Abstract

The automotive industry has been experiencing a significant transition toward electrified powertrains in recent years. A torsional model of a common type of electric vehicle (EV) drivetrains is proposed to demonstrate certain dynamic behaviors that are unique to such high-speed applications. This two-stage helical gear drive train is supported by three shafts and connects the electric motor to the vehicle axle. The gear mesh interfaces are modeled by periodically time-varying stiffnesses subjected to backlash and displacement excitations to represent gear tooth errors and modifications. In addition to these internal excitations, torque fluctuations caused by electric motor are included as the external excitations. Two different operating conditions are studied here: (i) steady-state response as the vehicle is operated under steady torque conditions and (ii) transient response during EV system transitions between the drive and regenerative (regen) braking modes of operation. The torsional model predictions are verified through comparisons to simulations from a deformable-body contact model. Parameter sensitivity studies are performed to demonstrate nonlinear behavior of a helical gear train caused by external torque fluctuations as well as the interactions between external and internal excitations. Finally, drivetrain structural modes are shown to respond to drive-regen transitions resulting in certain transient (vibro-impact) behavior with elevated dynamic mesh forces.

1 Introduction

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 [13]. 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=25 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 [1,2,615]. 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 [1517]. 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 [2023], 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 ωeωm. 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 [2931] 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.

2 Discrete Torsional Model

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.

Fig. 1
Discrete dynamic model of a countershaft electric vehicle drivetrain
Fig. 1
Discrete dynamic model of a countershaft electric vehicle drivetrain
Close modal

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].

External torque applied by the electric motor Te(t) is defined in the Fourier series form as
(1)

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.

Relative gear mesh displacements along the lines of action of gear meshes m1 and m2 are given respectively as rg1θ1(t)+rg2θ23(t) and rg3θ23(t)+rg4θ4(t). They consist of gear mesh deflections qj(t) and geometric deviations εj(t) such that
(2a)
(2b)
Here, εj(t) (j=1,2) are displacement functions that represent geometric deviations of tooth surfaces from the ideal involute geometry position (commonly known as the unloaded static transmission error, representing the deviations caused profile manufacturing errors or intentional tooth profile modifications as the gears rolled in contact under no load). εj0 for precise gears with no tooth modifications. Further, relative rotational displacement between the output gear and vehicle inertia is denoted by
(2c)

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.

Force due to stiffness of the mj mesh Fmj[t,qj(t)] along its line of action can be approximated as a first-order variation about the static deflection qsj(t) at given load F¯mj (F¯m1=T¯e/rg1 and F¯m2=T¯erg2/(rg1rg3)) as [36]
(3a)
Here, kj(t) is time-varying gear mesh stiffness at given static mesh force F¯mj and it is the slope of mesh force versus static deflection qsj(t) curve, i.e., kj(t)=Fmj/qj|qj=qsj(t). Functions kj(t), qsj(t), εj(t) can be expanded using K-term truncated Fourier series
(3b)
(3c)
(3d)

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].

Nonlinear version of mesh force description Eq. (3a) in the presence of backlash takes the form
(4a)
(4b)
(4c)
where piecewise linear functions gj and hj capture the backlash nonlinearity of gear mesh mj. With this, equations of motion of the model of Fig. 1(a) are written in matrix form using q1(t), q2(t), and ξ(t) as coordinates to obtain
(5a)
(5b)
(5c)
(5d)
(5e)
(5f)
(5g)
(5h)
(5i)
(5j)
The damping in Eq. (5a) is defined in the Rayleigh's form as
(5k)
As will be shown later, drive-regen transients with backlash nonlinearity can excite both high-frequency gear modes and the low frequency output side drivetrain mode. Equation (5) in its general form can capture both. In the drive mode of the operation, high-frequency gear modes can be assumed to be isolated from the rest of the output-side drivetrain, given relatively low value of kt. Then the mean torques applied on the drive train by the motor and the output shaft are balanced such that [rg2rg4/(rg1rg3)]T¯ektξss=0 where ξss is the steady-state rotational deflection at T¯e. With this, steady-state behavior of the system during the drive mode can be carried out with a 2-dof version of the model whose equations of motion given as
(6)
where I2=rg2rg42T¯e/(I4rg1rg3). A further simplification can be made on the time-variant part of the stiffness matrix. Assuming that kj(t)k¯j, nonlinear time-invariant version of Eq. (6) can be written as
(7)

2.1 Solution Methodology.

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.

A recently developed piecewise-linear solution [20] scheme is well-suited to deal with such NTI systems where K[t,q(t)]K[q(t)], eliminating the numerical issues stemming from the nonsmooth nature of the backlash nonlinearity. This methodology considers the motion within each linear contact regime, namely, drive-side contact qj>bj, backlash |qj|<bj, and coast-side contact qj<bj for each mesh. Linear analytical solutions within each regime are patched at the switching manifolds |qj|=bj to solve for transition times. For a double-mesh gear train, a total of 32=9 contact states exist. For a given contact state, a linear system then can be defined within a time interval t[tw*,tw+1*] with tw* being the w-th transition time
(8a)
(8b)
(8c)
(8d)

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.

The excitation functions kj(t), εj(t), qsj(t), and Te(t) must be defined for the model upfront. External input torque function with its ripples and reversals (during drive-regen transitions) are straightforward to define. Assuming harmonic input torque fluctuations (i.e., Te(t)=T¯e+Tasinωet), the fluctuation amplitude of the Ta is defined as a certain percentage of T¯e as a property of the electric motor [28], while this percentage changes with T¯e. The T̂=Ta/T¯e ratio reduces with increasing T¯e. As an example, T̂ might be 5% at peak T¯e value while it increases to 7.5% at half of the peak T¯e and to 10% at one-tenth of the peak T¯e. In addition to this Ta(T¯e) function, the internal excitations εj(t) (at zero torque) and qsj(t) (of T¯e) must also be defined. While a deformable-body model defines εj(t) and qsj(t) implicitly through a gear mesh load distribution analysis, the proposed discrete model relies on a quasi-static load distribution model [39,40] to predict qsj(t) in the form of Eq. (3b) at a given T¯e. With this, the stiffness of the gear mesh mj is estimated as
(9)

2.2 Model Verification.

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.

Table 1

Basic design parameters for gear pairs used to generate deformable body model

ParameterGear 1Gear 2Gear 3Gear 4
Number of teeth20823381
Transverse module3.1753.17533
Transverse pressure angle (deg)25252525
Helix angle (deg)29292929
Major diameter68.89265.7104248
Minor diameter56.84253.6892.7236.7
Trans. circ. tooth thickness4.614.614.364.36
Face width40403838
Center distance161.93171.00
Total backlash0.3380.240
ParameterGear 1Gear 2Gear 3Gear 4
Number of teeth20823381
Transverse module3.1753.17533
Transverse pressure angle (deg)25252525
Helix angle (deg)29292929
Major diameter68.89265.7104248
Minor diameter56.84253.6892.7236.7
Trans. circ. tooth thickness4.614.614.364.36
Face width40403838
Center distance161.93171.00
Total backlash0.3380.240

All dimensions are in mm unless specified.

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)7were 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 I1=0.012kg.m2, I23=0.129kg.m2, and Ig4=0.1kg.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=[10.98] 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 fefn1 and fefn2, 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:

Fig. 2
Comparison of forced response predictions of discrete NTI, discrete NTV, and deformable body models for unmodified gear train at T¯e=200 Nm, T̂=0.3, and Ze=12: (a) first mesh deflection q1rms and (b) second mesh deflection q2rms. Vertical dashed lines mark primary resonance frequencies.
Fig. 2
Comparison of forced response predictions of discrete NTI, discrete NTV, and deformable body models for unmodified gear train at T¯e=200 Nm, T̂=0.3, and Ze=12: (a) first mesh deflection q1rms and (b) second mesh deflection q2rms. Vertical dashed lines mark primary resonance frequencies.
Close modal
Table 2

Numerical values of the excitation parameters of the example system of Table 1 according to Eqs. (3b) and (3c) with εj(t)0

T¯e=70 Nm
k¯jκj1,κj2,κj3ϕj1,ϕj2,ϕj3
j =162913.07, 4.34, 2.840.25, −1.14, 0.64
j =276214.07, 2.1, 1.01−1.44, −1.92, 2.00
T¯e=70 Nm
k¯jκj1,κj2,κj3ϕj1,ϕj2,ϕj3
j =162913.07, 4.34, 2.840.25, −1.14, 0.64
j =276214.07, 2.1, 1.01−1.44, −1.92, 2.00
δ¯jδj1,δj2,δj3ψj1,ψj2,ψj3
j =13.870.08, 0.03, 0.02−2.89, 2.00, −2.50
j =28.390.16, 0.02, 0.011.70, 1.25, −1.12
δ¯jδj1,δj2,δj3ψj1,ψj2,ψj3
j =13.870.08, 0.03, 0.02−2.89, 2.00, −2.50
j =28.390.16, 0.02, 0.011.70, 1.25, −1.12
T¯e=200 Nm
k¯jκj1,κj2,κj3ϕj1,ϕj2,ϕj3
j =166812.89, 1.19, 2.840.24, −0.14, 0.77
j =283111.43, 4.74, 1.79−1.55, −1.33, −0.91
T¯e=200 Nm
k¯jκj1,κj2,κj3ϕj1,ϕj2,ϕj3
j =166812.89, 1.19, 2.840.24, −0.14, 0.77
j =283111.43, 4.74, 1.79−1.55, −1.33, −0.91
δ¯jδj1,δj2,δj3ψj1,ψj2,ψj3
j =110.410.20, 0.02, 0.04−2.90, 3.04, −2.37
j =221.980.30, 0.13, 0.051.59, 1.81, 2.22
δ¯jδj1,δj2,δj3ψj1,ψj2,ψj3
j =110.410.20, 0.02, 0.04−2.90, 3.04, −2.37
j =221.980.30, 0.13, 0.051.59, 1.81, 2.22

These values were predicted using Ref. [40] under quasi-static conditions. k¯j and κjk are in N/μm, δ¯j and δjk are in μm, and the phase angles ϕjk and ψjk are in radians.

  • 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¯jkj(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 fm1fn1, fm1fn2, fm1fn1, and fm2fn2, 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 fefn1 and fefn2 with softening type behavior due to intermittent contact losses. This nonlinear behavior is particularly severe for the q1rms and q2rms peaks at fefn1 with two stable motions within Ω1[6.15,7.2] krpm. Meanwhile, a smaller jump with an asymmetric peak defines fefn2 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 fefn1. 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.

Fig. 3
Predicted gear mesh deflections q1(t) and mesh forces Fm1(t) at Ω1=6150 rpm for point A in Fig. 2: (a) NTI model, (b) NTV model, and (c) deformable-body model. (a1, b1, c1) time histories of q1(t) signals, (a2, b2, c2) FFT of q1(t) signals, and (a3, b3, c3) time histories of Fm1(t) signals.
Fig. 3
Predicted gear mesh deflections q1(t) and mesh forces Fm1(t) at Ω1=6150 rpm for point A in Fig. 2: (a) NTI model, (b) NTV model, and (c) deformable-body model. (a1, b1, c1) time histories of q1(t) signals, (a2, b2, c2) FFT of q1(t) signals, and (a3, b3, c3) time histories of Fm1(t) signals.
Close modal
Fig. 4
Predicted gear mesh deflections q2(t) and mesh forces Fm2(t) at Ω1=6150 rpm for point A in Fig. 2: (a) NTI model, (b) NTV model, and (c) deformable-body model. (a1, b1, c1) time histories of q2(t) signals, (a2, b2, c2) FFT of q2(t) signals, and (a3, b3, c3) time histories of Fm2(t) signals.
Fig. 4
Predicted gear mesh deflections q2(t) and mesh forces Fm2(t) at Ω1=6150 rpm for point A in Fig. 2: (a) NTI model, (b) NTV model, and (c) deformable-body model. (a1, b1, c1) time histories of q2(t) signals, (a2, b2, c2) FFT of q2(t) signals, and (a3, b3, c3) time histories of Fm2(t) signals.
Close modal

3 Steady-State Dynamic Response

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).

3.1 Influence of Input Torque Parameters.

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 fmjfn1 and fmjfn2 due to qsj(t). As the higher harmonic amplitudes of qsj(t) are much lower, the resonance peaks associated with them (2fmjfn1, 2fmjfn2, 3fmjfn1, 3fmjfn2) 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 fefn1 and fefn2 (at Ω1=6980 and 9180 rpm). Further increasing the torque fluctuation ratio to T̂=0.3 makes the resonances at fefn1 and fefn2 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 fefn1 corresponding to the mode shape qn1=[10.98], 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 fefn2 corresponding to the mode shape qn2=[10.81] where the second mesh does not allow separations of the first mesh to grow.

Fig. 5
Forced response predictions of discrete NTI model for the unmodified gear train at T¯e=70 Nm, T̂={0,0.2,0.3}, and Ze=12: (a) q1rms and (b) q2rms. Vertical dashed lines mark primary resonance frequencies.
Fig. 5
Forced response predictions of discrete NTI model for the unmodified gear train at T¯e=70 Nm, T̂={0,0.2,0.3}, and Ze=12: (a) q1rms and (b) q2rms. Vertical dashed lines mark primary resonance frequencies.
Close modal

Next, the forced response curves for T¯e=70 and 200 Nm are compared in Fig. 6, given T̂=0.25 and Ze=12. The overall shapes of the response curves for each T¯e are very similar in qualitatively while the response amplitudes near resonances of fefn1 and fefn2 are seen to increase with T¯e. At both T¯e levels, the same degree of tooth separations and softening-type nonlinearities is observed, indicating that the ratio T̂=Ta1/T¯e dictates tooth separations under external excitations [2,41].

Fig. 6
Forced response curves of modified gear train at T̂=0.3 Nm, Ze=12, and T¯e={70,200} Nm: (a) first mesh deflection q1rms and (b) second mesh deflection q2rms
Fig. 6
Forced response curves of modified gear train at T̂=0.3 Nm, Ze=12, and T¯e={70,200} Nm: (a) first mesh deflection q1rms and (b) second mesh deflection q2rms
Close modal

Above examples showed rather weak interactions between internal excitations at frequencies ωm1 and ωm2, and the external excitation at ωe when Ze=12. Figures 7 and 8 examine response curves with different torque fluctuation orders of Ze=8, 15, and 36 to ascertain whether the same holds true for different Ze values. In Fig. 7, T̂=0.1 and T¯e=200 Nm such that resonances at fefn1 and fefn2 are both linear. In this case, qjrms is simply superposition of the linear responses to gear mesh and external torque excitations with no sign of any nonlinear interactions. Changing the value of Ze (EV motor toque ripple order) only shifts the fefn1 and fefn2 resonance peaks. In Fig. 8 for T̂=0.3, however, the response becomes nonlinear at the peaks associated with the torque fluctuations. When Ze=36, nonlinear torque fluctuation resonances occur at lower speeds and there is no sign of interactions with internal excitation resonances which remain as low-amplitude, linear peaks at higher speeds. For Ze=15, the fefn1 and fefn2 peaks are only shifted to the right while their shapes and amplitude are similar to those of the Ze=36 curve. Meanwhile, when Ze=8, excitation frequencies of fh2 and fe become closer, contributing to the same resonance peaks at Ω111 and 14.2 krpm. These peaks occur when fm2fefn1 and fm2fefn2 are both nonlinear and higher in amplitude. This suggests that an EV motor design whose orders coincide with one of the gear mesh orders might cultivate interactions between external and internal excitations to cause undesirable response characteristics.

Fig. 7
Influence of torque fluctuation order Ze={8,15,36} at T¯e=200 Nm and T̂=0.1: (a) first mesh deflection q1rms and (b) second mesh deflection q2rms. Vertical dashed lines mark primary resonance frequencies when fmj≈fn1,fn2.
Fig. 7
Influence of torque fluctuation order Ze={8,15,36} at T¯e=200 Nm and T̂=0.1: (a) first mesh deflection q1rms and (b) second mesh deflection q2rms. Vertical dashed lines mark primary resonance frequencies when fmj≈fn1,fn2.
Close modal
Fig. 8
Influence of torque fluctuation order Ze={8,15,36} at T¯e=200 Nm and T̂=0.3: (a) first mesh deflection q1rms and (b) second mesh deflection q2rms. Vertical dashed lines mark primary resonance frequencies when fmj≈fn1,fn2.
Fig. 8
Influence of torque fluctuation order Ze={8,15,36} at T¯e=200 Nm and T̂=0.3: (a) first mesh deflection q1rms and (b) second mesh deflection q2rms. Vertical dashed lines mark primary resonance frequencies when fmj≈fn1,fn2.
Close modal

To demonstrate this interaction between fm2 and fe, Fig. 9 shows frequency spectra obtained during speed down simulations within shaded regions of A, B, and C on Fig. 8 for Ze=8, 15, and 36, respectively. In both Figs. 9(a) and 9(b), a dominant harmonic amplitude at Ze is observed. This amplitude increases to 23 μm until it jumps down to the lower solution branch. Figure 9(c) for Ze=36, however, shows that energy of the harmonic component at fe is modulated with its amplitude remaining below 11 μm. This can be attributed to the fact that nonlinearity introduces different combinations of those two close excitation frequencies fm2 and fe in the form of sidebands. Those additional nonlinear forcing excitations occurring at the close vicinity of fn1 seem to pull energy from the dominant harmonic component at fe. Figure 10(a) shows the resultant long period motion q1(t) for point A on Fig. 9(c) at Ω1=10,650 rpm along with the first mesh force Fm1(t) in Fig. 10(b). The motion in Fig. 10 repeats itself after each τf=2π/ωf that corresponds to 164 torque fluctuation periods of τe. Corresponding spectrum in Fig. 10(c) shows how energy is distributed over the sidebands about fe.

Fig. 9
Comparison of predicted frequency spectra of q1(t) for selected regions A to C on Fig. 8 when fe≈fn1. First mesh order Z1=20, the second mesh order  Z1Z3/Z2=8.04, and torque ripple order: (a) Ze=36, (b) Ze=15, and (c) Ze=8.
Fig. 9
Comparison of predicted frequency spectra of q1(t) for selected regions A to C on Fig. 8 when fe≈fn1. First mesh order Z1=20, the second mesh order  Z1Z3/Z2=8.04, and torque ripple order: (a) Ze=36, (b) Ze=15, and (c) Ze=8.
Close modal
Fig. 10
A long period motion at Ω1=10,650 rpm for point A on Fig. 9: (a) predicted gear mesh deflections q1(t), (b) mesh force Fh1(t), and (c) FFT of q1(t)
Fig. 10
A long period motion at Ω1=10,650 rpm for point A on Fig. 9: (a) predicted gear mesh deflections q1(t), (b) mesh force Fh1(t), and (c) FFT of q1(t)
Close modal

3.2 Influence of Tooth Modifications.

For various durability and noise reasons, helical gear tooth surfaces are modified in practice by removing additional material from the involute tooth contact surfaces. These modifications (tip relief, profile crown, lead crown, etc.) can be designed to minimize ej(t) at a given design load [3]. Up to this point, simulations shown in Figs. 210 considered tooth modification variation A in Table 3, formed by gears that are unmodified. In order to study the influence of tooth modifications in the presence of external excitations, two different sets of modifications are applied to both gear meshes in the forms of profile crown (λ) and lead crown (η) as specified in Table 3 as modifications B and C. The corresponding excitation parameters defining qsj(t) and εj(t) as well as k¯j are listed in Table 4 at T¯e=200 Nm. It is observed for these modified gear meshes that εj(t)0. Furthermore, k¯1 and k¯2 values change with modifications, suggesting that modifications alter the natural frequencies fn1 and fn2 of gear trains [3]. Figure 11 compares forced response curves obtained for the example gear train with modifications A, B, and C at T¯e=200 Nm, T̂=0.3 and Ze=12. The influence of the modifications is observed to be limited to the regions where resonances of the internal excitations take place (at fm1fn1, fm1fn2, fm1fn1, and fm2fn2) where modifications reduce the resonance amplitudes notably. This is directly related to the reduction of ej(t) amplitudes with modifications. Meanwhile, high-amplitude nonlinear resonance peaks caused by external torque fluctuations remain unaffected, indicating that conventional means of reducing helical gear dynamic response through optimizing tooth profiles has no influence on dynamics induced by the external torque fluctuations.

Fig. 11
Forced response predictions of discrete NTI model at T¯e=200Nm, T̂=0.3, and Ze=12 for gear trains A–C defined in Table 3: (a) first mesh deflection q1rms and (b) second mesh deflection q2rms.
Fig. 11
Forced response predictions of discrete NTI model at T¯e=200Nm, T̂=0.3, and Ze=12 for gear trains A–C defined in Table 3: (a) first mesh deflection q1rms and (b) second mesh deflection q2rms.
Close modal
Table 3

Profile crown (λ) and lead crown (η) modifications of gear train tooth modification variations A, B, and C

Gear trainProfile crown λ1 (mesh-1)Lead crown η1 (mesh 1)Profile crown λ2 (mesh-2)Lead crown η2 (mesh 2)
A0000
B5577
C10101414
Gear trainProfile crown λ1 (mesh-1)Lead crown η1 (mesh 1)Profile crown λ2 (mesh-2)Lead crown η2 (mesh 2)
A0000
B5577
C10101414

All dimensions are in μm.

Table 4

Numerical values of the excitation parameters of gear trains B and C of Table 3 at T¯e=200 Nm according to Eqs. (3b) and (3c)

Tooth modifications B
k¯jκj1,κj2,κj3ϕj1,ϕj2,ϕj3
j =166112.84, 2.89, 5.240.27, −0.78, 0.71
j =282614.89, 1.96, 1.941.88, −1.02, −0.77
Tooth modifications B
k¯jκj1,κj2,κj3ϕj1,ϕj2,ϕj3
j =166112.84, 2.89, 5.240.27, −0.78, 0.71
j =282614.89, 1.96, 1.941.88, −1.02, −0.77
δ¯jδj1,δj2,δj3ψj1,ψj2,ψj3
j =110.510.20, 0.05, 0.08−2.88, 2.36, −2.43
j =222.140.40, 0.06, 0.05−1.27, 2.12, 2.37
δ¯jδj1,δj2,δj3ψj1,ψj2,ψj3
j =110.510.20, 0.05, 0.08−2.88, 2.36, −2.43
j =222.140.40, 0.06, 0.05−1.27, 2.12, 2.37
υj1,υj2,υj3ϑj1,ϑj2,ϑj3
j =10.26, 0.02, 0.080.26, −0.09, 0.68
j =20.30, 0.08, 0.021.87, −0.99, −0.83
υj1,υj2,υj3ϑj1,ϑj2,ϑj3
j =10.26, 0.02, 0.080.26, −0.09, 0.68
j =20.30, 0.08, 0.021.87, −0.99, −0.83
Tooth modifications C
k¯jκj1,κj2,κj3ϕj1,ϕj2,ϕj3
j =16322.93, 10.15, 7.04−2.89, −1.04, −2.23
j =281515.03, 1.02, 2.35−1.46, −1.39, 1.70
Tooth modifications C
k¯jκj1,κj2,κj3ϕj1,ϕj2,ϕj3
j =16322.93, 10.15, 7.04−2.89, −1.04, −2.23
j =281515.03, 1.02, 2.35−1.46, −1.39, 1.70
δ¯jδj1,δj2,δj3ψj1,ψj2,ψj3
j =110.990.05, 0.18, 0.120.26, 2.10, 0.91
j =222.420.41, 0.03, 0.071.68, 1.76, −1.44
δ¯jδj1,δj2,δj3ψj1,ψj2,ψj3
j =110.990.05, 0.18, 0.120.26, 2.10, 0.91
j =222.420.41, 0.03, 0.071.68, 1.76, −1.44
υj1,υj2,υj3ϑj1,ϑj2,ϑj3
j =10.19, 0.21, 0.120.26, −1.00, −2.19
j =20.63, 0.09, 0.06−1.47, −1.37, 1.68
υj1,υj2,υj3ϑj1,ϑj2,ϑj3
j =10.19, 0.21, 0.120.26, −1.00, −2.19
j =20.63, 0.09, 0.06−1.47, −1.37, 1.68

These values were predicted using Ref. [40] under quasi-static conditions. k¯j and κjk are in N/μm, δ¯j, δjk, and υjk are in μm, and the phase angles λjk, ψjk, and ϑjk are in radians.

3.3 Influence of Input and Idler Inertia.

The final set of data examines the influence of input and idler inertias on the dynamic response at T¯e=200 Nm, T̂=0.25 and Ze=12. Figure 12(a) varies input inertia about its nominal value I1* such that I1/I1*=0.8, 1, and 1.5. As expected, increasing input side inertia shifts all resonance speeds to the left as it decreases the natural frequencies fn1 and fn2. While the influence of input inertia on internal excitation-related resonances at fmjfn1,fn2 is limited except for the shift in resonance speeds, it significantly alters the external torque fluctuation-related resonances at fefn1,fn2. Increasing I1 reduces the amplitude of vibrations making the motion linear at fefn2 as external torque fluctuations are subjected to input gear only. The opposite is true for idler inertia I23 in Fig. 12(b) as increasing I23 elevates the amplitude of nonlinear responses caused by torque fluctuations as well as enlarging the speed range for the nonlinear response for fefn1.

Fig. 12
(a) influence of varying input inertia I1/I1*={0.8,1,1.5} and (b) varying idler inertia I23/I23*={0.8,1,1.5} on first mesh deflection q1rms at T¯e=200 Nm, T̂=0.25, and Ze=12.
Fig. 12
(a) influence of varying input inertia I1/I1*={0.8,1,1.5} and (b) varying idler inertia I23/I23*={0.8,1,1.5} on first mesh deflection q1rms at T¯e=200 Nm, T̂=0.25, and Ze=12.
Close modal

4 Transient Behavior During Drive-Regen Transitions

Section 3 demonstrated the steady-state dynamics of the drive train as governed by Eq. (7). This assumes harmonic form of the external torque fluctuations of Eq. (1). Another form of stepwise variations of external mean torques occurs during transitions between drive-regen operating modes of EVs. This can be defined in its simplest form by
(10)

where t* is the torque reversal instant. Here, the mean torque during the regen mode is negative (T¯r<0) while it T¯d>0 during the drive mode. This stepwise torque reversal event causes gear meshes to lose contact on the drive side, travel along the backlash zone and initiate contact along the coast side, in the process producing a transient response with vibro-impacts. This steep and sudden change in external torque excites both low-frequency drivetrain modes fd as well as gear modes fnj (j=1,2). This transient behavior can be studied using the proposed model in its general form represented by Eq. (5).

Representative example simulations will be presented here with a total torsional driveline compliance of kt=16.8 kN/rad resulting in fd=45 Hz. Rayleigh's damping coefficients are set to α=10 and β=28(10)7, corresponding to the damping ratios for all three modes within 1–2%. First example considers the torque reversal of T¯d=T¯r=200 Nm with no torque ripple (i.e., Td=Tr=0 in Eq. (10)). This drive-regen transition is assumed to occur when the drivetrain was in steady-state operation at Ω1=9000rpm. Figure 13 shows predicted time histories qj(t) as well as the relative output gear vibration ξ(t). In the absence of torque ripple, both displacements have very small amplitude vibrations about the static equilibrium point [δ¯1,δ¯2,ξss] due to gear mesh excitations when t<t*. Following the transition at t=t*, both meshes separate from the drive side, travel across the backlash zone, and impact the regen (coast) side. A number of single-sided impacts (SSI) are observed for q1(t) following the contact at the regen side. The initial rebounds of q2(t) from the regen side, meanwhile, are seen to travel all the way back to the drive side to create four cycles of double-sided impact motions. Output gear vibrations in Fig. 13(c) are seemed to be dominated by the driveline mode at a period of τd=2π/fd. For t>t*, these low-frequency driveline vibrations are evident in q1(t) and q2(t) as well. It is also noted here that the overall response display rattle type vibro-impacts observed in conventional powertrains. The periodicity of the vibro-impacts is, however, dictated by the low-frequency driveline mode. Several recent theoretical and experimental studies [2124] showed that this type of vibro-impact behavior can be predicted accurately. The corresponding gear mesh forces are plotted in Fig. 14 to show that both meshes experience multiple fully-released, impact- induced force spikes at amplitudes three times higher than the steady-state mesh force values. This suggests that the proposed model could be a tool for studying the fatigue consequences of such transitions.

Fig. 13
A drive-regen transient response for T¯d=−T¯r=200 Nm, T̂=0, and Ω1=9000 rpm for gear train A: (a) q1(t), (b) q2(t), and (c) ξ(t). Vertical dashed line denotes the drive-regen transition instant t*. Horizontal dashed lines are the backlash boundaries.
Fig. 13
A drive-regen transient response for T¯d=−T¯r=200 Nm, T̂=0, and Ω1=9000 rpm for gear train A: (a) q1(t), (b) q2(t), and (c) ξ(t). Vertical dashed line denotes the drive-regen transition instant t*. Horizontal dashed lines are the backlash boundaries.
Close modal
Fig. 14
Mesh force time histories corresponds to the drive-regen transition of Fig. 13; (a)Fm1(t) and (b) Fm2(t). Vertical dashed line denotes the drive-regen transition instant t*.
Fig. 14
Mesh force time histories corresponds to the drive-regen transition of Fig. 13; (a)Fm1(t) and (b) Fm2(t). Vertical dashed line denotes the drive-regen transition instant t*.
Close modal

Steady-state dynamic behavior presented earlier with the torque ripple T̂ indicated that nonlinear speed ranges with two coexisting stable motions are a reality (e.g., Ω1[6,7] krpm in Fig. 2). Initial conditions determine whether the resultant steady-state response dynamics would be an impacting (upper branch) or no-impact (lower branch) motion. The final simulation here considers a drive-regen transition from the steady-state upper-branch motion of Fig. 2 at Ω1=6800 rpm with T¯d=T¯r=200 Nm and T̂=Td/T¯d=Tr/T¯r=0.3. Figure 15 shows the transient q1(t) and q2(t) during this transition along with the corresponding spectrograms. In Fig. 15(a,1), q1(t) is seen to be a SSI motion before the drive-regen transition and the spectrogram of q1(t) in Fig. 15(a,2) shows a dominant harmonic at fe suggesting that the primary excitation here is the external torque fluctuations. Prior to the transition, q2(t) in Fig. 15(b,1) is still a NI motion without any tooth separations. Drive-regen transition causes a similar transient vibro-impact motion again with a slower dynamic content at τd when t[1,1.15] s. When the motions settle on the regen side, both q1(t) and q2(t) are seen to converge to the lower branch motions of the steady-state response with no impacts. This demonstrates that the drive-regen transitions can alter the steady-state dynamics within a nonlinear resonance zone, acting as agent for jump-downs or jump-ups between the coexisting stable motions.

Fig. 15
A drive-regen transient response for T¯d=−T¯r=200 Nm, T̂=0.3, and Ω1=6,800 rpm for the gear train A. (a1,b1) q1(t) and q1(t), and (b1,b2) spectrograms of q1(t) and q1(t). Vertical dashed line denotes the drive-regen transition instant t*.
Fig. 15
A drive-regen transient response for T¯d=−T¯r=200 Nm, T̂=0.3, and Ω1=6,800 rpm for the gear train A. (a1,b1) q1(t) and q1(t), and (b1,b2) spectrograms of q1(t) and q1(t). Vertical dashed line denotes the drive-regen transition instant t*.
Close modal

5 Conclusions

A discrete dynamic model of a double-mesh EV drivetrains was proposed for studying both steady-state and transient behaviors. Predictions from both discrete NTI and NTV variations of the discrete models were compared to those from a deformable-body contact dynamics model predictions to verify their accuracy. Given its accuracy and computational efficiency, NTI variation of the model was exercised with a piecewise solution method to investigate the dynamic response in the presence of both internal and external excitations. Interactions between torque ripple and internal gear mesh excitations were predicted to be weak when the excitations orders are well separated. For such cases, gear tooth modifications were shown to have no influence on resonance peaks caused by the torque ripple. While a helical gear drivetrain acts as a linear time-invariant manner when torque fluctuations are absent, such fluctuations were shown to cause tooth separations and softening type nonlinear behavior at the resonance associated with them. When torque fluctuation and gear orders are close enough to excite the same natural mode, nonlinearity was shown to produce long-period motions with additional side bands. The ratio of motor inertia and idler-axis inertia was found to be an influential parameter determining the transition to the nonlinear regions. At the end, drive-regen transitions in the form of stepwise torque variations were shown to excite high-frequency gear modes as well as the low frequency output side drivetrain mode with the slowest one dominating the transient response. Those transients were predicted to cause additional separations and trailing impacts with amplified gear mesh forces. They were also shown to trigger switching between the between impacting and no-impact motions if the drive-regen transition takes place when the system operated at two-motion region.

Data Availability Statement

The authors attest that all data for this study are included in the paper.

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