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 [1–3]. One such excitation is the periodic variation of gear mesh stiffness due to fluctuation of the total number of loaded tooth pairs between n and ( for conventional spur gears, and helical gears depending on the total contact ratio). Besides , tooth profile errors and designed tooth modifications result in an internal displacement excitation known as the transmission error [3]. Both and are periodic at the gear mesh (tooth-passing) frequency of , where is the number of teeth and 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 () 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 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 resulting in a time-varying transmitted gear mesh forces . While internal excitations apply at , the frequencies 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 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 . 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 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, and are comparable in EV applications, requiring inclusion of both external () and internal ( and ) excitations simultaneously to capture the steady-state response. Such a model is a must to aid specification of the 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 . In the regenerative braking (regen) mode, the mean torque is negative, and the motors acts now as a generator. In the regen mode, , where . 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.
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 rad/s (up to 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 and (base) radius . 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 . 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 . On the output shaft s3, , and represent equivalent torsional stiffness and damping of the rest of the drivetrain components, and is the vehicle inertia.
Each gear mesh interface mj () is modeled here by a time-varying mesh stiffness , a linear viscous damper , and a displacement excitation that represents the static transmission error under loaded conditions (including both the unloaded transmission error and deflections under static conditions ). Internal excitations and are both periodic functions at gear mesh frequency . Each gear mesh is subject to a clearance of representing the gear backlash amplitudes along the lines of action. In the model of Fig. 1, gears g1–g4 have angular displacement of , and where to are the nominal speeds of the shafts, and , , and are the vibratory components of the angular displacements. The vehicle inertia is typically orders of magnitude larger than gear inertias so that vehicle inertia can be assumed to follow the kinematic trajectory with no vibratory component () [38].
Here, is the fundamental frequency of the electric motor torque fluctuations, is the electric motor torque fluctuation order (an integer), is the mean torque and, and are the amplitude and the phase angle of the k-th harmonic component of the torque fluctuation.
Use of , , and as coordinates in place of rotational displacements , , , allows for elimination of the rigid-body mode at zero frequency, reducing the resultant model to a 3-dof definite one.
where is the mesh frequency of mesh mj ( and where , , and are numbers of teeth of gears 1–3), and (,,) and (,) are the amplitudes and phase angles of the k-th harmonic component of , , and , 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].
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, and , along with the time-varying mesh stiffnesses 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 () along with the static deflection function 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 . Readers are referred to Refs. [22,27] for further details of the piecewise-linear solution methodology.
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 81, 33, and 82 corresponding to an overall speed reduction ratio of 10.06:1.
Parameter | Gear 1 | Gear 2 | Gear 3 | Gear 4 |
---|---|---|---|---|
Number of teeth | 20 | 82 | 33 | 81 |
Transverse module | 3.175 | 3.175 | 3 | 3 |
Transverse pressure angle (deg) | 25 | 25 | 25 | 25 |
Helix angle (deg) | 29 | 29 | 29 | 29 |
Major diameter | 68.89 | 265.7 | 104 | 248 |
Minor diameter | 56.84 | 253.68 | 92.7 | 236.7 |
Trans. circ. tooth thickness | 4.61 | 4.61 | 4.36 | 4.36 |
Face width | 40 | 40 | 38 | 38 |
Center distance | 161.93 | 171.00 | ||
Total backlash | 0.338 | 0.240 |
Parameter | Gear 1 | Gear 2 | Gear 3 | Gear 4 |
---|---|---|---|---|
Number of teeth | 20 | 82 | 33 | 81 |
Transverse module | 3.175 | 3.175 | 3 | 3 |
Transverse pressure angle (deg) | 25 | 25 | 25 | 25 |
Helix angle (deg) | 29 | 29 | 29 | 29 |
Major diameter | 68.89 | 265.7 | 104 | 248 |
Minor diameter | 56.84 | 253.68 | 92.7 | 236.7 |
Trans. circ. tooth thickness | 4.61 | 4.61 | 4.36 | 4.36 |
Face width | 40 | 40 | 38 | 38 |
Center distance | 161.93 | 171.00 | ||
Total backlash | 0.338 | 0.240 |
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 and , displacement excitations , and backlash . 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 , where M and K are the mass and stiffness matrices, instead of defining a damping coefficient at the contact interface. Damping parameters of and were used in both deformable body model simulations and to describe 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 mm and mm, and inertias of , , and were considered.
For this comparison, the forced response of variation A of the example gear train was predicted within krpm at Nm and ( Nm) and . This variation is formed by unmodified gears (no profile or lead modifications, i.e., perfect involute shapes) such that 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 Hz and Hz with corresponding mode shape vectors of and . Figure 2 compares root-mean-square (rms) values of mesh deflections 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 and , 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 (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:
Nm | |||
---|---|---|---|
j = 1 | 629 | 13.07, 4.34, 2.84 | 0.25, −1.14, 0.64 |
j = 2 | 762 | 14.07, 2.1, 1.01 | −1.44, −1.92, 2.00 |
Nm | |||
---|---|---|---|
j = 1 | 629 | 13.07, 4.34, 2.84 | 0.25, −1.14, 0.64 |
j = 2 | 762 | 14.07, 2.1, 1.01 | −1.44, −1.92, 2.00 |
j = 1 | 3.87 | 0.08, 0.03, 0.02 | −2.89, 2.00, −2.50 |
j = 2 | 8.39 | 0.16, 0.02, 0.01 | 1.70, 1.25, −1.12 |
j = 1 | 3.87 | 0.08, 0.03, 0.02 | −2.89, 2.00, −2.50 |
j = 2 | 8.39 | 0.16, 0.02, 0.01 | 1.70, 1.25, −1.12 |
Nm | |||
---|---|---|---|
j = 1 | 668 | 12.89, 1.19, 2.84 | 0.24, −0.14, 0.77 |
j = 2 | 831 | 11.43, 4.74, 1.79 | −1.55, −1.33, −0.91 |
Nm | |||
---|---|---|---|
j = 1 | 668 | 12.89, 1.19, 2.84 | 0.24, −0.14, 0.77 |
j = 2 | 831 | 11.43, 4.74, 1.79 | −1.55, −1.33, −0.91 |
j = 1 | 10.41 | 0.20, 0.02, 0.04 | −2.90, 3.04, −2.37 |
j = 2 | 21.98 | 0.30, 0.13, 0.05 | 1.59, 1.81, 2.22 |
j = 1 | 10.41 | 0.20, 0.02, 0.04 | −2.90, 3.04, −2.37 |
j = 2 | 21.98 | 0.30, 0.13, 0.05 | 1.59, 1.81, 2.22 |
These values were predicted using Ref. [40] under quasi-static conditions. and are in , and are in μm, and the phase angles and 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 ( with ) 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 , , , and , 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 and with softening type behavior due to intermittent contact losses. This nonlinear behavior is particularly severe for the and peaks at with two stable motions within krpm. Meanwhile, a smaller jump with an asymmetric peak defines 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 and mesh forces for the upper branch motion at krpm are compared in Figs. 3 and 4 for 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 in Figs. 3(a1)–3(c1) all display time segments when tooth separations occur within forcing period (i.e., travels below 1, indicating that tooth pairs are separated and the motion is in the backlash zone). Corresponding Fourier spectra are dominated by the order since this motion is near The time periods of tooth separations are also demonstrated by zero-valued flat regions of the corresponding in Figs. 3(a3)–3(c3). While the first mesh exhibits tooth separations, 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 in Fig. 2(b) with two stable coexisting motions within krpm.
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 and , the simulations at each speed increments are performed long enough to capture at least three fundamental forcing periods (, where 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 and 200 Nm. Figure 5 presents the influence of torque fluctuation ratio at Nm and . Under a constant mean load condition, , each response curve exhibits low-amplitude linear resonances peaks at and due to . As the higher harmonic amplitudes of are much lower, the resonance peaks associated with them (, , , ) are insignificant in Fig. 5. When , the response curves in the vicinity of these internal excitation resonances remains unchanged while two additional linear resonance peaks are introduced at and (at and rpm). Further increasing the torque fluctuation ratio to makes the resonances at and 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 corresponding to the mode shape , and are in-phase such that they reach their maximum values at close times. The opposite is true for the second peak at corresponding to the mode shape where the second mesh does not allow separations of the first mesh to grow.
Next, the forced response curves for and 200 Nm are compared in Fig. 6, given and . The overall shapes of the response curves for each are very similar in qualitatively while the response amplitudes near resonances of and are seen to increase with . At both levels, the same degree of tooth separations and softening-type nonlinearities is observed, indicating that the ratio dictates tooth separations under external excitations [2,41].
Above examples showed rather weak interactions between internal excitations at frequencies and , and the external excitation at when . Figures 7 and 8 examine response curves with different torque fluctuation orders of , 15, and 36 to ascertain whether the same holds true for different values. In Fig. 7, and Nm such that resonances at and are both linear. In this case, 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 (EV motor toque ripple order) only shifts the and resonance peaks. In Fig. 8 for , however, the response becomes nonlinear at the peaks associated with the torque fluctuations. When , 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 , the and peaks are only shifted to the right while their shapes and amplitude are similar to those of the curve. Meanwhile, when , excitation frequencies of and become closer, contributing to the same resonance peaks at and 14.2 krpm. These peaks occur when and 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.
To demonstrate this interaction between and , Fig. 9 shows frequency spectra obtained during speed down simulations within shaded regions of A, B, and C on Fig. 8 for 15, and 36, respectively. In both Figs. 9(a) and 9(b), a dominant harmonic amplitude at is observed. This amplitude increases to 23 until it jumps down to the lower solution branch. Figure 9(c) for , however, shows that energy of the harmonic component at is modulated with its amplitude remaining below 11 . This can be attributed to the fact that nonlinearity introduces different combinations of those two close excitation frequencies and in the form of sidebands. Those additional nonlinear forcing excitations occurring at the close vicinity of seem to pull energy from the dominant harmonic component at . Figure 10(a) shows the resultant long period motion for point A on Fig. 9(c) at rpm along with the first mesh force in Fig. 10(b). The motion in Fig. 10 repeats itself after each that corresponds to 164 torque fluctuation periods of . Corresponding spectrum in Fig. 10(c) shows how energy is distributed over the sidebands about .
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 at a given design load [3]. Up to this point, simulations shown in Figs. 2–10 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 () and lead crown () as specified in Table 3 as modifications B and C. The corresponding excitation parameters defining and as well as are listed in Table 4 at Nm. It is observed for these modified gear meshes that . Furthermore, and values change with modifications, suggesting that modifications alter the natural frequencies and of gear trains [3]. Figure 11 compares forced response curves obtained for the example gear train with modifications A, B, and C at Nm, and . The influence of the modifications is observed to be limited to the regions where resonances of the internal excitations take place (at , , , and ) where modifications reduce the resonance amplitudes notably. This is directly related to the reduction of 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.
Gear train | Profile crown (mesh-1) | Lead crown (mesh 1) | Profile crown (mesh-2) | Lead crown (mesh 2) |
---|---|---|---|---|
A | 0 | 0 | 0 | 0 |
B | 5 | 5 | 7 | 7 |
C | 10 | 10 | 14 | 14 |
Gear train | Profile crown (mesh-1) | Lead crown (mesh 1) | Profile crown (mesh-2) | Lead crown (mesh 2) |
---|---|---|---|---|
A | 0 | 0 | 0 | 0 |
B | 5 | 5 | 7 | 7 |
C | 10 | 10 | 14 | 14 |
All dimensions are in .
Tooth modifications B | |||
---|---|---|---|
j = 1 | 661 | 12.84, 2.89, 5.24 | 0.27, −0.78, 0.71 |
j = 2 | 826 | 14.89, 1.96, 1.94 | 1.88, −1.02, −0.77 |
Tooth modifications B | |||
---|---|---|---|
j = 1 | 661 | 12.84, 2.89, 5.24 | 0.27, −0.78, 0.71 |
j = 2 | 826 | 14.89, 1.96, 1.94 | 1.88, −1.02, −0.77 |
j = 1 | 10.51 | 0.20, 0.05, 0.08 | −2.88, 2.36, −2.43 |
j = 2 | 22.14 | 0.40, 0.06, 0.05 | −1.27, 2.12, 2.37 |
j = 1 | 10.51 | 0.20, 0.05, 0.08 | −2.88, 2.36, −2.43 |
j = 2 | 22.14 | 0.40, 0.06, 0.05 | −1.27, 2.12, 2.37 |
j = 1 | 0.26, 0.02, 0.08 | 0.26, −0.09, 0.68 | |
j = 2 | 0.30, 0.08, 0.02 | 1.87, −0.99, −0.83 |
j = 1 | 0.26, 0.02, 0.08 | 0.26, −0.09, 0.68 | |
j = 2 | 0.30, 0.08, 0.02 | 1.87, −0.99, −0.83 |
Tooth modifications C | |||
---|---|---|---|
j = 1 | 632 | 2.93, 10.15, 7.04 | −2.89, −1.04, −2.23 |
j = 2 | 815 | 15.03, 1.02, 2.35 | −1.46, −1.39, 1.70 |
Tooth modifications C | |||
---|---|---|---|
j = 1 | 632 | 2.93, 10.15, 7.04 | −2.89, −1.04, −2.23 |
j = 2 | 815 | 15.03, 1.02, 2.35 | −1.46, −1.39, 1.70 |
j = 1 | 10.99 | 0.05, 0.18, 0.12 | 0.26, 2.10, 0.91 |
j = 2 | 22.42 | 0.41, 0.03, 0.07 | 1.68, 1.76, −1.44 |
j = 1 | 10.99 | 0.05, 0.18, 0.12 | 0.26, 2.10, 0.91 |
j = 2 | 22.42 | 0.41, 0.03, 0.07 | 1.68, 1.76, −1.44 |
j = 1 | 0.19, 0.21, 0.12 | 0.26, −1.00, −2.19 | |
j = 2 | 0.63, 0.09, 0.06 | −1.47, −1.37, 1.68 |
j = 1 | 0.19, 0.21, 0.12 | 0.26, −1.00, −2.19 | |
j = 2 | 0.63, 0.09, 0.06 | −1.47, −1.37, 1.68 |
These values were predicted using Ref. [40] under quasi-static conditions. and are in , , , and are in μm, and the phase angles , , and 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 Nm, and . Figure 12(a) varies input inertia about its nominal value such that , 1, and 1.5. As expected, increasing input side inertia shifts all resonance speeds to the left as it decreases the natural frequencies and . While the influence of input inertia on internal excitation-related resonances at is limited except for the shift in resonance speeds, it significantly alters the external torque fluctuation-related resonances at . Increasing reduces the amplitude of vibrations making the motion linear at as external torque fluctuations are subjected to input gear only. The opposite is true for idler inertia in Fig. 12(b) as increasing elevates the amplitude of nonlinear responses caused by torque fluctuations as well as enlarging the speed range for the nonlinear response for .
4 Transient Behavior During Drive-Regen Transitions
where is the torque reversal instant. Here, the mean torque during the regen mode is negative () while it 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 as well as gear modes (). 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 kN/rad resulting in Hz. Rayleigh's damping coefficients are set to and , corresponding to the damping ratios for all three modes within 1–2%. First example considers the torque reversal of Nm with no torque ripple (i.e., in Eq. (10)). This drive-regen transition is assumed to occur when the drivetrain was in steady-state operation at rpm. Figure 13 shows predicted time histories as well as the relative output gear vibration . In the absence of torque ripple, both displacements have very small amplitude vibrations about the static equilibrium point due to gear mesh excitations when . Following the transition at 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 following the contact at the regen side. The initial rebounds of 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 . For , these low-frequency driveline vibrations are evident in and 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 [21–24] 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.
Steady-state dynamic behavior presented earlier with the torque ripple indicated that nonlinear speed ranges with two coexisting stable motions are a reality (e.g., 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 rpm with Nm and . Figure 15 shows the transient and during this transition along with the corresponding spectrograms. In Fig. 15(a,1), is seen to be a SSI motion before the drive-regen transition and the spectrogram of in Fig. 15(a,2) shows a dominant harmonic at suggesting that the primary excitation here is the external torque fluctuations. Prior to the transition, 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 when s. When the motions settle on the regen side, both and 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.
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.