#### 4.1. Suspension Performance

Firstly, the frequency response of the MR regenerative suspension system is analyzed. For nonlinear systems, constant amplitude harmonic excitation will cause a large acceleration excitation signal at high frequencies and saturation of suspension components. Therefore, the segmented harmonic excitation signal shown in Equation (11) is used to meet the requirement of limiting the amplitude of the displacement harmonic signal at high frequencies.

where

${a}_{m}$ is the single-frequency harmonic amplitude.

${f}_{T}$ is the turning frequency. The natural frequency band of the car body and wheels is covered, setting

${a}_{m}=2\mathrm{cm}$,

${f}_{T}=2.1\mathrm{Hz}$.

Using the system frequency domain response about sprung mass acceleration transmission rate

T_{as}, unsprung mass acceleration transmission rate

T_{au}, suspension dynamic travel transmission rate

T_{dr}, and tire’s dynamic load coefficient (DLC), the overall suspension performance of vehicle suspension ride comfort and operational stability is evaluated, including resonance suppression, vibration isolation, suspension travel limit, and tire grip [

11]. It can be seen from

Figure 4 that the first resonance point of

T_{as} is 1.5 Hz, which meets the requirements of passenger car stability.

T_{au} is slightly distorted due to simulation reasons. Both

T_{dr} and DLC show that the TPMLM does not significantly change the handling performance of MR suspension.

Vehicles often encounter various impact roads during driving, which is a short time and high intensity. The smooth pulse signal excitation shown in Equation (12) is used [

14].

In the equation, ${a}_{m}$ is the amplitude of the smooth pulse signal, ${\omega}_{0}$is the fundamental frequency, and $\mu $ is the pulse stiffness coefficient. A larger $\mu $ value will cause the excitation to produce larger shock and acceleration signals.

Figure 5 shows the time-domain transient response of

${a}_{s}$,

${a}_{u}$,

${x}_{r}$, and

${F}_{t}$ under smooth pulse excitation. The stabilization time of each response adjustment of the system is within 1 s. Due to the addition of TPMLM, small-scale oscillations appear in the initial stage of impulsive excitation, especially the sprung mass acceleration

${a}_{s}$, but the response amplitude does not increase significantly.

The random road is established based on the filtered white noise method, which is used to simulate the wheel excitation input when the vehicle is running on the actual road and gives a general analysis of the performance of the suspension system. The differential equation of road roughness is

where

$q\left(t\right)$ is the function of road roughness.

$W\left(t\right)$ is the Gaussian white noise signal with power spectral density (PSD) of 1.

${n}_{00}$ is the cutoff frequency of the road spatial frequency,

${n}_{00}=0.011{\text{}\mathrm{m}}^{-1}$,

${n}_{0}$ is the reference spatial frequency of the road.

${n}_{0}=0.1{\mathrm{m}}^{-1}$.

${G}_{q}\left({n}_{0}\right)$is the PSD of the road under the reference spatial frequency

${n}_{0}$, related to pavement classification and speed

$u$ in steady-state. This experiment selects

$u=20\mathrm{m}/\mathrm{s}$,

${G}_{q}\left({n}_{0}\right)=64\times {10}^{-6}{\text{}\mathrm{m}}^{3}$.

Figure 6 shows

${a}_{u}$,

${x}_{r},$ and

${F}_{t}$ under random road excitation. It can be seen that the first resonance point of the MR regenerative suspension system is 1.5 Hz, and the second resonance point is about 9.5 Hz, which is consistent with the transmission rate under the above variable amplitude harmonic excitation. It is also indicated that the addition of TPMLM does not significantly change the system stability.

#### 4.2. Regenerative Characteristics

Under the single-frequency harmonic road excitation in Equation (14), above smooth road excitation, and random road excitation, the output regeneration characteristics are observed. The average output power

${\overline{P}}_{m}$ and the effective power value

P are employed as the evaluation indicators, as shown in Equations (15) and (16).

where

${a}_{m}$ is excitation amplitude,

$f$ is frequency. The typical frequencies observed here are 1.5 Hz near the resonant frequency of the sprung mass, 6 Hz at the intermediate frequency, and 15 Hz near the resonant frequency of the unsprung mass.

The average output power of the regenerative motor

The instantaneous output power is shown in

Figure 7,

Figure 8 and

Figure 9.

Table 2 lists the corresponding average output power and power effective value. It can be seen that the effective value of the output power at 1.5 Hz under harmonic excitation reaches about 500 W. As the frequency is increased, the regenerated power is greatly increased. On the impact road, the regenerative time is very short, but the effective value of output power is also about 50 W. On a general road, the recovered power is only 1.5 W on a flat A-class road. With the increase of road unevenness, the output power increases observably, which reaches about 46 W on a C-class road. The above results show that under the excitation of typical frequency harmonics, the MR regenerative suspension has a considerable performance of power regeneration.