# Modeling and Analysis of Amplitude-Frequency Characteristics of Torsional Vibration for Automotive Powertrain

Authors: Jinli Xu; Jiwei Zhu; Feifan Xia
Journal: Shock and Vibration
 (2020)

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## Abstract

In the present paper, the amplitude-frequency characteristics of torsional vibration are discussed theoretically and experimentally for automotive powertrain. A bending-torsional-lateral-rocking coupled dynamic model with time-dependent mesh stiffness, backlash, transmission error etc. is proposed by the lumped-mass method to analysis the amplitude-frequency characteristic of torsional vibration for practical purposes, and equations of motive are derived. The Runge–Kutta method is employed to conduct a sweep frequency response analysis numerically. Furthermore, a torsional experiment is performed and validates the feasibility of the theoretical model. As a result, some torsional characteristics of automotive powertrain are obtained. The first three-order nature torsional frequencies are predicted. Torsional behaviors only affect the vibration characteristics of a complete vehicle at low-speed condition and will be reinforced expectedly while increasing torque fluctuation. Gear mesh excitations have little effects on torsional responses for such components located before mesh point but a lot for ones behind it. In particular, it is noted that the torsional system has a stiffness-softening characteristic with respect to torque fluctuation.


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## Body

## 1. Introduction

The front-engine, rear-wheel drive layout (FR) vehicle has a complicated powertrain consisting of engine, clutch, transmission, drive shaft, rear axle, and tire, as shown in Figure1, which indicates that more dynamic behaviors occur. In a typical dynamic operating condition, the fluctuation of engine output torque, U-joint dynamic characteristics will induce torsional vibration of the whole powertrain. Internal excitations originating from inherent characteristics of hypoid gear set also are responsible for the transverse-torsional-rocking coupled vibration of rear axle. These vibrations are eventually transmitted to the vehicle body through intermediate support and suspension to act as the main aspects of excitations causing vibration and noise for the complete vehicle. In order to improve the dynamic performances for vehicle, a number of research studies had been performed.Figure 1
Schematic of the automotive powertrain system discussed in this paper.Vesali et al. [1] obtained the relationship between input and output angular speed for different U-joint with variations in structure. Lu et al. [2] proposed a dynamic model with clearance by the lumped-mass method to discuss dynamics of a cross shaft type universal joint and analyze the effects of clearance on output speed and dynamic torque response. Wu et al. [3] optimized the phase differences between adjacent U-joints, which reduced the torsional vibration of transmission shaft. A classical model was proposed by Porter [4] in which the transmission system containing a U-joint was modeled as a torsional system with one degree-of-freedom. Asokanthan and Wang [5] introduced a torsional model with two degree-of-freedom to investigate stability and bifurcation performances by using the maximum Lyapunov exponent method based on Porter’s model. Farshidianfar et al. [6] developed a lumped-mass model for a driveline forced by torsional excitation to analyze the formation mechanism of noise. Friction was considered in Abd Elmaksoud’s paper [7], whose investigation indicated that the friction moment is a main aspect of excitation-inducing torsional vibration for the automobile driveline. Juang et al. [8] discussed effects of the sliding-tube-type driveshaft on a complete vehicle by experimental and finite element method. They reached a conclusion that fluctuation of nonlinear contact force in sliding mechanism accounts for the complete vehicle vibration. Coutinho and Tamagna [9] analyzed the bending natural frequency of half-shaft to prevent from resonance under power excitation. The effects of intermediate support on complete vehicle vibration are discussed in Yiting Kang’s work [10]. Xu et al. [11] and Zhang et al. [12] analyzed dynamics of transmission shaft-rear driving axle based on ADAMS and experimental demonstration, and the coupling effects between transmission shaft and rear axle were discussed. Xu et al. [13] and Xu et al. [14] developed several different dynamic models considering coupling interaction between automobile transmission shaft and main reducer. In their studies, effects of bearing stiffness, backlash, intermediate support, etc. on the system were discussed numerically. It is noted that the key of analytical solution for the main reducer is the development of the dynamic model of hypoid gear, with time-varying meshing stiffness, damping, clearance, bear, and other line or nonlinear factors, which inherits the research studies on spur and helical gear pair conduct by Kahraman and Singh [15–17], Cai and Hayashi [18], and Velex et al. [19–21], and remarkable achievements can be found within documents of Lim’s team [22–26] and other researchers.As introduced in preceding section, a few investigations treating engine, clutch, transmission, transmission shaft, and rear axle as a whole system were conducted mathematically. Although many models have been introduced in the literature, very few considered the influence of transverse and rocking vibration on torsional performances theoretically for discussing torsional characteristics of powertrain. In the present paper, the amplitude-frequency characteristics of torsional vibration for automotive powertrain are analyzed theoretically and experimentally considering the interactions between powertrain and the main reducer gear system.As for the layout of the present paper, it mainly contains five sections besides introduction. In the second section, a lumped-parameter dynamic model with 29 degree-of-freedom is proposed considering the transverse, torsional, and rocking vibration, and equations of motion are derived. In the third section, the introduced model is adopted to discuss the amplitude-frequency characteristics of torsional vibration for the coupling system by sweep frequency response analysis numerically. In the fourth part, an experiment is performed to validate the feasibility of the considered model and analyze the effect of torque fluctuation on the amplitude-frequency characteristics of torsional vibration. As the ﬁnal part, a conclusion is offered.


## 2. Dynamic Modeling of Powertrain

As shown in Figure1, the powertrain discussed in this paper consists of engine, transmission, drive shaft system, and rear axle. Considering the complexity of the system, a series of simplifying processes are adopted to establish the dynamic model of the system by using lumped-parameter method. Thereby, a generalized model with twenty-nine degree-of-freedom (29DOF) is proposed as shown in Figure 2, in which U-joint dynamic actions, intermediate support stiffness, time-varying mesh stiffness, gear backlash, static transmission error, and other factors are considered.Figure 2
The dynamic model of the powertrain shown in Figure1: (a) dynamic model of powertrain without rear axle; (b) torsional model of rear axle; (c) dynamic model of the main reducer in rear axle.
(a)(b)(c)The generalized coordinate vector of the dynamic model is given by(1)S=θ1,θ2,θ3,θ4,θ5,θ6,θ7,θ8,θ9,θ10,θ11,θ12,θ13,θ14,θ15,θ16,θ17,θpy,θpz,θgx,θgz,ym,zm,xp,yp,zp,xg,yg,zgΤ,where θii=1−5 represents the torsional displacement of the equivalent element as their coordinates; ymandzm are the translations of the intermediate support along the axes y and z;xpx,xpy,andxpz are the translations of the pinion along the axes x,y,andz;xgx,xgy, andxgz are the translations of the gear along the axes x,y,andz;θpx is the torsional displacement of the pinion around the x axis; θpyandθpz are the rocking displacements of the pinion around the yandz axes; and θgxandθgz are the rocking displacements of the gear around the xandz axes, respectively.The equivalent mass and moment of inertia are denoted bymiandJi.


### 2.1. Engine

In order to satisfy limitations of experimental bench, parameters of engine are replaced by ones of input motor, which is processed as two equivalent moments of inertia corresponding to the equivalent moments of inertia of the motor, flywheel, and clutch.Output torque of engine, as a main external excitation of the system, is written as in Fourier series form:(2)TD=Tm+∑i=1NTdicosiωft+φi,where Tm is mean output torque; Tdi is the amplitude of ith harmonic term; ωf is output speed of engine; and φi is phase angel of ith harmonic term.Based on previous research [14], only mean component and the second-order harmonic term are considered here:(3)TD=Tm+Td2cos2ωft+φ2.


### 2.2. Transmission

The transmission is simplified as a chained system with six equivalent moments of inertia, andi1 is introduced to describe the drive ratio of transmission. Furthermore, gears are assumed as rigid, and gear mesh characteristics, frictions, etc. are neglected. Without loss of generality, the case of i1 = 1.350 is employed as an example in the present work.In order to simplify a multimesh gear train into a chained system, a shaft should be chosen as the main body firstly as illustrated in Figure3, in which Ji (i = 1–4) and Ji′i=3−4 are equivalent moments of inertia and Ki (i = 1–4) and Ki′i=3−4 denote equivalent stiffness of gear shaft.Figure 3
The schematic diagram of equivalent principle for simplifying a multimesh gear train into a chained system.Kinetic energy of output shaft for the original system and chained system can be calculated as follows:(4)E=J3ω22+J4ω222,E′=J3′ω12+J4′ω122,where E is the kinetic energy of output shaft for the original system; J3 and J4 are the equivalent moments of inertia for the original system, which are derived by dynamic balance theory; ω1 and ω2 are the angular velocities; E′ is the kinetic energy of output shaft for the original system; and J3′andJ4′ are equivalent moments of inertia for the chained system.From the conservation law of energy, the following relationships are obtained:(5)E=E′,J3′=J3i2;J4′=J4i2,where i is drive ratio.From the conservation of strain energy, an equation is derived:(6)12K3′φ12=12K3φ22,where K3′ and K3 are torsional stiffness of output shaft for the chained system and original system, respectively, and φ1andφ2 are torsional angels of driving and driven shaft.Owing to the assumption concerning a gear as rigid, the gear meshing stiffness,K2′ and K2, is treated as infinite quantities. The following equations are derived:(7)K3′=K3i2,i=φ1φ2,T2′=T2i,where T2′ is the input torque of output shaft and the T2 is the output torque of input shaft.According to preceding derivation, the equivalent moments of inertia for the chained system are calculated by(8)J3=Ia1+Ia22,J4=Ia22+Ig1,J5=Ib1+Ig2+Ib2/2i2:12,J6=Ib2/2+Ig3+Ib3+Ig4+Ig11/i11:42+Ib4+Ig5+⋯+Ig12/i12:52+Ib5+Id+Ib6+Ig8+Ig9+Ig15/i15:92/i9:82+Ib7i2:12,J7=Ic1+Id+Ig10+Ic2+Ic3+Ic4/2+Idi10:32⋅i2:12,J8=Ic4/2+Ig13+Ig6/i6:132+Ic5+Ig14+Ig7/i7:142+Ic6+Ic7+Idi10:32⋅i2:12,where Iajj=1,2 is the equivalent moment of inertia of the jth node of input shaft; Ibjj=1,2,…,7 is the equivalent moment of inertia of the jth node of middle shaft; Icjj=1,2,…,7 is the equivalent moment of inertia of the jth node of output shaft; Igjj=1,2,…,15 is the equivalent moment of inertia of jth gear; Id is the equivalent moment of inertia of the synchronizer ring; and im:n is the drive ratio of the mating gear pair.


### 2.3. Drive Shaft System

Drive shaft system comprises intermediate drive shaft, main drive shaft, three universal joints, and intermediate support. In this paper, the rotational inertia of shaft is equivalent to the universal joint by the lumped-mass method averagely. The lubrication, clearance, friction, manufacturing, and assembly errors, and flexibility of universal joint are not considered.In order to describe the dynamics of a U-joint, a set of parameters are introduced as follows:(9)Ai=1+cos2αi2cosαi;Bi=1−cos2αi2cosαi;i=1,2,3,where αii=1,2,3 is the intersection angel between shafts linked by a universal joint (U-joint).According to kinematics analysis, the following equations are derived:(10)θ9=arctantanθ8cosα1,θ9′=θ8′A1−B1cos2θ8,θ9″=θ8″A1−B1cos2θ8−2B1sin2θ8A1−B1cos2θ82θ8′2,T2=A1−B1cos2θ8T1,θ11=arctantanθ10cosα2,θ11′=θ10′A2−B2cos2θ10,θ11″=θ10″A2−B2cos2θ10−2B2sin2θ10A2−B2cos2θ102θ10′2,T4=A2−B2cos2θ10T3,θ13=arctantanθ12cosα3,θ13′=θ12′A3−B3cos2θ12,θ13″=θ12″A3−B3cos2θ12−2B3sin2θ12A3−B3cos2θ122θ12′2,T6=A3−B3cos2θ12T5,where α1 is the intersection angel between output shaft of transmission and intermediate drive shaft; α2 is the intersection angel between main drive shaft and intermediate drive shaft; α3 is the intersection angel between main drive shaft and input shaft of rear axle.The equivalent moments of inertia for intermediate drive shaft and main drive shaft are calculated by(11)J9=J10=12i12Jids,J11=J12=12i12Jmds,where i1 is the drive ratio of transmission; Jids is the moment of inertia of intermediate drive shaft; and Jmds is the moment of inertia of main shaft.Equivalent torsional stiffnessk and damping coefficient c of shaft are given by(12)k=Mϕ=GapπD4−d432L,c=2ζkJaJbJa+Jb,where ζ is the damping ratio and JaandJb are the equivalent moment of inertia of shaft end.


### 2.4. Rear Axle

In this paper, the main reducer system is simplified as a geared rotor-bearing system with bending-torsional-lateral-rocking coupled vibration. Pinion shaft is simplified as two lumped masses linked by a spring-damping pair with infinite stiffness, and the differential assembly is modeled as a rigid part with a concentrated mass. Half-shafts are described by two mass-spring-damping pairs. Otherwise, pinion and gear are treated as rigid bodies within torsional vibration analysis.The equivalent moments of inertia for rear axle are calculated by(13)J13=1/5Jpsi12,J14=4/5Jpsi12,J15=Jd+0.5Jhsi12i22,J16=J17=0.5Jhs+Jhsi12i22,where i1 (i1 = 1.350) is the drive ratio of transmission, i2 (i2 = 4.1) is the drive ratio of the main reducer; Jps is the moment of inertia of pinion shaft; Jd is the moment of inertia of differential assembly; and Jhs is the moment of inertia of half-shaft.Equivalent stiffness of bearing is expressed as [27](14)Kr=7.253l10.8z10.9cos2.0α1⋅Fa00.1cos0.1α1,Ka=29.011l10.8cos1.9α1⋅Fa00.1,where Kr is the equivalent radial stiffness of bearing; Ka is the equivalent axial stiffness of bearing; l1 is the nominal contact length; z1 is the number of rolling element;α1 is the pressure angle; and Fa0 is the pretightening force.In addition, a new model for pinion shaft is proposed, which is different from previous works. Owing to negligible elastic deformation, pinion shaft is modeled as rigid body. Bearing is modeled as a spring-damping pair. Without loss of generality, the derivation procedure on displacements of the mounting point in thex-o-y plane is used as an example.In Figure4, Xpx and Xpy are introduced to represent translational displacements for centroid of pinion shaft; θpz denotes rocking displacement around the z axis.Figure 4
The dynamic model of pinion shaft.From the geometry deformation relationship and proposed concepts, the displacement for mounting point is calculated by(15)xpy1=xpy−lp1tanθpz,xpy2=xpy−lp2tanθpz.Thereby, the following expressions are derived:(16)xpy1′=xpy′−lp1θpz′cos2θpz,xpy2′=xpy′−lp2θpz′cos2θpz.Static transmission erroret [22–26, 28] is simulated in Fourier series form by(17)et=em+∑i=1NeAicosiωmt+φe,where em is the average value of transmission error; eAi are the amplitude of the ith order harmonic; φe are the initial phase angle of the ith order harmonic; and ωm is the meshing frequency.The dynamic meshing stiffness is the periodically time-varying parameter with respect to the mesh frequency as shown in Figure5, which can be simulated by means of a Fourier expansion [13, 14, 22–26]:(18)kmt=km+∑i=1NkAicosiωmt+φk,where km is the mean value of meshing stiffness, kAi is the fluctuation amplitude of ith order meshing stiffness, ωm is the mesh frequency, and φk is the initial phase of meshing stiffness.Definingme denotes mean mass of pinion and gear:(19)me=11/mp+1/mg=mpmgmp+mg.The following equation can be derived:(20)ωn=kmme,where ωn is the equivalent natural frequency of pinion and gear.Equivalent mesh damping coefficient is calculated, which is different from Wang and Lim’s analysis [25]:(21)cm=ξ∗2meωn=2ξkm1/mp+1/mg,where ζ is the damping ratio, generally set as a range of 0.03 to 0.17 [25]; mpandmg are equivalent mass of pinion and gear, respectively.The relative displacement along the normal direction at the meshing pointxn is given as follows:(22)xn=xpx+Eyθpz−xgx+Ezθgy+lg3θgz⋅cosαnsinβmcosδ1+sinαnsinδ1+xpy+lp3θpz−xgy+Ezθgx−Exθgz⋅−cosαnsinβmsinδ1+sinαncosδ1+xpz−Rpθpx+lp3θpy−xgz−lg3θgx−Rgθgy⋅cosαncosβm−et=ε1xpx−xgx−Ezθgy+Eyθpz−lg3θgz+ε2xpy−xgy−Ezθgx+lp3θpz+Exθgz+ε3xpz−xgz−Rpθpx+lg3θgx+lp3θpy+Rgθgy−et.The backlash function is expressed as [23](23)fxn=xn−b,xn≥b,0,−b≤xn≤b,xn+b,xn≤−b,where b is half of gear backlash.The dynamic meshing force along the line of actionFn can be calculated by(24)Fn=kmtfxn+cmxn′,where cm represents the value of equivalent mesh damping coefficient.The components of dynamic mesh force along the coordinate directions are expressed as(25)Fx=Fncosαnsinβmcosδ1+Fnsinαnsinδ1,Fy=−Fncosαnsinβmsinδ1+Fnsinαncosδ1,Fz=Fncosαncosβm,where δ1 is cone angle of the pinion; αn is the normal pressure angle, and βm is the helix angle at the midpoint of the pinion.In additional, the following equations are derived by introducing the parameterεi:(26)ε1=cosαnsinβmcosδ1+sinαnsinδ1,ε2=−cosαnsinβmsinδ1+sinαncosδ1,ε3=cosαncosβm.Substituting equations (26) into (25), the expressions of dynamic mesh force applied on pinion and gear can be written as(27)Fpx=−Fgx=ε1Fn,Fpy=−Fgy=ε2Fn,Fpz=−Fgz=ε3Fn,where the subscript p is the label for pinion and the subscript g is the label for gear.Mesh impact due to teeth separation is neglected in this paper.


### 2.5. Equations of Motion

From the preceding derivations, the equations of motion considering bending-torsional-lateral-rocking vibration are expressed as(28)J1θ1″+c1θ1′−θ2′+k1θ1−θ2=TD,J2θ2″+c1θ2′−θ1′+k1θ2−θ1+c2θ2′−θ3′+k2θ2−θ3=0,J3θ3″+c2θ3′−θ2′+k2θ3−θ2+c3θ3′−θ4′+k3θ3−θ4=0,J4θ4″+c3θ4′−θ3′+k3θ4−θ3+c4θ4′−θ5′+k4θ4−θ5=0,J5θ5″+c4θ5′−θ4′+k4θ5−θ4+c5θ5′−θ6′+k5θ5−θ6=0,J6θ6″+c5θ6′−θ5′+k5θ6−θ5+c6θ6′−θ7′+k6θ6−θ7=0,J7θ7″+c6θ7′−θ6′+k6θ7−θ6+c7θ7′−θ8′+k7θ7−θ8=0,J8θ8″+c7θ8′−θ7′+k7θ8−θ7=−T1i1,J9θ9″+c8θ9′−θ10′+k8θ9−θ10=T2i1,J10θ10″+c8θ10′−θ9′+k8θ10−θ9=−T3i1,J11θ11″+c9θ11′−θ12′+k9θ11−θ12=T4i1,J12θ12″+c9θ12′−θ11′+k9θ12−θ11=−T5i1,J13θ13″+c10θ13′−θ14′+k10θ13−θ14=T6i1,J14θ14″+c10θ14′−θ13′+k10θ14−θ13=−FpzEyi1,J15θ15″+c11θ15′−θ16′+k11θ15−θ16+c12θ15′−θ17′+k12θ15−θ17=FgzEx+FgxEzi1i2,J16θ16″+c11θ16′−θ15′+k11θ16−θ15=−TRLi1i2,J17θ17″+c12θ17′−θ15′+k12θ17−θ15=−TRRi1i2,mmym″+cmyym′+kmyym=Fmy,mmzm″+cmzzm′+kmzzm=Fmz,mpxpx″+kpxxpx+cpxxpx′=−Fpx,mpxpy″+kpy1xpy−lp1tanθpz+cpy1xpy′−lp1θpz′cos2θpz+kpy2xpy−lp2tanθpz+cpy2xpy′−lp2θpz′cos2θpz=Fpy,mpxpz″+kpz1xpz−lp1tanθpy+cpz1xpz′−lp1θpy′cos2θpy+kpz2xpz−lp2tanθpy+cpz2xpz′−lp2θpy′cos2θpy=Fpz,Jpyθpy″−lp1kpz1xpz−lp1tanθpy−lp1cpz1xpz′−lp1θpy′cos2θpy−lp2kpz2xpz−lp2tanθpy−lp2cpz2xpz′−lp2θpy′cos2θpy=Fpzlp3,Jpzθpz″−lp1kpz1xpy−lp1tanθpy−lp1cp1xpy′−lp1θpz′cos2θpz−lp2kpy2xpy−lp2tanθpz−lp2cpy2xpy′−lp2θpz′cos2θpz=−FpxEy+Fpylp3,mgxgx″+kgx1xgx−lg1tanθgz+cgx1xgx″−lg1θgz′cos2θgz+kgx2xgx+lg2tanθgz+cgx2xgx′+lg2θgz′cos2θgz=Fgx,mgxgy″+kgyxgy+cgyxgy″=−Fgymgxgz″+kgz1xgz+lg1tanθgx+cgz1xgz′+lg1θgx′cos2θgx+kgz2xgz−lg2tanθgx+cgz2xgz′−lg2θgx′cos2θgx=−Fgz,Jgxθgx″+lg1kgz1xgz+lg1tanθgx+lg1cgz1xgz′+lg1θgx′cos2θgx−lg2kgz2xgz−lg2tanθgx+lg2cgz2xgz′−lg2θgx′cos2θgx=Fgzlg3−FgyEz,Jgzθgz″+lg1kgx1xgx+lg1tanθgz+lg1cgx1xgx′+lg1θgz′cos2θgz−lg2kgx2xgx−lg2tanθgz+lg2cgx2xgx′−lg2θgz′cos2θgz=Fgxlg3−FgyEx,,where TD is the input torque of motor; T1 is the output torque of transmission; T2andT3 are the input and output torque for intermediate drive shaft; T4andT5 are the input and output torque for main drive shaft; T6 is the input torque of rear axle; kii=1,2,…,7 are the equivalent torsional stiffness; cii=1,2,…,7 are the equivalent torsional damping; kii=8,9 are the equivalent torsional stiffness; cii=8,9 are the equivalent torsional damping; mm is the mass of transmission shaft; kii=10,11,12 are the equivalent torsional stiffness for pinion shaft and differential axle; cii=10,11,12 are the equivalent torsional damping for pinion shaft and differential axle; cmyandcmz are the equivalent damping of intermediate support along axes y and z; kmyandkmz are the equivalent stiffness of intermediate support along axes y and z; FmyandFmz are components of additional force Fi along axes y and z for intermediate support; Eii=x,y,z are coordinates of meshing point along axes x, y, and z; and TRLandTRR are load torque for wheels.


### 2.6. Numerical Solution Procedure

The fourth-order adaptive step Runge–Kutta method which is generally applicable to strong nonlinearity is adopted in the present work to solve the equations. In order to program, a new state space is needed.In consideration of equations (12)—(14), it can be derived that kinetic equations of 3 universal joints reduce the number of degree-of-freedom from 29 to 26 for the whole system. The following state space coordinate vector also describes the dynamic behaviors of the system:(29)Ss=θ1,θ2,θ3,θ4,θ5,θ6,θ7,θ8,θ10,θ12,θ14,θ15,θ16,θ17θpy,θpz,θgx,θgz,ym,zm,xp,yp,zp,xg,yg,zgΤ.In order to eliminate the difference in magnitude between parameters, which will reduce the solvability and even lead to no convergence for solving equations of motion mentioned above,b is defined as the characteristic length and a new time parameter τ is introduced as follows:(30)τ=tωn,where t denotes time.For the sake of nominalizing equations, a generalized state space coordinate vector with 26 parameters is proposed to characterize dimensionless equations as(31)st=sr1,sr2,sr3,sr4,sr5,sr6,sr7,sr8,sr10,sr12,sr16,sr17,srpx,srpy,srpz,srgx,srgy,srgz,smy,smz,spx,spy,spz,sgx,sgy,sgzT.The following equations are derived as(32)sri=θi×Rib,i=1∼8,10,12,16,17,srpx=θ14×Rpxb,srgy=θ15×Rgyb,srij=θij×Rijb,i=p,g;j=x,y,z,six=xib,i=p,g,siy=yib,i=m,p,g,siz=zib,i=m,p,g,where b is the half of backlash and Ri is the equivalent gyration radius, which can be calculated by the following equation:(33)Ri=Jimi,i=1∼8,10,12,16,17,Rij=Jijmi,i=p,g;j=x,y,z,where Ji and Jij are the equivalent moment of inertia and mi is equivalent mass.A set of dimensionless parameters are defined as(34)xnτ=xnb;fxnτ=fxntb;ci,j,s=cimjRjRsωn;ki,j,s=kimjRjRsωn2;cmi=cmmiωn;kmi=kmmiωn2;Tx,j=TxmjRjωn2b;lijτ=lijb.As equations (30), (32), and (34) are substituted into equation (28), the dimensionless equations of motion can be derived.The equivalent inertia parameters are obtained in UG, and stiffness parameters of powertrain are calculated by using the relationships given in modeling section as listed in Table1. Subsequently, equivalent damping coefficients can be obtained. The parameters of the main reducer can be calculated by substituting system parameters listed in Table 2 into equations (17)–(31). The solution procedures for dimensionless parameter are carried out by using preceding calculated parameters and the relationships proposed in previous parts.Table 1
The equivalent parameters of powertrain.
Equivalent mass (kg)Equivalent moment of inertia (kg·m2)Equivalent stiffness (N/m)m1=2J1=0.091125k1=10023.75m2=10J2=0.3801735k2=1640.25m3=0.6150J3=0.00008103k3=27500m4=1.3798J4=0.001001k4=1.0e11m5=2.3581J5=0.001323k5=479000m6=11.4323J6=0.00249k6=1.0e11m7=4.5323J7=0.001865k7=21400m8=7.5567J8=0.002098—m9=1.284J9=0.001695k8=43921.799m10=1.284J10=0.001695—m11=1.3125J11=0.00173k9=41933.635m12=1.3125J12=0.00173—m13=0.4254J13=0.00002772k10=1.0e11m14=1.7016J14=0.0001109km=9.6965e8m15=13.75J15=0.034177—m16=8.45J16=9.9k11=12032.114m17=8.45J17=9.9k12=12032.114Table 2
The system parameters of the main reducer.
ParameterPinionGearNumber of teeth1041Module (mm)4.00Mean pressure angle (deg.)2020Helix angle at midpoint of the pinion (deg.)5027.13Cone angle (deg.)23.1265.14Tooth width (mm)40.6634.52Tooth addendum (mm)3.596.44Pinion outset (mm)−38Mounting distance (mm)11264Pitch radius (mm)40108Backlash (mm)0.1536Distance from pinion reference point to thex-o-z plane for pinion (mm)40Distance from pinion reference point to they-o-z plane for gear (mm)101Distance between projection point and centroid of pinion shaft (mm)lp3 = 10lg3 = 5Distance between bearing reference point and centroid of pinion shaft (mm)lp1 = 100lg1 = 90lp2 = 17.5lg2 = 93Equivalent mass (kg)mg = 2.127mp = 13.75Jpx = J14Jgx = 0.050315Equivalent moment of inertia (kg·m2)Jpy = 0.002982Jgx = J15Jpz = 0.002982Jpz = 0.037551The exciting and configuration parameters are set asθ1=1500rpm, Td2 = 100 N·m, km = 9.697e8 N/m, i1 = 1.35, i2 = 4.1, 2b = 0.15 mm, em = 0, α1 = 1.695 degree, α2 = 1.3044 degree, and α3 = 1.554 degree, respectively. Without significant loss of accuracy, the fundamental harmonic form is adopted for time-varying mesh stiffness and static transmission error function whose initial phases are set at 0, respectively.


## 2.1. Engine

In order to satisfy limitations of experimental bench, parameters of engine are replaced by ones of input motor, which is processed as two equivalent moments of inertia corresponding to the equivalent moments of inertia of the motor, flywheel, and clutch.Output torque of engine, as a main external excitation of the system, is written as in Fourier series form:(2)TD=Tm+∑i=1NTdicosiωft+φi,where Tm is mean output torque; Tdi is the amplitude of ith harmonic term; ωf is output speed of engine; and φi is phase angel of ith harmonic term.Based on previous research [14], only mean component and the second-order harmonic term are considered here:(3)TD=Tm+Td2cos2ωft+φ2.


## 2.2. Transmission

The transmission is simplified as a chained system with six equivalent moments of inertia, andi1 is introduced to describe the drive ratio of transmission. Furthermore, gears are assumed as rigid, and gear mesh characteristics, frictions, etc. are neglected. Without loss of generality, the case of i1 = 1.350 is employed as an example in the present work.In order to simplify a multimesh gear train into a chained system, a shaft should be chosen as the main body firstly as illustrated in Figure3, in which Ji (i = 1–4) and Ji′i=3−4 are equivalent moments of inertia and Ki (i = 1–4) and Ki′i=3−4 denote equivalent stiffness of gear shaft.Figure 3
The schematic diagram of equivalent principle for simplifying a multimesh gear train into a chained system.Kinetic energy of output shaft for the original system and chained system can be calculated as follows:(4)E=J3ω22+J4ω222,E′=J3′ω12+J4′ω122,where E is the kinetic energy of output shaft for the original system; J3 and J4 are the equivalent moments of inertia for the original system, which are derived by dynamic balance theory; ω1 and ω2 are the angular velocities; E′ is the kinetic energy of output shaft for the original system; and J3′andJ4′ are equivalent moments of inertia for the chained system.From the conservation law of energy, the following relationships are obtained:(5)E=E′,J3′=J3i2;J4′=J4i2,where i is drive ratio.From the conservation of strain energy, an equation is derived:(6)12K3′φ12=12K3φ22,where K3′ and K3 are torsional stiffness of output shaft for the chained system and original system, respectively, and φ1andφ2 are torsional angels of driving and driven shaft.Owing to the assumption concerning a gear as rigid, the gear meshing stiffness,K2′ and K2, is treated as infinite quantities. The following equations are derived:(7)K3′=K3i2,i=φ1φ2,T2′=T2i,where T2′ is the input torque of output shaft and the T2 is the output torque of input shaft.According to preceding derivation, the equivalent moments of inertia for the chained system are calculated by(8)J3=Ia1+Ia22,J4=Ia22+Ig1,J5=Ib1+Ig2+Ib2/2i2:12,J6=Ib2/2+Ig3+Ib3+Ig4+Ig11/i11:42+Ib4+Ig5+⋯+Ig12/i12:52+Ib5+Id+Ib6+Ig8+Ig9+Ig15/i15:92/i9:82+Ib7i2:12,J7=Ic1+Id+Ig10+Ic2+Ic3+Ic4/2+Idi10:32⋅i2:12,J8=Ic4/2+Ig13+Ig6/i6:132+Ic5+Ig14+Ig7/i7:142+Ic6+Ic7+Idi10:32⋅i2:12,where Iajj=1,2 is the equivalent moment of inertia of the jth node of input shaft; Ibjj=1,2,…,7 is the equivalent moment of inertia of the jth node of middle shaft; Icjj=1,2,…,7 is the equivalent moment of inertia of the jth node of output shaft; Igjj=1,2,…,15 is the equivalent moment of inertia of jth gear; Id is the equivalent moment of inertia of the synchronizer ring; and im:n is the drive ratio of the mating gear pair.


## 2.3. Drive Shaft System

Drive shaft system comprises intermediate drive shaft, main drive shaft, three universal joints, and intermediate support. In this paper, the rotational inertia of shaft is equivalent to the universal joint by the lumped-mass method averagely. The lubrication, clearance, friction, manufacturing, and assembly errors, and flexibility of universal joint are not considered.In order to describe the dynamics of a U-joint, a set of parameters are introduced as follows:(9)Ai=1+cos2αi2cosαi;Bi=1−cos2αi2cosαi;i=1,2,3,where αii=1,2,3 is the intersection angel between shafts linked by a universal joint (U-joint).According to kinematics analysis, the following equations are derived:(10)θ9=arctantanθ8cosα1,θ9′=θ8′A1−B1cos2θ8,θ9″=θ8″A1−B1cos2θ8−2B1sin2θ8A1−B1cos2θ82θ8′2,T2=A1−B1cos2θ8T1,θ11=arctantanθ10cosα2,θ11′=θ10′A2−B2cos2θ10,θ11″=θ10″A2−B2cos2θ10−2B2sin2θ10A2−B2cos2θ102θ10′2,T4=A2−B2cos2θ10T3,θ13=arctantanθ12cosα3,θ13′=θ12′A3−B3cos2θ12,θ13″=θ12″A3−B3cos2θ12−2B3sin2θ12A3−B3cos2θ122θ12′2,T6=A3−B3cos2θ12T5,where α1 is the intersection angel between output shaft of transmission and intermediate drive shaft; α2 is the intersection angel between main drive shaft and intermediate drive shaft; α3 is the intersection angel between main drive shaft and input shaft of rear axle.The equivalent moments of inertia for intermediate drive shaft and main drive shaft are calculated by(11)J9=J10=12i12Jids,J11=J12=12i12Jmds,where i1 is the drive ratio of transmission; Jids is the moment of inertia of intermediate drive shaft; and Jmds is the moment of inertia of main shaft.Equivalent torsional stiffnessk and damping coefficient c of shaft are given by(12)k=Mϕ=GapπD4−d432L,c=2ζkJaJbJa+Jb,where ζ is the damping ratio and JaandJb are the equivalent moment of inertia of shaft end.


## 2.4. Rear Axle

In this paper, the main reducer system is simplified as a geared rotor-bearing system with bending-torsional-lateral-rocking coupled vibration. Pinion shaft is simplified as two lumped masses linked by a spring-damping pair with infinite stiffness, and the differential assembly is modeled as a rigid part with a concentrated mass. Half-shafts are described by two mass-spring-damping pairs. Otherwise, pinion and gear are treated as rigid bodies within torsional vibration analysis.The equivalent moments of inertia for rear axle are calculated by(13)J13=1/5Jpsi12,J14=4/5Jpsi12,J15=Jd+0.5Jhsi12i22,J16=J17=0.5Jhs+Jhsi12i22,where i1 (i1 = 1.350) is the drive ratio of transmission, i2 (i2 = 4.1) is the drive ratio of the main reducer; Jps is the moment of inertia of pinion shaft; Jd is the moment of inertia of differential assembly; and Jhs is the moment of inertia of half-shaft.Equivalent stiffness of bearing is expressed as [27](14)Kr=7.253l10.8z10.9cos2.0α1⋅Fa00.1cos0.1α1,Ka=29.011l10.8cos1.9α1⋅Fa00.1,where Kr is the equivalent radial stiffness of bearing; Ka is the equivalent axial stiffness of bearing; l1 is the nominal contact length; z1 is the number of rolling element;α1 is the pressure angle; and Fa0 is the pretightening force.In addition, a new model for pinion shaft is proposed, which is different from previous works. Owing to negligible elastic deformation, pinion shaft is modeled as rigid body. Bearing is modeled as a spring-damping pair. Without loss of generality, the derivation procedure on displacements of the mounting point in thex-o-y plane is used as an example.In Figure4, Xpx and Xpy are introduced to represent translational displacements for centroid of pinion shaft; θpz denotes rocking displacement around the z axis.Figure 4
The dynamic model of pinion shaft.From the geometry deformation relationship and proposed concepts, the displacement for mounting point is calculated by(15)xpy1=xpy−lp1tanθpz,xpy2=xpy−lp2tanθpz.Thereby, the following expressions are derived:(16)xpy1′=xpy′−lp1θpz′cos2θpz,xpy2′=xpy′−lp2θpz′cos2θpz.Static transmission erroret [22–26, 28] is simulated in Fourier series form by(17)et=em+∑i=1NeAicosiωmt+φe,where em is the average value of transmission error; eAi are the amplitude of the ith order harmonic; φe are the initial phase angle of the ith order harmonic; and ωm is the meshing frequency.The dynamic meshing stiffness is the periodically time-varying parameter with respect to the mesh frequency as shown in Figure5, which can be simulated by means of a Fourier expansion [13, 14, 22–26]:(18)kmt=km+∑i=1NkAicosiωmt+φk,where km is the mean value of meshing stiffness, kAi is the fluctuation amplitude of ith order meshing stiffness, ωm is the mesh frequency, and φk is the initial phase of meshing stiffness.Definingme denotes mean mass of pinion and gear:(19)me=11/mp+1/mg=mpmgmp+mg.The following equation can be derived:(20)ωn=kmme,where ωn is the equivalent natural frequency of pinion and gear.Equivalent mesh damping coefficient is calculated, which is different from Wang and Lim’s analysis [25]:(21)cm=ξ∗2meωn=2ξkm1/mp+1/mg,where ζ is the damping ratio, generally set as a range of 0.03 to 0.17 [25]; mpandmg are equivalent mass of pinion and gear, respectively.The relative displacement along the normal direction at the meshing pointxn is given as follows:(22)xn=xpx+Eyθpz−xgx+Ezθgy+lg3θgz⋅cosαnsinβmcosδ1+sinαnsinδ1+xpy+lp3θpz−xgy+Ezθgx−Exθgz⋅−cosαnsinβmsinδ1+sinαncosδ1+xpz−Rpθpx+lp3θpy−xgz−lg3θgx−Rgθgy⋅cosαncosβm−et=ε1xpx−xgx−Ezθgy+Eyθpz−lg3θgz+ε2xpy−xgy−Ezθgx+lp3θpz+Exθgz+ε3xpz−xgz−Rpθpx+lg3θgx+lp3θpy+Rgθgy−et.The backlash function is expressed as [23](23)fxn=xn−b,xn≥b,0,−b≤xn≤b,xn+b,xn≤−b,where b is half of gear backlash.The dynamic meshing force along the line of actionFn can be calculated by(24)Fn=kmtfxn+cmxn′,where cm represents the value of equivalent mesh damping coefficient.The components of dynamic mesh force along the coordinate directions are expressed as(25)Fx=Fncosαnsinβmcosδ1+Fnsinαnsinδ1,Fy=−Fncosαnsinβmsinδ1+Fnsinαncosδ1,Fz=Fncosαncosβm,where δ1 is cone angle of the pinion; αn is the normal pressure angle, and βm is the helix angle at the midpoint of the pinion.In additional, the following equations are derived by introducing the parameterεi:(26)ε1=cosαnsinβmcosδ1+sinαnsinδ1,ε2=−cosαnsinβmsinδ1+sinαncosδ1,ε3=cosαncosβm.Substituting equations (26) into (25), the expressions of dynamic mesh force applied on pinion and gear can be written as(27)Fpx=−Fgx=ε1Fn,Fpy=−Fgy=ε2Fn,Fpz=−Fgz=ε3Fn,where the subscript p is the label for pinion and the subscript g is the label for gear.Mesh impact due to teeth separation is neglected in this paper.


## 2.5. Equations of Motion

From the preceding derivations, the equations of motion considering bending-torsional-lateral-rocking vibration are expressed as(28)J1θ1″+c1θ1′−θ2′+k1θ1−θ2=TD,J2θ2″+c1θ2′−θ1′+k1θ2−θ1+c2θ2′−θ3′+k2θ2−θ3=0,J3θ3″+c2θ3′−θ2′+k2θ3−θ2+c3θ3′−θ4′+k3θ3−θ4=0,J4θ4″+c3θ4′−θ3′+k3θ4−θ3+c4θ4′−θ5′+k4θ4−θ5=0,J5θ5″+c4θ5′−θ4′+k4θ5−θ4+c5θ5′−θ6′+k5θ5−θ6=0,J6θ6″+c5θ6′−θ5′+k5θ6−θ5+c6θ6′−θ7′+k6θ6−θ7=0,J7θ7″+c6θ7′−θ6′+k6θ7−θ6+c7θ7′−θ8′+k7θ7−θ8=0,J8θ8″+c7θ8′−θ7′+k7θ8−θ7=−T1i1,J9θ9″+c8θ9′−θ10′+k8θ9−θ10=T2i1,J10θ10″+c8θ10′−θ9′+k8θ10−θ9=−T3i1,J11θ11″+c9θ11′−θ12′+k9θ11−θ12=T4i1,J12θ12″+c9θ12′−θ11′+k9θ12−θ11=−T5i1,J13θ13″+c10θ13′−θ14′+k10θ13−θ14=T6i1,J14θ14″+c10θ14′−θ13′+k10θ14−θ13=−FpzEyi1,J15θ15″+c11θ15′−θ16′+k11θ15−θ16+c12θ15′−θ17′+k12θ15−θ17=FgzEx+FgxEzi1i2,J16θ16″+c11θ16′−θ15′+k11θ16−θ15=−TRLi1i2,J17θ17″+c12θ17′−θ15′+k12θ17−θ15=−TRRi1i2,mmym″+cmyym′+kmyym=Fmy,mmzm″+cmzzm′+kmzzm=Fmz,mpxpx″+kpxxpx+cpxxpx′=−Fpx,mpxpy″+kpy1xpy−lp1tanθpz+cpy1xpy′−lp1θpz′cos2θpz+kpy2xpy−lp2tanθpz+cpy2xpy′−lp2θpz′cos2θpz=Fpy,mpxpz″+kpz1xpz−lp1tanθpy+cpz1xpz′−lp1θpy′cos2θpy+kpz2xpz−lp2tanθpy+cpz2xpz′−lp2θpy′cos2θpy=Fpz,Jpyθpy″−lp1kpz1xpz−lp1tanθpy−lp1cpz1xpz′−lp1θpy′cos2θpy−lp2kpz2xpz−lp2tanθpy−lp2cpz2xpz′−lp2θpy′cos2θpy=Fpzlp3,Jpzθpz″−lp1kpz1xpy−lp1tanθpy−lp1cp1xpy′−lp1θpz′cos2θpz−lp2kpy2xpy−lp2tanθpz−lp2cpy2xpy′−lp2θpz′cos2θpz=−FpxEy+Fpylp3,mgxgx″+kgx1xgx−lg1tanθgz+cgx1xgx″−lg1θgz′cos2θgz+kgx2xgx+lg2tanθgz+cgx2xgx′+lg2θgz′cos2θgz=Fgx,mgxgy″+kgyxgy+cgyxgy″=−Fgymgxgz″+kgz1xgz+lg1tanθgx+cgz1xgz′+lg1θgx′cos2θgx+kgz2xgz−lg2tanθgx+cgz2xgz′−lg2θgx′cos2θgx=−Fgz,Jgxθgx″+lg1kgz1xgz+lg1tanθgx+lg1cgz1xgz′+lg1θgx′cos2θgx−lg2kgz2xgz−lg2tanθgx+lg2cgz2xgz′−lg2θgx′cos2θgx=Fgzlg3−FgyEz,Jgzθgz″+lg1kgx1xgx+lg1tanθgz+lg1cgx1xgx′+lg1θgz′cos2θgz−lg2kgx2xgx−lg2tanθgz+lg2cgx2xgx′−lg2θgz′cos2θgz=Fgxlg3−FgyEx,,where TD is the input torque of motor; T1 is the output torque of transmission; T2andT3 are the input and output torque for intermediate drive shaft; T4andT5 are the input and output torque for main drive shaft; T6 is the input torque of rear axle; kii=1,2,…,7 are the equivalent torsional stiffness; cii=1,2,…,7 are the equivalent torsional damping; kii=8,9 are the equivalent torsional stiffness; cii=8,9 are the equivalent torsional damping; mm is the mass of transmission shaft; kii=10,11,12 are the equivalent torsional stiffness for pinion shaft and differential axle; cii=10,11,12 are the equivalent torsional damping for pinion shaft and differential axle; cmyandcmz are the equivalent damping of intermediate support along axes y and z; kmyandkmz are the equivalent stiffness of intermediate support along axes y and z; FmyandFmz are components of additional force Fi along axes y and z for intermediate support; Eii=x,y,z are coordinates of meshing point along axes x, y, and z; and TRLandTRR are load torque for wheels.


## 2.6. Numerical Solution Procedure

The fourth-order adaptive step Runge–Kutta method which is generally applicable to strong nonlinearity is adopted in the present work to solve the equations. In order to program, a new state space is needed.In consideration of equations (12)—(14), it can be derived that kinetic equations of 3 universal joints reduce the number of degree-of-freedom from 29 to 26 for the whole system. The following state space coordinate vector also describes the dynamic behaviors of the system:(29)Ss=θ1,θ2,θ3,θ4,θ5,θ6,θ7,θ8,θ10,θ12,θ14,θ15,θ16,θ17θpy,θpz,θgx,θgz,ym,zm,xp,yp,zp,xg,yg,zgΤ.In order to eliminate the difference in magnitude between parameters, which will reduce the solvability and even lead to no convergence for solving equations of motion mentioned above,b is defined as the characteristic length and a new time parameter τ is introduced as follows:(30)τ=tωn,where t denotes time.For the sake of nominalizing equations, a generalized state space coordinate vector with 26 parameters is proposed to characterize dimensionless equations as(31)st=sr1,sr2,sr3,sr4,sr5,sr6,sr7,sr8,sr10,sr12,sr16,sr17,srpx,srpy,srpz,srgx,srgy,srgz,smy,smz,spx,spy,spz,sgx,sgy,sgzT.The following equations are derived as(32)sri=θi×Rib,i=1∼8,10,12,16,17,srpx=θ14×Rpxb,srgy=θ15×Rgyb,srij=θij×Rijb,i=p,g;j=x,y,z,six=xib,i=p,g,siy=yib,i=m,p,g,siz=zib,i=m,p,g,where b is the half of backlash and Ri is the equivalent gyration radius, which can be calculated by the following equation:(33)Ri=Jimi,i=1∼8,10,12,16,17,Rij=Jijmi,i=p,g;j=x,y,z,where Ji and Jij are the equivalent moment of inertia and mi is equivalent mass.A set of dimensionless parameters are defined as(34)xnτ=xnb;fxnτ=fxntb;ci,j,s=cimjRjRsωn;ki,j,s=kimjRjRsωn2;cmi=cmmiωn;kmi=kmmiωn2;Tx,j=TxmjRjωn2b;lijτ=lijb.As equations (30), (32), and (34) are substituted into equation (28), the dimensionless equations of motion can be derived.The equivalent inertia parameters are obtained in UG, and stiffness parameters of powertrain are calculated by using the relationships given in modeling section as listed in Table1. Subsequently, equivalent damping coefficients can be obtained. The parameters of the main reducer can be calculated by substituting system parameters listed in Table 2 into equations (17)–(31). The solution procedures for dimensionless parameter are carried out by using preceding calculated parameters and the relationships proposed in previous parts.Table 1
The equivalent parameters of powertrain.
Equivalent mass (kg)Equivalent moment of inertia (kg·m2)Equivalent stiffness (N/m)m1=2J1=0.091125k1=10023.75m2=10J2=0.3801735k2=1640.25m3=0.6150J3=0.00008103k3=27500m4=1.3798J4=0.001001k4=1.0e11m5=2.3581J5=0.001323k5=479000m6=11.4323J6=0.00249k6=1.0e11m7=4.5323J7=0.001865k7=21400m8=7.5567J8=0.002098—m9=1.284J9=0.001695k8=43921.799m10=1.284J10=0.001695—m11=1.3125J11=0.00173k9=41933.635m12=1.3125J12=0.00173—m13=0.4254J13=0.00002772k10=1.0e11m14=1.7016J14=0.0001109km=9.6965e8m15=13.75J15=0.034177—m16=8.45J16=9.9k11=12032.114m17=8.45J17=9.9k12=12032.114Table 2
The system parameters of the main reducer.
ParameterPinionGearNumber of teeth1041Module (mm)4.00Mean pressure angle (deg.)2020Helix angle at midpoint of the pinion (deg.)5027.13Cone angle (deg.)23.1265.14Tooth width (mm)40.6634.52Tooth addendum (mm)3.596.44Pinion outset (mm)−38Mounting distance (mm)11264Pitch radius (mm)40108Backlash (mm)0.1536Distance from pinion reference point to thex-o-z plane for pinion (mm)40Distance from pinion reference point to they-o-z plane for gear (mm)101Distance between projection point and centroid of pinion shaft (mm)lp3 = 10lg3 = 5Distance between bearing reference point and centroid of pinion shaft (mm)lp1 = 100lg1 = 90lp2 = 17.5lg2 = 93Equivalent mass (kg)mg = 2.127mp = 13.75Jpx = J14Jgx = 0.050315Equivalent moment of inertia (kg·m2)Jpy = 0.002982Jgx = J15Jpz = 0.002982Jpz = 0.037551The exciting and configuration parameters are set asθ1=1500rpm, Td2 = 100 N·m, km = 9.697e8 N/m, i1 = 1.35, i2 = 4.1, 2b = 0.15 mm, em = 0, α1 = 1.695 degree, α2 = 1.3044 degree, and α3 = 1.554 degree, respectively. Without significant loss of accuracy, the fundamental harmonic form is adopted for time-varying mesh stiffness and static transmission error function whose initial phases are set at 0, respectively.


## 3. Numerical Results

### 3.1. Analysis of Coupled Torsional Vibration

In order to analysis coupled interaction of automotive powertrain, Fourier spectrums of torsional vibration response are calculated at the initial parameters as shown in Figure6. Two peaks and their sidebands are observed. The torsional vibrational responses are governed by the component of 50 Hz which equals with exciting frequency of input torque as shown in equation (3) (ωf = 1500/60 ∗ 2 = 50 Hz). This implies that engine input excitation is responsible for torsional vibration and a valuable approach to decreasing torsional vibration of powertrain is inferred. In addition, the component with a frequency of around 185 Hz is observed obviously for each considered part. As one knows, the meshing frequency undergoing this condition is presumably 185.185 Hz (f = 1500/60/1.35 = 185.185 Hz) which is quite in agreement with the frequency of the second peak. Coupled vibration occurs between the torsional system and hypoid gear set, and inner excitation caused by gear pair meshing process which is affected by clearance and time-varying meshing stiffness [25] accounts a lot for torsional vibration at 185 Hz of powertrain and affects torsional responses slightly. After simply calculation, bandwidths reaching the same value of around 37 Hz of sidebands around two peaks are got. U-joint is a transmission mechanism with nonconstant angular speed which will excite a doubled frequency torsional vibration of input as shown in equations (12)–(14). Under this condition, the doubled frequency excited by U-joint is calculated:(35)fu−joint=ni1∗60∗2=15001.35∗60∗2=37.04.It can be conjectured that modulation phenomenon appears when automotive powertrain works between excitations attributed to input, U-joint, and gear pair. As one can see in Figure6, compared torsional response of gear with that located front of meshing point, a lower value of torsional angular is observed, and it is inferred that gear mesh excitations reduce torsional vibration for components located behind meshing point. It is noted that taking gear mesh behaviors into account is quite necessary to when a torsional characteristic analysis is conducted.Figure 5
Schematic diagram of time-varying meshing stiffness on hypoid gear.Figure 6
The Fourier spectrum of torsional responses: (a) output shaft of transmission; (b) output end of the drive shaft system; (c) torsional displacement of the pinion along the axisx; (d) torsional displacement of the gear along the axis y.
(a)(b)(c)(d)


### 3.2. Amplitude-Frequency Response of Torsional Vibration for Automotive Powertrain

The previous works [29] analyzed torsional characteristics of the coupled system consisted of drive shaft and rear axle, which denoted that resonance of drive shaft at the first two-order nature frequency accounts for dramatic torsional vibration responses undergoing high-rotation-speed condition. However, it cannot reveal the torsional vibration mechanism of automotive under low- and middle-speed condition.In order to obtain a more realistic understanding for amplitude-frequency response of automotive powertrain undergoing middle-speed condition theoretically, the preceding equations of motion are adopted. A sweep frequency analysis procedure is carried out in a range from 1000 rpm to 3500 rpm, which, respectively, corresponds to frequency internal of 16.6–58.3 Hz for input speed, 33.33–116.7 Hz for input excitation, and 123.46–432.09 Hz for meshing excitation. The response at input end of rear axle is taken as an example in this work.Figure7 shows the amplitude-frequency response, in which abscissa denotes torsion speed of input and ordinate is RMS value of angular displacement. As one can see, three obvious peaks which correspond to input speed at about 1000 rpm, 1600 rpm, and 2800 rpm, respectively, are observed. Owing to the assumption that the dramatic responses are attributed to resonance of the system, the first three-order nature frequency respect to the second-order term of input torque fluctuation is proposed at 33.3 Hz, 53.3 Hz, and 93.3 Hz successively. It is noted that on the basis of previous analysis, these three peaks also correspond to frequencies of 24.7 Hz, 39.5 Hz, and 69.1 Hz for the U-joint excitation and 123.4 Hz, 197.5 Hz, and 345.6 Hz for the meshing excitation. Preliminary works [29] showed that resonance is attributed to input excitation of drive shaft and the meshing excitation of hypoid gear pair with higher exciting frequency than input will not excite prominent resonance behaviors of drive shaft as illustrated in Figure 8, i.e., meshing excitation has slight influence on the torsional system, which reaches the same conclusion with this work. According to analysis in Section 3.1, the sidebands respect to each exciting frequency are (8.6–58) Hz at 33.3 Hz, (13.8–92.8) Hz at 53.3 Hz, and (24.2–162.4) Hz at 93.3 Hz for input torque, and (98.7–148.1) Hz at 123.4 Hz, (158–237) Hz at 197.5 Hz, and (276.5–414.7) Hz at 345.6 Hz for meshing behavior of the gear pair, respectively. As listed in Table 3, the first peak can be attributed to the first two-order resonances of drive shaft excited by sideband around 123.4 Hz. Sidebands at 53.3 Hz and 197.5 Hz result in the second-order resonance, which generates the second peak. Moreover, the second-order resonances excited by the sideband at 93.3 Hz and third-order one caused by sideband at 345.6 Hz explain the appearance of the third peak. A typical phenomenon is observed that the torsional vibration reduces a lot while one compares the amplitude of the third peak with another two, which indicates that torsional behaviors of automotive powertrain affect the vibration characteristics of the complete vehicle under a low- or middle-input-speed condition. In order to validate the preceding result, an experimental study is necessary.Figure 7
Amplitude-frequency response at input end of rear axle.Figure 8
Influence of input rotation speed of drive shaft on vibration of the main reducer [29].Table 3
The first six-order nature frequency of drive shaft [29].
Order123456Frequency (Hz)130.9140.2415.2442.2636.2906.6


## 3.1. Analysis of Coupled Torsional Vibration

In order to analysis coupled interaction of automotive powertrain, Fourier spectrums of torsional vibration response are calculated at the initial parameters as shown in Figure6. Two peaks and their sidebands are observed. The torsional vibrational responses are governed by the component of 50 Hz which equals with exciting frequency of input torque as shown in equation (3) (ωf = 1500/60 ∗ 2 = 50 Hz). This implies that engine input excitation is responsible for torsional vibration and a valuable approach to decreasing torsional vibration of powertrain is inferred. In addition, the component with a frequency of around 185 Hz is observed obviously for each considered part. As one knows, the meshing frequency undergoing this condition is presumably 185.185 Hz (f = 1500/60/1.35 = 185.185 Hz) which is quite in agreement with the frequency of the second peak. Coupled vibration occurs between the torsional system and hypoid gear set, and inner excitation caused by gear pair meshing process which is affected by clearance and time-varying meshing stiffness [25] accounts a lot for torsional vibration at 185 Hz of powertrain and affects torsional responses slightly. After simply calculation, bandwidths reaching the same value of around 37 Hz of sidebands around two peaks are got. U-joint is a transmission mechanism with nonconstant angular speed which will excite a doubled frequency torsional vibration of input as shown in equations (12)–(14). Under this condition, the doubled frequency excited by U-joint is calculated:(35)fu−joint=ni1∗60∗2=15001.35∗60∗2=37.04.It can be conjectured that modulation phenomenon appears when automotive powertrain works between excitations attributed to input, U-joint, and gear pair. As one can see in Figure6, compared torsional response of gear with that located front of meshing point, a lower value of torsional angular is observed, and it is inferred that gear mesh excitations reduce torsional vibration for components located behind meshing point. It is noted that taking gear mesh behaviors into account is quite necessary to when a torsional characteristic analysis is conducted.Figure 5
Schematic diagram of time-varying meshing stiffness on hypoid gear.Figure 6
The Fourier spectrum of torsional responses: (a) output shaft of transmission; (b) output end of the drive shaft system; (c) torsional displacement of the pinion along the axisx; (d) torsional displacement of the gear along the axis y.
(a)(b)(c)(d)


## 3.2. Amplitude-Frequency Response of Torsional Vibration for Automotive Powertrain

The previous works [29] analyzed torsional characteristics of the coupled system consisted of drive shaft and rear axle, which denoted that resonance of drive shaft at the first two-order nature frequency accounts for dramatic torsional vibration responses undergoing high-rotation-speed condition. However, it cannot reveal the torsional vibration mechanism of automotive under low- and middle-speed condition.In order to obtain a more realistic understanding for amplitude-frequency response of automotive powertrain undergoing middle-speed condition theoretically, the preceding equations of motion are adopted. A sweep frequency analysis procedure is carried out in a range from 1000 rpm to 3500 rpm, which, respectively, corresponds to frequency internal of 16.6–58.3 Hz for input speed, 33.33–116.7 Hz for input excitation, and 123.46–432.09 Hz for meshing excitation. The response at input end of rear axle is taken as an example in this work.Figure7 shows the amplitude-frequency response, in which abscissa denotes torsion speed of input and ordinate is RMS value of angular displacement. As one can see, three obvious peaks which correspond to input speed at about 1000 rpm, 1600 rpm, and 2800 rpm, respectively, are observed. Owing to the assumption that the dramatic responses are attributed to resonance of the system, the first three-order nature frequency respect to the second-order term of input torque fluctuation is proposed at 33.3 Hz, 53.3 Hz, and 93.3 Hz successively. It is noted that on the basis of previous analysis, these three peaks also correspond to frequencies of 24.7 Hz, 39.5 Hz, and 69.1 Hz for the U-joint excitation and 123.4 Hz, 197.5 Hz, and 345.6 Hz for the meshing excitation. Preliminary works [29] showed that resonance is attributed to input excitation of drive shaft and the meshing excitation of hypoid gear pair with higher exciting frequency than input will not excite prominent resonance behaviors of drive shaft as illustrated in Figure 8, i.e., meshing excitation has slight influence on the torsional system, which reaches the same conclusion with this work. According to analysis in Section 3.1, the sidebands respect to each exciting frequency are (8.6–58) Hz at 33.3 Hz, (13.8–92.8) Hz at 53.3 Hz, and (24.2–162.4) Hz at 93.3 Hz for input torque, and (98.7–148.1) Hz at 123.4 Hz, (158–237) Hz at 197.5 Hz, and (276.5–414.7) Hz at 345.6 Hz for meshing behavior of the gear pair, respectively. As listed in Table 3, the first peak can be attributed to the first two-order resonances of drive shaft excited by sideband around 123.4 Hz. Sidebands at 53.3 Hz and 197.5 Hz result in the second-order resonance, which generates the second peak. Moreover, the second-order resonances excited by the sideband at 93.3 Hz and third-order one caused by sideband at 345.6 Hz explain the appearance of the third peak. A typical phenomenon is observed that the torsional vibration reduces a lot while one compares the amplitude of the third peak with another two, which indicates that torsional behaviors of automotive powertrain affect the vibration characteristics of the complete vehicle under a low- or middle-input-speed condition. In order to validate the preceding result, an experimental study is necessary.Figure 7
Amplitude-frequency response at input end of rear axle.Figure 8
Influence of input rotation speed of drive shaft on vibration of the main reducer [29].Table 3
The first six-order nature frequency of drive shaft [29].
Order123456Frequency (Hz)130.9140.2415.2442.2636.2906.6


## 4. Experimental Analysis

A test bench which can be used to test the vibration behavior of a transmission system used for four-wheel drive vehicle belonging to the China Automotive Technology and Research Center is adopted to perform this experiment. There are five motors in the test bench, and three of them are used in this experiment. One of them drives automotive powertrain, which can generate a high frequency input torque with fluctuation varied in accordance with necessary, and others are used as loads. In this test, two photoelectric sensors and two magnetoelectric sensors are used to detect torsional vibration signals, and other five accelerometers are mounted to monitor transverse vibration. Installation site for each sensor can be found in Figure9. Signals are sampled and processed by LMS Test.Lab.Figure 9
The experiment bench and arrangement of sensors.In order to validate the feasibility of the theoretical model, the input torque simulated by input motor and loads are consistent with parameters employed in numerical analysis (mean value and the second-order term are contained only, and fluctuation amplitude is set at 100 N·M for input torque). Within the experiment, the output speed of motor varies from 1000 rpm to 3500 rpm to conduct a sweep frequency response analysis experimentally.As illustrated in Figure10, the amplitude-frequency responses of torsional vibration have the similar feature for different parts of powertrain. Peaks can be observed for carves at a similar input speed. It is inferred that a presume resonance occurs at each peak. As a result, the first three-order nature torsional frequencies can be predicted. Meanwhile, the same results are reached. As shown in Figure 10, each curve has a higher value nearby the first and second peak. With the further increase in frequency, it reaches a lower plateau value. Torsional behaviors have an effect on the vibration characteristics under a low-input-speed condition. Furthermore, the part located behind the mesh point, such as output end of half-shaft, has a lower RMS value than transmission and drive shaft, which can be attributed to reduction effect of mesh excitation on torsional vibration.Figure 10
Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100 N·M.The experimental result of input end of rear axle is compared with the numerical one. As shown in Figure11, a quite consistency is observed and the first three-order frequencies are inferred at 35 Hz (1050 rpm), 55.7 Hz (1670 rpm), and 88.3 Hz (2650 rpm) for experimental result by the same way mentioned in preceding section. Compared with numerical predictions, an acceptable difference of about 5% (the difference is 5.1% at 33.3 Hz, 4.5% at 53.3 Hz, and 5.36% at 93.3 Hz) can be obtained. These indicate that the theoretical model is feasible for predicting nature torsional frequencies. Moreover, numerical torsional displacement is a little lower than the experimental one at the same input speed which is below about 2450 rpm, and a little higher with the further increase. This can be presumably attributed to assumptions made in this present work with respect to the complexity of the system and the limitations of the lumped-mass method.Figure 11
Comparison of numerical and experimental results for input end of rear axle.The previous section indicates numerically that excitation of input dominates torsion vibration of the system and gear set affects responses slightly. In order to validate the correctness of numerical analysis and feasibility decreasing torsional vibration by reducing input excitation, effect of input torque fluctuation on torsional response is studied experimentally. The mean value of input torque is set at 150 N·M, and fluctuation of the second-order term is set at 25 N·M, 50 N·M, 75 N·M, 100 N·M, 125 N·M, 150 N·M, 175 N·M, and 200 N·M in order.Together Figures12 and 13 expectably show that the torsional vibration is reinforced while increasing torque fluctuation. Meanwhile an obvious increase in amplitude occurs at the second peak when fluctuation is increased from 175 N·M to 200 N·M, which indicates that a limitation should be considered. The accurate value can be determined by a further study. In summary reducing torque fluctuation is a beneficial approach for vibration damping. Furthermore, it is noted that such a nonlinear phenomenon as natural frequency shift occurs while changing torque fluctuation. Taking the second-order torsional frequency as an example, the frequency value decreases with elevating fluctuation of input torque, which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuation. This result can be accounted for a number of nonlinearities, such as friction, clearance, and sliding, and the existence of which sophisticates system nonlinear characteristics.Figure 12
Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations.Figure 13
Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations.


## 5. Conclusion

In the present paper, a lumped-parameter dynamic model with 29 degree-of-freedom is proposed considering the transverse, torsional, and rocking coupled vibration for a better understanding of the automotive powertrain system dynamics and equations of motion are derived. The numerical result shows that gear mesh excitations have little effects on torsional response for such components located before the mesh point as transmission, transmission shaft, and pinion but significantly reduce the torsional responses for ones behind the mesh point. The amplitude-frequency characteristics of torsional vibration graphically indicate that torsional behaviors of automotive powertrain only affect the vibration characteristics of complete vehicle under low-speed condition, and predict the first three-order nature torsional frequencies of the whole system. An experimental examination is performed and validates the feasibility of the considered model. The torsional vibration is reinforced while increasing torque fluctuation. In addition, a stiffness softening characteristic is observed for torsional vibration experimentally. In order to reach a better understanding in amplitude-frequency response at low- and middle-input-speed condition, a procedure to analyze mode shapes of automotive powertrain will be conducted in future works.



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*Source: 6403413-2020-08-29.xml*