Research on the Mechanical Mechanism of the Shuffle Problem of Electric Vehicles and the Sensitivity to Clearances

Abstract

In order to address the shuffle problem of the electric powertrain system that occurs at the moment of torque reversing, a multibody dynamics model of the powertrain system, with the measured motor torque applied as the input loading, has been established to analyze the generating mechanism of the rotating speed ripple of the drive system which is regarded the root of shuffle. The influence on speed ripple from cumulative gap size and motor torque has been investigated. The model was validated by showing good agreement between the simulated speed response and the measured data. Perturbance on each backlash was performed in the simulation to reveal the sensitivity of the speed ripple on the size of the backlash. Much higher speed-to-gap sensitivities have been observed for the low-speed engagement pairs than the high-speed engagement pairs, indicating that compressing the backlashes of the former could achieve more NVH (noise, vibration, harshness, i.e., NVH) performance improvement.

1. Introduction

Compared with conventional vehicles, battery electric vehicles have the advantages of faster dynamic response, a shorter acceleration time of 0 to 100 km, and a quieter interior environment. They bring a new driving experience to drivers. Given that there is no internal combustion engine in an electric vehicle, the environment inside an electric vehicle is quieter compared to that of conventional vehicles. However, this hardly mitigates the potential problem or challenge with respect to NVH for the powertrain system. Some of them are even harder to control, such as shock caused by transient impact, abnormal sound, and other problems. The rapid torque response characteristics of electric vehicles and the unique regenerative energy recovery function of the powertrain bring several challenges to the transient NVH control of the powertrain [1]. Given that the response of motor torque is sensitive, torsional vibration is prone to occur in the powertrain system under rapid and massive excitation caused by driving torque and disturbance [2,3], which adversely influences the driving comfort. On the one hand, the environment of electric vehicles inside is quieter than that of fuel vehicles when the masking effect of background noise on abnormal noise is weakened; as a result, the abnormal noise caused by gap impact is easier to detect. On the other hand, faster power-to-torque response and new system features, braking and coasting energy recovery, will enable the drivetrain to operate under more frequent and intense torque shocks, then will increase the frequency of transient shocks and make them more difficult to control.

Electric vehicle drivetrain meshing pairs have a certain gap. During open or close throttle conditions, the torque switches from negative to positive or positive to negative, which leads to the transmission clearance releasing and closing in the opposite direction; the voids’ closure instant impact may stimulate the torsional vibration transmission system, causing mutations and transient wave transmission speed [4,5]. This causes a transient shuffle in the front and back directions of the vehicle, a process that can also cause the perceived dull clunk sound inside the vehicle. Anibal Valenzuela [6] analyzed the single-stage deceleration transmission system. The positioning of the radius and length of the transmission shaft and the gear clearance of the extinguisher were the main factors affecting the torsional vibration of the system.

Xia Y [7] established a seven-degree-of-freedom lumped parameters model with new piecewise nonlinear clearance elements based on nonlinear Hertz contact theory. The transient vibro-impact phenomena of the vehicle driveline system during fast disengagement of the clutch are numerically simulated. The phase plane revealed the phenomenon of multiple impacts and rebounds in each transient impact and showed the relationship between the relative contact displacement and velocity.

Robert A. Krenz [8] quantified the performance of the vehicle’s impact problem under the condition of pedal or loose throttle from the perspective of a test, which provided data reference for the transient calibration strategy of the engine control system. In this study, three main factors affecting the severity of the vibration and abnormal sound are analyzed which are the rate of change of torque, the cumulative clearance of driveline, and the flexibility of driveline. The smaller the zero-crossing torque change rate is, the less obvious the impact and abnormal sound is. However, a low rate of torque change will affect the dynamic response of the vehicle. Therefore, a balance between NVH and vehicle response is needed. The smaller the cumulative clearance of the transmission system, the smaller the alarm and abnormal sound. Under the condition of high transmission flexibility and under damping, the abnormal sound level will be improved, but the vibration severity will increase. It is more difficult to control driveline oscillation in battery electric vehicles than in conventional fuel vehicles. On the one hand, the motor responds faster than the internal combustion engine, and the regenerative braking torque of the motor is usually much greater than the starting mechanism torque of traditional cars, which also leads to more severe oscillation. On the other hand, in the regenerative and friction hybrid braking process, the intervention of friction braking increases the interference on the load side, which further affects the clearance and increases the complication of oscillation compensation control [9,10].

Most studies on abnormal sound and shock of battery electric vehicles only consider the influence of the flexibility of the drivetrain and compensate for the rotational speed or torque through the control algorithm. Valenzuela A [11] and Makkapati V P [12] used feedforward calibration and acceleration feedback, respectively, to suppress the torsional vibration of the transmission system during fast closed-loop control. For high-speed transmission systems with a resonance frequency of more than 100 Hz, Muszynski R [13] adopted an improved PID controller to suppress the torsional vibration of the transmitter during closed-loop control. H. Kawamura [14] developed a jitter control system for the electric Nissan Leaf, and the jitter control system consists of a feedforward compensator with an inverse filter and a feedback compensator with a complete zeroing method. F. Bottiglione [15] adopted a state estimation feedback control system based on an extended Kalman filter to compensate for the influence of semi-axis dynamics. Zhao Z [16] carried out dynamic simplified modeling of a composite power shunt hybrid power system by considering the elasticity of the torsional shock absorber, drive shaft, and tire. Based on the simplified model, he designed a feedforward corrector, a wheel speed observer, and an active damping control strategy. The above two anti-vibration strategies can effectively suppress the speed fluctuation of the system, reduce the vehicle impact, and improve ride comfort.

In a practical transmission system, the clearance of the meshing pair is unavoidable, and the nonlinear characteristics of the clearance also play an important role in the impact noise and vibration. Lv C [9,10] analyzed the influence of powertrain clearance and flexibility on drivability under regenerative braking conditions by establishing a powertrain system model containing nonlinear clearance and flexibility as well as a hydraulic brake module. In order to further improve the maneuverability and hybrid braking performance of electric vehicles, a hierarchical structure active control algorithm based on mode switching is being developed, which could compensate for the influence of clearance and flexibility. The simulation showed that the control algorithm could significantly improve the maneuverability and hybrid braking performance of the vehicle. Zhang J [17] established a nonlinear drive system model including traction motor, half shaft, drive train backlash, and wheel. Then, the double extended Kalman filter method was used to estimate the vehicle mass and half axle torsion angle. Based on the estimated information, a similar hierarchical active oscillation compensation method was proposed to reduce the vibration of contact and clearance modes. M. Ravichandran [18] developed a similar mode transition control system based on driveline state estimation, which switched between contact mode and backlash mode according to recognition conditions. The proposed method was applied to electric vehicle transmission systems, and the simulation showed that the control system could meet the needs of the driver quickly and also effectively reduce the impact noise and shock problems of the vehicle.

In the past, a one-dimensional mechanical model was used to analyze the impact problem, and the cumulative clearance of the transmission system was equivalent to a single angle clearance, which is a simplification in engineering. Reddy P [19] developed a full-order physics-based model and used this model to develop a reduced-order model (ROM), which captured the main dynamics that influenced the shuffle and clunk phenomena. Simulation results showed that the ROM replicated the behavior of the FOM with less than 5% error when predicting shuffle frequency. Existing studies have shown that clearance at different positions of the transmission chain has significantly different impacts on impact problems [20]. Compression of accumulated clearance is a potential way to improve the shock problem, but this approach is also limited by cost, process, and reliability risks. When controlling the clearance size in engineering, it is often necessary to comprehensively consider and balance the improvement effect of NVH, cost, and reliability risk, so it is necessary to clarify the shock sensitivity to clearance at different positions.

In this study, simulations and experiments have been conducted to analyze the reasons, mechanical process, and sensitivity to clearances at different positions of the transmission chain of battery electric vehicle drivetrain shock. In Section 2, based on the theory of impact shock mechanics, a multibody dynamic model of impact shock was established to complete the simulation and mechanical analysis of the speed and torque response of the power transmission system under throttle conditions and to reveal the cause of impact shock. Section 3 verifies the simulation based on vehicle test data. In Section 4, the sensitivity of speed fluctuation excited by impact to the clearance of the transmission chain is analyzed. On the basis of the dynamic model, the meshing pair gap was changed in turn. After eliminating the high-speed pair meshing gap, the motor speed fluctuation was about 95% of the original value. After eliminating the low-speed pair meshing gap, the motor speed fluctuation dropped to about 18% of the original value, indicating an obvious improvement. This paper provides a theoretical basis for the control of impact shock of electric vehicles and reveals the variation of the sensitivity of speed variation on different clearances of the powertrain system.

2. Mechanical Mechanism of Shock Shudder Problem

2.1. Simplified Mechanical Model of Shock Shudder

The structure of a typical battery-electric vehicle power transmission system is shown in Figure 1, mainly composed of the electric drive assembly, suspension system, left and right half shafts, and wheels. The electric drive assembly casing was connected with the subframe through a three-point suspension structure. The rotor shaft of the motor is connected with the input shaft of the reducer through a spline pair, and the latter and the output shaft of the reducer transmit torque through two pairs of gear pairs to achieve the effect of deceleration and torque increase. The torque from the output shaft is transmitted through a set of differential gears to the two drive axles, which drive the wheels. The differential gear and the transmission half shaft and the transmission half shaft, and the wheel are connected by splines.

Most of the previous studies on the shock shudder problem used the simplified mechanical model of double inertia shown in Figure 2 [9,17], which equivalently divided the rotational inertia of the entire power transmission system into two parts, namely the motor inertia and the wheel-end inertia in the figure. Among them, the inertia of the relevant rotating components of the reducer gear and differential gear was equivalent to the motor inertia J_m, and the wheel rotational inertia was represented by the wheel end inertia J_w. The backlash contribution of the entire drive train was modeled by a single backlash with an angle of 2α. A half-shaft element with torsional stiffness k_hs and damping property c_hs was used to simulate the flexibility effect of the whole system. The model assumed that the two half-shafts are of equal length and that the output torques of the left and right half-shafts are equal.

The dynamics of the motor inertia can be expressed as:

$$J_m{\ddot{\theta }}_m+b_m{\dot{\theta }}_m=T_m-\frac{1}{i}2T_{hs}$$

(1)

where b_m is the viscous damping of the motor rotor, i is the total speed ratio of the reducer, and T_hs is the half-shaft torque.

There is a nonlinear gap connection on the half shaft, and the nonlinear equation is satisfied between the torque and the relevant angle.

$$T_{hs}=k_{hs}{\theta }_s+c_{hs}{\dot{\theta }}_s$$

(2)

$${\theta }_s={\theta }_d-{\theta }_b$$

(3)

$${\theta }_d={\theta }_1-{\theta }_3, {\theta }_b={\theta }_2-{\theta }_3$$

(4)

where θ_d is the torsion angle at both ends of the half shaft (including the gap), θ_b is the position angle of the state of the gap, θ₁ = θ_m/i is the position angle of the half shaft close to the side of the reducer, θ₃ = θ_w is the half shaft close to the wheel end position angle on one side. The nonlinear model of the clearance position angle can be expressed:

$${\dot{\theta }}_b=\{\begin{array}{ll}\max (0,{\dot{\theta }}_d+\frac{k_{hs}}{c_{hs}}({\theta }_d-{\theta }_b)),\hfill & {\theta }_b=-\alpha \hfill \\ {\dot{\theta }}_d+\frac{k_{hs}}{c_{hs}}({\theta }_d-{\theta }_b),\hfill & \left|{\theta }_b\right|<\alpha \hfill \\ \min (0,{\dot{\theta }}_d+\frac{k_{hs}}{c_{hs}}({\theta }_d-{\theta }_b)),\hfill & {\theta }_b=\alpha \hfill \end{array}$$

(5)

The kinematic equation of wheel inertia is:

$$J_w{\ddot{\theta }}_w+b_w{\dot{\theta }}_w=T_{hs}-T_{hb}-T_{bx}$$

(6)

where the road load is decomposed into the damping term b_w and the exogenous longitudinal moment T_bx. T_hb is the mechanical torque generated by the brake [9].

The above-mentioned dual-inertia theoretical model equations are simple in form and clear in constitutive relations and are often used to study the active problem of shock shudder and active control algorithms. However, this model ignores the influence of factors such as the shell flexibility of the electric drive system, the flexibility of the suspension bushing, and the low-order torsional vibration mode of the half-shaft on the dynamic response. In addition, the model simplifies the drive chain mesh clearance to a single clearance and cannot assess how sensitive the shock response is to clearance at different internal locations. In order to make up for the above deficiencies, Section 2.2 builds a multibody dynamic model of the power transmission system based on this mechanics theory, incorporating factors such as the flexibility of the shell structure, the flexibility of the suspension bushing, and the half-shaft mode into the model, and separates the internal gaps of different meshing pairs. Modeling analysis of the above-mentioned dual-inertia theoretical model equations were simple in form and clear in constitutive relations and were often used to study the active problem of shock shudder and active control algorithms. However, this model ignored the influence of factors such as the shell flexibility of the electric drive system, the flexibility of the suspension bushing, and the low-order torsional vibration mode of the half-shaft on the dynamic response. In addition, the model simplified the drive chain mesh clearance to a single clearance and could not assess how sensitive the shock response is to clearance at different internal locations. In order to make up for the above deficiencies, Section 2.2 built a multibody dynamic model of the power transmission system based on this mechanics theory, incorporating factors such as the flexibility of the shell structure, the flexibility of the suspension bushing, and the half-shaft mode into the model, and separated the internal gaps of different meshing pairs.

2.2. Multibody Dynamics Model

The drive motor is a permanent magnet synchronous type with a maximum power of 160 kW, the reducer is a single-speed two-stage reduction, gear parameters are detailed in

Table 1. Gear parameters of reducer.

	The First Gear Pair		The Secondary Gear Pair
Items	Driving Gear	Driven Gear	Driving Gear	Driven Gear
Tooth number	27	79	20	77
Normal module (mm)	1.54	1.54	2.42	2.42
Face width (mm)	47	45	50	48
Pitch diameter (mm)	44.2	128.4	51.5	198.2
Root diameter (mm)	41.3	124.1	46.3	190.6
Base diameter (mm)	41.8	122.3	48.2	183.7
Pressure angle (°)	20	20	20	20
Helix angle (°)	20	20	20	20

, and the total speed ratio i (the ratio of the motor speed n_m to the wheel end speed n_w) is about 11. The schematic diagram of the power transmission system is shown in Figure 3. For this system, the electric drive assembly, transmission axle shafts, and wheels together form a torsional vibration system. The simulation of the shock shudder problem mainly focused on the speed change and torsional vibration characteristics of the power transmission system aftershock; the speed fluctuation response caused by the shock depended on its low-order torsional vibration mode. In order to make the model capture the torsional vibration characteristics of the system as accurately as possible, when building up a multibody dynamics model with SIMPACK, the motor shaft, reducer shafting, left and right half shafts, and other shaft structures were modeled as flexible bodies in this model, depending on the complexity of the geometry of each shaft structure. A parametric flexible shaft model or a finite element modal polycondensation shaft model was established in the software. For example, the motor shaft, the input shaft of the reducer, the intermediate shaft, etc., with simple structure and geometry, all adopted the parameterized flexible shaft model, and the output shaft of the reducer with complex structure and geometry adopted the finite element polycondensation model. After the finite element model was established, modal polycondensation was performed and imported into the multibody dynamics model for simulation. For the gears fixed on each shaft, the main structure was modeled as a rigid body. The bearings of the transmission system were simulated by bearing force elements.

The electric drive assembly shell, suspension, shafting, and wheels form a coupled dynamic system. In order to consider the flexibility and mounting stiffness of the electric drive shell as much as possible and its influence on the dynamic process of the shock, the electric drive shell is modeled as a flexible body with finite element polycondensation, and each mounting bushing is modeled as a spring unit with a certain stiffness. One end of the spring unit is connected to the adjacent node on the electric drive housing, and one end is fixed (the displacement is 0, that is, the flexibility of the subframe is ignored), and the stiffness of the suspension spring unit is the linear segment stiffness value of the suspension bushing cushion. The wheels are simplified as two rigid bodies with the rotational inertia of IW, and the moving inertia of the whole vehicle is equivalent to the rotational inertia of the two wheels to consider the load effect of the vehicle inertia on the power transmission system. The wheel inertia is calculated as follows:

$$I_W=I_w+I_v$$

(7)

where I_w is the moment of inertia of the wheel itself, and I_v is the moment of inertia obtained by the equivalent translation inertia of the whole vehicle. When the speed of a vehicle of mass M is V, the angular velocity of the wheel rotating around the axle is ω, which can be calculated according to the kinetic energy equivalent principle 1/2MV² = 1/2I_vω²_*2.

$$I_v=\frac{1}{2}M{R}^2$$

(8)

where R is the wheel radius.

As shown in Figure 3, the model simulated the transmission chain clearance (mesh backlash) in 5 positions, namely the spline pair clearance between the motor shaft and the input shaft, the first gear pair clearance, the secondary gear pair clearance, the differential gear clearance, transmission half shaft clearance. In order to simplify the model, the equivalent simplification of the half shaft clearance was carried out here, that is, the half shaft universal joint ball cage clearance, the spline pair clearance between the half shaft and the reducer gear, and the spline pair clearance between the half shaft and the wheel. Equivalent to modeling a single gap (gap number 5 in the figure). The gap size in the model was taken as the median value of the design gap tolerance zone of each meshing pair.

2.3. Result Analysis

The motor torque curve of the real vehicle under the condition of repeatedly stepping on and releasing the accelerator on a straight road was used as the simulation torque input and loaded onto the motor shaft of the model described in Section 2.2 for multibody dynamics simulation. A set of accelerator pedal opening signals and torque curves corresponding to typical stepping and releasing conditions are shown in Figure 4.

It can be seen from Figure 4 that the opening of the accelerator pedal increased after the accelerator pedal was stepped on, and the motor output positive torque. At this time, the motor converted the electrical energy into kinetic energy to propel the vehicle forward. After the accelerator was released, the accelerator pedal opening rapidly dropped to zero, and the vehicle quickly entered the coasting energy recovery state. At this time, the motor generated electricity to convert mechanical energy into electrical energy, and the corresponding torque was negative. Figure 4 also shows the motor shaft speed response curve simulated by the above model.

It can be seen from Figure 4 that when the motor torque was positive, its rotational speed tended to increase; on the contrary, when the torque was negative, the motor rotational speed gradually decreased. When the torque changed from negative to positive and from positive to negative, the motor shaft speed had obvious sudden changes and transient fluctuations. This reflected the phenomenon of clearance impact and torsional vibration of the power transmission system under the action of alternating positive and negative torque. At the vehicle level, it was manifested as transient impact vibration and front and rear swaying of the body. When the impact is severe, it might also cause a noticeable impact. The positive and negative of the torque were used to represent the direction of the torque transmitted by each axis. The direction of the motor torque in the driving condition when the accelerator was depressed was defined as positive, and the direction of the motor torque in the power generation condition after the accelerator was released was defined as negative. For other shafts, except for the differential gear shaft, when the direction of the torque transmitted by it was the same as the positive direction of the motor shaft torque, its direction was defined as positive; otherwise, it was negative. When the torque of the meshing pair was zero, the clearance of the meshing pair was in a disengaged state, so the opening and closing state of the meshing pair could be judged by the torque curve of the meshing pair.

In order to analyze the reasons for the sudden change and fluctuation of the speed of the transmission system when the torque was switched between positive and negative, the torque and motor shaft speed curves transmitted by each meshing pair were compared and analyzed (corresponding to the gaps 1 to 5 in Figure 3) at the moment when the accelerator was stepped on. The results are shown in Figure 5. The torque curves corresponding to the first and second-level meshing pairs in the figure were the torques transmitted by the driven wheel (large gear), and the following was consistent with this.

It can be seen from Figure 5 that the torque transmitted by the transmission pairs at all levels switched in the direction of torque around t = 24.7 s, and the gaps at each position were disengaged and closed during the corresponding period. The first time the gap of the meshing pair was disengaged was around 24.698 s, and the moment when the last set of gaps was closed in reverse occurred at around 24.717 s, and then the torque of each meshing pair increased rapidly. In the time period of about 20 ms between the above-mentioned two times, the torque of each meshing pair remained at 0 or fluctuated around 0. Taking the torque of the half-shaft spline pair as an example, its torque experienced a process from negative to 0 and then to positive around this time, indicating that the half-shaft spline pair was experiencing the process of engagement–disengagement–reverse re-engagement at this time.

The simulation in Figure 5 shows that the torque of the motor shaft–input shaft spline pair, the first-stage gear pair, and the second-stage gear pair experienced obvious unilateral fluctuations in the time period of 24.707~24.717 s, indicating that these meshing pairs had more clearances for secondary anti-tooth percussion. In fact, the transmission chain between the motor shaft and the wheel was in a physical disengagement state during the above short period of time, leaving the motor in a no-load state. At this time, each gear and spline could also be approximately regarded as a no-load state. Gears or splines that were under no-load and rotating motion were easily knocked against adjacent meshing objects by slight disturbance, but the vibration energy caused by these knocks was low, which usually manifested as local, transient, high-frequency vibration on the gear or spline component. Whether the crisp knocking sound caused by such knocks could be perceived by passengers in the car was related to the size of the meshing clearance, system damping, and the NVH transmission path of the entire vehicle. Considering the above working conditions, no appreciable high-frequency knocking noise was observed in the vehicle, and the no-load anti-tooth surface knocking phenomenon would not be further analyzed later.

According to the motor shaft speed data obtained from the simulation, the speed started to rise sharply from around t = 24.70 s, rising by about 180 rpm in about 20 ms, then dropped sharply and oscillated attenuates. The magnitude of speed fluctuation between a peak and the adjacent valley was 231 rpm. The time period corresponding to the rising edge of the rotational speed was highly matched with the zero-crossing time of the torque of each meshing pair. The starting time was close to the opening time of the first gap, and the ending time of the rising edge was consistent with the closing time of the last group of gaps.

It can be seen from the above results that at the moment before the accelerator pedal was stepped on, the vehicle was in a state of coasting energy recovery; the motor was under negative torque and was in a state of generating electricity. The kinetic energy of the vehicle was gradually reduced, and each gap in the transmission system was in a closed state. When the driver stepped on the accelerator pedal, the motor instantly generated positive torque, and during the process of switching between positive and negative torque, the torque direction of each rotating component was reversed. The originally closed gaps began to disengage one by one and hit the anti-meshing surfaces. When the gap was disengaged, the torque transmitted by the meshing pair was zero. When the transmission system still had the meshing gap in the disengaged state, the motor was in the no-load state. Therefore, when the motor torque rose rapidly, the rotor speed of the motor increased sharply until the last set of meshing gaps in the transmission chain were closed, and the motor rotor shaft and the wheel end load (including the effect of the body inertia load) was connected again when the “rebound” of the rotational motion occurred. At the moment before the relative clearance was disengaged, the rotating part of the transmission chain close to the motor rotor obtained a certain kinetic energy increment due to the short-term no-load acceleration. After all clearances were closed, these kinetic energy increments were rapidly converted into the transmission system. The release of these elastic potential energies caused torsional vibration of the transmission system, resulting in rotational speed fluctuations. The impact of the transmission system entered its low-order torsional vibration mode, and the quasi-periodic characteristics of rotational speed oscillation attenuation reflected the first-order torsional vibration of the transmission system.

It can be seen from the above analysis that the height of the rising edge of the rotational speed determined the severity of the impact and also determined the magnitude of the rotational speed fluctuation caused by the impact. All 5 groups of clearances in the corresponding model in Figure 5 were compressed to 1/2 of the original value, and the same torque time history was used for loading.

The time that each shaft system was in the no-load state was reduced from the original 20 ms to about 13 ms, the rising edge curve of the motor speed during the corresponding period became shorter, and the speed fluctuation caused by the gap impact was reduced from the original 231 rpm to 145 rpm. It could be seen that reducing the cumulative gap value of the meshing pair of the transmission chain, thereby shortening the closing time of the cumulative gap of the transmission system, could reduce the amplitude of the shafting speed increase, thereby improving the peak-to-peak value of the shafting speed fluctuation.

The amplitude of the loading torque in the corresponding simulation model in Figure 5 was reduced to 1/2 of the original value, and the gap of each meshing pair remained the same as the original value.

When the loading torque was halved, within a short period of time (about 26 ms) from the disengagement of the first set of gaps to the closing of all gaps, the gradient of the motor speed rise decreased. The kinetic energy gain of the shaft was reduced, so the impact-induced rev fluctuations dropped to 186 rpm. In the case of a certain cumulative gap of the transmission system, reducing the average torque of the motor during the no-load period could reduce the amplitude of the motor speed increase when the cumulative gap of the transmission system was closed, thereby improving the peak-to-peak value of the shafting speed fluctuation.

Comparing the results in Figure 4, Figure 5, Figure 6 and Figure 7, it can be seen that when the accumulated clearance or torque decreased, the rotational speed fluctuation caused by the impact decreased significantly, but the rotational speed fluctuation was not proportional to the accumulated clearance and torque.

3. Test Verification

In order to validate the accuracy of the simulation model proposed in Section 2, a torsional vibration test was conducted on an electric vehicle equipped with the powertrain system mentioned above by repetitively pressing and releasing the acceleration pedal. The vehicle was driven under 25 km/h, and the accelerator pedal was repetitively pressed and released. The pedal position, motor torque, and motor speed were obtained via CAN bus.

When the motor torque curve was fed into the multibody simulation model described in Section 2.2, the curve of the speed response of the motor shaft could be obtained. Hence, the accuracy of the model could be validated by comparing the predicted speed fluctuation with experiment results.

Figure 8 shows the measured and simulated curves of the motor speed when the accelerator was repeatedly stepped on and released over a period of time. It can be seen from Figure 8 that the motor speed signal obtained by the simulation analysis was in good agreement with the test speed curve.

Figure 9 shows a partial amplification of the motor speed curve at the moment of stepping on the accelerator near t = 30 s. The results showed that the transient fluctuation characteristics of the simulated speed were consistent with the measured curve characteristics, and the simulation of the mechanical model could well reproduce the impact caused by torsional vibration.

It can be seen from Figure 8 and Figure 9 that the simulated rotational speed fluctuation caused by the gap impact was higher than the measured rotational speed fluctuation, and the error mainly came from the cumulative gap difference between the simulated model and the physical model. As mentioned in Section 2.3, under the same torque control strategy, the magnitude of the rotational speed fluctuation was strongly correlated with the size of the accumulated gap. In fact, the clearance of each transmission pair of the power transmission system was within a certain design tolerance zone, and the cumulative clearance of the transmission chain was bound to be within a certain range. In this simulation model, each gap size was taken as the median value of its design gap tolerance zone. Therefore, the gap error between the simulation model and the physical model was bound to exist, and the error between the simulation and the measured rotational speed fluctuation was also unavoidable.

4. Gap Sensitivity Analysis

Based on the above analysis, it can be seen that the compression clearance was a method to improve the shock shudder from the source of structural design. However, it should be noted that the compression clearance might bring increased process difficulty, poor lubrication conditions, and increased cost. If the sensitivity of the shock shudder problem in terms of different clearances could be clearly defined before the gap control, then when the design was improved, the high-sensitivity clearance could be controlled based on the sensitivity results so as to obtain the best NVH performance at the minimum cost.

This section analyzes the sensitivity of the impact-induced rotational speed fluctuation to the clearance based on the previous dynamic model. On the basis of the original dynamic model, the meshing pair clearances 1 to 5 revealed in Figure 3 were disturbed in turn, with each clearance being eliminated at a time (the clearance size is set to 0). The clearance sizes of the other groups were kept consistent with the original model. The simulation compared the clearance changes in motor speed caused by impacts before and after disturbance. Figure 10 shows the simulation of the motor speed after eliminating the motor shaft and input shaft spline, the first gear pair, and the secondary gear pair clearance, respectively. After eliminating the above-mentioned clearance, the speed curve showed a very small change, and the peak-to-peak value of the speed fluctuation was within 10 rpm. Figure 11 shows the simulation of the motor speed after eliminating the differential gear pair clearance and the half shaft clearance, respectively. The results showed that compressing the two sets of clearances could significantly reduce the impact-induced rotational speed fluctuations. At the same time, the speed curve obtained by compressing the differential gear pair clearance and the half shaft clearance is shown by the green dotted line in Figure 11. The speed fluctuation was reduced to about 18% of the original level, and the unevenness caused by the clearance impact was significantly improved.

The clearance sensitivity is defined as the ratio of the variation of the motor speed fluctuation to the clearance compression; it can be expressed as:

$$S_i=\Delta n_i/\Delta c_i$$

(9)

where S_i is the sensitivity of the ith clearance, Δn_i is the variation of speed fluctuation caused by compression of the clearance, and Δc_i is the compression of the clearance. Considering the sensitivity, comparative analysis of different types of clearances such as gear pair and spline pair, Δc_i is defined as the circumferential angle clearance with the dimension of divided. The ratio of the sensitivity of each clearance calculated by Equation (9) is S₁:S₂:S₃:S₄:S₅ = 0.9:3:11:36:100.

The above results show that the speed fluctuation caused by impact was more sensitive to the gap of the meshing pair on the low-speed shaft of the transmission system. The mechanism analysis of Section 2.3 showed that under the condition of a constant torque control strategy, the speed fluctuation caused by impact was positively correlated with the time length required for motor rotation to eliminate the cumulative gap.

It can be seen from the above results that the severity of the speed fluctuation caused by the impact was more sensitive to the meshing pair clearance on the low-speed shaft of the transmission system. From the mechanism analysis in Section 2.3, it can be seen that when the torque control strategy remained unchanged, the degree of rotational speed fluctuation caused by the impact was positively related to the time length required for the motor’s rotational motion to eliminate the accumulated clearance.

For the clearance on a certain axis, the time required for the motor’s rotational motion to cover the clearance per unit angle was proportional to the rotational speed of the axis. The speed of the half shaft and the differential gear shaft associated with it was approximately 1/11 of the speed of the motor shaft and approximately 1/4 of the speed of the intermediate shaft of the reducer (the ratio is determined by the speed ratio). Therefore, the contribution of the meshing pair clearances on these low-speed shafts to the duration of the “speed rising edge” was significantly greater than that on the high-speed shafts, which caused the speed fluctuations to be more sensitive to the clearances on the low-speed shafts.

5. Discussion of Control Method

Compression clearance is a method to improve shock shudder from the source of structural design. The results of the clearance sensitivity analysis in Section 4 show that the control of the mesh pair clearance on the low-speed shaft of the transmission system could bring better NVH performance gains. However, it should be noted that the meshing clearance is unavoidable. The design must reserve enough clearance for each pair of gears, splines, and universal joints to avoid assembly difficulties, poor lubrication, and reliability risks caused by too small clearances. In addition, excessively strict clearance control standards would inevitably put forward higher requirements for the machining accuracy of parts, which may also cause the cost to be out of control. When formulating a clearance control scheme, an appropriate balance must be made in terms of NVH performance, process, reliability, and cost.

Adjusting the motor torque control strategy is another effective method to improve shock shudder/shock pulsation. The analysis of Figure 5 in Section 2.3 shows that reducing the average torque of the motor during the “no-load” period could reduce the height of the rising edge of the motor speed during the clearance disconnection period of the transmission system, thereby reducing the degree of speed fluctuation. In the actual application process, the motor torque control strategy could be adjusted to reduce the slope of the torque curve near the zero-crossing moment of the motor torque so as to minimize the pull-up speed of the motor speed in the no-load period and reduce the impact degree. However, this kind of torque smoothing strategy often sacrifices the speed of the vehicle’s dynamic response, which makes the accelerator pedal appear hysteretic in torque response. Active control methods [17,18] could effectively reduce this cost. This method adjusted the control mode by estimating the state of the system and adopted the torque following and speed following control strategies for the meshing pair contact mode and the tooth clearance mode, respectively, which could effectively reduce the shock and abnormal noise of the vehicle. At present, most of these kinds of control algorithms are in the theoretical research stage.

6. Conclusions

In this study, a multibody dynamic model for the shock shudder problem of the battery electric vehicle powertrain is established. The experimental verification showed that the model could predict the sudden change of the motor speed and the fluctuation response caused by the clearance shock. The study revealed the causes of the shaft speed fluctuations, analyzed the sensitivity of the speed fluctuations to the clearance at different positions, and drew the following conclusions.

(1) Under the action of positive and negative reverse torque, the power transmission system was affected by the clearance of the meshing pair. The impact caused sudden changes and fluctuations in the shafting speed. When the transmission chain had a meshing clearance and was in a disengaged state, its rotational speed changed abruptly under the action of torque. When the last set of clearances came into contact with the meshing surfaces (that is, close again), it caused torsional vibration and speed fluctuations in the transmission system.

(2) The degree of rotational speed fluctuation caused by the impact was strongly related to the size of the accumulated clearance of the transmission chain and the motor torque control strategy. Compressing the size of the accumulated clearance or reducing the average torque of the motor could effectively alleviate the impact degree and reduce the amount of speed fluctuation, but the relationship between the amount of speed fluctuation and the size of the accumulated clearance and torque was not strictly proportional.

(3) By comparing the clearance sensitivity, it could be seen that the sensitivity of the rotational speed fluctuation to the low-speed pair clearance was significantly greater than that of the high-speed pair clearance. Compressing the clearance size of the low-speed pair could significantly reduce the speed fluctuation.