3.2.1. Frequency Response Matrix

When analyzing the vibration of a vehicle model, the general work to be done is to find the vibration transfer function, which can characterize the amplitude and phase of the vehicle model under different frequency excitations. The vibration transfer matrix of the HIIRS-equipped half-car model is solved as follows.

When the road roughness is used as input, Equation (10) can be obtained as

$$[s^2M + s\overline{\mathcal{C}}(s) + \mathcal{K}]Y(s) = F\_x(s) \tag{41}$$

where *Fx*(*s*) = → *F*(*s*) → *ξ* (*s*) is the force exerted by the road on the tire, → *ξ* (*s*) = [*ξ<sup>l</sup>* , *ξ<sup>r</sup>* , 0, 0] *T* is the road excitation. → *F* is a 4 × 4 matrix whose elements are zero except for the first two diagonals, which are → *F*11(s) = *ktl* + s*ctl* and → *F*22(*s*) = *ktr* + s*ctr*.

Equation (41) can be written as

$$B(s)Y(s) = \overline{F}(s)\overline{\xi}(s) \tag{42}$$

Now, the frequency response matrix of the half-car system can be defined as

$$H\_{\mathcal{Y}}(s) = \frac{\mathcal{Y}(s)}{\overline{\mathcal{Y}}(s)} = \mathcal{B}^{-1}(s)\overline{\mathcal{F}}(s) \tag{43}$$

With *s = jω*, the frequency response matrix describes the system displacement response to any excitations. Therefore, as long as the excitation frequency is known, the HIIRS system equation can be completely determined, and the vibration analysis can be carried out just like other linear vehicle models.

#### 3.2.2. The Response to Random Road Excitation

Frequency domain analysis, similar to time domain analysis, is an important method for studying vehicle system vibration. In the frequency domain analysis map, the independent variable is the frequency and the dependent variable is the amplitude of the signal to be analyzed. In Section 2, the model of the mechanical hydraulic coupling system is established. Now, the input spectrum of the road is necessary to obtain the response of the HIIRS system to the excitation of the road.

Road input is the source of the vehicle vibration system. Whether the impact of the vehicle on the road can be obtained, obtaining accurate road information is the key. In the frequency domain, the power spectral density function is generally used to describe the vibration of the random vibration system. The road power spectrum density function is mainly used in vehicle dynamic response, optimal control of suspension, calculation of road load, etc. The vibration response of the vehicle can be evaluated through the road roughness power spectrum and the dynamic characteristics of the vehicle system. If the frequency response of the suspension is waiting to be solved on the basis of the above transfer function, a model of the road input is also needed. Since the car is excited by the unevenness of the road through the tire contacting the ground, it can be known from the random vibration theory that its vibration response is a smooth random vibration. The research object of this paper is a half-car model. The dynamic response characteristics can be obtained by determining the road input spectrum of the left and right wheels.

Suppose that the spatial frequency n represents the road self-power spectral density, *Sq*(*n*) and *n* = <sup>1</sup> *λ* holds. When the vehicle is driving on the road at speed *u*, it has

$$f = \mathfrak{u}n\tag{44}$$

where *f* is the time frequency. When the vehicle speed does not change, the time-domain frequency bandwidth ∆*f* has the following relationship with the corresponding spatial domain frequency bandwidth ∆*n*

$$
\Delta f = \mu \cdot \Delta n \tag{45}
$$

The power spectral density *Sq*(*n*) at the frequency in the spatial domain can be expressed as

$$S\_q(n) = \lim\_{\Delta n \to 0} \frac{\sigma\_{q \sim \Delta n}^2}{\Delta n} \tag{46}$$

where *σ* 2 *q*∼∆*n* is the energy of the road power spectrum in the frequency domain bandwidth ∆*n*. When the vehicle speed is constant, the harmonic components of the road roughness displacement contained in the time band ∆*f* corresponding to the spatial frequency band ∆*n* are the same, so the road power spectral density in the time domain is

$$\mathcal{S}\_{q}(f) = \lim\_{\Delta f \to 0} \frac{\sigma\_{q \sim \Delta n}^{2}}{\Delta f} \tag{47}$$

By Equations (44)–(47), *Sq*(*f*) can be written as

$$\mathcal{S}\_{\eta}(f) = \frac{1}{u} \mathcal{S}\_{\eta}(n) \tag{48}$$

Under normal circumstances, *Sq*(*n*) can be directly calculated. The selection of *n* in the equation is related to the speed and frequency of the driving vehicle. If the condition that 5 m/s < *u*< 50 m/s and 0.5 Hz < *f* < 50 Hz are satisfied at the same time, there is 0.01*m*−<sup>1</sup> < *n* < 10*m*−<sup>1</sup> .

It can be seen from the experiment that *Sq*(*n*) can be written as

$$S\_{\emptyset}(n) = c n^{-2w} \tag{49}$$

where w is a coefficient ranging from 1–1.25 and is generally taken as 1. The value of *c* is related to the road surface level, shown in Table 4.

**Table 4.** The value of c for various road surface level.


When the frequency index is taken as 2, the road power spectral density *Gq*(*f*) satisfies

$$\mathcal{G}\_q(f) = \mathcal{G}\_q(n\_0) n\_0^2 \frac{u}{f^2} \tag{50}$$

where the reference spatial frequency *n*<sup>0</sup> is taken as 0.1. According to Equations (49) and (50), we can obtain

$$S\_q(f) = c \frac{u}{f^2} \tag{51}$$

Considering the actual coherence of the left and right wheels, the road input spectral density matrix of the half-vehicle model is

$$\mathbf{S} = \begin{bmatrix} \mathbf{S}\_D & \mathbf{S}\_X \\ \mathbf{S}\_X & \mathbf{S}\_D \end{bmatrix} \tag{52}$$

where *S<sup>D</sup>* is the self-power spectral density of the road excitation

$$S\_D = \frac{1}{\mathfrak{u}} S\_\mathfrak{q}(n) = \frac{1}{\mathfrak{u}} \mathfrak{c} n^{-2w} \tag{53}$$

The road cross power spectral density *S<sup>X</sup>* is

$$S\_X = \frac{1}{\mu} \left[ 2 \text{c} \left( \frac{\pi L}{n} \right)^w / \Gamma(\mathbf{w}) \right] l\_w(2\pi Ln) \tag{54}$$

where *L* is the left and right wheel track, *J<sup>w</sup>* is the second-class modified Bessel function of order *w*, and Γ(w) is the gamma function.

When the left and right wheels are excited by the road roughness transfer function, the relationship between the response spectrum and the input spectrum in the frequency domain is as

$$\mathcal{S}\_i(f) = [H\_{i1}^\* \ H\_{i2}^\*] \mathcal{S} \begin{bmatrix} H\_{i1} \\ H\_{i2} \end{bmatrix} \tag{55}$$

where \* represents the conjugate complex number, *Si*(*f*) represents the power spectrum of the *i*-th output, *Hi*<sup>1</sup> and *Hi*<sup>2</sup> , respectively, represent the transfer function of the *i*-th output to the first and second inputs.

When *<sup>c</sup>* <sup>=</sup> <sup>64</sup> <sup>×</sup> <sup>10</sup>−<sup>8</sup> (Class-B road) and the vehicle speed is 36, 72 and 108 Km/h, the power spectrum density of bounce acceleration and the power spectrum density of roll acceleration of the HIIRS system are shown, respectively, in Figure 7.

**Figure 7.** The power spectrum density of the acceleration of an HIIRS-equipped vehicle when driven at 36, 72 and 108 km/h on Class-B road.

When <sup>c</sup> <sup>=</sup> <sup>256</sup> <sup>×</sup> <sup>10</sup>−<sup>8</sup> (C-level road) and the vehicle speed is 36, 72 and 108 km/h, the power spectrum density of bounce acceleration and the power spectrum density of roll acceleration of the HIIRS system are shown, respectively, in Figure 8.

**Figure 8.** The power spectrum density of the bounce/roll acceleration of an HIIRS-equipped vehicle when driven at 36, 72 and 108 km/h on Class-C road.

Figures 7 and 8 show the power spectrum density of the bounced acceleration and roll acceleration of the vehicle body. The natural frequency and root mean square (RMS) of the acceleration response under Class-B and Class-C roads are in Table 5. It indicates that the natural frequency of bounce is around 1.5 Hz, while the roll frequency is around 2.19 Hz. In addition, an increase in speed on the same road will lead to an increase in

the acceleration power spectrum density. At the same speed, an improvement in road conditions will reduce the acceleration power spectrum.


**Table 5.** The natural frequency and RMS acceleration response of the HIIRS-equipped vehicle.

No matter what kind of road surface excitation, the HIIRS system has the same modal law: no matter the speed is low, medium or high, the natural frequency of the roll mode always maintains at around 2.19 Hz, and the natural frequency of the bounce mode varies when the vehicle speed increases. The natural frequency is 1.48 Hz at low speed, 1.50 Hz at medium speed and 1.51 Hz at high speed. This indicates that when the vehicle speed increases, the HIIRS system can provide greater rigidity. In terms of amplitude, the RMS bounce acceleration at medium speed increased by 143.66% compared with that at low speed, while the RMS bounce acceleration at high speed increased by 63.54% compared with that at medium speed. The RMS roll acceleration at medium speed increased by 149.59% compared with that at low speed, while the RMS roll acceleration at high speed increased by 67.63% compared with that at medium speed. From this point of view, it can be seen that as the vehicle speed increases, the RMS acceleration also increases, but the amount of increase at high speed is less than that at low speed. To a certain extent, it indicates that the anti-bounce and anti-roll capabilities of the HIIRS system are more greatly enhanced at high speeds.

Figure 9 indicates that the acceleration power spectral density curve of the HIIRS system has the same trend with the traditional suspension. The difference is that the peak value of the HIIRS system is much lower than that of the traditional suspension. Additionally, the improvement is more noticeable in the vertical acceleration than the roll acceleration.

**Figure 9.** Comparison of bounce/roll acceleration between the HIIRS-equipped vehicle and the traditional vehicle when driven at 36, 72 and 108 km/h on Class-B road.

Figure 9 compared the power spectrum density of bounce/roll acceleration between the HIIRS-equipped vehicle and the traditional vehicle, when the vehicles are driven on a Class-B road (c <sup>=</sup> <sup>64</sup> <sup>×</sup> <sup>10</sup>−<sup>8</sup> ). In the traditional vehicle, the stiffness and damping are, respectively, 80000 and 4400 Ns/m. The corresponding results are also summarized in Table 6.

**Table 6.** The comparison of the natural frequency and RMS bounce/roll acceleration between the HIIRS system and traditional suspension system.


From the natural frequency aspect, the roll natural frequency of the HIIRS system and traditional suspension system are maintained at 2.19 and 2.04 Hz, respectively, while the bounce natural frequency of HIIRS ranges from 1.48 to 1.51 HZ and that of traditional suspension is about 1.48 HZ. This indicates that when the road surface and speed are the same, the HIIRS system can provide greater stiffness, especially when the vehicle rolls, since the roll natural frequency of the HIIRS system is higher than that of the traditional suspension.

From the amplitude aspect, Figure 9 shows that the response of the HIIRS is lower than the traditional suspension in the whole frequency range, and the HIIRS can greatly reduce the peak value of the responses. Table 6 shows the RMS of bounce and roll acceleration of the HIIRS system can, respectively, reach 64.91 and 12.38% lower than the traditional suspension when the vehicle is driven at 72 km/h on Class-B road. That is, the HIIRS system provides superior ride comfort and handling stability to traditional suspension systems.

#### **4. Energy Harvesting Power of the HIIRS**

Section 3 demonstrated the vibration characteristics of the HIIRS, and this section focuses on the energy harvesting characteristics.

In the energy harvesting circuit, the relationship between the various physical quantities can be written as 

$$\begin{array}{c} P = I^2 R\_{\varepsilon} \\ I = \frac{\mathcal{U}\_{\text{emf}}}{\mathcal{R}\_{\varepsilon} + \mathcal{R}\_{\text{in}}} \\ \mathcal{U}\_{\text{emf}} = k\_{\varepsilon} \omega \\ \omega = 2\pi \frac{\mathcal{Q}\_{\text{M}}}{\mathcal{Q}\_{\text{m}}} \eta\_{\text{v}} \end{array} \tag{56}$$

where *P* is the energy harvesting power, *R<sup>e</sup>* is the external resistance, *I* is the current, *Uem f* is the induced electromotive force, *Rin* is the circuit resistance, *k<sup>e</sup>* is the speed constant of the generator, *Q<sup>M</sup>* is the motor inlet flow, *Q<sup>m</sup>* is the motor displacement, *η<sup>v</sup>* is the volumetric efficiency.

In Equation (56), only *Q<sup>M</sup>* is the unknown quantity, and thus, the goal is to solve *QM*.

According to Figure 4 and Equation (15), the state quantity (pressure, flow) of the motor inlet is

$$T\_{X6} = T\_{X5 \to X6} T\_{X4 \to X5} T\_{X3 \to X4} T\_{X2 \to X3} T\_{X1 \cdot X2} T\_{X1} \tag{57}$$

State quantity at node 1 is *TX*<sup>1</sup> = *P*1 *Q*<sup>1</sup> , *P*<sup>1</sup> and *Q*<sup>1</sup> can be calculated according to Equations (7) and (8).

According to Equation (56), the transfer function *HX*<sup>61</sup> between the motor inlet flow rate and *TX*<sup>1</sup> can be known. Equations (7) and (8) provide the transfer function *HX*1*<sup>Y</sup>*

between *TX*<sup>1</sup> and the displacement vector *Y*. Therefore, the transfer function between the motor inlet flow and the displacement vector *Y* in Equation (1) is

$$H\_{Q\_M Y} = H X\_{61} \cdot H X\_{1Y} \tag{58}$$

Then, the power spectral density of the flow rate at the motor inlet, *GQ<sup>M</sup>* , can be calculated as

$$G\_{\mathcal{Q}\_M} = H\_{\mathcal{Q}\_M Y}^2 G\_Y \tag{59}$$

where *G<sup>Y</sup>* is the power spectral density of Y.

The power spectral density has the following relationship with the amplitude

$$G\_{\mathbb{Q}\_M} = A\_{\mathbb{Q}\_M}^2 / f\_s \tag{60}$$

where *AQ<sup>M</sup>* represents the amplitude of *AQ<sup>M</sup>* , and *f<sup>s</sup>* represents the frequency bandwidth.

In Section 3, we obtained the power spectral density of bounce and roll acceleration under random road input and the frequency response matrix of the displacement vector *Y*. If the method of solving acceleration power spectral density is extended to displacement, with the known power spectral density of the road displacement vector *Y*, the power spectral density of the flow rate at the motor inlet under various random road input can be obtained. Then, the corresponding amplitude can be solved with Equation (60), and the time domain flow rate at the motor inlet can be solved by performing inverse Fourier transform. The time domain flow rate is then substituted into Equation (56), and the energy harvesting power can be determined.

The energy regenerative power of a half-car with HIIRS system on a Class-C road at the speed of 36, 72 and 108 km/h is shown in Figure 10. The average value of energy regenerative power at this road surface excitation is shown in Table 7. It shows that the energy harvesting power can reach 655.90 W for an off-road vehicle when it is driven on a Class-C road at 108 km/h.

**Figure 10.** The energy harvesting power of the HIIRS-equipped half-vehicle when driven at various speeds on Class-C road.

**Table 7.** Energy harvesting power.


## **5. Conclusions**

This paper studied the vibration isolation and energy harvesting characteristics of a novel hydraulic integrated interconnected regenerative suspension (HIIRS). The model in the frequency domain was established. Both free and forced vibration analysis were carried out and compared with a traditional suspension. The comparison showed that the RMS bounce and roll acceleration of the HIIRS system was, respectively, 64.91 and 12.38% lower than the traditional suspension when the vehicle was driven at 72 km/h on a Class-B road. With the frequency domain model of the HIIRS, an approach for calculating the energy harvesting power was also presented. The calculated energy harvesting power was 186.93, 416.40 and 656.90 W, when the vehicle speed was 36, 72 and 108 km/h. In summary, the HIIRS system can significantly enhance the vehicle ride comfort and handling stability while harvesting vibration energy to achieve an energy-saving purpose.

**Author Contributions:** Conceptualization, S.G.; Funding acquisition, S.G.; Investigation, L.C.; Methodology, S.G. and J.Z.; Software, L.C.; Supervision, S.G. and S.H.; Validation, X.W. and S.H.; Writing—original draft, S.G. and L.C.; Writing—review & editing, S.G., J.Z. and S.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by National Natural Science Foundation of China (51905394) and Hubei Natural Science Foundation of China (2019CFB202).

**Acknowledgments:** The authors are grateful to the support from Hubei Key Laboratory of Advanced Technology for Automotive Components, Hubei Collaborative Innovation Centre for Automotive Components Technology.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript or in the decision to publish the results.

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