An Investigation of Dynamic Behavior of Electric Vehicle Gear Trains

Gear systems have two fundamentally different excitation types, each causing characteristically different vibratory and acoustic response. The first type of gear system excitations, known as “internal” excitations, relates to the kinematics, geometry, and contact mechanics of the gear mesh interface [1–3]. One such excitation is the periodic variation of gear mesh stiffness $km(t)$ due to fluctuation of the total number of loaded tooth pairs between n and $n+1$ (⁠$n=1$ for conventional spur gears, and $n=2−5$ helical gears depending on the total contact ratio). Besides $km(t)$⁠, tooth profile errors and designed tooth modifications result in an internal displacement excitation $ε(t)$ known as the transmission error [3]. Both $km(t)$ and $ε(t)$ are periodic at the gear mesh (tooth-passing) frequency of $ωm=Ziωi$⁠, where $Zi$ is the number of teeth and $ωi$ is the rotational speed of gear i. These internal excitations were shown to cause primary and super-harmonic resonances as well as subharmonic resonances [1,2,4] with a softening-type nonlinear behavior due to intermittent tooth separation in presence of backlash [5]. Majority of gear dynamics experiments and models available in the literature has dealt with the response of low-contact ratio spur gear pairs (⁠$n=1$⁠) caused by the internal excitation with the spur gear pair behaving as a nonlinear time-varying (NTV) system under steady-state (constant torque and speed) conditions [1,2,6–15]. Contrarily, helical gear pairs with higher contact ratios (higher n) were shown experimentally to act in a linear time-invariant (LTI) manner as the parametric variation of $km(t)$ is somewhat subdued [3,14], in line with published LTI models of helical gear systems [15–17]. Regardless of whether linear or nonlinear, acoustic response of a gear pair to these internal excitations under constant torque conditions is primarily a tonal noise, known as gear whine.

The second type of excitations is caused by fluctuations of input or output torque $Ti(t)$ resulting in a time-varying transmitted gear mesh forces $Fm(t)$⁠. While internal excitations apply at $ωm$⁠, the frequencies $ωe$ of these external excitations in conventional applications are very low. Driven by an internal combustion engine, the torques delivered by the crankshaft to the transmission [18,19], the timing gear drive [20–23], or the engine balancer [24] fluctuate at $ωe$ that is only a few integer multiples of the rotational frequency. While some earlier gear pair dynamics studies included both external and internal excitations [2,25], such studies were not further pursued since $ωe≪ωm$⁠. Instead, external excitations were studied alone in the scheme of vibro-impact (rattle) studies for applications such gear systems. These vibro-impact studies considered behavior away from gear train resonances, and hence, they did not include the internal excitations, still comparing well to the experiments and yielding practical rattle metrics that correlated to the resultant rattle noise [26,27].

This split of gear dynamic studies into two groups based on the excitation types, those focusing on internal excitations (gear whine noise) and the others concerned with external excitations (rattle noise), pose issues toward a comprehensive analysis of electric vehicle (EV) drives. Specifically:

• The electric motors often produce high-frequency torque fluctuations (depending on the combination of stator/rotor poles, as many times as $Ze=36$ or 48 per input rotation [28]). While their amplitudes are modest (say within 5–10% of the mean torque), they are at frequencies capable of exciting the natural modes of the gear train. In other words, $ωe=Zeω1$ and $ωm$ are comparable in EV applications, requiring inclusion of both external (⁠$Te(t)$⁠) and internal (⁠$km(t)$ and $e(t)$⁠) excitations simultaneously to capture the steady-state response. Such a model is a must to aid specification of the $Zi$ values as well as EV motor design specifications in terms of the torque fluctuation amplitudes and frequency.

• An electric motor in an EV system operates in two modes. In the drive mode, the motor is the input that applies a torque in the form $Te(t)=T¯d+Td(t)$⁠. In the regenerative braking (regen) mode, the mean torque is negative, and the motors acts now as a generator. In the regen mode, $Te(t)=T¯r+Tr(t)$⁠, where $T¯r<0$⁠. Transitions between these two modes might occur frequently. This physically means for gear meshes that, while they are operating in a steady-state manner with contacts along one set of gear tooth flanks, they must move their contacts to the opposite flanks by traveling across the backlash zone. Such transitions should result in certain transient (vibro-impact) behavior that can again only be captured by a model equipped with both types of excitations.

Despite the extensive high-frequency gear dynamics and gear vibro-impact studies centered around conventional powertrain applications, only a few studies focused on dynamics of EV drivetrains. They were mostly limited to either noise radiated from the electric motor casing due to rotating electromagnetic forces [29–31] or influence of higher speeds on gear related excitations [32,33]. Westphal et al. [34,35] proposed a methodology that minimizes the contact and root stresses of a stepped planetary EV drive through tooth flank modifications at the operating speed ranges of EV drives. Wellmann et al. [36] suggested that electromagnetic torque orders should be separated from gear mesh orders to avoid plausible interactions and modulation effect. There are also commercial software packages available to simulate dynamic behavior of electromechanical powertrains as reviewed in Ref. [37]. While being versatile tools, they mostly approach the problem at the system level to address the needs on electric motor controls, vehicle, and powertrain dynamics with limited attention given to gear dynamics perspective.

In this study, a low-order torsional dynamic model of a typical EV transmission will be developed by including both external and internal excitations. Equations of motion will be solved to obtain the steady-state response as well as transient motions during drive-regen-drive transitions. The predicted steady-state response will be verified through comparisons to a deformable-body model. In addition, drive-regen-drive transitions starting from various (resonance or off-resonance) steady-state response conditions will be predicted. Various parameter sensitivity studies will be presented at the end to highlight key system parameters and trends toward better design practices.

A reduced order model of a typical countershaft EV gear train is shown in Fig. 1. This three-shaft layout (shafts s1–s3) has been commonly used in various EV applications. The system consists of four gears (g1–g4) and disks representing electric motors and vehicle inertias. The motor on axis s1 has a wide rotational speed range of $ω1⊂[0,2100]$ rad/s (up to $Ω1=602πω1=20$ krpm). Two gear meshes m1 (mesh of gears g1 and g2) and m2 (mesh of gears g3 and g4) are used in series to reduce the rotational speed about 10 times at the output side. In Fig. 1, each gear is represented by a rigid disk of inertia $Igi$ and (base) radius $rgi$⁠. The motor is typically connected to the input gear g1 rigidly. As such, the moments of inertia of the electric motor and gear g1 can be lumped in the model such that $I1=Ig1+Ie$⁠. Likewise, gears g2 and g3 on shaft s2 are typically laid next to each other with little or no gap in between such that they can be assumed to share the same degree-of-freedom in the model with a combined inertia of $I23=Ig2+Ig3$⁠. On the output shaft s3, $kt$⁠, and $ct$ represent equivalent torsional stiffness and damping of the rest of the drivetrain components, and $Iv$ is the vehicle inertia.

Each gear mesh interface mj (⁠$j=1,2$⁠) is modeled here by a time-varying mesh stiffness $kj(t)$⁠, a linear viscous damper $cj$⁠, and a displacement excitation $ej(t)$ that represents the static transmission error under loaded conditions (including both the unloaded transmission error $εj(t)$ and deflections under static conditions $qsj(t)$⁠). Internal excitations $kj(t)$ and $ej(t)$ are both periodic functions at gear mesh frequency $ωm$⁠. Each gear mesh is subject to a clearance of $2bj$ representing the gear backlash amplitudes along the lines of action. In the model of Fig. 1, gears g1–g4 have angular displacement of $θ¯g1(t)=θ1(t)+Ω1t$⁠, $θ¯g2(t)=θ¯g3(t)=θ¯g23(t)=θ23(t)+Ω2t$ and $θ¯g4(t)=θ4(t)+Ω3t$ where $Ω1$ to $Ω3$ are the nominal speeds of the shafts, and $θ1(t)$⁠, $θ23(t)$⁠, and $θ4(t)$ are the vibratory components of the angular displacements. The vehicle inertia $Iv$ is typically orders of magnitude larger than gear inertias so that vehicle inertia can be assumed to follow the kinematic trajectory $θ¯v(t)=Ω3t$ with no vibratory component (⁠$θv(t)=0$⁠) [38].

Here, $ωe=Ze(2π60Ω1)$ is the fundamental frequency of the electric motor torque fluctuations, $Ze$ is the electric motor torque fluctuation order (an integer), $T¯e$ is the mean torque and, $Tak$ and $γk$ are the amplitude and the phase angle of the k-th harmonic component of the torque fluctuation.

Use of $q1(t)$⁠, $q2(t)$⁠, and $ξ(t)$ as coordinates in place of rotational displacements $θ¯1(t)$⁠, $θ¯23(t)$⁠, $θ¯4(t)$⁠, $θ¯v(t)$ allows for elimination of the rigid-body mode at zero frequency, reducing the resultant model to a 3-dof definite one.

where $ωmj$ is the mesh frequency of mesh mj (⁠$ωm1=Z1ω1$ and $ωm2=Z1Z3ω1/Z2$ where $Z1$⁠, $Z2$⁠, and $Z3$ are numbers of teeth of gears 1–3), and (⁠$κjk$⁠,$δjk$⁠,$υjk$⁠) and (⁠$ϕjk$⁠,$ψjk,ϑjk$⁠) are the amplitudes and phase angles of the k-th harmonic component of $kj(t)$⁠, $qsj(t)$⁠, and $εj(t)$⁠, respectively. These three functions can be computed upfront using a conventional gear load distribution model under quasi-static conditions [39,40]. Time-dependent Fourier series expansion here assumes that small vibrations about the nominal speed would not cause significant phase modulations [1,2].

Provided that the system acts torsionally, the fidelity of the model proposed above relies heavily on representation of the gear mesh interfaces using discrete stiffness and damping elements along with a displacement excitation. Above formulation uses unloaded static transmission error and static deflection functions, $εj(t)$ and $qsj(t)$⁠, along with the time-varying mesh stiffnesses $kj(t)$ as the excitations of the j-th gear mesh, resulting in a NTV model. Semi-analytical solution methods such as multiterm harmonic balance method [2,41] are not desirable here as the system is subject to multiple excitations that at noncommensurate frequencies. Direct numerical integration solution of Eq. (6) requires that the nonsmooth transitions during tooth separations are handled properly [21,42,43]. Several studies (e.g., Refs. [4,44]) showed on a spur gear pair that employs above form of a discrete gear mesh model compares well with experiments as well as deformable models where such discretization is not required [7,45].

An alternate discretization of the gear mesh interface can be realized by employing the average value of the gear mesh stiffness (⁠$kj(t)≈k¯j$⁠) along with the static deflection function $qsj(t)$ predicted under loaded conditions as given in Eq. (7). This nonlinear time-invariant (NTI) approach was first proposed by Ozguven and Houser [46]. This approach was shown to be deficient for spur gears since they have significant mesh stiffness fluctuations causing parametric instabilities [1,4,44]. Yet, it was considered to be reasonably accurate, through modeling and experimental studies, for high-contact ratio helical gear pairs [3,14,15] since the mesh stiffness fluctuation amplitudes are typically low for such gearing.

where K and H matrices are defined for a given contact state. Closed-form analytical solutions to linear system of Eq. (8) can be easily obtained as H(t) and F(t) matrices consist of constant and harmonic components only. Then, the contact transition points can be obtained numerically by imposing $qj(tw+1*)=±bj$⁠. Readers are referred to Refs. [22,27] for further details of the piecewise-linear solution methodology.

In the absence of experimental data to validate the proposed discrete dynamic model, in its NTV form governed by Eq. (6) or in its NTI form of Eq. (7), an example system of Table 1 is analyzed using both the NTV and NTI versions of model, and their predictions are compared to the predictions of a high-fidelity deformable-body contact model [40]. The EV drivetrain system defined in Table 1 is representative of such applications while it does not correspond to any specific commercial product. Numbers of teeth of gears g1–g4 of this example system are $Zi=20,$ 81, 33, and 82 corresponding to an overall speed reduction ratio of 10.06:1.

The proposed discrete model of Fig. 1 represents of the complex gear mesh interface by a set of discrete system components $kj(t)$ and $cj$⁠, displacement excitations $ej(t)$⁠, and backlash $2bj$⁠. As done in Refs. [5,46] for a spur gear pair, the proposed discrete model of the example EV drive will be compared to a commercial deformable-body contact mechanics model [42] for its verification. This deformable-body model employs finite elements away from the contact interface to account for deformations of gear bodies and the support structures, and a semi-analytical formulation to model the contact interfaces. It requires significant computational effort, especially for dynamic simulations that are performed in time domain using Newmark numerical integration scheme. The primary energy dissipation mechanism in the deformable body model is internal damping, defined by a damping matrix $C=αM+βK$⁠, where M and K are the mass and stiffness matrices, instead of defining a damping coefficient at the contact interface. Damping parameters of $α=479$ and $β=1.4(10)−7$were used in both deformable body model simulations and to describe $C$ matrix of Eq. (6), per Refs. [4,44], which showed good correlations to comparable experiments with these damping values. For the discrete model simulations, the backlash clearance amplitude of $2b1=0.338$ mm and $2b2=0.240$ mm, and inertias of $I 1=0.012 kg.m2$⁠, $I23=0.129 kg.m2$⁠, and $Ig4=0.1 kg.m2$ were considered.

For this comparison, the forced response of variation A of the example gear train was predicted within $Ω1⊂[4,20]$ krpm at $T¯e=200$ Nm and $T̂=0.3$ (⁠$Ta=60$ Nm) and $Ze=12$⁠. This variation is formed by unmodified gears (no profile or lead modifications, i.e., perfect involute shapes) such that $εj(t)=0$ with other discrete model parameters are listed in Table 2. The natural frequencies of variation A at this torque value are obtained from the linearized version of Eq. (6) as $fn1=1474$ Hz and $fn2=1930$ Hz with corresponding mode shape vectors of $qn1=[1−0.98]$ and $qn2=[10.81]$⁠. Figure 2 compares root-mean-square (rms) values of mesh deflections $qjrms$ obtained by solving NTI and NTV versions of the discrete model as well as deformable body model [40]. As expected, external excitation causes resonance peaks when $fe≈fn1$ and $fe≈fn2$⁠, which in this case exhibit softening type nonlinear behavior since tooth separations occur. These coexisting multiple stable solutions can be captured by varying the nominal speeds $Ωi$ (i = 1–3) and performing speed-up and speed-down simulations with the solutions of the previous speed increment serving as the initial conditions. The following general observations can be made from Fig. 2:

• Predictions of both NTI and NTV versions of the discrete model agree well with each other as well as the predictions of the deformable-body model [42]. This indicates that the gear mesh discretization scheme of the discrete NTV model (Eq. (6)) is accurate in capturing helical gear dynamics. Furthermore, it shows that the NTI version of the model (⁠$K[t,q(t)]≈K[q(t)]$ with $k¯j≈kj(t)$⁠) is equally accurate. Considering the simulation of the NTI model to obtain the predictions in Fig. 2 took about 3 min on a 3.40 GHz 4-core processor (plus 7 min preliminary quasistatic analysis to obtain excitation parameters of Table 2) in comparison to nearly 23 days for the deformable body simulations of the same, the proposed discrete model can be deemed suitable for extensive parametric studies and multiple design evaluations.

• For internal excitation related resonances at $fm1≈fn1$⁠, $fm1≈fn2$⁠, $fm1≈fn1$⁠, and $fm2≈fn2$⁠, the response peaks are seen to be linear, agreeing with the assertion that high-contact ratio helical gears act linearly under the sole influence of internal excitations (i.e., under constant torque conditions) [3,10,14]. Both variations of the discrete model capture the impact of internal excitations accurately.

• All models predict nonlinear torque-fluctuation-induced primary resonances at $fe≈fn1$ and $fe≈fn2$ with softening type behavior due to intermittent contact losses. This nonlinear behavior is particularly severe for the $q1rms$ and $q2rms$ peaks at $fe≈fn1$ with two stable motions within $Ω1⊂[6.15,7.2]$ krpm. Meanwhile, a smaller jump with an asymmetric peak defines $fe≈fn2$ resonances. Contrary to the perception that helical gears should act in a linear manner, Fig. 2 indicates that they can act nonlinearly, especially near the resonance frequencies associated with external excitations.

Predicted time histories of $qj(t)/bj$ and mesh forces $Fmj(t)$ for the upper branch motion at $Ω1=6.15$ krpm are compared in Figs. 3 and 4 for $j=1$ and 2, respectively. In both figures, row (a) presents the deformable body model predictions, while rows (b) and (c) are predictions of the discrete model in its NTV and NTI forms. Predictions of all three models of the response time histories agree well as well as the instances of tooth separations and the corresponding frequency spectra. Predicted $q1(t)$ in Figs. 3(a1)–3(c1) all display time segments when tooth separations occur within forcing period $τe=2π/ωe$ (i.e., $qj(t)/bj$ travels below 1, indicating that tooth pairs are separated and the motion is in the backlash zone). Corresponding Fourier spectra are dominated by the $fe=ωe/2π$ order since this motion is near $fe≈fn1.$ The time periods of tooth separations are also demonstrated by zero-valued flat regions of the corresponding $Fm1(t)$ in Figs. 3(a3)–3(c3). While the first mesh exhibits tooth separations, $q2(t)$ is a no-impact motion as shown in Figs. 4(a1)–4(c1). This is because the first mesh is the one that is subjected to torque fluctuations directly. Interactions between the first mesh experiencing tooth separations and the second mesh remaining in contact still lead to jump discontinuities of $q2rms$ in Fig. 2(b) with two stable coexisting motions within $Ω1⊂[6.15,7.2]$ krpm.

Given its good agreement with the deformable-body model, this section employs the NTI version of the proposed discrete model along with the accompanying piecewise-linear solution methodology to investigate the combined influences of internal and external excitations under varying operating conditions and for different system parameters. As the gear train is subject to excitations at distinct frequencies of $ωm1,$$ωm2$ and $ωe$⁠, the simulations at each speed increments are performed long enough to capture at least three fundamental forcing periods $τf$ (⁠$τf=2π/ωf$⁠, where $ωf=2π60Ω1GCD[Z1,Z1Z3Z2,Ze]$ is the fundamental frequency and GCD is the greatest common divisor).

Here, variation A of the gear train is considered at $T¯e=70$ and 200 Nm. Figure 5 presents the influence of torque fluctuation ratio $T̂∈{0,0.2,0.3}$ at $T¯e=70$ Nm and $Ze=12$⁠. Under a constant mean load condition, $T̂=0$⁠, each response curve exhibits low-amplitude linear resonances peaks at $fmj≈fn1$ and $fmj≈fn2$ due to $qsj(t)$⁠. As the higher harmonic amplitudes of $qsj(t)$ are much lower, the resonance peaks associated with them (⁠$2fmj≈fn1$⁠, $2fmj≈fn2$⁠, $3fmj≈fn1$⁠, $3fmj≈fn2$⁠) are insignificant in Fig. 5. When $T̂=Ta1/T¯e=0.1$⁠, the response curves in the vicinity of these internal excitation resonances remains unchanged while two additional linear resonance peaks are introduced at $fe≈fn1$ and $fe≈fn2$ (at $Ω1=6980$ and $9180$ rpm). Further increasing the torque fluctuation ratio to $T̂=0.3$ makes the resonances at $fe≈fn1$ and $fe≈fn2$ nonlinear with a softening-type behavior due to tooth separations. It is also noted that these nonlinear resonance peaks associated with the first and second mode differ. This can be explained by the corresponding mode shapes and the interactions between gear meshes under impacting conditions. For the first peak at $fe≈fn1$ corresponding to the mode shape $qn1=[1−0.98]$⁠, $q1(t)$ and $q2(t)$ are in-phase such that they reach their maximum values at close times. The opposite is true for the second peak at $fe≈fn2$ corresponding to the mode shape $qn2=[10.81]$ where the second mesh does not allow separations of the first mesh to grow.