Regenerative braking has become a standard function in every modern electric car, yet the very actuator that makes energy recovery possible—the permanent-magnet synchronous machine coupled to a geared driveline—is also the dominant source of the torque excitation that degrades ride comfort. During a regenerative event, the braking torque generated by the machine is transmitted through the gearbox and the half-shafts to the wheels, and this transmission path is intrinsically nonlinear: the machine produces a rich harmonic torque spectrum, the meshing gears operate with backlash, and the shafts behave as flexible elements with very weak damping. The combination of these effects produces strong torsional excitation, and once the electrical torque is reversed at the instant of mode switching, the driveline is set into a low-frequency oscillation that is perceived by the occupants as a longitudinal shock and a prolonged rocking motion of the electric car.
In this work I take the position that the problem can be solved without any change to the mechanical architecture of the electric car. Instead of stiffening the shafts or preloading the gears, I quantify how the machine’s torsional characteristics shift the natural frequencies of the driveline, I examine the additional excitation produced by gear backlash and shaft flexibility, and I show that the whole system behaves as a strongly under-damped two-degree-of-freedom oscillator. On this basis I design a feedforward–feedback damping compensator that uses an estimate of the half-shaft torque as its feedback signal and injects a correcting torque into the machine command. The strategy raises the equivalent damping of the electric car driveline, flattens the resonance peaks, and converts a violent acceleration transient into a smooth transition, while leaving the energy-recovery capability of the electric car essentially unchanged.
1. Modelling of the Electric Car Powertrain
The analysis is carried out on a front-wheel-drive electric car whose traction motor is connected to a two-stage helical gearbox and, through two half-shafts, to the driven wheels. Three sub-models are required: an electromagnetic model of the machine, a torsional model of the geared driveline including backlash and shaft compliance, and a longitudinal vehicle model that converts wheel torque into acceleration. The three sub-models are then coupled into a single simulation environment so that the electrical and mechanical nonlinearities act simultaneously, as they do in a real electric car.
1.1 Electromagnetic model of the permanent-magnet synchronous machine
The traction machine is a non-linear, strongly coupled system whose output torque is never perfectly constant. The prototype considered here is an 8-pole, 48-slot permanent-magnet synchronous machine; its main parameters are listed in Table 1. To represent the nonlinear behaviour faithfully, the machine is described in the rotor reference frame through its flux-linkage model, in which the stator voltage equations are written as
$$
\begin{cases}
u_d = R_s i_d + \dfrac{\mathrm{d}\psi_d}{\mathrm{d}t} – \omega_e \psi_q \\[6pt]
u_q = R_s i_q + \dfrac{\mathrm{d}\psi_q}{\mathrm{d}t} + \omega_e \psi_d
\end{cases}
$$
where $\omega_e$ is the electrical angular velocity, $R_s$ is the stator resistance, and $\psi_d$, $\psi_q$ are the flux linkages along the direct and quadrature axes. The flux linkages themselves are given by
$$
\begin{cases}
\psi_d = L_d i_d + \psi_f \\
\psi_q = L_q i_q
\end{cases}
$$
so that the electromagnetic torque becomes
$$
T_e = \frac{3}{2} P_n \left[ \psi_f i_q + \left( L_d – L_q \right) i_d i_q \right]
$$
with $P_n$ the number of pole pairs. The mechanical dynamics of the rotor are governed by
$$
J_m \frac{\mathrm{d}\omega_m}{\mathrm{d}t} = T_e – T_L – B \omega_m
$$
in which $J_m$ is the rotor inertia, $B$ the viscous damping coefficient, $T_L$ the load torque delivered by the driveline, and $\omega_m$ the mechanical angular velocity of the rotor. Because the machine is fed by a finite-bandwidth inverter and because the air-gap field is not purely sinusoidal, the torque $T_e$ contains a series of harmonics whose amplitudes are far from negligible; these harmonics are the primary broadband excitation entering the electric car driveline.
| Parameter | Symbol | Value |
|---|---|---|
| Maximum power | $P_{max}$ | 80 kW |
| Maximum torque | $T_{max}$ | 180 N·m |
| Base speed | $n_b$ | 4 000 r/min |
| DC bus voltage | $U_{dc}$ | 300 V |
| Stator resistance | $R_s$ | 0.008 Ω |
| d-axis inductance | $L_d$ | 0.047 mH |
| q-axis inductance | $L_q$ | 0.142 mH |
| Permanent-magnet flux | $\psi_f$ | 0.083 Wb |
| Number of pole pairs | $P_n$ | 4 |
1.2 Torsional model of the geared driveline
The gearbox consists of two helical stages. Contact stiffness, mesh damping and backlash are all retained, and the mesh force is written in a piecewise form that distinguishes the driving flank, the backlash region and the coast flank:
$$
F_g =
\begin{cases}
k_g \delta + c_g \dot{\delta}, & \delta > \alpha \\[4pt]
0, & -\alpha \le \delta \le \alpha \\[4pt]
k_g \delta + c_g \dot{\delta}, & \delta < -\alpha
\end{cases}
$$
where $\delta$ is the relative mesh displacement, $\alpha$ is half of the total backlash, $k_g$ is the mesh stiffness and $c_g$ is the mesh damping. The mesh displacement and the mesh stiffness are related to the geometry of the helical pair through
$$
\delta = r_b \left( \theta_1 – \theta_2 \right) \cos\beta, \qquad k_g = k_n \cos\beta
$$
with $r_b$ the base-circle radius, $\beta$ the helix angle and $k_n$ the normal contact stiffness. Each shaft, including the machine shaft, the intermediate shaft, the half-shaft and the tyre, is represented as a spring–damper element. For the half-shaft, for example,
$$
T_{ds} = k_{ds} \left( \theta_{fd} – \theta_{ds} \right) + c_{ds} \left( \dot{\theta}_{fd} – \dot{\theta}_{ds} \right)
$$
where $k_{ds}$ and $c_{ds}$ are the torsional stiffness and damping of the shaft and $\theta_{fd}$, $\theta_{ds}$ are the input and output angular displacements. The gear parameters are collected in Table 2 and the driveline parameters in Table 3. It is worth noting that the half-shaft damping is only 9.5 N·m·s/rad, which is small compared with the equivalent inertia it has to stabilize; this single number is the origin of most of the comfort problems observed in the electric car.
| Parameter | First stage | Second stage | ||
|---|---|---|---|---|
| Driving gear | Driven gear | Driving gear | Driven gear | |
| Number of teeth | 36 | 144 | 36 | 84 |
| Helix angle (deg) | 20 | 20 | 20 | 20 |
| Normal pressure angle (deg) | 20 | 20 | 20 | 20 |
| Base-circle radius (mm) | 34 | 136 | 34 | 79.5 |
| Parameter | Symbol | Value |
|---|---|---|
| Intermediate shaft stiffness | $k_m$ | 85 943 N·m/rad |
| Machine shaft stiffness | $k_i$ | 100 000 N·m/rad |
| Half-shaft stiffness | $k_a$ | 6 208 N·m/rad |
| Tyre stiffness | $k_o$ | 18 000 N·m/rad |
| Machine shaft damping | $c_i$ | 2 N·m·s/rad |
| Intermediate shaft damping | $c_m$ | 0.9 N·m·s/rad |
| Half-shaft damping | $c_a$ | 9.5 N·m·s/rad |
| Tyre damping | $c_o$ | 286.56 N·m·s/rad |
| Machine equivalent inertia | $J_m$ | 0.03 kg·m² |
| First-stage driving gear inertia | $J_1$ | 0.000 16 kg·m² |
| First-stage driven gear inertia | $J_2$ | 0.001 25 kg·m² |
| Second-stage driving gear inertia | $J_3$ | 0.000 31 kg·m² |
| Second-stage driven gear inertia | $J_4$ | 0.007 kg·m² |
| Tyre equivalent inertia | $J_o$ | 0.63 kg·m² |
| Vehicle equivalent inertia | $J_v$ | 144 kg·m² |
1.3 Longitudinal dynamics of the electric car
The wheel torque is converted into vehicle motion through the longitudinal force balance. The resistance forces are the rolling resistance, the aerodynamic drag, the grade force and the ground braking force, and the equation of motion of the electric car is
$$
\begin{cases}
\dfrac{\mathrm{d}v}{\mathrm{d}t} = \dfrac{1}{m} \left( F_f + F_a + F_i + F_{xb} \right) \\[6pt]
F_f = m g f \cos\alpha \\[4pt]
F_a = \dfrac{1}{2} C_d A \rho \left( v + v_w \right)^2 \\[4pt]
F_i = m g \sin\alpha \\[4pt]
F_{xb} = F_{bf} + F_{br}
\end{cases}
$$
where $v$ is the vehicle speed, $v_w$ the wind speed, $f$ the rolling-resistance coefficient, $\alpha$ the road gradient, $C_d$ the aerodynamic drag coefficient, $A$ the frontal area, and $F_{bf}$, $F_{br}$ the braking forces at the front and rear wheels. Because the electric car studied here is front-wheel driven, the front-wheel braking force is the sum of the hydraulic contribution and the regenerative contribution:
$$
F_{bf} = F_{zr} + \frac{T_{reg}\, i_0\, i_g\, \eta_t}{R}
$$
in which $F_{zr}$ is the hydraulic braking force at the front wheel, $T_{reg}$ the regenerative torque of the machine, $i_0$ and $i_g$ the two gear ratios, $\eta_t$ the driveline efficiency and $R$ the rolling radius. The vehicle parameters are summarized in Table 4.
| Parameter | Symbol | Value |
|---|---|---|
| Vehicle mass | $m_c$ | 1 327 kg |
| Wheelbase | $L$ | 2.69 m |
| Rolling radius | $R$ | 0.3 m |
| Total reduction ratio | $i$ | 9.3 |
1.4 Integrated simulation model
The three sub-models are coupled in a single simulation platform, so that the electromagnetic torque of the machine, the backlash of the two gear stages, the compliance of the four shaft elements and the longitudinal dynamics of the electric car all act simultaneously. The simulation reproduces a complete manoeuvre: the accelerator is fully depressed at 2 s and held for 5 s, after which the pedal is released and the machine is commanded to a constant regenerative torque of −100 N·m for a further 5 s. This manoeuvre excites the driveline both in traction and in regenerative mode, and it contains the mode transition that is the object of the present study.
2. Evaluation Metrics and Modal Characteristics
Two metrics are used throughout. The first is the torque ripple rate, which measures the dynamic quality of the torque delivered to the half-shaft:
$$
\delta_T = \frac{T_{max} – T_{min}}{T_{ref}}
$$
where $T_{max}$, $T_{min}$ and $T_{ref}$ are the maximum, minimum and mean values of the torque over the evaluation window. The second is the root-mean-square jerk, which quantifies the comfort perceived by the occupants of the electric car:
$$
J_{rms} = \sqrt{ \frac{1}{T} \int_0^{T} \left( \frac{\mathrm{d}a_z}{\mathrm{d}t} \right)^2 \mathrm{d}t }
$$
with $a_z$ the longitudinal acceleration and $T$ the length of the evaluation window. When an external excitation acts on the driveline, the dynamic response is governed by the inherent properties of the system, which are obtained from the generalized eigenvalue problem associated with the mass matrix $M$ and the stiffness matrix $K$:
$$
\begin{cases}
M \ddot{X} + K X = 0 \\
\left| M^{-1} K – \omega_n^2 E \right| = 0
\end{cases}
$$
Solving this problem for the full eleven-inertia model of the electric car gives the six natural angular frequencies listed in Table 5. The first three modes lie at 1.15 Hz, 16.79 Hz and 30.88 Hz, and they dominate the low-frequency behaviour of the driveline; the remaining three modes lie above 1.6 kHz and belong to the high-frequency band associated with gear meshing and shaft elasticity. For the comfort problem addressed here, only the three low-frequency modes matter, because they are the modes that can be excited by the torque harmonics of the machine in the normal speed range of the electric car.
| Order | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Angular frequency (rad/s) | 7.20 | 105.53 | 105.53 | 10 485.07 | 27 331.42 | 54 613.76 |
| Frequency (Hz) | 1.15 | 16.79 | 30.88 | 1 668.75 | 4 349.93 | 8 692.05 |
3. Influence of the Nonlinear Excitations
3.1 Torque harmonics of the machine and critical speeds
The harmonic components of the electromagnetic torque act as the initial disturbance source. When the frequency of a given harmonic coincides with a modal frequency of the driveline, the mechanical transmission path amplifies the response and the dynamic load in the shaft rises sharply. The harmonic frequency of the machine torque grows linearly with rotor speed, so that every harmonic crosses every natural frequency at a well-defined rotor speed, generally called the critical speed:
$$
n_c = \frac{60 f_n}{r}
$$
where $f_n$ is the natural frequency of the driveline and $r$ is the order of the torque harmonic. For the machine considered here, the 24th and 48th orders carry the largest amplitudes, and they are therefore used for the critical-speed analysis. The intersections of these two order lines with the six natural-frequency lines are summarized in Table 6. The 24th order meets the fourth natural frequency at 4 171.95 r/min, corresponding to 47.21 km/h, and meets the fifth natural frequency at 10 874.83 r/min, corresponding to 120 km/h. The 48th order reaches the fourth natural frequency at 2 085.96 r/min (23.58 km/h), the fifth at 5 437.41 r/min (61.51 km/h), and the sixth at 10 865.12 r/min (120 km/h). Most of these critical speeds fall inside the everyday operating range of the electric car, which means that the resonance risk cannot be avoided by simply restricting the operating envelope. The resonance band must instead be shifted by modifying the effective damping of the driveline.
| Harmonic order $r$ | Natural frequency excited | Critical speed (r/min) | Equivalent vehicle speed (km/h) |
|---|---|---|---|
| 24 | Fourth, 1 668.75 Hz | 4 171.95 | 47.21 |
| 24 | Fifth, 4 349.93 Hz | 10 874.83 | 120.00 |
| 48 | Fourth, 1 668.75 Hz | 2 085.96 | 23.58 |
| 48 | Fifth, 4 349.93 Hz | 5 437.41 | 61.51 |
| 48 | Sixth, 8 692.05 Hz | 10 865.12 | 120.00 |
Using the half-shaft torque as the vibration indicator and sweeping the machine speed, the speed–frequency–amplitude representation of the response shows that the first three low-frequency modes dominate the response of the electric car and that they are responsible for the large dynamic loads observed in the half-shaft. The time-domain behaviour confirms the same picture. During traction, the half-shaft torque rises to 180 N·m and stays at that plateau while the electric car accelerates at its maximum rate. Once the machine speed exceeds the base speed, the torque enters the constant-power region and begins to fall from about 1.4 s. At 6 s the system enters the regenerative braking mode. In both the launch and the braking transition, the zoomed responses show a delay in the torque, because the meshing teeth must first traverse the backlash before contact is re-established. At the instant of the regenerative transition the machine torque and speed both oscillate violently, and the oscillation amplitude is larger than in the aggressive launch case.
3.2 Influence of gear backlash
Backlash is the key source of meshing impact. When the teeth pass from one flank to the other, the lost motion produces an instantaneous impact force, and the driveline responds with a torsional oscillation and with audible noise. The effect becomes more pronounced at high speed or high torque, and it directly affects the comfort and the driving feel of the electric car. To quantify this effect I ran the model with backlash values of 0°, 3°, 5°, 7° and 10° while keeping the half-shaft stiffness and damping unchanged. As the backlash increases, the amplitude of the half-shaft torque oscillation grows steadily, although the growth rate is moderate, and the response delay of the driveline increases as well. The backlash therefore acts as an amplitude multiplier on the impact and as a pure delay on the torque transmission, and both effects are detrimental to comfort during the mode transition of the electric car.
3.3 Influence of half-shaft damping
The half-shaft damping coefficient is one component of the overall driveline damping. If the driveline is regarded as a single vibration system, then the half-shaft damping contributes directly to the total damping of that system, and increasing it should therefore reduce the oscillation. To verify this, I varied the half-shaft damping over 10, 50, 100, 150 and 200 N·m·s/rad while keeping the backlash and the half-shaft stiffness fixed. The results are unambiguous: when the damping rises from 10 to 200 N·m·s/rad, the torque ripple rate falls from 290.50 % to 91.24 %, a reduction of 68.59 %, and the decay time shortens from 1.29 s to 0.28 s, a reduction of 78.29 %. The system therefore reaches steady state much faster, and the oscillation amplitude of the half-shaft torque decreases monotonically with increasing damping. This result is the physical justification for the control strategy developed in the next section: if the damping of the electric car driveline cannot be raised mechanically, it must be raised actively.
4. Reduced-Order Model and Transfer Characteristics
4.1 Two-degree-of-freedom simplification
For controller design the eleven-inertia model is too detailed, and it is reduced to the two-degree-of-freedom representation in which the machine and the gears on the motor side form one inertia and the wheel, tyre and vehicle form the other, connected by an equivalent spring–damper element. The equations of motion of the reduced model are
$$
\begin{cases}
J_{me} \dot{\omega}_m = T_m – \dfrac{T_{hs}}{i_g} \\[6pt]
J_{ve} \dot{\omega}_v = T_{hs} – T_L \\[6pt]
T_{hs} = k_e \left( \dfrac{\theta_m}{i_g} – \theta_v \right) + c_e \left( \dfrac{\omega_m}{i_g} – \omega_v \right)
\end{cases}
$$
where $i_g$ is the total reduction ratio, $\omega_v$ the half-shaft angular velocity and $T_L$ the load torque referred to the wheel side. The equivalent inertias are obtained by referring every rotating element to its own side of the gearbox:
$$
\begin{cases}
J_{me} = J_m + J_1 + \dfrac{J_2 + J_3}{i_1^2} + \dfrac{J_4}{i_1^2 i_2^2} \\[8pt]
J_{ve} = J_o + J_v
\end{cases}
$$
with $i_1$ and $i_2$ the ratios of the first and second gear stages. The equivalent stiffness and damping of the reduced model follow from energy equivalence and are written as the series combination of the individual shaft elements:
$$
k_e = \left( \frac{1}{k_i} + \frac{1}{i_1^2 k_m} + \frac{1}{i_1^2 i_2^2 k_a} \right)^{-1}, \qquad
c_e = \left( \frac{1}{c_i} + \frac{1}{i_1^2 c_m} + \frac{1}{i_1^2 i_2^2 c_a} \right)^{-1}
$$
Substituting the numerical values of Table 3 into these expressions shows that the equivalent damping $c_e$ is dominated by the half-shaft damping $c_a$, which is precisely why the parametric study of the previous section produced such a strong effect. The state vector is chosen as $x = [\omega_m,\ \omega_v,\ \theta]^{\mathsf{T}}$, and the reduced model is cast in state-space form as
$$
\dot{x} = A x + B u + d, \qquad y = C x
$$
$$
A =
\begin{bmatrix}
-\dfrac{c_e}{J_{me} i_g^2} & \dfrac{c_e}{J_{me} i_g} & -\dfrac{k_e}{J_{me} i_g} \\[10pt]
\dfrac{c_e}{J_{ve} i_g} & -\dfrac{c_e}{J_{ve}} & \dfrac{k_e}{J_{ve}} \\[10pt]
\dfrac{1}{i_g} & -1 & 0
\end{bmatrix},
\quad
B = \begin{bmatrix} \dfrac{1}{J_{me}} \\ 0 \\ 0 \end{bmatrix},
\quad
d = \begin{bmatrix} 0 \\ -\dfrac{T_L}{J_{ve}} \\ 0 \end{bmatrix},
\quad
C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}
$$
where $u = T_m$ is the control input and $y = \omega_m$ is the measured output.
4.2 Parametric transfer function
From the state-space representation, the transfer function between the machine torque and the machine angular velocity can be written in parametric form as
$$
G_{ma\_T}(s) = \frac{\omega_m(s)}{T_m(s)} = \frac{b_2 s^2 + b_1 s + b_0}{s^2 + 2 \zeta \omega_n s + \omega_n^2}
$$
with the numerator coefficients and the modal parameters given by
$$
b_2 = \frac{1}{J_{me}}, \qquad
b_1 = \frac{c_e}{J_{me} J_{ve}}, \qquad
b_0 = \frac{k_e}{J_{me} J_{ve}}
$$
$$
\omega_n = \sqrt{ \frac{k_e \left( J_{me} i_g^2 + J_{ve} \right)}{J_{me} J_{ve} i_g^2} }, \qquad
\zeta = \frac{c_e \left( J_{me} i_g^2 + J_{ve} \right)}{2 J_{me} J_{ve} i_g^2 \omega_n}
$$
Substituting the numerical values of the electric car driveline into these expressions gives the explicit transfer function
$$
G_{ma\_T}(s) = \frac{33.3 s^2 + 1.5 s + 2090}{s^2 + 5.2 s + 3590}
$$
The poles of this transfer function are located at $s = -2.6 \pm 59.86\,\mathrm{i}$. They are very close to the imaginary axis, and the damping ratio obtained from the denominator is
$$
\zeta = \frac{5.2}{2 \sqrt{3590}} \approx 0.044 < 1
$$
which confirms that the driveline of the electric car behaves as a strongly under-damped system. The natural frequency of the reduced model is $\omega_n \approx 59.9$ rad/s, that is about 9.5 Hz, and it lies in the band where the low-order torque harmonics of the machine and the first torsional modes of the driveline overlap. The under-damped character of the electric car is therefore not a marginal detail of the model: it is the root cause of the prolonged oscillation observed after every regenerative mode transition.
5. Design of the Damping Compensation Controller
5.1 Rationale
The modal analysis above shows that the damping ratio is the single parameter that governs both the peak amplitude of the oscillation and the settling time. When the damping ratio reaches its critical value, the system returns to equilibrium as fast as possible without any periodic oscillation. A proportional–integral–derivative (PID) controller is attractive here because its three terms act directly on the error and on the error derivative, and the derivative term in particular is equivalent to adding a virtual damping to the closed loop. By tuning the PID gains appropriately, the equivalent damping ratio of the electric car driveline can be raised without touching a single mechanical component, and the resonance peak at the low-frequency band can therefore be suppressed.
$$
G_c(s) = K_p + \frac{K_i}{s} + K_d s
$$
The controller is organised as a feedforward–feedback composite structure. The feedforward path carries the base regenerative torque demand, so that the steady-state braking capability of the electric car is preserved exactly. The feedback path carries the damping correction, which is computed from the estimated half-shaft torque. Because the correction is added to the torque command rather than to the mechanical system, no structural parameter of the electric car is modified, and the strategy can be calibrated entirely in software.
5.2 Estimation of the half-shaft torque
The feedback signal is the half-shaft torque. Measuring it directly would require a torque transducer on each shaft, which is not acceptable for a production electric car. Instead, the torque is reconstructed from the speed signals that are already available from the machine resolver and the wheel-speed sensors. The half-shaft is modelled as a torsional spring in parallel with a viscous damper, so that the transmitted torque is the sum of an elastic and a damping contribution:
$$
T_{hs} = T_{elastic} + T_{damping}
$$
The elastic contribution follows from Hooke’s law applied to the relative twist of the shaft:
$$
T_{elastic} = k_{hs} \theta_{hs} = k_{hs} \int \left( \frac{\omega_m}{i_g} – \omega_v \right) \mathrm{d}t
$$
and the damping contribution follows from the viscous-damping assumption applied to the relative speed:
$$
T_{damping} = c_{hs} \left( \omega_{in} – \omega_v \right) = c_{hs} \left( \frac{\omega_m}{i_g} – \omega_v \right)
$$
Combining the two contributions gives the estimator actually implemented in the controller:
$$
\hat{T}_{hs} = k_{hs} \int \left( \frac{\omega_m}{i_g} – \omega_v \right) \mathrm{d}t + c_{hs} \left( \frac{\omega_m}{i_g} – \omega_v \right)
$$
where $k_{hs}$ and $c_{hs}$ are the torsional stiffness and damping of the half-shaft and $\omega_{in} = \omega_m / i_g$ is the angular velocity at the input end of the half-shaft. The estimator is simple, it uses only signals that already exist on the electric car, and it introduces no additional sensor cost or wiring. The integrator is initialized at the moment of the mode transition so that the estimated torque starts from the correct steady-state value and no artificial transient is injected into the loop.
5.3 Controller tuning
The three gains are tuned on the reduced-order model and then verified on the full nonlinear simulation of the electric car. The integral gain is set to zero, because the feedforward path already guarantees zero steady-state error in braking torque and an integral term would only slow down the mode transition. The proportional gain sets the bandwidth of the correction loop, while the derivative gain sets the amount of virtual damping injected into the driveline. Increasing the derivative gain raises the effective damping ratio and flattens the resonance peak, but an excessively large derivative gain amplifies the noise present in the estimated torque and produces torque chatter. The final gains are listed in Table 7.
| Gain | Symbol | Value |
|---|---|---|
| Proportional gain | $K_P$ | 15.71 |
| Integral gain | $K_I$ | 0 |
| Derivative gain | $K_D$ | 0.79 |
6. Simulation Validation
6.1 Vehicle-level response
The effectiveness of the strategy is verified on the full nonlinear model of the electric car under the manoeuvre described in Section 1.4. During the acceleration phase from 1 s to 6 s, the controlled electric car reaches a slightly higher speed than the uncontrolled one: the maximum speed rises from 71.07 km/h to 71.66 km/h. During the regenerative phase from 6 s to 11 s, the controlled torque profile is smoother, and the vehicle speed therefore lags slightly behind the uncontrolled profile. At 11 s the uncontrolled electric car has decelerated to 10.17 km/h, whereas the controlled electric car has decelerated to 12.33 km/h. Comparing the slopes of the two speed traces over 6–11 s shows that the deceleration rates are essentially identical, so the intervention of the controller has almost no effect on the energy-recovery efficiency of the electric car. This is an important result: the improvement in comfort is obtained without a measurable penalty in regenerative energy.
The half-shaft speed difference is the clearest indicator of the torsional oscillation. Without control, the maximum speed difference reaches 268.31 rad/s; with control, it falls to 44.61 rad/s. This is a reduction of 83.4 %, and the half-shaft speed trace becomes visibly smoother and much less oscillatory. The vehicle longitudinal acceleration follows the same trend. Without control it rises sharply at the beginning of the transition, peaks at −9.58 m/s² and then decays with a superimposed oscillation. With control the acceleration profile is smooth and its fluctuation amplitude is greatly reduced. Evaluating the root-mean-square jerk over the 6–11 s window gives 51.59 m/s³ without control and 11.81 m/s³ with control, a reduction of 77.1 %. In terms of passenger perception, this corresponds to the difference between a distinct fore-aft shock and an almost imperceptible transition.
6.2 Torque-level response
The torque traces confirm the same conclusion. Without control, the half-shaft torque ripple rate reaches 353.8 % and the half-shaft torque continues to oscillate for about 1 s after the mode transition. With control, the machine torque changes smoothly and the torque oscillation disappears; the half-shaft torque ripple rate falls to 206.57 %, a reduction of 41.61 % with respect to the uncontrolled case. This result demonstrates that the proposed strategy attenuates the torque pulsation caused by the simultaneous action of gear backlash, shaft flexibility and the electromagnetic torque harmonics of the machine. The controller therefore acts on the multiple nonlinear excitation sources at once, rather than on a single isolated nonlinearity.
| Indicator | Without control | With control | Change |
|---|---|---|---|
| Half-shaft torque ripple rate | 353.8 % | 206.57 % | −41.61 % |
| Maximum half-shaft speed difference | 268.31 rad/s | 44.61 rad/s | −83.4 % |
| Root-mean-square jerk (6–11 s) | 51.59 m/s³ | 11.81 m/s³ | −77.1 % |
| Peak longitudinal acceleration | −9.58 m/s² | Smooth transition | Oscillation removed |
| Maximum speed during traction | 71.07 km/h | 71.66 km/h | +0.8 % |
| Speed at the end of braking | 10.17 km/h | 12.33 km/h | Deceleration slope unchanged |
6.3 Frequency-domain interpretation
The speed–frequency–amplitude representation of the half-shaft response before and after compensation provides the mechanistic explanation of the time-domain results. Before compensation, three sharp resonance peaks appear at 1.15 Hz, 16.79 Hz and 30.88 Hz, which are exactly the first three natural frequencies of the electric car driveline identified in Table 5. After compensation, these three peaks are flattened and the amplitude response in their vicinity approaches zero. The controller has therefore added enough equivalent damping to move the operating point of the electric car away from the resonance band, and the driveline no longer amplifies the excitation that reaches it. Because the damping is created electronically, the effect is obtained without any change to the shaft diameters, the gear geometry or the bearing preload, and it can be retuned for a different load condition or a different electric car variant purely in software.
6.4 Robustness considerations
Three practical aspects deserve attention. First, the estimator relies on the difference between the machine speed divided by the total ratio and the wheel speed; both signals are available at high resolution on any production electric car, and the differentiation required for the damping term is performed on a filtered signal to avoid noise amplification. Second, the derivative gain is deliberately kept moderate, as noted above, so that the compensator remains stable when the half-shaft stiffness changes with temperature or with the wear of the joints. Third, because the compensator is placed in parallel with the base torque command, a fault in the compensation path can be detected by comparing the commanded torque with the estimated torque, and the electric car can revert to the uncompensated regenerative strategy, which remains safe although less comfortable. These features make the strategy compatible with the functional-safety requirements that apply to the braking function of a road-going electric car.
7. Conclusions
I have investigated the comfort degradation that occurs in an electric car when the driveline switches from traction to regenerative braking, and I have proposed and validated an active compensation strategy that removes the problem without modifying the mechanical architecture. The main conclusions are as follows.
1) A detailed electromechanical model of the electric car driveline was constructed, comprising the permanent-magnet synchronous machine, the two-stage helical gearbox and the longitudinal vehicle dynamics, and retaining the machine torque harmonics, the gear backlash and the shaft flexibility as explicit nonlinearities. The model shows that the 24th and 48th torque harmonics of the machine coincide with the natural frequencies of the electric car at 4 171.95 r/min and 10 874.83 r/min for the 24th order, and at 2 085.96 r/min, 5 437.41 r/min and 10 865.12 r/min for the 48th order, several of which lie inside the normal operating range. Increasing backlash raises both the impact amplitude and the response delay. Increasing the half-shaft damping from 10 to 200 N·m·s/rad reduces the torque ripple rate from 290.50 % to 91.24 %, a reduction of 68.59 %, and shortens the decay time from 1.29 s to 0.28 s, confirming that damping is the decisive parameter.
2) The mechanism by which the nonlinearities degrade comfort was identified. The machine torque harmonics, the gear backlash and the shaft flexibility act together, and during the regenerative transition the backlash lost motion delays the torque response while the shaft flexibility produces a low-frequency torsional oscillation that persists because of the very weak damping of the driveline. The reduced-order analysis shows that the electric car driveline is an under-damped system with a damping ratio of about 0.044, and this is the fundamental reason for the comfort deterioration.
3) A PID damping compensation strategy based on the estimated half-shaft torque was proposed and validated. Using a feedforward–feedback architecture, the controller raises the equivalent damping ratio of the electric car without altering any structural parameter, and it suppresses several nonlinear excitation mechanisms simultaneously. In the mode-switching window from 6 s to 11 s, the half-shaft torque ripple rate falls from 353.8 % to 206.57 %, a reduction of 41.61 %; the maximum half-shaft speed difference falls from 268.31 rad/s to 44.61 rad/s, a reduction of 83.4 %; and the root-mean-square jerk falls from 51.59 m/s³ to 11.81 m/s³, a reduction of 77.1 %. The longitudinal acceleration changes from a violent fluctuation with a peak of −9.58 m/s² into a smooth transition. The resonance peaks at 1.15 Hz, 16.79 Hz and 30.88 Hz are flattened to near zero, and the deceleration slope during regenerative braking is unchanged, so the energy-recovery performance of the electric car is preserved.
The broader implication is that the comfort of an electric car during regenerative braking is governed not only by the traction machine but by the interaction between the machine, the gearbox and the shafts. Treating the electric car driveline as an under-damped electromechanical oscillator, and damping it electronically rather than mechanically, offers a low-cost route to a smoother and more refined braking feel. Future work will extend the strategy to blended hydraulic–regenerative braking, where the coordination between the friction brake and the machine introduces an additional switching nonlinearity, and to the case of a distributed-drive electric car in which each wheel is driven by an independent machine and the torsional modes of the individual half-shafts must be damped without exciting the vehicle-level modes.