [12, 13], and EIMG has different dynamic characteristics compared with mechanical gear systems. Because EIMG system has a lower magnetic coupling stiffness, its natural frequencies are much lower than those of mechanical gear systems. As a result, low-frequency resonance can easily occur. When such resonance occurs, the amplitudes of the EIMG system’s components slowly attenuate, degrading the dynamics of the system. To overcome this problem, Frank et al. added damper windings to synchronous generators to suppress the oscillations caused by transients [14]. Montague et al. used servo control in magnetic gears to overcome the disadvantages of low stiffness [15].As EIMG is a relatively new type of magnetic gear, its dynamics should be thoroughly studied to control the behavior and improve the service performance of such a system. This study develops a dynamic model and differential equations for EIMG system by considering the nonlinearity of the magnetic coupling stiffness. The main resonances and superharmonic resonances produced by a torque wave are discussed, and these provide the foundation for the parameter optimization and performance improvement of the EIMG system.


## 2. Nonlinear Dynamic Model of EIMG

Figure1 shows the inner rotor, inner and outer stator, and inner and outer ferromagnetic pole-pieces (FP) that comprise an EIMG system. All components are concentric, and the EIMG has four air gaps. Windings made of insulated wire are mounted in the dew-drop slot of the inner stator. PMs are positioned uniformly on the inner surface of the outer stator and the inner and outer surfaces of the inner rotor. The inner and outer FP comprise permeable and nonpermeable magnetic materials layered at regular intervals. FP modulate the magnetic fields in the two air gaps next to them, thus ensuring that the number of pole pairs of PMs on the inner rotor and outer stator agrees with the number of pole pairs of the space harmonic flux density of the air gaps.Figure 1
Topology of electromechanical integrated magnetic gear.A three-phase alternating current is supplied to the inner stator coils to generate a rotary magnetic field with a main harmonic ofp1. After being modulated by the inner FP, the main harmonic p2 of the magnetic field in the second air gap is consistent with the number of pole pairs of PMs on the inner surface of the inner rotor. Therefore, the inner rotor can be actuated, thereby realizing the first drive. When the inner rotor runs, a magnetic field with a main harmonic of p3 in the third air gap is generated and modulated by the outer FP. As a result, the main harmonic p4 of the magnetic field in the fourth air gap becomes equal to the number of pole pairs of PMs on the outer stator. Then, the second drive is achieved.To realize equal pole coupling among the EIMG components,p1,  p2,  p3, and p4 have the following constraint relations:(1)p12=p1+p2,p34=p3+p4,where p1 is the number of pole pairs of the three-phase alternating current and p4,  p2, and p3 are the number of pole pairs of PMs on the stator and the inner and outer surfaces of the inner rotor, respectively.The magnetic coupling torques among the components are calculated as follows:(2)T1=Tc1sin⁡p1θ12+θ10,T3=Tc3sin⁡p2θ23+θ10=Tc3sin⁡p3θ34+θ20,T5=Tc5sin⁡p4θ45+θ20,where T1,  T3, and T5 are the torques on the inner stator, inner rotor, and outer stator, respectively; Tc1,  Tc3, and Tc5 are the maximum torques on the inner stator, inner rotor, and outer stator, respectively; θ10 and θ20 are the relative rotation angles between the inner stator and inner rotor and between the inner rotor and outer stator, respectively; and θ12,  θ23,  θ34, and θ45 are the relative torsional displacements between the inner stator and inner FP, inner FP and inner rotor, inner rotor and outer FP, and outer FP and outer stator, respectively.Figure2 shows the dynamic model of the EIMG system. This model contains four subsystems: inner stator/inner FP subsystem, inner FP/inner rotor subsystem, inner rotor/outer FP subsystem, and outer FP/outer stator subsystem. Because the magnetic coupling stiffness is much lower than the supporting stiffness [12], only torsional displacements are considered, and transverse vibration displacements are assumed to be negligible.Figure 2
Nonlinear dynamic model of electromechanical integrated magnetic gear.
(a)
Inner stator/inner FP subsystem (b)
Inner FP/inner rotor subsystem (c)
Inner rotor/outer FP subsystem (d)
Outer FP/outer stator subsystemThe tangential magnetic coupling forces among the components are calculated as follows:(3)FsI=T1R1=Tc1sin⁡p1θ1-θ2+θ10R1,FII=T3R3=Tc3sin⁡p2θ2-θ3+θ10R3,FIo=T3R3=Tc3sin⁡p3θ3-θ4+θ20R3,Foo=T5R5=Tc5sin⁡p4θ4-θ5+θ20R5,where FsI,  FII,  FIo, and Foo are the tangential magnetic coupling forces between the inner stator and inner FP, inner FP and inner rotor, inner rotor and outer FP, and outer FP and outer stator, respectively; R1,  R3, and R5 are the equivalent radius of gyration of the inner stator, inner rotor, and outer stator, respectively; and θk (k=1,2,3,4,5) is the torsional angular displacement of the inner stator, inner FP, inner rotor, outer FP, and outer stator, respectively.For convenience, the torsional angular displacements are all replaced by their corresponding torsional displacements as(4)x=x1x2x3x4x5T,where xk (k=1,2,3,4,5) is the torsional displacement of the inner stator, inner FP, inner rotor, outer FP, and outer stator, respectively, and xk=Rkθk.The magnetic coupling forces among the components are functions of the relative torsional displacements. Whent=0, namely, xi=0, the magnetic coupling forces are expanded using the Taylor series as(5)FIs=h1+a1x1+b1x2+c1x12+d1x22+e1x1x2,FII=h2+a2x2+b2x3+c2x32+d2x22+e2x2x3,FIo=h3+a3x3+b3x4+c3x32+d3x42+e3x3x4,Foo=h4+a4x4+b4x5+c4x52+d4x42+e4x4x5,where h1=Tcssin⁡θ10/R1,  a1=Tcsp1cos⁡θ10/R13,  b1=-Tcsp1cos2⁡θ10/R1R22,  c1=Tcsp12cos2⁡θ10/2R15,  d1=Tcsp12cos2⁡θ10/2R1R24, and e1=-Tcsp12cos2⁡θ10/R13R22;  h2=TcIsin⁡θ10/R3,  a2=TcIp2cos⁡θ10/R3R22,  b2=-TcIp2cos⁡θ10/R33,  c2=TcIp22cos2⁡θ10/2R3R24,  d2=TcIp22cos2⁡θ10/2R35, and e2=-TcIp22cos2⁡θ10/R33R22;  h3=TcIsin⁡θ20/R3,  a3=TcIp3cos⁡θ20/R33,  b3=-TcIp3cos⁡θ20/R3R42,  c3=TcIp32cos2⁡θ20/2R35,  d3=TcIp32cos2⁡θ20/2R3R44, and e3=TcIp32cos2⁡θ20/R33R42; and h4=Tcosin⁡θ20/R5,  a4=Tcop4cos⁡θ20/R42R5,  b4=-Tcop4cos⁡θ20/R53,  c4=Tcop42cos2⁡θ20/2R53R42,  d4=Tcop42cos2⁡θ20/2R55, and  e4=-Tcop42cos2⁡θ20/R42R53.The magnetic coupling forces among the components are nonlinear functions of the relative torsional displacements. The constant components of the Taylor series expansions of the magnetic coupling forces will lead to relative static displacements rather than dynamic displacements. Therefore,hi is neglected. Based on Newton’s second law of motion, the nonlinear dynamic differential equations of an EIMG system can be expressed as follows:(6)M1x¨1+csx˙1+a1x1+b1x2+c1x12+d1x22+e1x1x2+ksx1=0,M2x¨2+cIfx˙2-a1x1+b1x2+c1x12+d1x22+e1x1x2+a2x2+b2x3+c2x32+d2x22+e2x2x3+kIfx2=0,M3x¨3+cIx˙3-a2x2+b2x3+c2x32+d2x22+e2x2x3+a3x3+b3x4+c3x32+d3x42+e3x3x4=0,M4x¨4+cofx˙4-a3x3+b3x4+c3x32+d3x42+e3x3x4+a4x4+b4x5+c4x52+d4x42+e4x4x5=ΔTcos⁡ωetR4,M5x¨5+cox˙5-a4x4+b4x5+c4x52+d4x42+e4x4x5+kox5=0,where m1 and m2 are the masses of the inner stator and inner FP, respectively; M1 (M1=J1/R12) and M2 (M2=J2/R22) are the equivalent masses of the inner stator and inner FP along their torsional vibration direction, respectively; J1 (J1=m1·R12/2) and J2 (J2=m2·R22/2) are the mass moments of inertia of the inner stator and inner FP, respectively; m3 and M3 (M3=J3/R32) are the mass and equivalent mass of the inner rotor, respectively; J3 (J3=m3·R32/2) is the mass moment of inertia of the inner rotor; m4 and M4 (M4=J4/R42) are the mass and equivalent mass of the outer FP, respectively; J4 (J4=m4·R42/2) is the mass moment of inertia of the outer FP; m5 and M5 (M5=J5/R52) are the mass and equivalent mass of the outer stator, respectively; J5 is the mass moment of inertia of the outer stator; ks,  kIf, and ko are the torsional support stiffness values of the inner stator, inner FP, and outer stator, respectively; cs, cIf, cI, cof, and co are the damping coefficients of the inner stator, inner FP, inner rotor, outer FP, and outer stator, respectively (cs=ζs2m1a1ks/a1+ks,  cIf=ζIf2m2a1a2kIf/a1+kIf+a2,  cI=ζI2m3a2a3/a2+a3,  cof=ζof2m4a3a4/a3+a4, and  co=ζo2m5a4ko/a4+ko, where ζs,  ζIf,  ζI,  ζof, and ζo are the damping ratios of the inner stator, inner FP, inner rotor, outer FP, and outer stator, resp.).Equation (6) can be rewritten in matrix form as(7)Mx¨+cx˙+kx=F+ΔF.The mass matrixM, displacement vector x, damping matrix c, stiffness matrix k, load vector F, and equivalent load vector ΔF have the following forms:(8)M=diag⁡M1M2M3M4M5x0=x1x2x3x4x5T,c=diag⁡cscIfcIcofco,k=a1+ksb1000-a1a2-b1+kIfb2000-a2a3-b2b3000-a3a4-b3b4000-a4ko-b4,F=F1F1F1F1F1T,ΔF=000ΔTcos⁡ωetR40TF1=-c1x12+d1x22+e1x1x2F2=c1x12+d1x22+e1x1x2-c2x32+d2x22+e2x2x3F3=c2x32+d2x22+e2x2x3-c3x32+d3x42+e3x3x4F4=c3x32+d3x42+e3x3x4-c4x52+d4x42+e4x4x5F5=c4x52+d4x42+e4x4x5,where ΔT is the amplitude of the torque fluctuation and ωe is the excitation frequency of the torque wave.Equation (7) can be normalized as(9)x¨N+cNx˙N+kNxN=FN,where xN,  kN, and FN are the normal displacement vector, normal stiffness matrix, and normal equivalent load vector, respectively; these are expressed as follows:(10)xN=xN1xN2xN3xN4xN5T,kN=diag⁡kN1kN2kN3kN4kN5=diag⁡ω1ω2ω3ω4ω5,FN=FN1FN2FN3FN4FN5T,FNk=AN4kΔTcos⁡ωetR4+AN1iF1+AN2iF2+AN3iF3+AN4iF4+AN5iF5,where ωi is the natural frequency of the linear system derived for the EIMG.Because the damping matrixc is diagonal, the elements on the primary diagonal of the normal damping matrix cN are much larger than the other elements. Therefore, the normal damping matrix cN in (9) is simplified into the following diagonal matrix:(11)cN=diag⁡cN1cN2cN3cN4cN5.


## 3. Main Resonance

A multiscale method is used to solve (9). To balance the effects of the damping forces and nonlinear elements and express them in the same perturbation equation, the following assumptions are made:(12)xNk=xNk0T0,T1+εxNk1T0,T1+⋯,ΔT=εΔT′,cNk=εcNk′.When the wave frequency of the output torque is close to the natural frequency of the outer FP rotational mode, the following assumption is introduced:(13)ωe=ω4+εσ1.By substituting (12) and (13) into (9) and setting the power coefficients of the small parameters ε on both sides of the equations to be equal, the following differential equations are obtained:Zero power(14)D02xNk0+ωk2xNk0=0First power(15)D02xNk1+ωk2xNk1=-cN1′D0xNk0-2D0D1xNk0+FNk0,where (16)FNk0=AN4kΔT′cos⁡ωetR4+AN1kF10+AN2kF20+AN3kF30+AN4kF40+AN5kF50,F10=-c1x102+d1x202+e1x10x20,F20=c1x102+d1x202+e1x10x20-c2x302+d2x202+e2x20x30,F30=c2x302+d2x202+e2x20x30-c3x302+d3x402+e3x30x40,F40=c3x302+d3x402+e3x30x40-c4x502+d4x402+e4x40x50,F50=c4x502+d4x402+e4x40x50.Under the normal coordinate system, the general solution of (14) is expressed as(17)xNk0=AkT1eiωkT0+cc.Under the regular coordinate system, the solutions of (14) are(18)x10=α1A1T1eiω1T0+α2A2T1eiω2T0+α3A3T1eiω3T0+α4A4T1eiω4T0+α5A5T1eiω5T0+cc,x20=β1A1T1eiω1T0+β2A2T1eiω2T0+β3A3T1eiω3T0+β4A4T1eiω4T0+β5A5T1eiω5T0+cc,x30=γ1A1T1eiω1T0+γ2A2T1eiω2T0+γ3A3T1eiω3T0+γ4A4T1eiω4T0+γ5A5T1eiω5T0+cc,x40=μ1A1T1eiω1T0+μ2A2T1eiω2T0+μ3A3T1eiω3T0+μ4A4T1eiω4T0+μ5A5T1eiω5T0+cc,x50=η1A1T1eiω1T0+η2A2T1eiω2T0+η3A3T1eiω3T0+η4A4T1eiω4T0+η5A5T1eiω5T0+cc,where α1=AN11,  α2=AN12,  α3=AN13,  α4=AN14, and α5=AN15;  β1=AN21,  β2=AN22,  β3=AN23,  β4=AN24, and β5=AN25;  γ1=AN31,  γ2=AN32,  γ3=AN33,  γ4=AN34, and γ5=AN35;  μ1=AN41,  μ2=AN42,  μ3=AN43,  μ4=AN44, and μ5=AN45; and η1=AN51,  η2=AN52,  η3=AN53,  η4=AN54, and η5=AN55.Here,ANkj denotes the element in the kth row and jth column of the normal modal matrix AN:(19)AN=AN11AN12AN13AN14AN15AN21AN22AN23AN24AN25AN31AN32AN33AN34AN35AN41AN42AN42AN42AN42AN51AN52AN53AN54AN55.By substituting (18), (15) can be rewritten as(20)D02xNk1+ωk2xNk1=-cN1′iωkAkeiωkT0-2Ak′eiωkT0+iωkAkeiωkT0+AN4kΔT′cos⁡ωetR4+∑k=15∑j=15GkjAkAjeiωk+ωjT0+A-jeiωk-ωjT0+cc,where cc is the complex conjugate of the right-hand-side expression in (20) and Gkj is a constant associated with AN,  ck,  dk, and ek that is expressed as follows:(21)Gkj=-AN1kc1αkαj+d1βkβj+e1αkβj+AN2kc1αkαj+d1βkβj+e1αkβj-c2γkγj+d2βkβj+e2γkβj+AN3kc2γkγj+d2βkβj+e2γkβj-c3γkγj+d3μkμj+e3γkμj+AN4kc3γkγj+d3μkμj+e3γkμj-c4ηkηj+d4μkμj+e4ηmμj+AN5ic4ηkηj+d4μkμj+e4ηkμj.There are multiple frequency components in the right-hand-side expressions of (20), such as ±2ωk,  ±ωk±ωj, and ±ωe, where m=1,2,3,4,5 and k≠j. When there is no internal resonance, the secular terms can be eliminated to obtain(22)-cNm′Am-2A˙m=0,-cN4′iω4A4-2iω4A˙4+AN44ΔT′eiεσ1tR4=0,where m=1,2,3, and 5.Solutions of (22) are obtained as follows:(23)Am=Eme-cNkt/2,A4=E4e-cN4t/2-AN44ΔT′eiεσ1t+θ1Rofω4cN42+4ε2σ12,where Ek and E4 are constants related to the output torque, φ1 is a constant related to ε and σ, cos⁡φ1=2εσ1/cN4′2+4ε2σ12, and sin⁡φ1=cN4/cN4′2+4ε2σ12.Under the normal coordinate system, the zero-order approximate analytical solution of the EIMG system is expressed as(24)xNm0=Eme-cNm′t/2eiωmT0+cc,xN40=E4e-cN4′t/2eiω4T0-AN44ΔT′eiω4t+εσ1t+φ1R4ω4cN42+4ε2σ12+cc.After eliminating the secular terms, the first-order approximate solutions are obtained by substituting (23) and (24) into (20):(25)xNm1=AN44ΔT′ejωetR4ωm2-ωe2+xN41,xN41=∑m=1,2,3,5∑j=1,2,3,5GmjEmEje-cNj′+cNm′t/2eiωm±ωjtωm2-ωm±ωj2+∑m=1,2,3,5Gm4EmE4e-cN4′+cNmt/2eiωm±ω4tωm2-ωm±ω42-X4Eme-cNm′t/2eiωm+ω4t±εσ1t+φ1ωm2-ωm+ω4+εσ12+∑j=1,2,3,5G4jEjE4e-cN4′+cNj′t/2eiω4±ωjT0ωm2-ω4±ωj2-Xe-cNj′t/2eiω4+ωjt±εσ1t+φ1ωm2-ω4+ωj+εσ12+E42e-cN4′te2iω4tω42-2ω42+X4E4e-cN4′t/2eiωm+ω4t+εσ1t+φ1ω42-2ω4+εσ12-X4E4e-cN4′t/2ei2ω4t+εσ1t+φ1ω42-2ω4+εσ12-X42ei2ω4t+2εσ1t+2φ1ω42-2ω4+2εσ12+cc,where(26)X4=AN44ΔT′R4ω4cN4′2+4ε2σ12;Gmj=-AN1ic1αmαj+d1βmβj+e1αmβj+AN2ic1αmαj+d1βmβj+e1αmβj-c2γmγj+d2βmβj+e2γmβj+AN3ic2γmγj+d2βmβj+e2γmβj-c3γmγj+d3μmμj+e3γmμj+AN5ic4ηmηj+d4μmμj+e4ηmμj+AN4ic3γmγj+d3μmμj+e3γmμj-c4ηmηj+d4μmμj+e4ηmμj.E m will approach zero with increasing time because of the damping among components. The stable responses of the first-order approximate analytic solution of the EIMG system can be obtained by substituting (24) and (25) into (12):(27)xNm=εAN44ΔT′eiωetR4ωm2-ωe2+εX42ei2ω4t+2εσ1t+2φ1ω42-2ω4+2εσ12+cc,xN4=-AN44ΔT′eiω4t+εσ1t+θ1R4ω4cN4′2+4ε2σ12+εX42ei2ω4t+2εσ1t+2φ1ω42-2ω4+2εσ12+cc.Based on (27), approximate analytic solutions of the EIMG system under the regular coordinate system are given by(28)x=ANxN.When the torque wave frequency approaches a certain frequencyωk of the derived EIMG system, resonance will occur. In this situation, stable responses of the EIMG system under the normal coordinate system are calculated as(29)xNj=εAN4jΔT′eiωetR4ωj2-ωe2+εXj2ei2ωjt+2εσ1t+2φ1ωj2-2ωj+2εσ12+cc,xNk=-AN4jΔT′eiωjt+εσ1t+θ1R4ωjcNj′2+4ε2σ12+εXj2ei2ωjt+2εσ1t+2φ1ωj2-2ωj+2εσ12+cc,where Xk=AN44ΔT′/R4ωkcNk′2+4ε2σ12.


## 4. Superharmonic Resonance

When the wave frequency of the output torque is close to half of the natural frequency of the outer FP rotational mode, the following assumption is introduced:(30)xNj=xNj0T0,T1+εxNj1T0,T1+⋯,2ωe=ω4+εσ2,cNj=εcNj′.By substituting (30) into (9) and setting the power coefficients of the small parameters ε on both sides of the equation to be equal, the following differential equations are obtained:Zero power(31)D02xNj0+ωj2xNj0=FNj0′First power(32)D02xNk1+ωk2xNk1=-cN1′iωkAkeiωkT0-2Ak′eiωkT0+iωkAkeiωkT0+Lke2jωeT0+Lk∑m=15HkAmeiωm+ωeT0+A-mei-ωm+ωeT0+∑k=15∑j=15GkjAkAjeiωk+ωjT0+A-jeiωk-ωjT0+cc,where (33)FNj0=AN4jΔTcos⁡ωetR4,FNj0′′=AN1jF10+AN2jF20+AN3jF30+AN4jF40+AN5jF50,Hk=AN1ic1αm+d1βmρ2+e1βmρ1+αmρ2+AN2ic1αm+d1βmρ2+e1βmρ1+αmρ2-c2γmρ3-d2βmρ2-e2βmρ3+γmρ2+AN3ic2γmρ3+d2βmρ2+e2βmρ3+γmρ2-c3γmρ3-d3μmρ4-e3γmρ4+μmρ3+AN4ic3γmρ3+d3μmρ4+e3γmρ4+μmρ3-c4γmρ5-d4μmρ4-e4ηmρ4+μmρ5+AN5ic4γmρ5+d4μmρ4+e4ηmρ4+μmρ5Lk=AN1ic1ρ12+d1ρ22+e1ρ1ρ2+AN2ic1ρ12+d1ρ22+e1ρ1ρ2-c2ρ32+d2ρ22+e2ρ2ρ3+AN3ic2ρ32+d2ρ22+e2ρ2ρ3-c3ρ32+d3ρ42+e3ρ3ρ4+AN4ic3ρ32+d3ρ42+e3ρ3ρ4-c4ρ52+d4ρ42+e4ρ4ρ5+AN5ic4ρ52+d4ρ42+e4ρ4ρ5.Under the normal coordinate system, the general solution of (31) is expressed as(34)xNj0=AjT1eiωjT0+BjeiωeT0+cc,where Bj=AN4jΔT/R4ωj2-ωe2.Under the regular coordinate system, solutions of (31) are calculated as(35)x10=α1A1T1eiω1T0+α2A2T1eiω2T0+α3A3T1eiω3T0+α4A4T1eiω4T0+α5A5T1eiω5T0+ρ1ejωeT0+cc,x20=β1A1T1eiω1T0+β2A2T1eiω2T0+β3A3T1eiω3T0+β4A4T1eiω4T0+β5A5T1eiω5T0+ρ2ejωeT0+cc,x30=γ1A1T1eiω1T0+γ2A2T1eiω2T0+γ3A3T1eiω3T0+γ4A4T1eiω4T0+γ5A5T1eiω5T0+ρ3ejωeT0+cc,x40=μ1A1T1eiω1T0+μ2A2T1eiω2T0+μ3A3T1eiω3T0+μ4A4T1eiω4T0+μ5A5T1eiω5T0+ρ4ejωeT0+cc,x50=η1A1T1eiω1T0+η2A2T1eiω2T0+η3A3T1eiω3T0+η4A4T1eiω4T0+η5A5T1eiω5T0+ρ5ejωeT0+cc,where ρ1=AN11B1+AN12B2+AN13B3+AN14B4+AN15B5, ρ2=AN21B1+AN22B2+AN23B3+AN24B4+AN25B5, ρ3=AN31B1+AN32B2+AN33B3+AN34B4+AN35B5, ρ4=AN41B1+AN42B2+AN43B3+AN44B4+AN45B5, and ρ5=AN51B1+AN52B2+AN53B3+AN54B4+AN55B5.After substituting (35) into (32), there are multiple frequency components such as 2ωj, ±ωj±ωk, 2ωe, and ±ωj±ωe, where k=1,2,3,4,5 and k≠j. When there is no internal resonance, the secular terms can be eliminated from (32) to obtain(36)-cNmAm-2A˙m=0,-cN4iω4A4eiω4t-2iω4A˙4eiω4t+L4e2iωeT0=0.The solutions of (36) are written as(37)Am=Eme-cNm′t/2,A4=E4e-cN4′t/2-L4eiεσ2t+φ2ω4cN42+4ε2σ22,where φ2 is a constant related to ε and σ, cos⁡φ2=2εσ2/cN4′2+4ε2σ22, and sin⁡φ2=cN4/cN4′2+4ε2σ22.Because of the damping in the EIMG system, the initial vibration displacements will decrease to zero. Under the normal coordinate system, the zero-order stable analytical solutions are given by(38)xNm0=BmeiωeT0,xN40=-L4eiω4t+εσ2t+θ2ω4cN4′2+4ε2σ22+B4eiωeT0+cc.After eliminating the secular terms, the first-order approximate solutions are obtained by substituting (38) into (32):(39)xNm1=V4e2iωetωm2-4ωe2+TmL42e2iω4t+εσ2t+φ2ω42cN4′2+4ε2σ22ωm2-4ω4+εσ22-UmL4ω4cN4′2+4ε2σ22eiω4+ωe+εσ2t+φ2ωm2-ω4+ωe+εσ22+eiωe-ω4+εσ2t+φ2ωm2-ωe-ω4+εσ22,xN41=T4L42e2iω4t+εσ2t+φ2ω42cN4′2+4ε2σ22ω42-4ω4+εσ22+V4e2iωetω42-4ωe2-U4L4ω4cN4′2+4ε2σ22eiω4+ωe+εσ2t+φ2ω42-4ω4+ωe+εσ22+eiωe-ω4+εσ2t+φ2ω42-4ωe-ω4+εσ22,where Ti=AN1iP1+AN2iP1+P2+AN3iP2+P3+AN4iP3+P4+AN5iP5,  Ui=AN1iQ1+AN2iQ1+Q2+AN3iQ2+Q3+AN4iQ3+Q4+AN5iQ5,  Vi=AN1iS1+AN2iS1+S2+AN3iS2+S3+AN4iS3+S4+AN5iS5,  P1=-c1μ12+d1μ22+e1μ1μ2,  P2=-c2μ32+d2μ22+e2μ2μ3,  P3=-c3μ32+d3μ42+e3μ3μ4, and P4=-c4μ52+d4μ42+e4μ4μ5;  Q1=-2c1μ1ρ1+2d1μ2ρ2+e1ρ1μ2+ρ2μ1,  Q2=-2c2μ3ρ3+2d2μ2ρ2+e2ρ2μ3+ρ3μ2,  Q3=-2c3μ3ρ3+2d3μ4ρ4+e3ρ3μ4+ρ4μ3, and Q4=-2c4μ5ρ5+2d4μ4ρ4+e4ρ4μ5+ρ5μ4; and S1=-c1ρ12+d1ρ22+e1ρ1ρ2,  S2=-c2ρ32+d2ρ22+e2ρ2ρ3,  S3=-c3ρ32+d3ρ42+e3ρ3ρ4, and S4=-c4ρ52+d4ρ42+e4ρ4ρ5.Under the normal coordinate system, the approximate solutions are then calculated by substituting (38) and (39) into (29):(40)xNm=BmeiωeT0+εVme2iωetωm2-4ωe2+εTmY42e2iω4t+εσ2t+φ2ω42cN4′2+4ε2σ22ωm2-4ω4+εσ22-εUmL4ω4cN4′2+4ε2σ22eiω4+ωe+εσ2t+φ2ωm2-ω4+ωe+εσ22+eiωe-ω4+εσ2t+φ2ωm2-ωe-ω4+εσ22+cc,xN4=B4eiωeT0-L4eiω4t+εσ2t+φ2ω4cN4′2+4ε2σ22+εT4L42e2iω4t+εσ2t+φ2ω42cN4′2+4ε2σ22ω42-4ω4+εσ22+εV4e2iωetω42-4ωe2-εU4L4ω4cN4′2+4ε2σ22eiω4+ωe+εσ2t+φ2ω42-ω4+ωe+εσ22+eiωe-ω4+εσ2t+φ2ω42-ωe-ω4+εσ22+cc.When the wave frequency of the torque wave is close to half of the natural frequency of the EIMG system, namely,2ωe=ωj+εσ2, the approximate solutions of the EIMG system under the normal coordinate system are(41)xNm=BmeiωeT0+εVme2iωetωm2-4ωe2+εTiLi2e2iωkt+εσ2t+φ2ωk2cNk′2+4ε2σ22ωm2-4ωk+εσ22-εUmLiωkcNk′2+4ε2σ22eiωk+ωe+εσ2t+φ2ωm2-ωk+ωe+εσ22+eiωe-ωk+εσ2t+φ2ωm2-ωe-ωk+εσ22+cc,xNi=BkeiωeT0-Lkeiωkt+εσ2t+φ2ωkcNk2+4ε2σ22+εTkLk2e2iωkt+εσ2t+φ2ωk2cNk′2+4ε2σ22ωk2-4ωk+εσ22+εVke2iωetωk2-4ωe2-εUkLkωkcNk′2+4ε2σ22eiωk+ωe+εσ2t+φ2ωk2-ωk+ωe+εσ22+eiωe-ωk+εσ2t+φ2ωk2-ωe-ωk+εσ22+cc.The superharmonic resonance responses of the EIMG system under the regular coordinate system are calculated using (28).The initial displacementsEk are obtained from the static solution of the linear differential equations derived for the EIMG system. The linear differential equations in (7) are expressed as(42)Mx¨+kx=F0,where F0=T1/RsT2/RIfT3/RIT4/RofT5/RoT and T2, T4 are the output torques on the inner and outer FP, respectively.By setting the mass matrix in (42) equal to zero and only considering the static loads, the static displacement equations can be expressed under the assumption that the deformations of components caused by gravity and the stiffness waves are negligible. In this case, we have(43)kx0=F0,where x0=E1E2E3E4E5T.


## 5. Resonance of Example EIMG System and Discussion

A finite element model (FEM) of the example EIMG system described in Table1 can be built using Ansys; a diagram of the system is presented in Figure 3. In this model, the inner stator and the permeable magnetic of the inner and outer FP consist of silicon steel (23TW250) sheets. The back irons of the inner rotor and outer stator are also made of 23TW250. The nonpermeable magnets of the inner and outer FP are made of epikote, which has similar material characteristics to air. The remanence and coercivity of the PMs on the inner rotor and outer stator are coincident (see Table 1). In this study, we consider linear and unsaturated magnetic circuits. The air gaps have denser grids because these areas have greater magnetic reluctance and higher variation gradients of magnetic flux density than the PMs and back irons. The torque profiles calculated for all components are displayed in Figure 4.Table 1
Parameters of example EIMG system.
Number of pole pairs on inner stator 2 Number of pole pairs on inner surface of inner rotor 5 Number of pole pairs on outer surface of inner rotor 4 Number of pole pairs on outer stator 17 Number of inner FP 7 Number of outer FP 21 Outer radius of inner stator/mm 98 Inner radius of inner FP/mm 100 Outer radius of inner FP/mm 130 Thickness of PMs on inner rotor/mm 10 Inner radius of inner rotor yoke/mm 152 Outer radius of inner rotor yoke/mm 182 Inner radius of outer FP/mm 204 Outer radius of outer FP/mm 234 Thickness of PMs on outer stator/mm 10 Inner radius of outer stator yoke/mm 256 Outer radius of outer stator yoke/mm 276 Axial length/mm 40 Remanence of PMs/T 1.3 Coercive force of PMs/KOe 11.6 Magnitude of currents on inner stator/A 20Figure 3
FEM of the example EIMG system.Figure 4
Torque profiles of the example EIMG system.
(a) (b)Figure4 indicates that the torques on all components vary sinusoidally with respect to the phase angle. The wave period is determined by the number of pole pairs of the currents on the inner stator or the number of pole pairs of the PMs on the inner rotor and outer stator; that is, Ti=2π/pi. The torque profiles can be expressed as Tci=sin⁡(piθij+θi0), where θij is the relative torsional displacement between two adjacent components. When the outer torque on the outer FP is less than the maximum Tc3 and takes a constant value, θi0 is also constant. As such, Tci is a (nonlinear) sinusoidal function of the relative torsional displacements.Table2 shows the corresponding dynamic parameters of the example EIMG system.Table 2
Characteristic parameters of example EIMG system.
k s (N/m) k I f (N/m) K o (N/m) θ 10 (rad) θ 20 (rad) M 1 (kg) M 2 (kg) 8.6444 × 105 3.1446 × 105 4.5467 × 105 5π/12 π/4 0.64 0.657  M 3 (kg) M 4 (kg) M 5 (kg) ζs ζIf ζI ζof  5.4 3.0 3.85 0.057 0.0653 0.019 0.0173  ζo T 1 (N⋅m) T 2 (N⋅m) T 3 (N⋅m) T 4 (N⋅m) T 5 (N⋅m) ΔT (N⋅m)  0.0793 22 77 55 288.75 233.75 1By substituting the dynamic parameters of the example system into (25) and (28), the main resonance responses of the EIMG system can be determined. When the wave frequency is close to the natural frequency of the outer FP rotational mode, the transient responses of the components are shown in Figure 5.Figure 5
Transient responses of EIMG system whenωe≈ω4.
(a) (b) (c)Figure5 indicates that the output torque causes larger initial static displacements among the components of the EIMG system. The system damping ensures that the transient responses will gradually transition to the steady state. This process takes approximately 6 s. When resonance occurs, there are multiple frequency components in the transient responses, and the steady-state responses have a particular frequency.Because of the magnetic coupling among components in the EIMG system, the magnetic coupling stiffness is much smaller than the mechanical support stiffness. Moreover, because there is no contact among the components (namely, no frictional damping), the resonance amplitudes of the transient responses will be larger and will slowly decay to zero. This results in some deterioration in the dynamic characteristics.Considering the effects of damping, the resonance will gradually be close to stabilization. When the wave frequencyωe is close to the natural frequency ωi and half of the corresponding frequency ωi/2 of the derived nonlinear EIMG system, the steady-state response curves are as shown in Figures 6 and 7.Figure 6
Main resonance whenωe≈ωi.
(a)
ω e ≈ ω 1 (b)
ω e ≈ ω 2 (c)
ω e ≈ ω 3  (d)
ω e ≈ ω 4 (e)
ω e ≈ ω 5Figure 7
Main resonance whenωe≈ωi/2.
(a)
ω e ≈ ω 1 / 2 (b)
ω e ≈ ω 2 / 2 (c)
ω e ≈ ω 3 / 2  (d)
ω e ≈ ω 4 / 2 (e)
ω e ≈ ω 5 / 2Figures6 and 7 show that strong resonance will occur when the wave frequency is close to the natural frequency or half of the natural frequency of the derived EIMG system. The dominant frequency in the superharmonic resonance is the natural frequency, rather than the wave frequency or half of the natural frequency. The resonance amplitudes are much larger when the wave frequency is close to the natural frequency of the inner rotor rotational mode and outer FP rotational mode than when the wave frequency is close to the natural frequency of the inner FP rotational mode and inner stator rotational mode.When resonance occurs and the displacement of a certain degree of freedom reaches its maximum, the resonance amplitudes are very different. For instance, when the wave frequency is close to the natural frequency of the outer FP rotational mode, the torsional displacements of the outer FP attain their maximum. Furthermore, the displacements of the adjacent components, namely, the inner rotor and outer stator, will also be larger; however, the displacements of the interval components, namely, the inner FP and stator, will be much smaller.Similarly, when the wave frequency is close to twice the natural frequency of the derived EIMG system, strong subharmonic resonance will occur because of the nonlinear magnetic coupling stiffness. The dynamic behavior is similar to the superharmonic response. When the outer stator outputs torque with a wave frequency close to the natural frequency of the derived EIMG system, strong resonance will occur. Furthermore, the amplitudes of the various components will be very different.Because there is no contact among the components, the relative influences among adjacent (interval) components are larger (smaller) than those in a mechanical gear system. Because of the output torque on the outer FP waves, the equivalent load is greatest on the outer FP. Therefore, the largest resonance occurs when the wave frequency is close to the natural frequency of the outer FP rotational mode. Because the magnetic coupling stiffness is much smaller than the support stiffness, the natural frequencies of the inner rotor and outer FP rotational mode are much larger. When the current frequency is smaller, the EIMG can be used in applications that easily admit low frequencies, high torque, and low-frequency resonance. When the current frequency is larger, low-frequency main resonances may not occur; however, superharmonic resonances can easily occur.


## 6. Conclusion

EIMG systems output torque as a result of the magnetic coupling among components. The magnetic coupling stiffness is much smaller than the mesh stiffness of mechanical gear systems and exhibits distinct nonlinearity. Thus, there are quadratic nonlinearities in the EIMG dynamic system. When the wave frequency is close to the natural frequency or twice/half the natural frequency of the derived EIMG system, strong resonance will occur. The dominant frequency is the natural frequency, and the resonance amplitudes exhibit large differences when superharmonic resonance occurs. Because of the smaller magnetic coupling stiffness and damping, the transient responses slowly decay to zero. In order to accelerate the decay rate of the transient responses and avoid the resonances, the vibration control strategy of the EIMG system should be analyzed. These can provide the foundation for the parameter optimization and performance improvement of the EIMG system.



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*Source: 9821870-2018-03-14.xml*