# Nonlinear Forced Response of Electromechanical Integrated Toroidal Drive to Coupled Excitation

Authors: Lizhong Xu; Fen Wang
Journal: The Scientific World Journal
 (2012)

---

## Abstract

The electric excitation and the parameter excitation from mesh stiffness fluctuation are analyzed. The forced response equations of the drive system to the coupled excitations are presented. For the exciting frequencies far from and near natural frequencies, the forced responses of the drive system to the coupled excitations are investigated. Results show that the nonlinear forced responses of the drive system to the coupled excitations change periodically and unsteadily; the time period of the nonlinear forced responses depends on the frequencies of the electric excitation, the mesh parameter excitation, and the nonlinear natural frequencies of the drive system; in order to improve the dynamics performance of the drive system, the frequencies of the electric excitations should not be taken as integral multiple of the mesh parameter exciting frequency.


---

## Body

## 1. Introduction

Toroidal drive can transmit large torque in a very small size and is suitable for technical fields such as aviation and space flight [1–3]. As electrical and control techniques are utilized in mechanical engineering field widely, generalized composite drives become advancing edge of the mechanical science. So far, types of the generalized composite drives with integrated structure are still very limited.The electromagnetic harmonic drive [4] and piezoelectric harmonic drive [5] are active drives in which the meshing forces between flexible gear and rigid one are controlled by electromagnetic force or piezoelectric one, and drive and power are integrated. Based on researching toroidal drive [6], the authors presented a kind of active generalized composite drive: electromechanical integrated toroidal drive. In the drive, the toroidal drive, power, and control are integrated [7].The drive consists of four basic elements (Figure1) (a) the central worm, (b) radially positioned planets, (c) a toroidal shaped stator, and (d) a rotor, which forms the central output shaft upon which the planets are mounted. The central worm is fixed and coils are mounted in helical grooves of its surface. The planets have permanent magnets instead of teeth. The N and S polar permanent magnets are mounted alternately on a planet. And the stator has helical permanent magnets instead of helical teeth. In the same manner as planet, the N and S polar helical permanent magnets are mounted alternately on the stator.Figure 1
The electromechanical integrated toroidal drive.If a specific parameter relation is realized,N pole of one element will correspond to S pole of the other one all along. The attractive forces between N and S pole of the different elements are driving forces and the meshes without contact are realized. When the alternating voltage source is connected to the coils of the worm, a toroidal circular field is formed. It drives several planets to rotate about their own axial. And by means of magnetic forces between teeth of the planet and stator, the rotor is driven to rotate about its own axial. Thus, a power of low speed and large torque is output.Compared with toroidal drive, the new drive is easy to produce, without wear, and does not need lubrication. It can be substituted for a servo system to simplify the structure of the existing electromechanical systems. Beside the above-mentioned fields that require compactness, the drive can be used in fields such as robots, which require accurate control.The electromechanical integrated toroidal drive consists of a mechanical system, an electrical system and a coupled part. The mechanical vibration may occur in the mechanical system, and the electrical current oscillation may occur in the electrical system. With the coupled part, the mechanical vibration and the electrical current oscillation will influence on each other. Hence, the drive system is an electromechanical coupled dynamics system. The electromechanical coupled dynamics was first proposed for the motor [8]. Then, the electromechanical coupled dynamics of the electromechanical system consisting of the motor and mechanical system driven by the motor was developed. An electromechanical coupled dynamics model of the electromechanical system consisting of the several motors and mechanical system driven by these motors was proposed. Using the model, the natural frequency of the electromechanical system is analyzed [9]. The authors investigated nonlinear forced response of electromechanical integrated toroidal drive to voltage excitations [10]. However, not only the electric excitation occurs, but also the parameter excitations from mesh stiffness fluctuations occur. The nonlinear forced responses of the drive system to the coupled excitations consisting of the electric excitation and the parameter excitations have not been investigated yet. These nonlinear forced responses have important influence on the operating performance of the drive system. To design, evaluate, and control dynamics behavior of the drive system effectively, the nonlinear forced responses of the drive system to the coupled excitations should be developed.In this paper, the electric excitation and the parameter excitation from mesh stiffness fluctuation are analyzed. The forced response equations of the drive system to the coupled excitations are presented. For the exciting frequencies far from and near natural frequencies, the forced responses of the drive system to the coupled excitations are investigated. The work can be used to predict the noise and dynamic load and are useful in maximizing the power density of the drive and reducing noise radiation.


## 2. Electric Excitation of the Drive System

The magnetic energy storage of the electric system for the drive system is(1)W=12∑i=1nλisIis,
where λis is magnetic linkage of the worm coils, λis=∑j=1nLijIis, Lij is inductances of the ith phase worm coils, Iis is current of the ith phase worm coils, and n is phase number of the worm coils.From (1), the electromagnetic torque on the planet is given as
(2)Tp=-∂W∂θ=-12∑j=1n∑i=1ndLijdθIisIjs,
where θ is the relative rotating angle between planet and worm.The torqueTp consists of the static torque Tp0 and the dynamic torque δTp. The current Iis consists of the static component Ii and the dynamic one δii. The angle θ consists of the static angle θ0 and the dynamic angle δθ. The torque Tp is written in series form as
(3)Tp=Tp0+δTp=12∑j=1n‍∑i=1n[(δLijδθ)θ=θ0IiIj+(δ2Lijδθ2)θ=θ0IiIjδθ+(δ3Lijδθ3)θ=θ0IiIj(δθ)2+(δLijδθ)θ=θ0Iiδij+(δLijδθ)θ=θ0Ijδii+(δLijδθ)θ=θ0Ii(δij)2+(δLijδθ)θ=θ0Ij(δii)2+⋯].
If δij=δii=0, from (3), one can give
(4)Tp0=12∑j=1n‍∑i=1n(δLijδθ)θ=θ0IiIj,(5)δTp=12∑j=1n‍∑i=1n[(δ2Lijδθ2)θ=θ0IiIjδθ+(δ3Lijδθ3)θ=θ0IiIj(δθ)2+⋯].LetΔFwpi denote the dynamic magnetic meshing force between a planet tooth and worm, ΔFwpi=δTp/R. From (5), neglecting the high-order terms, one knows. 						(6)ΔFwpi=-Rδθ2R2∑i=1n‍∑j=1n[(δ2Lijδθ2)θ=θ0IiIj+(δ3Lijδθ3)θ=θ0IiIjδθ].Here,ζwpi=Rδθ; it is the dynamic relative displacement between planet and worm. Thus, (6) can be changed into the following form:
(7)ΔFwpi=(kwpi+Δkwpi)ζwpi.
Here, kwpi=(1/2R2)∑i=1n∑i=1n(δ2Lij/δθ2)θ=θ0IisIjs, it is considered as linear electromagnetic mesh stiffness between a tooth of the planet and worm; Δkwpi=-(1/2R2)∑i=1n∑j=1n(δ3Lij/δθ3)θ=θ0IiIjδθ, it is nonlinear component of the electromagnetic mesh stiffness.Equation (3) shows that current fluctuation can produce electromagnetic torque fluctuation. The torque fluctuation from current fluctuation can be considered as equivalent exciting torque ΔTe. Let the current periodically change as below:
(8)δii=δij=ΔIcos(ωet),
where ΔI is the magnitude of the fluctuation current, ωe is the frequency of the fluctuation current, and t is time.Substituting (8) into (3), neglecting high-order terms, yields
(9)ΔTe=∑j=1n‍∑i=1n‍(δLijδθ)θ=θ0IiΔIcosωet.LetΔFe denote the equivalent exciting force between a planet tooth and worm; thus
(10)ΔFe=1R∑j=1n‍∑i=1n‍(δLijδθ)θ=θ0IiΔIcosωet.


## 3. Parametric Excitation from Stiffness Fluctuation

In operation of the drive system, the number of meshing tooth pairs between the worm and the planets is variable. It can cause fluctuation of the mesh stiffness between them. It is dependent on the numberz1 of planet teeth and the conditional face angles of the worm ϕv. At ϕv=100° and z1=8, the changes of mesh stiffness along with the mesh tooth pair number are shown in Figure 2. In Figure 2, a typical stiffness variation through a mesh cycle of the drive system is given. The mesh stiffness at more contact regions is higher than that at less contact regions. If the drive rotates at appreciable speed, this time-varying stiffness will be a major excitation source of the drive system.Figure 2
Changes of the mesh sifness.The mesh stiffness between a planet and worm is considered to consist of its mean valuek- and time-varying one Δk(t). The average mesh stiffness between them through one periodic time can be given by
(11)k-wpi=4π{∫0π/36k3dθ+∫π/362π/9k2dθ+∫2π/9π/4k3dθ},
where k2=2(kwpi+Δkwpi) and k3=3(kwpi+Δkwpi); they are two and three teeth mesh stiffness, respectively.The periodical time-varying portion of the mesh stiffness can be defined in the Fourier series form as(12)Δk(t)=∑n=1∞Δkncosnωpt,
where Δkn=(2/l)∫0lk(t)ωpcosnωptdt, l is the period of the stiffness fluctuation. In Figure 2, l=π/4.For worm and planet, the periodical time-varying portion of the mesh stiffness is(13)Δknwpi=2l{∫0π/36(k3-k-wpi)ωpcosnωptdt+∫π/362π/9(k2-k-wpi)ωpcosnωptdt+∫2π/9π/4(k3-k-wpi)ωpcosnωptdt}=8π[(k3-k-wpi)(sinπ36+sinπ4-sin2π9)+(k2-k-wpi)(sin2π9-sinπ36)].Substituting (13) into (12) yields
(14)Δkwpi(t)=∑n=1∞Δknwpicosnωt.


## 4. Forced Response Equation to Coupled Excitation

The dynamic model for the electromechanical integrated toroidal drive (see Figure3) allows rotor and each planet to rotate about their own rotating axes and allows each planet to translate in xi and zi directions. The rotations are replaced by the corresponding translational mesh displacements as uj=rjθj, j=1,…,m,r (here, m is planet number, θj the rotation of planet or rotor, rj is the rolling circle radius for planet and the radius of the circle passing through planet centers for the rotor). A displacement vector qj and a mass matrix mj are defined for each planet j as qj=[ujxjzj]T and mj=Diag[Jj/rj2mjmj]. Here, Jj and mj are polar mass moment of inertia and mass for planet j, respectively. Mr(Mr=Jr/rr2) is equivalent mass of rotor corresponding to its displacement ur. Thus, the motion equations of the drive system are
(15)Jiri2u¨i+(kwpi+Δkwpi)pwpisinγwpi+ΔFesinγwpi-kspipspicosγspi=0,mix¨i+kcpxixi=0,miz¨i-(kwpi+Δkwpi)pwpicosγwpi-ΔFecosγwpi-kspipspisinγspi+kcpzipcpzi=0,Mru¨r-∑i=1mkcpzipcpzi=-Trrr(i=1tom),
where pwpi, pspi and pcpzi are relative displacements between planet-i and worm, stator, or rotor, respectively, pwpi=uisinγwpi-zicosγwpi, pspi=-uicosγspi-zisinγspi and pcpzi=zi-ur. γwpi and γspi are lead angles at contact points between planet-i and worm or stator, respectively, tanγwpi=1/[iwp(a/R-1)], and tanγspi=1/[isp(a/R+1)]. Here, a is center distance between worm and planet R is reference circle radius of planet iwp and isp are speed ratios between planet and worm or stator, respectively. kwpi and kspi are mesh stiffness between planet-i and worm or stator, respectively, kczi and kcxi are planet support stiffness in zi and xi directions, respectively, and Tr is torque transmitted by rotor.Figure 3
Dynamic model for a four-planet electromechanical integrated toroidal drive.Equation (15) can be written in matrix form as
(16)MX¨+KX=F+εΔF+εΔFe,
where X and F are displacement and static load vectors; respectively,X={q1⋯⋯qmur}T,F={0⋯⋯0-Tr/rr}T;M and K are mass and stiffness matrix, respectively,M=diag[m1⋯⋯mmMr],(17)K=[kwp1+ksp1+kcp1···0···0kc1·····················kwpi+kspi+kcpi····kci·········kwpm+kspm+kcpmkcmsymmetric∑i=1mkczi],kwpi=kwpi[sin2γwpi0-cosγwpisinγwpi000-cosγwpisinγwpi0cos2γwpi],kspi=kspi[cos2γspi0sinγspicosγspi000sinγspicosγspi0sin2γspi],kcpi=[0000kcxi000kczi],kci={00-kczi}T.Δ
F is nonlinear component of the force vector, ΔFe is equivalent force vector caused by current fluctuation:Consider(18)ΔF=(Bui2+Cui3)[-sinγwpi0cosγwpi-sinγwpi0cosγwpi⋯0]T,ui=Rδθ,ε=rR,it is a disturbing parameter,B=z1∑i=1m∑j=1mAtan(z1θ-ϕv/np)IiIj/2Rr, C=-z12∑i=1m∑j=1mAIiIj/2Rr.(19)ΔFe=Dcos(ωet)[sinγwpi0cosγwpisinγwpi0cosγwpisinγwpi0cosγwpi⋯0]T,D=12R∑i=1n∑j=1n(∂Lij∂θ)θ=θ0IiΔI.The total displacement of the each component consists of the static displacement and the dynamic one. Hence, the displacement vector is(20)X=X-+ΔX,
where X- and ΔX are static and dynamic displacement vectors; respectively,
(21)ΔX={Δq1⋯⋯ΔqmΔur}T,Δqj=[ΔujΔxjΔzj]T.Considering the fluctuation of the mesh stiffness caused by the changes of the mesh tooth pair number, the stiffness matrix can be expressed as(22)K=K-+ΔK(t),
where K- is the mean stiffness matrix, and ΔK(t) is time-varying stiffness matrix.Substituting (20)–(22) into (16) yields
(23)K-X-=F,(24)MΔX¨+K-ΔX=εΔF+εΔFe+εΔFp,
where ΔFp=-EΔK(t)X-; it is considered as the equivalent exciting force vector caused by mesh stiffness excitation; E=R/r.Equation (24) is just the forced response equation of the drive system to coupled excitation.


## 5. Solution of the Nonlinear Forced Response Equation

### 5.1. Far from Natural Frequencies

First, we consider to resolve the nonlinear forced response equation of the drive system to electric excitation (letΔFp=0).For simplicity purposes, (24) should be transmitted into equations independent on each other. Then, (24) is changed into the following form:
(25)ΔX¨N+K-NΔXN=εΔFN+εΔFeN,
where K-N is the diagonal mean stiffness matrix, respectively. ΔXN is transmitted dynamic displacement vector. ΔFeN and ΔFN are transmitted exciting force vectors of the forces ΔFe and ΔF, respectively.MatrixKN and vector ΔXN are given by KN=ANTKAN and ΔXN=ANTΔX. The transmitted nonlinear force vector ΔFN and transmitted equivalent exciting force vector ΔFeN can be given as below:
(26)ΔFN=ANTΔF=(Bui2+Cui3)ΔPN,ΔFeN=εDcos(ωet)ΔPeN,
where AN is the mode matrix of (24):
(27)AN=[A1(1)A2(1)⋯Am(1)A1(2)A2(2)⋯Am(2)⋯⋯⋯⋯A1(m)A2(m)⋯Am(m)].Atm=3, ΔPN=[PN1PN2⋯PNi⋯]T, PNi=-3AN1isinγwpi+3AN3icosγwpi(i = 1,4,…,3m-2), PNi=0(i=2,3,…,3m+1,i≠1,4,…,3m-2); ΔPeN=[PeN100PeN200PeN3000]T, PNe=3AN1isinγwpi + 3AN3icosγwpi(i=1,4,…,3m-2), PNei=0(i=2,3,…,3m+1,i≠1,4,…,3m-2).Let(28)ΔXN=X0+εX1+ε2X2+⋯,(29)ωi2=ω0i2(1+εσ1i+ε2σ2i+⋯)(i=1,2,3),
where ωi is natural frequency of ith order mode for the nonlinear drive system, and ω0i is natural frequency of ith order rotational mode for the linear drive system.Substituting (28) and (29) into (25), let sum of the coefficients with the same-order power of the parameter ε equal zero, following equations can be given
(30)x¨N0+ωi2xN0=0,x¨N1+ωi2xN1=-σ1x¨N0+PNBu0i2+PNCu0i3+Dcos(ωet)PNex¨N2+ωi2xN2=-σ1x¨N1-σ2x¨N0+PNB(2u0iu1i+σ1u0i2)+PNC(3u0i2u1i+σ1u0i3)+Dσ1cos(ωet)PNe….
Here, initial conditions are xN0i(0)=AN0i, x˙N0i(0)=0.The solution of zero-order equation under the above initial conditions is(31)x0i=A0icosωit(i=1,2,3).Substituting (31) into the second equation of (30) yields
(32)x¨N11+ω12xN11=-σ11x¨N01+PN1Bu0i2+PN1Cu0i3+Dcos(ωet)PNe,x¨N12+ω22xN12=-σ12x¨N02+PN2Bu0i2+PN2Cu0i3+Dcos(ωet)PNe,x¨N13+ω32xN13=-σ13x¨N03+PN3Bu0i2+PN3Cu0i3+Dcos(ωet)PNe.The rotational displacementui is
(33)ui=AN11xNi1+AN12xNi2+AN13xNi3.
Substituting (33) into (30) yields
(34)x¨N11+ω12xN11=-σ11x¨N01+PN1B(AN11xN01+AN12xN02+AN13xN03)2+PN1C(AN11xN01+AN12xN02+AN13xN03)3+Dcos(ωet)PNe1x¨N12+ω22xN12=-σ12x¨N02+PN2B(AN11xN01+AN12xN02+AN13xN03)2+PN2C(AN11xN01+AN12xN02+AN13xN03)3+Dcos(ωet)PNe2,x¨N13+ω32xN13=-σ13x¨N03+PN3B(AN11xN01+AN12xN02+AN13xN03)2+PN3C(AN11xN01+AN12xN02+AN13xN03)3+Dcos(ωet)PNe3.In order to remove secular item, let(35)σ11=-PN1Cω12×[34(AN11AN01)3+32(AN12AN02)2AN11AN01+32(AN13AN03)2AN11AN01],σ12=-PN2Cω22×[34(AN12AN02)3+32(AN11AN01)2AN12AN02+32(AN13AN03)2AN12AN02],σ13=-PN3Cω32×[34(AN13AN03)3+32(AN12AN02)2AN13AN03+32(AN11AN01)2AN13AN03].Substituting (35) into (34), the solutions of the first-order equations can be obtained. As the equations of the solutions are relatively complicated, it is not given here.In a same manner, the solution ofnth order equation can be obtained as well. Substituting these solutions into (28) and (29), the solutions of the regular nonlinear forced response equations and natural frequencies of the drive system can be given. Then, the real solutions of the nonlinear forced responses can be calculated as below:
(36)ΔX=ANΔXN.


### 5.2. Near Natural Frequencies

When the exciting frequency is near natural frequency, the nonlinear forced response equation can be resolved as below. Considering damping of the drive system, (25) can be changed into
(37)ΔX¨N+CNΔX˙N+KNΔXN=εΔFN+εΔFeN,
where CN is regular damping matrix, CN=ANTCAN=diag[CN1CN2CN3CN4] (C is damping matrix). CNi=2ζωi, ζ is relative damping coefficient.When exciting frequency is near natural frequency, the exciting frequency can be written as(38)ωei2=ω0i2(1+εσ1i)(i=1,2,3).Substituting (38) and (28) into (37), let sum of the coefficients with the same-order power of the parameter ε equal zero, the following equations can be given:
(39)X¨0+KNX0=0,X¨1+KNX1=-σ1iX¨0-2ζ1X˙0+ΔPN(Biui2+Ciui3)+ΔPeNDicos(ωeit+θ)⋯,
where CNi=2ζωi,
(40)ζ=εζ1,ωei2=ωi2(1+εσ1),ΔxNi=xN0i+εxN1i,B=Biωi2,C=Ciωi2,D=Diωi2.The solution of zero-order equation under the above initial conditions is(41)xN0i=AN0icosωeit(i=1,2,3).Substituting (41) into the second equation of (43) yields
(42)x¨N11+ωe12xN11=-σ1x¨N01-2ζ1x˙N01+PN1B1u012+PN1C1u013+D1PNe1cos(ωe1t+θ),x¨N12+ωe22xN12=-σ1x¨N02-2ζ1x˙N02+PN2B2u012+PN2C2u013+D2PNe2cos(ωe2t+θ),x¨N13+ωe32xN13=-σ1x¨N03-2ζ1x˙N03+PN3B3u012+PN3C3u013+D3PNe3cos(ωe3t+θ).Substituting rotational displacementui=AN11xNi1+AN12xNi2+AN13xNi3 into (42) yields
(43)x¨N11+ωe12xN11=-σ1x¨N01-2ζ1x˙N01+PN1B1(AN11xN01+AN12xN02+AN13xN03)2+PN1C1(AN11xN01+AN12xN02+AN13xN03)3+D1PNe1(cosωe1tcosθ+sinωe1tsinθ),x¨N12+ωe22xN12=-σ1x¨N02-2ζ1x˙N02+PN2B2(AN11xN01+AN12xN02+AN13xN03)2+PN2C2(AN11xN01+AN12xN02+AN13xN03)3+D2PNe2(cosωe2tcosθ+sinωe2tsinθ),x¨N13+ωe32xN13=-σ1x¨N03-2ζ1x˙N03+PN3B3(AN11xN01+AN12xN02+AN13xN03)2+PN3C3(AN11xN01+AN12xN02+AN13xN03)3+D3PNe3(cosωe3tcosθ+sinωe3tsinθ).In order to remove secular item, let(44)σ11AN01+PN1C1AN01P1′+D1PNe1cosθ=0,2ζ1AN01+D1PNe1sinθ=0,
where P1′=(1/AN01)[(3/4)(AN11AN01)3+(3/2)(AN12AN02)2AN11AN01+(3/2)(AN13AN03)2AN11AN01].From (44), it is known that
(45)(σ11+PN1C1P1′)2+(2ζ1)2=(D1PNe1AN01)2.Thus(46)si2=1-εPNiCiPi′-2ζ2±(Di′AN0i)2-4ζ2(1-εPNiCiPi′-ζ2),
where si=ωei/ωi and Di′=εDiPNei.From (46), the changes of the nonlinear vibrating magnitudes along with exciting frequencies can be given.


## 5.1. Far from Natural Frequencies

First, we consider to resolve the nonlinear forced response equation of the drive system to electric excitation (letΔFp=0).For simplicity purposes, (24) should be transmitted into equations independent on each other. Then, (24) is changed into the following form:
(25)ΔX¨N+K-NΔXN=εΔFN+εΔFeN,
where K-N is the diagonal mean stiffness matrix, respectively. ΔXN is transmitted dynamic displacement vector. ΔFeN and ΔFN are transmitted exciting force vectors of the forces ΔFe and ΔF, respectively.MatrixKN and vector ΔXN are given by KN=ANTKAN and ΔXN=ANTΔX. The transmitted nonlinear force vector ΔFN and transmitted equivalent exciting force vector ΔFeN can be given as below:
(26)ΔFN=ANTΔF=(Bui2+Cui3)ΔPN,ΔFeN=εDcos(ωet)ΔPeN,
where AN is the mode matrix of (24):
(27)AN=[A1(1)A2(1)⋯Am(1)A1(2)A2(2)⋯Am(2)⋯⋯⋯⋯A1(m)A2(m)⋯Am(m)].Atm=3, ΔPN=[PN1PN2⋯PNi⋯]T, PNi=-3AN1isinγwpi+3AN3icosγwpi(i = 1,4,…,3m-2), PNi=0(i=2,3,…,3m+1,i≠1,4,…,3m-2); ΔPeN=[PeN100PeN200PeN3000]T, PNe=3AN1isinγwpi + 3AN3icosγwpi(i=1,4,…,3m-2), PNei=0(i=2,3,…,3m+1,i≠1,4,…,3m-2).Let(28)ΔXN=X0+εX1+ε2X2+⋯,(29)ωi2=ω0i2(1+εσ1i+ε2σ2i+⋯)(i=1,2,3),
where ωi is natural frequency of ith order mode for the nonlinear drive system, and ω0i is natural frequency of ith order rotational mode for the linear drive system.Substituting (28) and (29) into (25), let sum of the coefficients with the same-order power of the parameter ε equal zero, following equations can be given
(30)x¨N0+ωi2xN0=0,x¨N1+ωi2xN1=-σ1x¨N0+PNBu0i2+PNCu0i3+Dcos(ωet)PNex¨N2+ωi2xN2=-σ1x¨N1-σ2x¨N0+PNB(2u0iu1i+σ1u0i2)+PNC(3u0i2u1i+σ1u0i3)+Dσ1cos(ωet)PNe….
Here, initial conditions are xN0i(0)=AN0i, x˙N0i(0)=0.The solution of zero-order equation under the above initial conditions is(31)x0i=A0icosωit(i=1,2,3).Substituting (31) into the second equation of (30) yields
(32)x¨N11+ω12xN11=-σ11x¨N01+PN1Bu0i2+PN1Cu0i3+Dcos(ωet)PNe,x¨N12+ω22xN12=-σ12x¨N02+PN2Bu0i2+PN2Cu0i3+Dcos(ωet)PNe,x¨N13+ω32xN13=-σ13x¨N03+PN3Bu0i2+PN3Cu0i3+Dcos(ωet)PNe.The rotational displacementui is
(33)ui=AN11xNi1+AN12xNi2+AN13xNi3.
Substituting (33) into (30) yields
(34)x¨N11+ω12xN11=-σ11x¨N01+PN1B(AN11xN01+AN12xN02+AN13xN03)2+PN1C(AN11xN01+AN12xN02+AN13xN03)3+Dcos(ωet)PNe1x¨N12+ω22xN12=-σ12x¨N02+PN2B(AN11xN01+AN12xN02+AN13xN03)2+PN2C(AN11xN01+AN12xN02+AN13xN03)3+Dcos(ωet)PNe2,x¨N13+ω32xN13=-σ13x¨N03+PN3B(AN11xN01+AN12xN02+AN13xN03)2+PN3C(AN11xN01+AN12xN02+AN13xN03)3+Dcos(ωet)PNe3.In order to remove secular item, let(35)σ11=-PN1Cω12×[34(AN11AN01)3+32(AN12AN02)2AN11AN01+32(AN13AN03)2AN11AN01],σ12=-PN2Cω22×[34(AN12AN02)3+32(AN11AN01)2AN12AN02+32(AN13AN03)2AN12AN02],σ13=-PN3Cω32×[34(AN13AN03)3+32(AN12AN02)2AN13AN03+32(AN11AN01)2AN13AN03].Substituting (35) into (34), the solutions of the first-order equations can be obtained. As the equations of the solutions are relatively complicated, it is not given here.In a same manner, the solution ofnth order equation can be obtained as well. Substituting these solutions into (28) and (29), the solutions of the regular nonlinear forced response equations and natural frequencies of the drive system can be given. Then, the real solutions of the nonlinear forced responses can be calculated as below:
(36)ΔX=ANΔXN.


## 5.2. Near Natural Frequencies

When the exciting frequency is near natural frequency, the nonlinear forced response equation can be resolved as below. Considering damping of the drive system, (25) can be changed into
(37)ΔX¨N+CNΔX˙N+KNΔXN=εΔFN+εΔFeN,
where CN is regular damping matrix, CN=ANTCAN=diag[CN1CN2CN3CN4] (C is damping matrix). CNi=2ζωi, ζ is relative damping coefficient.When exciting frequency is near natural frequency, the exciting frequency can be written as(38)ωei2=ω0i2(1+εσ1i)(i=1,2,3).Substituting (38) and (28) into (37), let sum of the coefficients with the same-order power of the parameter ε equal zero, the following equations can be given:
(39)X¨0+KNX0=0,X¨1+KNX1=-σ1iX¨0-2ζ1X˙0+ΔPN(Biui2+Ciui3)+ΔPeNDicos(ωeit+θ)⋯,
where CNi=2ζωi,
(40)ζ=εζ1,ωei2=ωi2(1+εσ1),ΔxNi=xN0i+εxN1i,B=Biωi2,C=Ciωi2,D=Diωi2.The solution of zero-order equation under the above initial conditions is(41)xN0i=AN0icosωeit(i=1,2,3).Substituting (41) into the second equation of (43) yields
(42)x¨N11+ωe12xN11=-σ1x¨N01-2ζ1x˙N01+PN1B1u012+PN1C1u013+D1PNe1cos(ωe1t+θ),x¨N12+ωe22xN12=-σ1x¨N02-2ζ1x˙N02+PN2B2u012+PN2C2u013+D2PNe2cos(ωe2t+θ),x¨N13+ωe32xN13=-σ1x¨N03-2ζ1x˙N03+PN3B3u012+PN3C3u013+D3PNe3cos(ωe3t+θ).Substituting rotational displacementui=AN11xNi1+AN12xNi2+AN13xNi3 into (42) yields
(43)x¨N11+ωe12xN11=-σ1x¨N01-2ζ1x˙N01+PN1B1(AN11xN01+AN12xN02+AN13xN03)2+PN1C1(AN11xN01+AN12xN02+AN13xN03)3+D1PNe1(cosωe1tcosθ+sinωe1tsinθ),x¨N12+ωe22xN12=-σ1x¨N02-2ζ1x˙N02+PN2B2(AN11xN01+AN12xN02+AN13xN03)2+PN2C2(AN11xN01+AN12xN02+AN13xN03)3+D2PNe2(cosωe2tcosθ+sinωe2tsinθ),x¨N13+ωe32xN13=-σ1x¨N03-2ζ1x˙N03+PN3B3(AN11xN01+AN12xN02+AN13xN03)2+PN3C3(AN11xN01+AN12xN02+AN13xN03)3+D3PNe3(cosωe3tcosθ+sinωe3tsinθ).In order to remove secular item, let(44)σ11AN01+PN1C1AN01P1′+D1PNe1cosθ=0,2ζ1AN01+D1PNe1sinθ=0,
where P1′=(1/AN01)[(3/4)(AN11AN01)3+(3/2)(AN12AN02)2AN11AN01+(3/2)(AN13AN03)2AN11AN01].From (44), it is known that
(45)(σ11+PN1C1P1′)2+(2ζ1)2=(D1PNe1AN01)2.Thus(46)si2=1-εPNiCiPi′-2ζ2±(Di′AN0i)2-4ζ2(1-εPNiCiPi′-ζ2),
where si=ωei/ωi and Di′=εDiPNei.From (46), the changes of the nonlinear vibrating magnitudes along with exciting frequencies can be given.


## 6. Results and Discussions

When exciting frequency is far from natural frequency, from the above equations, the nonlinear forced vibrations for the drive system are analyzed. The parameters of the numerical example are shown in Table1. Figure 4 shows changes of the forced vibrations of the transmitted variables along with nonlinear parameter ε. From Figure 4, the following are known.(1)
Under the coupled excitations, the nonlinear forced responses of the drive system change periodically and unsteadily. The time period of the nonlinear forced responses depends on the frequencies of the electric excitation the mesh parameter excitation, and the nonlinear natural frequencies of the drive system.(2)
The vibrating amplitudes of the nonlinear forced responses of the drive system to the coupled excitations are larger than those of the nonlinear forced responses to the single excitation. The vibrating amplitudes of the nonlinear forced responses of the planet are larger than that of the rotor. It is because the exciting frequencies are near to the vibrating frequency of the planet modes.(3)
The vibrating amplitudes of the tangent vibration for the planet are larger than that of the axial vibration for the planet, and the frequency of the tangent vibration for the planet is smaller than that of the axial vibration for the planet.(4)
The unstable periodic vibrations are harmful for the drive operation. In order to increase the dynamics performance of the drive system, the electric excitations should be avoided or the frequencies of the electric excitations should not be taken as integral multiple of the mesh parameter exciting frequency.Table 1
Parameters of the example system.
ε
ω
e (rad/s)
a
/
R
i
wpi
N
I
s (A)
r (mm)
I (A)
C
D
R (mm)
z
1
L (H)
0.25
1000
2
8
100
25
180
8707.2
−0.00066
100
8
1
×
10
-
3Nonlinear forced vibration to coupled excitations far from natural frequencies.
(a)
u
i
(b)
z
i
(c)
u
rWhen exciting frequency is near natural frequency, changes of the nonlinear vibrating magnitudes along with exciting frequencies and the drive parameters are given in Figure5. From Figure 5, the following are known.(1)
As exciting frequency increase, the vibrating magnitudes of all the modes increases, and at pointsi=ωei/ωi≈1, the vibrating magnitudes get to the maximum, and then they decrease with increasing exciting frequency. For different modes, the previously mentioned curves bend toward the direction of the exciting frequency increase. The results are typical nonlinear character of the drive system.(2)
For the first and third modes, as nonlinear parameterε increases, their vibrating magnitudes increase. For the second mode, as nonlinear parameter ε increases, its vibrating magnitude decreases. The vibrating magnitudes are large for the first and third modes and small for the second mode. The vibrating magnitude for the first mode is larger than that for the third mode.(3)
For a given exciting frequency, the vibrating amplitudes of the nonlinear forced responses of the drive system to the coupled excitations for modes 1 and 3 are larger than those of the nonlinear forced responses to the single excitation; for mode 2, the vibrating amplitude of the nonlinear forced responses of the drive system to the coupled excitations is smaller than that of the nonlinear forced responses to the single excitation.(4)
Under the coupled excitations, the nonlinearity has obvious effects on the relationship between the frequency and the amplitudes; it should be considered.Nonlinear forced vibration to coupled excitations near natural frequencies.
(a)
mode 1
(b)
mode 2
(c)
mode 3


## 7. Conclusions

In this paper, the electric excitation and the parameter excitation from mesh stiffness fluctuation are analyzed. The forced response equations of the drive system to the coupled excitations are presented. For the exciting frequencies far from and near to natural frequencies, the forced responses of the drive system to the coupled excitations are investigated. Results show the following.(1)
Under the coupled excitations, the nonlinear forced responses of the drive system change periodically and unsteadily.(2)
The time period of the nonlinear forced responses depends on the frequencies of the electric excitation the mesh parameter excitation, and the nonlinear natural frequencies of the drive system.(3)
In order to improve the dynamics performance of the drive system, the frequencies of the electric excitations should not be taken as integral multiple of the mesh parameter exciting frequency.



---

*Source: 743138-2012-09-25.xml*