# A Coupled Torsional-Transition Nonlinear Vibration and Dynamic Model of a Two-Stage Helical Gearbox Reducer for Electric Vehicles

Authors: Abdulhameed M. Y. Al-Tayari; Siyu Chen; Zhou Sun
Journal: Shock and Vibration
 (2020)

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## Abstract

A coupled torsional-transition nonlinear dynamic model of a two-stage helical gear (TSHG) reduction system for electric vehicles (EVs) is presented in this paper. The model consists of 16 degrees of freedom (DOF), which includes factors such as the nonlinearity of backlash, time-varying mesh stiffness (TVMS), mesh damping, supporting bearings, static transmission error (STE), and the torsional damping and stiffness of the intermediate shaft, in which the fourth-order Runge–Kutta numerical integration method was applied to solve the differential equations. With the help of bifurcation diagrams, time-domain histories diagrams, amplitude-frequency spectrums, phase plane diagrams, Poincaré maps, root-mean-square (RMS) curves, peak-peak values (PPVs), and Lyapunov exponents, the effects of pinion rotational speed, backlash, torsional stiffness, and torque fluctuation on the dynamic behavior of TSHG system are investigated. The stability properties of steady-state responses are investigated using Lyapunov exponents. The results reveal various types of dynamic evolution mechanisms and nonlinear phenomena such as periodic-one responses, quasiperiodic responses, jumps phenomena, and chaotic responses. The research presents useful results and information to vibration control and dynamic design of the TSHG transmission system used in EVs.


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## Body

## 1. Introduction

It is known that the power mechanism of EVs is composed of a power coupling system (transmission shafts, gear pairs, bearings, etc.), electrical control systems, motor, and other subsystems. As gear pairs are one of the main parts of EV power coupling system, the dynamic behavior of the gear pairs mechanism has a significant influence on power transmission and even the whole vehicle [1]. Therefore, the analysis of the dynamic characteristics of gear pairs is becoming essential in power transmission systems.Numbers of dynamical models for both helical and spur gear pairs systems have been presented; a mathematical model and computer simulation of a two-stage gearbox including translational-torsional vibration responses of gear pairs system were presented by Walter Bartelmus [2]. Based on ANSYS software, Yan and Liu [3] developed a new finite-element modeling method for helical gear transmission with multiple shafts considering the effects of bearing flexibilities and shafts. A dynamic model of a helical gear multiple-shafts reduction box was presented by Kubur et al. in [4, 5], which consists of a finite-element model of shaft structures combined with another three-dimensional discrete model of gear pairs. Moreover, Zhang et al. [6] presented an effective dynamic model of a multiple-shafts helical geared-rotor system with geometric eccentricity, bearing flexibility, and gear mesh, in which the eigenvalue solution and summation method were applied to estimate the forced responses and the natural frequencies of the geared system.Since the nonlinear characteristics of gear-pair system have been the most significant research fields, many studies on the dynamic model and response of two-stage gear systems have been published. For instance, Walha et al. [7] proposed a purely torsional spur gear system with three shafts connected by two spur gear pairs, in which the Newton–Raphson algorithm was used to examine the influence of backlash on two-stage spur gear (TSSG) system. Based on the variation of frequency response characteristic, Baek et al. [8] proposed a new technique to predict the contribution ratio of each stage backlash of the TSSG reducer; the validity of the used method was validated in a seeker gimbal with adequate results achieved. Since the dynamic coupling and TVMS between the helical gear pairs have a significant effect on the vibration characteristics of the helical gear system, Wang and Shi [9] proposed a systematic model to analyze the influence of TVMS and helix angle on gear pair system. In [10], Wang and Zhang developed a dynamics model of tensional-bending-swing-axial coupled motion for (TSHG) transmission considering the TVMS, backlash, and deviation, where a fifth-order Runge–Kutta method was used to solve the differential equations. An eighteen-DOF model considering the effects of the loading system, driving motor, and supporting bearings was proposed by Brethee et al. [11] to investigate the dynamic response of surface wear on gear including tooth friction and TVMS based on electrohydrodynamic lubrication (EHL) principles.Considering the effects of TVMS and nonlinear characteristics of bearing, a dynamical model of a TSHG box including the motion of housing was established by Xu et al. [12], and steady-state vibration response of the gearbox has been obtained based on the coupling gear-rotor-bearing dynamic model. Furthermore, experimental modal analysis was used by Patel and Pathan [13] to derive the natural frequencies of TSHG reducer considering the effect of TVMS on natural frequency, and the modal parameters were obtained by applying the Frequency Response Function (FRF) method. Walha et al. [14] developed a 12-DOF dynamic model to study the nonlinear dynamic responses of a TSSG system including mesh stiffness fluctuation, backlash, and bearing flexibility, in which the technique of linearization was used to decompose the system from nonlinear to linear. Ma et al. [15, 16] presented a 14-DOF dynamic model with an experimental study of TSSG to investigate the nonlinear dynamic response analysis of the TSSG space driving mechanism under large inertia load taking into consideration TVMS, damping, backlash, and transmission error. And Bin [17] used the model to analyze the effects of the profile modification parameters on the load transmission error, dynamic load coefficient, tooth profile error, and modification. Another 26 DOF of TSSG was established by Jia et al. [18] to study the dynamical modeling of multiple pairs of spur gears in mesh considering the effect of variable tooth stiffness, friction, geometrical errors, localized tooth crack, and pitch and profile errors on one gear.Systematic modeling and analysis of a TSSG box model with 12 and 26 DOF were used to describe the gear fault features when processed with harmonic wavelet transform (HWT) in [19, 20]. Furthermore, He et al. [21] proposed a TSSG with a twelve-DOF dynamic model to study the influences of gear eccentricity on transverse and torsional dynamic responses and the dynamic transmission errors. Dynamic behavior of a three-dimensional model TSHG system considering the effect of manufacturing defects was formulated by Walha et al. [22], in which the dynamic response was performed by the Newmark method. Dyk [23] examined the models of TSSG used by the discrete models and considering the effects of the intermediate shaft on dynamic loads in both two stages. Abboudi et al. [24] developed a lamped-mass dynamic model of TSHG with twelve DOF used in wind turbines, excited by the variability in wind resources and TVMS fluctuation; the differential equations’ motion of system was solved by the implicit Newmark algorithm. Beyaoui et al. [25] proposed a new methodology considering uncertainties in a gear transmission system of a horizontal-axis wind turbine, in which the dynamic equations of 12 DOF were solved by applying the polynomial chaos method and the ODE-45 MATLAB solver. A dynamic model for an automotive train system with 22 DOF was established by Ghorbel et al. [26] to study the kinetic, the vibration mode, and strain modal energies distributions taking into consideration the engine excitation, clutch, gearbox, and disc brake. In order to investigate the nonlinear dynamic response of the TSHG system coupled with an automotive clutch, Walha et al. [27] proposed a dynamic model of 27 DOF considering spline clearance, double-stage stiffness, and dry friction path, in which the equation of motion was solved by Runge–Kutta method.According to the literature reviews mentioned in Section1, it is indicated that various dynamical models have been presented to analyze the dynamics of two-stage gears pairs. Nevertheless, limited studies have addressed the effect of gear nonlinear dynamic response of TSHG box reduction used in EVs. For this reason, a nonlinear torsional-translational dynamic model of a TSHG reduction with a 16 DOF is derived based on Zheng’s method [6], in which the TVMS, mesh damping, support bearings, STE, backlash, and torsional stiffness and damping of the intermediate shaft are considered in this paper.Consequently, the dynamic model in Section2 and numerical experiments in Section 3 were carried out to demonstrate the nonlinear dynamic characteristics, where the equation of motion is solved by the Runge–Kutta method. Moreover, the effects of pinion rotational speed, backlash, torsional stiffness, and torque fluctuation on the dynamic behavior of TSHG reduction were also studied, which provide an essential understanding of the nonlinear dynamic features of TSHG reduction, as concluded in Section 4.


## 2. The Dynamic Model of TSHG System

A 16-DOF nonlinear dynamic model is presented to simulate the dynamic behavior of a TSHG reduction system, as shown in Figure1. As revealed by the geometric description in Table 1, the chosen gearbox system is a speed reducer. The system is formed by four helical gears p1,g1,p2, and g2; pinion and gear are denoted with the subscripts p and g as in i=p,g; the two stages of the system are denoted with subscripts 1 and 2 as in j=1,2, respectively. Each gear is represented as rigid blocks with 4 DOF (one rotation and three translations), in which Ωij,Iij, and rij represent the rotating speed, the moment of inertia, and the base circle radius of gears i and j, respectively. TD and TL represent the torque values applied to the pinion p1 and the gear g2, respectively. The nonlinear backlash function fxmj, STE ejt, TVMSkmjt, and mesh damping cmj are combined to describe the gear deformation during the meshing process [28]. The resilient elements of bearing supports are represented by the damping cvw and stiffness kvw coefficients, where v=1,2,3,4 indicates the four bearings in w=x,y, and z directions (LOA, OLOA, and axial direction), respectively. The intermediate shaft of the TSHG component is described by torsional stiffness K1 and damping C1 components.Figure 1
A simplified dynamic model of a TSHG system.Table 1
The geometric properties of the TSHG system.
PropertiesFirst stageSecond stagePinion (p1)Gear (g1)Pinion (p2)Gear (g2)Teeth numberzp1=25zg1=77zp2=29zg2=91Base circle radius (mm)rp1=21.4rg1=65.8rp2=30.6rg2=96.0Mass (kg)mp1=0.286mg1=2.496mp2=0.797mg2=7.068Working face width (mm)bp1=22.5bg1=21.0bp2=30.0bg2=28.0Helix angle (°)β1=21.25β2=16.65Pressure angle (°)ϕ1=18.5ϕ2=16.5Modulus (mm)M1=1.69M2=2.12The generalized coordinates vectors of the nonlinear dynamic model including 16 DOF can be defined as(1)qm=θp1,xp1,yp1,zp1,θg1,xg1,yg1,zg1,θp2,xp2,yp2,zp2,θg2,xg2,yg2,zg2T,where xij,yij, and zij represent the translations of pinion and gear along axes x,y, and z; θij represents the torsional displacement of pinion and gear around axis z.The STEejt is generally considered to be a periodic function of displacement, usually spreads out in Fourier series as fundamental frequency part of the harmonics [29], and can be expressed as(2)ejt=e0j+Ehjsinωhjt+ϕhj,where e0j,Ehj, and ϕhj are the constant amplitude of STE, the amplitude of STE, and phase angle, respectively. The excitation meshing frequency ωhj of gear pair is determined as shown in [30](3)ωhj=ΩpjZpj60,where Ωpj and Zpj are the rotational speed of pinion and teeth number.According to Ishikawa’s method, the TVMS is simplified as periodic waveforms under the meshing frequency and spreads out into Fourier series [31]; the nonlinear TVMS kmjt obtained using the Fourier expansion is written as follows:(4)kmjt=koj+Akjsinωhjt+ϕkj,where koj,Akj, and ϕkj are the mean value of meshing stiffness, the stiffness fluctuation amplitude, which equals koj×0.2, and phase angle, respectively. The mean value of the mesh damping cmj is expressed as(5)cmj=2ζjkmj∗rpj2rgj2IpjIgjrpj2Ipj+rgj2Igj.Here, the dampingζj ratio is calculated as Rayleigh damping [32, 33], in which the value generally ranges from 0.03 to 0.17; in this model, the ratio is ζj=0.1, where j=1,2 demonstrates the first and second stages.Whenxmj presents the relative meshing displacement of the gear under the influence of the gear backlash sij [34], the nonlinear backlash function is expressed as follows:(6)fxmj=xmj−sj,xmj>sj,0,−sj≤xmj≤sj,xmj+sj,xmj<sj.Here,2sj represents the total backlash.The gear mesh is expressed by a nonlinear TVMSkmjt and mesh damping cmj. The dynamic mesh force Fmj of gear pairs along the LOA can be expressed as(7)Fmj=kmjtfxmj+cmjx˙mj,Fpjx=Fmjsinαnjcosβj,Fgjx=−Fpjx,Fpjy=Fmjcosαnjcosβj,Fgjy=−Fpjy,Fpjz=−Fmjsinβj,Fgjz=−Fpjz,where Fpjx,Fpjy,Fpjz and Fgjx,Fgjy,Fgjz are the dynamic mesh forces of pinions and gears along with the coordinate directions x,y, and z, respectively. αnj and βj indicate the pressure angle and helix angle, respectively.The governing motion equations of the TSHG system shown in Figure1 are expressed in the state space form, which is solved by MATLAB ODE-45 solver and derived considering the following assumptions:(1)
Pinions and gears are modeled as rigid disks(2)
Both inputTD and load TL torque are applied to a system with constant values(3)
The gear teeth are considered to be fully involute; assembly and manufacturing errors are ignored(4)
Spring and damper are used to represent the torsional stiffness of the intermediate shafts in the midsection of the shaftIt is noted that the modeling method of Zhang in [6] takes into account the effects of geometric eccentricity on gears, though, in this study, the influence of eccentricity is neglected. According to the proposed concept, the rotational and translational motion equations of the TSHG reduction gear pairs are expressed by the following equations, respectively:(8)Ip1θ¨p1+Fm1rp1cosβ1=TD,Ig1θ¨g1+C1θ˙g1−θ˙p2+K1θg1−θp2+Fm1rg1cosβ1=0,Ip2θ¨p2−C1θ˙g1−θ˙p2−K1θg1−θp2−Fm2rp2cos−β1=0,Ig2θ¨g2−Fm2rg2cos−β2=−TL,(9)mp1x¨p1+cx1x˙p1+kx1xp1−Fm1cosβ1sin−αn1=0,mp1y¨p1+cy1y˙p1+ky1yp1+Fm1cosβ1cos−αn1=0,mp1z¨p1+cz1z˙p1+kz1zp1+Fm1sinβ1=0,mg1x¨g1+cx2x˙g1+kx2xg1+Fm1cosβ1sin−αn1=0,mg1y¨g1+cy2y˙g1+ky2yg1−Fm1cosβ1cos−αn1=0,mg1z¨g1+cz2z˙g1+kz2zg1−Fm1sinβ1=0,mp2x¨p2+cx3x˙p2+kx3xp2−Fm2cos−β2sinαn2−π=0,mp2y¨p2+cy3y˙p2+ky3yp2+Fm2cos−β2cosαn2−π=0,mp2z¨p2+cz3z˙p2+kz3zp2−Fm2sin−β2=0,mg2x¨g2+cx4x˙g2+kx4xg2+Fm2cos−β2sinαn2−π=0,mg2y¨g2+cy4y˙g2+ky4yg2−Fm2cos−β2cosαn2−π=0,mg2z¨g2+cz4z˙g2+kz4zg2+Fm2sin−β2=0.With those four DOF for each gear, the two-stage gear pairij has a total of 16 DOF that define the coupling among the TSHG system, where mij indicates the mass of i=p,g pinion p and gear g in the first and second stages j=1,2, respectively.The relative displacement of the first-stagexm1 and second-stage xm2 gear mesh along the LOA is defined as(10)xm1=−xp1+xg1sin−αn1+yp1−yg1cos−αn1+rp1θp1+rg1θg1cosβ1+zp1−zg1sinβ1−e1t,xm2=−xp2+xg2sinαn2−π+yp2−yg2cosαn2−π−rp2θp2+rg2θg2cos−β2−zp2−zg2sin−β2−e2t.Equation (8) can be combined and substituted into equation (10); thus, the relative displacement of the first and second stages became as follows:(11)x¨m1=−x¨p1+x¨g1sin−αn1+y¨p1−y¨g1cos−αn1−C1DIg1x˙b−K1DIg1xb+rp1TDIp1−Fm1rp12cosβ1Ip1−Fm1rg12cosβ1Ig1cosβ1+z¨p1−z¨g1sinβ1−e¨1t,x¨m2=−x¨p2+x¨g2sinαn2−π+y¨p2−y¨g2cosαn2−π−C1HIP2x˙b−K1HIP2xb+rg2TLIg2−Fm2rp22cos−β2Ip2−Fm2rg22cos−β2Ig2cos−β2−z¨p2−z¨g2sin−β2−e¨2t,where xb is the relative displacement of the first gear and second pinion and can be written as follows:(12)x¨b=−C1DIg1+C1HIp2x˙b−K1DIg1+K1HIp2xb−Fm1rg12cosβ1Ig1−Fm2rp22cos−β2Ip2,with B=rg1+rp2/2,D=rg1/B,andH=rp2/B.For analytical convenience, the dimensionless form of equations (9), (11), and (12) is obtained by assuming the following nondimensional parameters as(13)Xij=xijsj,Yij=yijsj,Zij=zijsj,Xb=xbsj,Ejτ=ejtsj,τ=ωnt.Here,ωnj=koj/mei represents the natural frequency of gear pairs and mei=IpiIgi/Ipirgi2+Igirpi2 is the equivalent mass of gear pairs. The small letter of each variable represents the derivative to time t, while the big letter represents the derivative with respect to dimensionless time τ.Therefore, the dimensionless form of the nonlinear backlash function in equation (7) becomes (14)fXmj=Xmjτ−1,Xmj>1,0,−1≤Xmj≤1,Xmjτ+1,Xmj<1.The equations of motion of the entire system in the dimensionless form can be expressed in matrix forms as(15)MQ¨m+CQ˙m+KQm=FQ.The dimensionless mass matrix, damping matrix, stiffness matrix, and the external excitation force vectors of the system are represented byM,C,K, and FQ, respectively. Consequently, the simplified coordinates vectors of the dimensional nonlinear dynamic model can be defined as Qm:(16)Qm=Xp1,Yp1,Zp1,Xg1,Yg1,Zg1,Xp2,Yp2,Zp2,Xg2,Yg2,Zg2,Xb,Xm1,Xm2T.The mass matrixM can be expressed by(17)M=100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100m1m2m3m4m5m6000000010000000m7m8m9m10m11m12001.Here,mi represents the mass of the block i=1,⋯,12 which can be expressed as follows:(18)m1=−cosβ1sin−αn1,m2=cosβ1cos−αn1,m3=sinβ1,m4=cosβ1sin−αn1,m5=−cosβ1cosαn1,m6=−sinβ1,m7=−cos−β2sinαn2−π,m8=cos−β2cosαn2−π,m9=−sin−β2,m10=cos−β2sinαn2−π,m11=−cos−β2cosαn2−π,m12=sin−β2.For the sake of simplicity, the stiffness matrixK and damping matrix C are expressed by ΛCK as follows:(19)ΛCK=Λ1000000000000Λ200Λ300000000000Λ4000Λ50000000000Λ60000Λ7000000000Λ800000Λ900000000Λ10000000Λ110000000Λ120000000Λ130000000Λ140000000Λ15000000Λ1600000000Λ1700000Λ18000000000Λ190000Λ200000000000Λ21000Λ2200000000000Λ2300Λ24000000000000Λ25Λ26Λ27000000000000Λ28Λ290000000000000Λ300Λ31,where the damping matrix C and stiffness matrix K are denoted in ΛCK with subscripts C and K, respectively. Here, Λi is the damping and stiffness of the block i=1,⋯,31, which can be expressed by equations (20) and (21) as follows:(20)k1=−kx1mp1ωn12,k2=Km1τcosβ1sin−αn1mp1ωn12,k3=−ky1mp1ωn12,k4=−Km1τcosβ1cos−αn1mp1ωn12,k5=−kz1mp1ωn12,k6=−Km1τsinβ1mp1ωn12,k7=−kx2mg1ωn12,k8=−Km1τcosβ1sin−αn1mg1ωn12,k9=−ky2mg1ωn12,k10=Km1τcosβ1cos−αn1mg1ωn12,k11=−kz2mg1ωn12,k12=Km1τsin−β1mg1ωn12,k13=−kx3mp2ωn22,k14=Km2τcos−β2sinαn2−πmp2ωn22,k15=−ky3mp2ωn22,k16=−Km2τcos−β2cosαn2−πmp2ωn22,k17=−kz3mp2ωn22,k18=Km2τsin−β2mp2ωn22,k19=−kx4mg2ωn22,k20=−Km2τcos−β2sinαn2−πmg2ωn22,k21=−ky4mg2ωn22,k22=Km2τcos−β2cosαn2−πmg2ωn22,k23=−kz4mg2ωn22,k24=−Km2τsin−β2mg2ωn22,k25=−K1DIg1ωn12+K1HIp2ωn22,k26=−Km1τrg12cosβ2Ig1ωn12,k27=−Km2τrp12cos−β2Ip2ωn22,k28=−K1Dcosβ1Ig1ωn12,k29=−Km1τcos2β1ωn12rp12Ip1+rg12Ig1,k30=−K1Hcos−β1Ip2ωn22,k31=−Km2τcos2−β2ωn22rp22Ip2+rg22Ig2,(21)c1=−cx1mp1ωn1,c2=Cm1cosβ1sin−αn1mp1ωn1,c3=−cy1mp1ωn1,c4=−Cm1cosβ1cos−αn1mp1ωn1,c5=−cz1mp1ωn1,c6=−Cm1sinβ1mp1ωn1,c7=−cx2mg1ωn1,c8=−Cm1cosβ1sin−αn1mg1ωn1,c9=−cy2mg1ωn1,c10=Cm1cosβ1cos−αn1mg1ωn1,c11=−cz2mg1ωn1,c12=Cm1sinβ1mg1ωn1,c13=−cx3mp2ωn2,c14=Cm2cos−β2sinαn2−πmp2ωn2,c15=−cy3mp2ωn2,c16=−Cm2cos−β2cosαn2−πmp2ωn2,c17=−cz3mp2ωn2,c18=Cm2sin−β2mp2ωn2,c19=−cx4mg2ωn2,c20=−Cm2cos−β2sinαn2−πmg2ωn2,c21=−cy4mg2ωn2,c22=Cm2cos−β2cosαn2−πmg2ωn2,c23=−cz4mg2ωn2,c24=−Cm2sin−β2mg2ωn2,c25=−C1DIg1ωn1+C1HIp2ωn2,c26=−Cm1rg12cosβ2Ig1ωn1,c27=−Cm2rp12cos−β2Ip2ωn2,c28=−C1Dcosβ1Ig1ωn1,c29=−Cm1cos2β1ωn1rp12Ip1+rg12Ig1,c30=−C1Hcos−β1Ip2ωn2,c31=−Cm2cos2−β2ωn2rp22Ip2+rg22Ig2.However, the vector forceFQ can be written as(22)FQ=0,0,0,0,0,0,0,0,0,0,0,0,0,rp1TDcosβ1Ip1s1ωn12−E1τ,rg2TLcos−β2Ig2s2ωn22−E2τT.


## 3. Results and Discussion

The basic parameters of the TSHG reduction gear pairs studied in this paper are shown in Table2. With those parameters, the set of the second-order differential equations in equation (16) are solved by the Runge–Kutta method. With the help of the FFT spectrum, phase portrait, Poincaré point section, time-domain history, RMS, PPVs curves, and Lyapunov exponents, the effects of the rotational speed of pinion Ωp1, backlash, torsional stiffness, and torque fluctuation on the dynamic behavior of TSHG components were studied by applying the numerical integration method considering different conditions. Generally, the dynamic responses are divided into five categories: chaotic, quasiperiodic, subharmonic, periodic harmonic, and periodic nonharmonic response. When the time history is nonperiodic and has unlimited nonrepeating points in the Poincaré section, the response is chaotic. While the response is quasiperiodic, the phase portrait has nonperiodic circles and the points of the Poincaré section formed a closed orbit. When multiple discrete points are formed in the Poincaré section and repeat themselves at the excitation frequencies, the responses become subharmonic response. The system is harmonic period when the phase portrait is circular and repeats itself in Poincaré section. At last, the response is nonharmonic periodic if the phase portrait is noncircular and repeats itself in Poincaré section [35]. The results in this section provide some optimization suggestions for the TSHG reduction gear-pair design.Table 2
The basic parameters of the TSHG reduction gear pairs.
ParametersFirst stageSecond stagePinion (p1)Gear (g1)Pinion (p2)Gear (g2)Moment of inertia (kg⋅mm2)Ip1=76.19Ig1=6000Ip2=437.2Ig2=36000Meshing damping (N⋅s/mm)cm1=0.8064×10−3cm2=3.2918×10−3Meshing stiffness (N/mm)km1=4.676×105km2=6.623×105Torque (Nmm)TD=250×103TL=245.70×103STE (μm)Eh1=20Eh2=20Bearing damping (N⋅s/mm)cij=0.5,i=x,y,z,j=1,2,3,4Bearing stiffness (N/mm)kij=3.5×104,i=x,y,z,j=1,2,3,4Torsional damping (N⋅mm⋅s/rad)C1=2091.6Torsional stiffness (N⋅mm/rad)K1=3.3134×1010


### 3.1. Effect of the Rotational Speed on the TSHG Dynamic Response

The rotational speed of pinionΩp1 is considered as one of the main parameters that affect the dynamical behavior of system transmission. In this section, the half backlash of gears is set to s1=s2=40μm; the other parameters of TSHG transmission system are given in Table 2. Figure 2 presents the forward bifurcation characteristic of the first stage Xm1 and second stage Xm2 in dimensionless displacement with regard to the rotating speed of pinion Ωp1 as the control parameter, which increases gradually from 1000 rpm to 20000 rpm, while Figure 3 presents the backward bifurcation characteristic with regard to the decreasing of Ωp1 from 20000 rpm to 1000 rpm. As shown in Figures 2 and 3, the system responses of both Xm1 and Xm2 under the variation of Ωp1 contain types of motion forms such as periodic-one motion marked with letter P, jumps phenomena, quasiperiodic motion marked with letter Q, and chaotic motion marked with letter C.Figure 2
Forward bifurcation diagrams of system usingΩp1 as control parameter; s1=s2=40μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 3
Backward bifurcation diagrams of system usingΩp1 as control parameter; s1=s2=40μm. (a) Xm1. (b) Xm2.
(a)(b)In the case of the foreword bifurcation, the systemXm1 behaves as a steady period-one motion at low speed and persists until Ωp1 reaches 7110.5528 rpm. The response of time-domain history shows a sine wave, one main peak amplitude is found in FFT spectrum, the phase plane has a single closed circle, a single point is found in Poincaré section, and these characteristics indicate that the response of Xm1 is harmonic periodic-one motion, as shown in Figure 4.Figure 4
The dynamic response of systemXm1 at 2500 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)The system keeps transforming between periodic motion and quasiperiodic motion several times untilΩp1 reaches 17040 rpm. However, the system is periodic-one motion except when Ωp1 is in the ranges of 7110–7493 rpm, 9974–10452 rpm, 11407–13221 rpm, and 15703–17040 rpm; the system experiences the quasiperiodic motion response. As illustrated in Figure 5, the phase plane diagram does not repeat itself, which forms closed orbit points in Poincaré section and the FFT spectrum is continuous; this means that the system response is in quasiperiodic motion. Meanwhile, a jump phenomenon can be observed when Ωp1 is around the resonant regions at 12744 rpm. When Ωp1 increases to 17040 rpm, the system response changes from quasiperiodic motion to chaotic and remains until reaching 18758 rpm. From Figure 6, it can be seen that system response does not repeat itself in any pattern, the FFT spectrum has continuous broadband, and the phase plane shows disorder circles with many discrete points found in Poincaré section. It is revealed from these properties that the system responses as chaotic motion at the range of 17040–18758 rpm. However, with the increase of pinion speed, the system abandons the chaotic region and returns to quasiperiodic motion response again which preserves from 18758 rpm to 20000 rpm.Figure 5
The dynamic response of systemXm1 at 16950 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 6
The dynamic response of systemXm1 at 17157 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)As for the backward bifurcation, considering the decreasing ofΩp1 from 20000 rpm to 1000 rpm continuously, a chaotic motion is determined at 20000–18759 rpm and 17613–17040 rpm. As illustrated in Figure 7, the phase plane diagram shows nonrepeated circles with many clustered points of Poincaré map and the diagram of time-domain history also shows a nonperiodic motion. Figure 8 reveals that the system transforms into quasiperiodic response at 18759–17613 rpm, 17040–15703 rpm, and 13221–15512 rpm, where a jump phenomenon occurs at 17613 rpm. The system behaves as periodic-one motion response at 15703–13221 rpm and undergoes the chaotic region at 13221–11979 rpm. The chaotic phenomenon causes the vibration and noise issues in the system, which should be avoided during the design. Meanwhile, the system leaves the chaotic region with a jump phenomenon occurring at 11979 rpm. However, the system characteristic responses are the same as the responses in the forwarding bifurcation when Ωp1 is less than 11979 rpm, which is verified by PPVs and RMS curves, as shown in Figures 9 and 10. At this range of speed, the system response of Xm1 is mostly periodic-one motion, as marked in bifurcation diagram with letter P and proved in Figure 11.Figure 7
The dynamic response of systemXm1 at 19522 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 8
The dynamic response of systemXm1 at 16600 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 9
PPVs of forward and backward behaviors of system;s1=s2=40μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 10
RMS curves of forward and backward behaviors of system ats1=s2=40μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 11
The dynamic response of systemXm1 at 10930 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)It is observed that, under the variation ofΩp1 between 1000 and 20000 rpm, the characteristic of system Xm2 in forward bifurcation shown in Figure 2(b) is similar to the one in backward bifurcation shown in Figure 3(b), which is verified by RMS and PPVs curves in Figures 9(b) and 10(b). For the sake of simplicity, both bifurcations are discussed in detail as one bifurcation. As illustrated in Figures 2(b) and 3(b), the system Xm2 behaves as a periodic-one motion response at 1000–1763 rpm, as illustrated in Figure 12. The system is in quasiperiodic response at 1763–2241 rpm and returns into periodic-one motion again at 2214–5869 rpm. Consequently, the system behaves as quasiperiodic motion and remains in the range of 5869–17040 rpm, where the quasiperiodic motion is indicated by the closed orbit formed in Poincaré section, as shown in Figure 13. And jumps phenomena occur around 3577 rpm and 17040 rpm. In the range from 17135 rpm to 17899 rpm, the system becomes irregular and turns rapidly into chaotic motion, as verified by the disorder circles of phase plane diagram and the discrete points of Poincaré map in Figure 14. In the meantime, with the increasing of the speed, the displacement range of chaotic motion narrows gradually. Eventually, the system behaves as a quasiperiodic motion at 17899–2000 rpm, as illustrated in Figure 15. The results mentioned above reveal that the dynamic behavior of Xm1 and Xm2 is steady at low-speed range. Therefore, it is concluded that the dynamic behavior transforms from linear to nonlinear response with the increases of Ωp1.Figure 12
The dynamic response of systemXm2 at 1500 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 13
The dynamic response of systemXm2 at 16370 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 14
The dynamic response of systemXm2 at 17709 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 15
The dynamic response of systemXm2 at 19000 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)From the corresponding PPVs and RMS curves, the conclusions of the relative displacement are obtained that there are two apparent bistable response regions in systemXm1 at the range of 11979–12839 rpm and 17613–20000 rpm, respectively. The chaotic motion ranges of Xm1 are indicated at 17040–18758 rpm as for forward case, but at 20000–18758 rpm, 17040–17613 rpm, and 11979–12457 rpm as for backward case, as shown in Figure 9(a), while the chaos response of Xm2 is indicated at 17135–17899 rpm, as illustrated in Figure 9(b). The response of the system Xm1 experiences a jump down phenomenon at 12744 rpm in the acceleration process and a jump up phenomenon at 11979 rpm and 17613 rpm in the deceleration process, as illustrated in Figure 10(a), while in system Xm2, jump up and down phenomena occur around 3577 rpm and 17040 rpm, as shown in Figure 10(b).


### 3.2. Effect of the Nonlinear Backlash on the TSHG Dynamic Response

According to the manufacturing accuracy level of gear pair, half of the backlash values of systemsXm1 and Xm2 are set to increase from s1=s2=40μm,60μm to 100μm. For the sake of simplicity, the backward bifurcation diagrams of systems under different backlash values are investigated, as illustrated in Figures 3, 16, and 17, respectively.Figure 16
Backward bifurcation diagrams of system;s1=s2=60μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 17
Backward bifurcation diagrams of system;s1=s2=100μm. (a) Xm1. (b) Xm2.
(a)(b)By comparing the bifurcation diagrams of both systemsXm1 and Xm2 under different backlash values, it is found that the range of chaotic motion in the rotational speed of pinion enlarges; the critical speed goes frontward along with the increasing of backlash value. In the meantime, the characteristics of chaotic motion are increased in both systems, which increases the difficulty of predicting the vibration of the gear pairs and leads to more noise possibility.The corresponding RMS and PPVs curves of both systems concerning the variation ofΩp1 under different backlash values are illustrated in Figures 9, 10, and 18–21. It is seen that the displacement amplitudes of the PPV and RMS of both systems Xm1 and Xm2 decrease along with the increase of the backlash, and the range of Ωp1 expands as the chaos range extends.Figure 18
PPVs of forward and backward bifurcation system;s1=s2=60μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 19
RMS curves of forward and backward bifurcation system;s1=s2=60μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 20
PPVs of forward and backward bifurcation system;s1=s2=100μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 21
RMS curves of forward and backward bifurcation system;s1=s2=100μm. (a) Xm1. (b) Xm2.
(a)(b)Considering the bifurcation diagrams, PPVs, and RMS curves of both systems, we conclude that, under the increases of backlash value, the characteristics of chaotic response could be enhanced which expand the chaotic motion range and enlarge the rotational speed range; as a result, this will lead to more possibility of noise vibration. Nevertheless, the backlash could also lower the amplitudes of RMS and PPVs and deescalate the vibration extension of the system. Consequently, a suitable value of backlash should be selected to meet the requirements of vibration amplitude and reduce the possibility of chaotic behavior extension [32–34].


### 3.3. Effect of the Torsional Stiffness on the TSHG Dynamic Response

The torsional stiffness of the intermediate shaft is one of the key parameters that affect the dynamic behavior of the transmission system. Therefore, it is essential to investigate the effect of the torsional stiffness of the intermediate shaft on the dynamic characteristics of the TSHG system. In this section, the backlash values of both systemsXm1 and Xm2 are set to s1=s2=40μm; the torsional stiffness K1 is set to increase as 3.3134×107N⋅mm/rad,3.3134×108N⋅mm/rad, and 3.3134×1010N⋅mm/rad, respectively.Figures22, 23, and 2 illustrate the corresponding forward bifurcation diagrams of the systems Xm1 and Xm2 with respect to the rotational speed of pinion under different torsional stiffness values. According to the comparison of the bifurcation diagrams under different torsional stiffness values, the substantial region of periodic, quasiperiodic, and chaotic responses including jump phenomena are observed. At high-speed range of 17000–19000 rpm, the system is stable and behaves as a periodic motion when K1=3.3134×107N⋅mm/rad. The dynamic response transits from periodic to quasiperiodic motion when K1=3.3134×108N⋅mm/rad and eventually turns into chaotic response as K1 increases to 3.3134×1010N⋅mm/rad. Therefore, the nonlinear system response becomes unstable and the chaotic behavior consequently expands as K1 increases.Figure 22
Forward bifurcation diagrams of system whenK1=3.3134×107N⋅mm/rad. (a) Xm1. (b) Xm2.
(a)(b)Figure 23
Forward bifurcation diagrams of system whenK1=3.3134×108N⋅mm/rad. (a) Xm1. (b) Xm2.
(a)(b)


### 3.4. Effect of the Torque Fluctuation on the TSHG Dynamic Response

According to the BMW i8 electric motor characteristics [36], the ideal torque fluctuation TD is set to vary from 44×103Nmm to 250×103Nmm under the rotational speed range of 1000–20000 rpm. In this section, the backlash values of both systems Xm1 and Xm2 are set to s1=s2=100μm, the torsional stiffness K1 is set to 3.3134×1010N⋅mm/rad, and the torque fluctuation TD is considered as a control parameter that varies along with the changing of rotational speed. From the corresponding backward bifurcation diagrams of both systems Xm1 and Xm2 illustrated in Figure 24, it is observed that the dynamic response and the displacement of both systems at the speed range of 1000–4000 rpm are stable when the torque fluctuation is 250×103Nmm. As the torque varies among 240−70×103Nmm, the corresponding displacement changes clearly at the speed range of 4000–13000 rpm. Then, the displacement is stable again under the torque variation among 70−44×103Nmm until the speed reaches 20000 rpm.Figure 24
Backward bifurcation diagrams of system under the motor torque fluctuation ofTD. (a) Xm1. (b) Xm2.
(a)(b)However, it is observed that chaotic motion for both systems exists at the speed range of 13000–20000 rpm when the torque fluctuates at low values between52−44×103Nmm and 68−66×103Nmm. Quasiperiodic motions are observed at the speed range of 7111–17000 rpm when the torque fluctuates between 131−124, 92−88, 80−70, and 67−52×103Nmm as for the system Xm1 and at the speed range of 6000–17000 rpm when the torque fluctuation range is around 153−68×103Nmm and 65−61×103Nmm as for the system Xm2. When the torque fluctuation is fixed at 250×103Nmm, the responses of system are stable one-periodic motion at the speed range of 1000–7000 rpm. As a result, comparing with Figure 17, the torque fluctuation at high-speed range extends the chaotic and quasiperiodic regions corresponding to the speed and expands their ranges corresponding to the displacement. The corresponding displacement response is stable at 250×103Nmm and at low range 70−44×103Nmm, but it has the greatest influence when the torque fluctuates among 240−70×103Nmm.The Lyapunov exponents shown in Figure25 are calculated based on the Gram–Schmidt QR-decomposition method [37]; corresponding to the chaotic motions of Xm1 in Figure 24(a), the maximum Lyapunov exponents are larger than zero around the range of 16000–20000 rpm. Besides, the largest Lyapunov exponents are closed to zero, around the range of 7110–7492 rpm, 9900–10400 rpm, and 11400–16800 rpm corresponding to the quasiperiodic motion in Figure 24(a). Similarly, corresponding to the chaotic motions of Xm2 in Figure 24(b), the maximal Lyapunov exponents are larger than zero around the range of 16000–20000 rpm, and the largest Lyapunov exponents are closed to zero around the range of 8000–16000 rpm, indicating the quasiperiodic motions in Figure 24(b).Figure 25
Lyapunov exponents of the system under the motor torque fluctuation ofTD. (a) Xm1. (b) Xm2.
(a)(b)


## 3.1. Effect of the Rotational Speed on the TSHG Dynamic Response

The rotational speed of pinionΩp1 is considered as one of the main parameters that affect the dynamical behavior of system transmission. In this section, the half backlash of gears is set to s1=s2=40μm; the other parameters of TSHG transmission system are given in Table 2. Figure 2 presents the forward bifurcation characteristic of the first stage Xm1 and second stage Xm2 in dimensionless displacement with regard to the rotating speed of pinion Ωp1 as the control parameter, which increases gradually from 1000 rpm to 20000 rpm, while Figure 3 presents the backward bifurcation characteristic with regard to the decreasing of Ωp1 from 20000 rpm to 1000 rpm. As shown in Figures 2 and 3, the system responses of both Xm1 and Xm2 under the variation of Ωp1 contain types of motion forms such as periodic-one motion marked with letter P, jumps phenomena, quasiperiodic motion marked with letter Q, and chaotic motion marked with letter C.Figure 2
Forward bifurcation diagrams of system usingΩp1 as control parameter; s1=s2=40μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 3
Backward bifurcation diagrams of system usingΩp1 as control parameter; s1=s2=40μm. (a) Xm1. (b) Xm2.
(a)(b)In the case of the foreword bifurcation, the systemXm1 behaves as a steady period-one motion at low speed and persists until Ωp1 reaches 7110.5528 rpm. The response of time-domain history shows a sine wave, one main peak amplitude is found in FFT spectrum, the phase plane has a single closed circle, a single point is found in Poincaré section, and these characteristics indicate that the response of Xm1 is harmonic periodic-one motion, as shown in Figure 4.Figure 4
The dynamic response of systemXm1 at 2500 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)The system keeps transforming between periodic motion and quasiperiodic motion several times untilΩp1 reaches 17040 rpm. However, the system is periodic-one motion except when Ωp1 is in the ranges of 7110–7493 rpm, 9974–10452 rpm, 11407–13221 rpm, and 15703–17040 rpm; the system experiences the quasiperiodic motion response. As illustrated in Figure 5, the phase plane diagram does not repeat itself, which forms closed orbit points in Poincaré section and the FFT spectrum is continuous; this means that the system response is in quasiperiodic motion. Meanwhile, a jump phenomenon can be observed when Ωp1 is around the resonant regions at 12744 rpm. When Ωp1 increases to 17040 rpm, the system response changes from quasiperiodic motion to chaotic and remains until reaching 18758 rpm. From Figure 6, it can be seen that system response does not repeat itself in any pattern, the FFT spectrum has continuous broadband, and the phase plane shows disorder circles with many discrete points found in Poincaré section. It is revealed from these properties that the system responses as chaotic motion at the range of 17040–18758 rpm. However, with the increase of pinion speed, the system abandons the chaotic region and returns to quasiperiodic motion response again which preserves from 18758 rpm to 20000 rpm.Figure 5
The dynamic response of systemXm1 at 16950 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 6
The dynamic response of systemXm1 at 17157 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)As for the backward bifurcation, considering the decreasing ofΩp1 from 20000 rpm to 1000 rpm continuously, a chaotic motion is determined at 20000–18759 rpm and 17613–17040 rpm. As illustrated in Figure 7, the phase plane diagram shows nonrepeated circles with many clustered points of Poincaré map and the diagram of time-domain history also shows a nonperiodic motion. Figure 8 reveals that the system transforms into quasiperiodic response at 18759–17613 rpm, 17040–15703 rpm, and 13221–15512 rpm, where a jump phenomenon occurs at 17613 rpm. The system behaves as periodic-one motion response at 15703–13221 rpm and undergoes the chaotic region at 13221–11979 rpm. The chaotic phenomenon causes the vibration and noise issues in the system, which should be avoided during the design. Meanwhile, the system leaves the chaotic region with a jump phenomenon occurring at 11979 rpm. However, the system characteristic responses are the same as the responses in the forwarding bifurcation when Ωp1 is less than 11979 rpm, which is verified by PPVs and RMS curves, as shown in Figures 9 and 10. At this range of speed, the system response of Xm1 is mostly periodic-one motion, as marked in bifurcation diagram with letter P and proved in Figure 11.Figure 7
The dynamic response of systemXm1 at 19522 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 8
The dynamic response of systemXm1 at 16600 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 9
PPVs of forward and backward behaviors of system;s1=s2=40μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 10
RMS curves of forward and backward behaviors of system ats1=s2=40μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 11
The dynamic response of systemXm1 at 10930 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)It is observed that, under the variation ofΩp1 between 1000 and 20000 rpm, the characteristic of system Xm2 in forward bifurcation shown in Figure 2(b) is similar to the one in backward bifurcation shown in Figure 3(b), which is verified by RMS and PPVs curves in Figures 9(b) and 10(b). For the sake of simplicity, both bifurcations are discussed in detail as one bifurcation. As illustrated in Figures 2(b) and 3(b), the system Xm2 behaves as a periodic-one motion response at 1000–1763 rpm, as illustrated in Figure 12. The system is in quasiperiodic response at 1763–2241 rpm and returns into periodic-one motion again at 2214–5869 rpm. Consequently, the system behaves as quasiperiodic motion and remains in the range of 5869–17040 rpm, where the quasiperiodic motion is indicated by the closed orbit formed in Poincaré section, as shown in Figure 13. And jumps phenomena occur around 3577 rpm and 17040 rpm. In the range from 17135 rpm to 17899 rpm, the system becomes irregular and turns rapidly into chaotic motion, as verified by the disorder circles of phase plane diagram and the discrete points of Poincaré map in Figure 14. In the meantime, with the increasing of the speed, the displacement range of chaotic motion narrows gradually. Eventually, the system behaves as a quasiperiodic motion at 17899–2000 rpm, as illustrated in Figure 15. The results mentioned above reveal that the dynamic behavior of Xm1 and Xm2 is steady at low-speed range. Therefore, it is concluded that the dynamic behavior transforms from linear to nonlinear response with the increases of Ωp1.Figure 12
The dynamic response of systemXm2 at 1500 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 13
The dynamic response of systemXm2 at 16370 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 14
The dynamic response of systemXm2 at 17709 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)Figure 15
The dynamic response of systemXm2 at 19000 rpm. (a) Time-domain response. (b) FFT spectrum. (c) Phase plane and Poincaré point.
(a)(b)(c)From the corresponding PPVs and RMS curves, the conclusions of the relative displacement are obtained that there are two apparent bistable response regions in systemXm1 at the range of 11979–12839 rpm and 17613–20000 rpm, respectively. The chaotic motion ranges of Xm1 are indicated at 17040–18758 rpm as for forward case, but at 20000–18758 rpm, 17040–17613 rpm, and 11979–12457 rpm as for backward case, as shown in Figure 9(a), while the chaos response of Xm2 is indicated at 17135–17899 rpm, as illustrated in Figure 9(b). The response of the system Xm1 experiences a jump down phenomenon at 12744 rpm in the acceleration process and a jump up phenomenon at 11979 rpm and 17613 rpm in the deceleration process, as illustrated in Figure 10(a), while in system Xm2, jump up and down phenomena occur around 3577 rpm and 17040 rpm, as shown in Figure 10(b).


## 3.2. Effect of the Nonlinear Backlash on the TSHG Dynamic Response

According to the manufacturing accuracy level of gear pair, half of the backlash values of systemsXm1 and Xm2 are set to increase from s1=s2=40μm,60μm to 100μm. For the sake of simplicity, the backward bifurcation diagrams of systems under different backlash values are investigated, as illustrated in Figures 3, 16, and 17, respectively.Figure 16
Backward bifurcation diagrams of system;s1=s2=60μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 17
Backward bifurcation diagrams of system;s1=s2=100μm. (a) Xm1. (b) Xm2.
(a)(b)By comparing the bifurcation diagrams of both systemsXm1 and Xm2 under different backlash values, it is found that the range of chaotic motion in the rotational speed of pinion enlarges; the critical speed goes frontward along with the increasing of backlash value. In the meantime, the characteristics of chaotic motion are increased in both systems, which increases the difficulty of predicting the vibration of the gear pairs and leads to more noise possibility.The corresponding RMS and PPVs curves of both systems concerning the variation ofΩp1 under different backlash values are illustrated in Figures 9, 10, and 18–21. It is seen that the displacement amplitudes of the PPV and RMS of both systems Xm1 and Xm2 decrease along with the increase of the backlash, and the range of Ωp1 expands as the chaos range extends.Figure 18
PPVs of forward and backward bifurcation system;s1=s2=60μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 19
RMS curves of forward and backward bifurcation system;s1=s2=60μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 20
PPVs of forward and backward bifurcation system;s1=s2=100μm. (a) Xm1. (b) Xm2.
(a)(b)Figure 21
RMS curves of forward and backward bifurcation system;s1=s2=100μm. (a) Xm1. (b) Xm2.
(a)(b)Considering the bifurcation diagrams, PPVs, and RMS curves of both systems, we conclude that, under the increases of backlash value, the characteristics of chaotic response could be enhanced which expand the chaotic motion range and enlarge the rotational speed range; as a result, this will lead to more possibility of noise vibration. Nevertheless, the backlash could also lower the amplitudes of RMS and PPVs and deescalate the vibration extension of the system. Consequently, a suitable value of backlash should be selected to meet the requirements of vibration amplitude and reduce the possibility of chaotic behavior extension [32–34].


## 3.3. Effect of the Torsional Stiffness on the TSHG Dynamic Response

The torsional stiffness of the intermediate shaft is one of the key parameters that affect the dynamic behavior of the transmission system. Therefore, it is essential to investigate the effect of the torsional stiffness of the intermediate shaft on the dynamic characteristics of the TSHG system. In this section, the backlash values of both systemsXm1 and Xm2 are set to s1=s2=40μm; the torsional stiffness K1 is set to increase as 3.3134×107N⋅mm/rad,3.3134×108N⋅mm/rad, and 3.3134×1010N⋅mm/rad, respectively.Figures22, 23, and 2 illustrate the corresponding forward bifurcation diagrams of the systems Xm1 and Xm2 with respect to the rotational speed of pinion under different torsional stiffness values. According to the comparison of the bifurcation diagrams under different torsional stiffness values, the substantial region of periodic, quasiperiodic, and chaotic responses including jump phenomena are observed. At high-speed range of 17000–19000 rpm, the system is stable and behaves as a periodic motion when K1=3.3134×107N⋅mm/rad. The dynamic response transits from periodic to quasiperiodic motion when K1=3.3134×108N⋅mm/rad and eventually turns into chaotic response as K1 increases to 3.3134×1010N⋅mm/rad. Therefore, the nonlinear system response becomes unstable and the chaotic behavior consequently expands as K1 increases.Figure 22
Forward bifurcation diagrams of system whenK1=3.3134×107N⋅mm/rad. (a) Xm1. (b) Xm2.
(a)(b)Figure 23
Forward bifurcation diagrams of system whenK1=3.3134×108N⋅mm/rad. (a) Xm1. (b) Xm2.
(a)(b)


## 3.4. Effect of the Torque Fluctuation on the TSHG Dynamic Response

According to the BMW i8 electric motor characteristics [36], the ideal torque fluctuation TD is set to vary from 44×103Nmm to 250×103Nmm under the rotational speed range of 1000–20000 rpm. In this section, the backlash values of both systems Xm1 and Xm2 are set to s1=s2=100μm, the torsional stiffness K1 is set to 3.3134×1010N⋅mm/rad, and the torque fluctuation TD is considered as a control parameter that varies along with the changing of rotational speed. From the corresponding backward bifurcation diagrams of both systems Xm1 and Xm2 illustrated in Figure 24, it is observed that the dynamic response and the displacement of both systems at the speed range of 1000–4000 rpm are stable when the torque fluctuation is 250×103Nmm. As the torque varies among 240−70×103Nmm, the corresponding displacement changes clearly at the speed range of 4000–13000 rpm. Then, the displacement is stable again under the torque variation among 70−44×103Nmm until the speed reaches 20000 rpm.Figure 24
Backward bifurcation diagrams of system under the motor torque fluctuation ofTD. (a) Xm1. (b) Xm2.
(a)(b)However, it is observed that chaotic motion for both systems exists at the speed range of 13000–20000 rpm when the torque fluctuates at low values between52−44×103Nmm and 68−66×103Nmm. Quasiperiodic motions are observed at the speed range of 7111–17000 rpm when the torque fluctuates between 131−124, 92−88, 80−70, and 67−52×103Nmm as for the system Xm1 and at the speed range of 6000–17000 rpm when the torque fluctuation range is around 153−68×103Nmm and 65−61×103Nmm as for the system Xm2. When the torque fluctuation is fixed at 250×103Nmm, the responses of system are stable one-periodic motion at the speed range of 1000–7000 rpm. As a result, comparing with Figure 17, the torque fluctuation at high-speed range extends the chaotic and quasiperiodic regions corresponding to the speed and expands their ranges corresponding to the displacement. The corresponding displacement response is stable at 250×103Nmm and at low range 70−44×103Nmm, but it has the greatest influence when the torque fluctuates among 240−70×103Nmm.The Lyapunov exponents shown in Figure25 are calculated based on the Gram–Schmidt QR-decomposition method [37]; corresponding to the chaotic motions of Xm1 in Figure 24(a), the maximum Lyapunov exponents are larger than zero around the range of 16000–20000 rpm. Besides, the largest Lyapunov exponents are closed to zero, around the range of 7110–7492 rpm, 9900–10400 rpm, and 11400–16800 rpm corresponding to the quasiperiodic motion in Figure 24(a). Similarly, corresponding to the chaotic motions of Xm2 in Figure 24(b), the maximal Lyapunov exponents are larger than zero around the range of 16000–20000 rpm, and the largest Lyapunov exponents are closed to zero around the range of 8000–16000 rpm, indicating the quasiperiodic motions in Figure 24(b).Figure 25
Lyapunov exponents of the system under the motor torque fluctuation ofTD. (a) Xm1. (b) Xm2.
(a)(b)


## 4. Conclusion

The torsional-translational model consists of 16 degrees of freedom (DOF), which includes factors such as the nonlinearity backlash, support bearings, STE, TVMS, and the torsional damping and stiffness of the intermediate shaft. According to the motion equations of system, the dynamical response ofXm1 and Xm2 is examined by means of the Runge–Kutta method; the effects of pinion rotational speed, backlash, torsional stiffness, and torque fluctuation are also studied on the dynamic responses.It was shown that the dynamic behavior ofXm1 and Xm2 is steady at low-speed range. Under the increases of Ωp1, the dynamic behavior transforms from linear to nonlinear and becomes unstable. Besides, the dynamic response of Xm2 under the variations of the Ωp1 has the least effect when the forward and backward bifurcations characteristics are compared, which tend to have the same dynamic responses and behaviors. With the help of bifurcation diagrams, PPVs, and RMS curves, the observed periodic-one responses, quasiperiodic responses, chaotic responses, and jumps phenomena were confirmed, which furthermore verifies the dynamic characteristics of system obtained from previous results according to FFT spectrum, phase portrait, time-domain history, and Poincaré section.The increase of the backlash value could change the dynamic responses significantly, enhance the chaotic characteristics, enlarge the rotational speed range, and widen chaos motion range, which leads to more possibility of noise vibration [32–34]. Moreover, the backlash could also lower the amplitudes of the RMS and PPVs and weaken the vibration extension of both systems. The choice of the dynamic parameters plays such a critical role in the controlling and designing of the gear-pair system. Hence, an appropriate value of backlash should be selected to meet the requirements of vibration amplitude and reduce the possibility of chaotic behavior extension.At high-speed range, the torsional stiffness of the intermediate shaftK1 has a great influence on the TSHG dynamic characteristic. As the torsional stiffness increases, the dynamic response transits from periodic to quasiperiodic motion and then eventually turns into chaotic response. Therefore, the nonlinear system response becomes unstable and the chaotic behavior consequently expands as K1 increases.The torque fluctuation at high-speed range could affect the dynamic responses, extend the chaotic and quasiperiodic regions corresponding to the speed, and expand their ranges corresponding to the displacement. The corresponding displacement response is stable at the fixed torque value250×103Nmm and at low torque range value 70−44×103Nmm, but it has the greatest influence when the torque fluctuates among 240−70×103Nmm. At high-speed range and at low torque fluctuation, both systems are unstable and in chaotic motions according to the maximum Lyapunov exponents.This paper mainly focuses on the theoretical study of the nonlinear responses of the TSHG transmission system for EVs. In order to improve the transmission performance and verify the theoretical results of TSHG pairs for EVs, the relevant experimental verification with more particular model will be carried out to accurately describe the TSHG reduction system, and deep study must be done in the future.



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*Source: 8838521-2020-08-30.xml*