Optimizing Hydro-Pneumatic Inerter Suspension for Improved Ride Comfort and Handling Stability in Engineering Vehicles Using Simulated Annealing Algorithm

Abstract

This study introduces a novel hydro-pneumatic inerter suspension (HPIS) system for engineering vehicles, aiming at enhancing ride comfort and handling stability. The research focuses on addressing the limitations of conventional suspension systems by incorporating an inerter element into the vehicle suspension. The unique aspects of HPIS, such as nonlinear stiffness and nonlinear damping characteristics of the hydro-pneumatic spring, are explored. Firstly, a half-car dynamic model of the HPIS suspension is established, and an improved simulated annealing algorithm is applied to optimize the suspension parameters. Then, we compare the dynamic performance of different HPIS structures, specifically parallel and series layouts. For practical analysis, a simplified three-element HPIS suspension model is used, and the suspension parameters are optimized by a simulated annealing algorithm at speeds of 10 m/s, 15 m/s, and 20 m/s. Key findings reveal that compared to the traditional suspension system of S0, the front and rear suspension working space of S1 decreased by 40%, 40.1%, 40.2% and 30.7%, 30.8%, 30.9%, while with the body acceleration and pitch acceleration deteriorated by 3.1%, 3.2%, 3.3% and 63.4%, 63.8%, 64.0%. However, the S2 can improve all the dynamic performance and offer better ride comfort and handling stability.

1. Introduction

The vehicle suspension, as a vital component of the vehicle chassis system, significantly influences the ride comfort, handling safety, and dynamic tire load [1]. Among the various types of suspension systems, hydro-pneumatic suspension stands out, and it combines hydraulic and pneumatic technologies to create a mature vehicle suspension mechanism [2]. In these systems, an inert gas acts as the elastic medium in the hydro-pneumatic springs, while oil is used for pressure transmission. This design takes advantage of the compressibility of gases and the inherent damping properties of hydraulic systems, ensuring the vehicle’s adaptability to various operational conditions [3]. Notably, hydro-pneumatic suspension autonomously adjusts the vehicle body height, adeptly handling variations from no-load to full-load states, thereby optimizing driving performance [4]. Hydro-pneumatic springs are mainly employed in heavy-duty vehicles and those operating under challenging conditions, such as engineering machinery, military vehicles, high-end automobiles, and racing cars [5,6]. Consequently, the application and optimization of hydro-pneumatic suspension have been fervently explored within scholarly discourse. Yang [7] employed an electro-hydraulic proportional valve to variably adjust the suspension system’s damping force, facilitating adaptable damping control. Jiao [8] created a reliability model for hydro-pneumatic suspension systems, addressing the challenges posed by environmental temperature changes and transient impacts. Zhao [9] combined integrating hydro-pneumatic suspension with mechanical elastic wheels to enhance vehicle ride comfort while preserving the advantages of non-pneumatic and anti-puncture wheel technology.

The advent of the inerter has revolutionized the design process of the traditional suspension system [10,11,12]. This innovation enhances the electromechanical analogy theory by introducing mechanical analogs to electrical capacitance, leading to the development of the ‘inerter-spring-damper’ (ISD) suspension system [13,14]. Research indicates that the ISD suspension exhibits superior vibration isolation [15,16,17]. Shen [18] employed a fractional-order electrical network to further enhance the vibration performance of ISD suspension. Baduidana [19] focused on optimizing inerter-based systems for vibration isolation and demonstrated that the systems significantly outperform traditional dynamic vibration absorbers in minimizing compliance and mobility transfer functions. In addition, the ISD suspension system integrates and significantly enhances two key functionalities, including ride comfort and handling stability. It not only elevates driving safety but also adapts to a diverse range of road conditions. The unique design of the ISD suspension system effectively minimizes vehicular vibrations across various road surfaces. This optimization is evident not only in the improved dynamic response of the vehicle but also in its capability to adapt to complex road conditions, underscoring the significant value and potential applications of the ISD suspension in the field of modern vehicle suspension technology [20,21]. Nie [22] presented a two-stage ISD structure suspension, and the results showed that it can restrain the vibrations of sprung and unsprung mass and improve ride comfort as well as road friendliness. Wang [23] investigated the use of adjustable inertance to enhance performance of suspension working space and dynamic tire load, while the performance in reducing vehicle body acceleration varies. Hu [24] explored the use of a skyhook inerter configuration in vehicle suspension design for enhanced ride comfort, with varying effectiveness depending on the suspension system’s static stiffness. However, evolving performance demands are steering suspension components from linear, fixed-characteristic models towards nonlinear variants [25,26], making nonlinear inerter and ISD suspensions a focal point of contemporary research.

Given the widespread application and proven effectiveness of hydro-pneumatic suspension technology [27,28,29,30,31], the integration of hydro-pneumatic suspension with inerter-spring-damper (ISD) systems presents a novel exploration. This integration is particularly significant in light of the nonlinear characteristics inherent in hydro-pneumatic suspension’s stiffness and damping. By combining a hydro-pneumatic spring with the ISD system, a new hybrid system emerges—the hydro-pneumatic inerter suspension (HPIS). This innovative HPIS aims to enhance vehicle performance by offering improved isolation from road inputs and better control over vibrations of both sprung and unsprung masses. The primary focus of HPIS is to deliver superior ride comfort and handling stability to a variety of road conditions, which is especially beneficial for engineering vehicles. The integration allows for more effective management of vehicle dynamics, ensuring smoother rides and enhanced handling stability. The concept of HPIS not only represents a significant leap in suspension technology but also opens up new possibilities for vehicle design, catering to the evolving demands for higher performance and comfort in modern transportation. The content layout of this paper is as follows.

Section 2 first introduces the selection of HPIS and analyzes its nonlinear stiffness and damping expression. In the third section, the half dynamic model of parallel and series HPIS is established, and the appropriate road excitation is selected. Then, in the fourth section, the performance parameters of ISD suspension models with two structures are optimized by the simulated annealing algorithm. Finally, in the fifth section, the time domain simulation results of the optimized HPIS are compared and evaluated, and the conclusion is drawn in the sixth section.

2. Selection of Nonlinear Hydro-Pneumatic Spring

Among the types of hydro-pneumatic suspension, the double-chamber and two-stage air chamber configuration offers a nuanced control mechanism and potentially broader adjustability. However, the single-chamber system possesses unique advantages that warrant consideration. One of the most compelling merits of the single-chamber design lies in its inherent simplicity. Compared to its double-chamber counterpart, it involves fewer components and connections, resulting in a significantly reduced likelihood of mechanical failures. This inherent simplicity not only elevates the system’s reliability and longevity but also enables it to respond swiftly to sudden changes in terrain or driving conditions. This rapid responsiveness contributes to a vehicle that excels in handling unpredictable situations and offers a smoother ride experience. For the scope of this research, our primary focus is on the single-chamber hydro-pneumatic spring. Its operational characteristics and advantages make it a suitable candidate for in-depth study and application in modern vehicles. A single-chamber hydro-pneumatic spring is selected, as shown in Figure 1.

2.1. Working Principle of Hydro-Pneumatic Suspension

The single-chamber hydro-pneumatic spring uses a floating piston to divide the working fluid and the gas chamber [3], ensuring no direct interaction between the two mediums. Upon the application of an external force, such as the dynamic loads encountered during vehicle motion, the piston is actuated upwards within chamber I, resulting in the displacement of hydraulic oil. This displacement escalates the hydraulic pressure within chamber I, labeled as P₁. The augmentation of pressure P₁ triggers the opening of the hydraulic valve, which in turn channels the hydraulic oil into chamber II. As chamber II becomes filled with hydraulic oil, the pressure within this chamber, labeled as P₂, correspondingly rises. The increased pressure P₂ exerts a downward force on the floating piston, thereby compressing the gas within chamber III, and as a result, the pressure within the gas chamber escalates. Subsequently, as the external force diminishes, the piston begins to retract downwards, causing a decrease in pressure P₁ within chamber I. The hydraulic valve responds to this change by allowing the hydraulic oil to exit chamber II. The decrement in P₂ results in the expansion of the compressed gas above, which imparts an upward force on the floating piston. This expansion within chamber III facilitates the hydraulic oil’s movement from chamber II back to chamber I. This system operates cyclically to absorb and dissipate kinetic energy from road irregularities, providing a smoother ride. Additionally, the assembly may be equipped with a height indicator to monitor and regulate the neutral position of the suspension system.

2.2. Analysis of Nonlinear Stiffness Characteristics

Assuming the cylinder body of the hydro-pneumatic suspension is fixed, let the displacement of the piston be x, with the compression stroke considered positive and the static equilibrium position as the origin. The force acting on the piston is denoted as F.

When the piston is in static equilibrium, its static equilibrium equation is:

$$F=P_1{A}_1$$

(1)

$$P_1=P_2=P_g$$

(2)

where P₁ is the chamber Ⅰ hydraulic pressure, P₂ is the chamber Ⅱ hydraulic pressure, P_g is the pressure at any time in chamber Ⅲ, A₁ is area of chamber Ⅰ, and F is external force.

The relationship between its displacement x and the displacement h of the floating piston is as follows:

$$A_{1}x=A_{2}h$$

(3)

The thermodynamic equation of state for a gas in a gas chamber is as follows:

$$P_g{V}_t^r=P_0{V}_0^r$$

(4)

$$P_g{(H_0-h)}^r=P_0{H}_0^r$$

(5)

where H₀ is the converted height of gas at static equilibrium, V₀ is the volume of gas at static equilibrium, P₀ is the gas pressure at the static equilibrium position, r is the gas variability index (GVI), and V_t is the volume of gas at any time.

The expression for the nonlinear stiffness of the hydro-pneumatic suspension is derived by substituting $\frac{A_1}{A_2}=k$:

$$F_g=\frac{A_1{P}_0}{{(1-kx/H_0)}^r}$$

(6)

where Equation (7) describes the load characteristic of the hydrocarbon suspension; the load F is derived from the displacement x to obtain the spring stiffness k, and F_g is the air spring force.

$$k=\frac{dF}{dx}=\frac{A_1{P}_{0}kr/H_0}{{(1-kx/H_0)}^{r+1}}$$

(7)

Equation (7) represents the expression for the nonlinear stiffness of the hydro-pneumatic suspension.

2.3. Analysis of Nonlinear Damping Characteristics

In hydro-pneumatic suspensions, damping generation primarily arises from three sources: fluid damping as the working fluid passes through the damping orifices, fluid damping at the accumulator outlet (chamber II), and frictional damping due to the relative movement between the piston and the cylinder wall. Among these, frictional damping is relatively minor and does not significantly contribute to the damping and vibration reduction during the hydro-pneumatic suspension’s operation. In contrast, fluid damping created as the fluid traverses the damping orifices is the primary focus of research. The fluid’s flow through these orifices is entirely turbulent, a fact substantiated by extensive experimental work. The hydro-pneumatic suspension disc-type damper examined in this study is depicted in Figure 2, featuring damping orifices designed as elongated circular through-holes. This configuration should be viewed as turbulent oil flow in elongated circular holes. Therefore, we apply the Haigen–Poiseuille equation to formulate the relationship between pressure drop and flow rate, subsequently deriving the expression for damping force.

$$\Delta p=\frac{0.3164}{R_e^{0.25}}·\frac{L}{D}·\frac{\rho Q^2}{2A^2}$$

(8)

$$R_e=\overline{v}d/V$$

(9)

Combining the above two equations, we obtain

$$\Delta p=0.1582\frac{L}{D^{1.25}{A}^{1.75}}\rho v^{0.25}{Q}^{1.75}$$

(10)

where $\overline{v}$ is the average velocity of liquid flow, $\Delta p$ is the pressure drop of the orifices, $L$ is the length of the orifices, $Q$ is the flow rate, $D$ is the caliber of the orifices, $\rho$ is the density of the working fluid, $R_e$ is the Reynolds number, $V$ is the kinematic viscosity of the oil, $A$ is the throttle area of the orifices, and $d$ is the cross-sectional diameter of the flow area.

Representing the total flow of a damper with ‘n’ parallel orifices using ‘Q’, the relationship between the total pressure drop and the flow rate of the damper can be expressed as

$$\Delta P_{total}=0.1582\rho v^{0.25}{Q}^{1.75}{\left[{({\displaystyle \sum _{i=1}^n\frac{D_i^{1.25}{A}_i^{1.75}}{L_i}})}^{\frac{1}{1.75}}\right]}^{-1.75}sign(\dot{x})$$

(11)

The pressure drop in the connecting piping between chamber I and chamber II can be expressed as

$$\Delta p_{32}=0.1582\frac{L_{tube}}{D_{tube}^{1.25}{A}_{tube}^{1.75}}\rho v^{0.25}{Q}^{1.75}sign(\dot{x})$$

(12)

where $L_{tube}$ is the length of the connecting piping, $D_{tube}$ is the caliber of the connecting piping, $A_{tube}$ is the flow area, $\Delta p_{32}$ is the pressure drop of the connecting piping, and $sign(\dot{x})$ is the signal function.

$$\begin{array}{l}{F}_p=0.1582A_1^{2.75}\rho v^{0.25}{C}_d{\dot{x}}^{1.75}sign(\dot{x})\\ F_p=C_p{\dot{x}}^{1.75}\\ C_p=0.1582A_1^{2.75}\rho v^{0.25}{C}_{d}sign(\dot{x})\end{array}$$

(13)

where C_d is the coefficient of the throttle orifice and F_p is the damping force.

By combining the above equations and simplifying them, we obtain the piston rod force F_d.

$$F_d=\frac{A_1{P}_0}{{(1-kx/H_0)}^r}-0.1582A_1^{2.75}\rho {\nu }^{0.25}{C}_d{\dot{x}}^{1.75}sign(\dot{x})$$

(14)

The first term represents the nonlinear stiffness characteristics of the hydro-pneumatic suspension, and the second term represents the nonlinear damping characteristics of the hydro-pneumatic suspension, with the damping force proportional to the 1.75 power of the piston speed. From the derivation of the equations, it can be seen that the stiffness and damping of the hydro-pneumatic suspension have nonlinear characteristics.

The important structural and physical parameters of the hydro-pneumatic springs are shown in

Table 1. Structural parameters of hydro-pneumatic springs.

Parameters	Symbolic	Value
chamber I area (m2)	A1	5.8 × 10−3
floating piston area (m2)	A2	13.2 × 10−3
converted height of gas at static equilibrium (m)	H0	0.237
density of working fluid (kg/m3)	ρ	850
liquid kinematic viscosity 50 °C (mm2/s)	v	20
gas variability index	r	1.2
gas pressure static equilibrium position (Pa)	P0	9.04 × 106
throttle coefficient of throttle orifice	Cd	0.7
the displacement of the floating piston (m)	h	——

.

3. Half-Car Model of HPIS

The four-degree of freedom half-car model serves as a comprehensive representation, capturing not only alterations in the centroid acceleration and velocity of the vehicle body but also the variations in the pitch angular acceleration and angular velocity of the vehicle body around its centroid axis [32]. For the purposes of this study, a simplified three-element HPIS model is employed for simulation. Its structural complexity is minimal, rendering the simulation results highly indicative. Moreover, this study establishes two types of HPIS models, one configured in parallel and the other in series.

3.1. Half-Car Model

Figure 3 shows a half-vehicle dynamics model, where m₁ and m₂ are the front/rear wheel mass; Z_r and Z_r′ are the input of front/rear suspension pavement; Z_u and Z_u′ are the front/rear wheel axle displacement; Z_s and Z_s′ are the displacement of the car body in the front/rear suspension; Z₃ is the displacement at the center of mass of the car body; θ is the body pitch angle; k₁₁ and k₁₂ are the front/rear tire stiffness; k₂₁ and k₂₂ are the stiffness of the front/rear suspension hydro-pneumatic spring; a is the distance from the center of mass to the front end of the car; and b is the distance from the center of mass to the end of the car. S is the different structural forms of damping and inerter, where c₁ and c₂ are the front/rear suspension damping coefficients; The red modules represent distinct inertance, with b₁ and b₂ referring to the inertance of the front/rear suspension.

The layouts of this suspension system are shown in Figure 3. Among them, when S is S0, the suspension structure is a traditional suspension structure, and its dynamic model is as follows:

$$\left\{\begin{array}{l}{m}_1{\ddot{z}}_1=k_{21}(Z_s-Z_u)+c_1({\dot{Z}}_s-{\dot{Z}}_u)-k_{11}(Z_u-Z_r)\\ m_2{\ddot{z}}_2=k_{22}({Z_s}^{\prime }-{Z_u}^{\prime })+c_2({{\dot{Z}}_s}^{\prime }-{{\dot{Z}}_u}^{\prime })-k_{12}({Z_u}^{\prime }-{Z_r}^{\prime })\\ m_3{\ddot{z}}_3=-[k_{21}(Z_s-Z_u)+c_1({\dot{Z}}_s-{\dot{Z}}_u)+k_{22}({Z_s}^{\prime }-{Z_u}^{\prime })+c_2({{\dot{Z}}_s}^{\prime }-{{\dot{Z}}_u}^{\prime })]\\ J\ddot{\theta }=a[k_{21}(Z_s-Z_u)+c_1({\dot{Z}}_s-{\dot{Z}}_u)]-b[k_{22}({Z_s}^{\prime }-{Z_u}^{\prime })+c_2({{\dot{Z}}_s}^{\prime }-{{\dot{Z}}_u}^{\prime })]\end{array}\right.$$

(15)

$$k_{2j}=\frac{A_1{P}_{0}kr/H_0}{{(1-kx/H_0)}^{r+1}} j=1,2$$

(16)

where x, the displacement of the piston, can be substituted with the relative displacement of the body to the wheels.

$$\left\{\begin{array}{l}x={z_1}^{\prime }-z_1\hspace{1em}{k}_{21}\\ x={z_2}^{\prime }-z_2\hspace{1em}{k}_{22}\hspace{1em}\end{array}\right.$$

(17)

$$c_i=0.1582A_1^{2.75}\rho v^{0.25}{C}_{d}sign({\dot{x}}_i) i=1,2$$

(18)

where ρ is the density of the working fluid, v is the liquid kinematic viscosity, C_d is the throttle coefficient of the throttle orifice, and $sign({\dot{x}}_i)$ is the signal function

$$sign({\dot{x}}_i)=\left\{\begin{array}{ll}1& {\dot{x}}_i>1\\ 0& {\dot{x}}_i>0\\ -1& {\dot{x}}_i<1\end{array}\right.$$

(19)

where ${\dot{x}}_i$ can be substituted for the relative displacement of the body and wheel.

$$\left\{\begin{array}{l}{\dot{x}}_1={{\dot{z}}_1}^{\prime }-{\dot{z}}_1\\ {\dot{x}}_2={{\dot{z}}_2}^{\prime }-{\dot{z}}_2\end{array}\right.$$

(20)

When S is S1, the suspension structure is parallel HPIS, and its dynamic model is:

$$\left\{\begin{array}{l}{m}_1{\ddot{z}}_1=k_{21}(Z_s-Z_u)+c_1({\dot{Z}}_s-{\dot{Z}}_u)+b_1({\ddot{Z}}_s-{\ddot{Z}}_u)-k_{11}(Z_u-Z_r)\\ m_2{\ddot{z}}_2=k_{22}({Z_s}^{\prime }-{Z_u}^{\prime })+c_2({{\dot{Z}}_s}^{\prime }-{{\dot{Z}}_u}^{\prime })+b_2({{\ddot{Z}}_s}^{\prime }-{{\ddot{Z}}_u}^{\prime })-k_{12}({Z_u}^{\prime }-{Z_r}^{\prime })\\ m_3{\ddot{z}}_3=-[k_{21}(Z_s-Z_u)+c_1({\dot{Z}}_s-{\dot{Z}}_u)+b_1({\ddot{Z}}_s-{\ddot{Z}}_u)+k_{22}({Z_s}^{\prime }-{Z_u}^{\prime })+c_2({{\dot{Z}}_s}^{\prime }-{{\dot{Z}}_u}^{\prime })+b_2({{\ddot{Z}}_s}^{\prime }-{{\ddot{Z}}_u}^{\prime })]\\ J\ddot{\theta }=a[k_{21}(Z_s-Z_u)+c_1({\dot{Z}}_s-{\dot{Z}}_u)+b_1({\ddot{Z}}_s-{\ddot{Z}}_u)]-b[k_{22}({Z_s}^{\prime }-{Z_u}^{\prime })+c_2({{\dot{Z}}_s}^{\prime }-{{\dot{Z}}_u}^{\prime })+b_2({{\ddot{Z}}_s}^{\prime }-{{\ddot{Z}}_u}^{\prime })]\end{array}\right.$$

(21)

When S is S2, the suspension structure is series HPIS, and Z_b and Z_b′ are the vertical displacement of the front/rear suspension inerter. Based on the half-car mathematical model of HPIS and the related knowledge of vehicle dynamics, the dynamic equation of HPIS can be obtained as follows [33]:

$$\left\{\begin{array}{l}{m}_1{\ddot{Z}}_u=k_{21}(Z_s-Z_u)+u_1-k_{11}(Z_u-Z_r)\\ m_2{{\ddot{Z}}_u}^{\prime }=k_{22}({Z_s}^{\prime }-{Z_u}^{\prime })+u_2-k_{12}({Z_u}^{\prime }-{Z_r}^{\prime })\\ m_3{\ddot{Z}}_3=-(k_{21}(Z_s-Z_u)+u_1+k_{22}({Z_s}^{\prime }-{Z_u}^{\prime })+u_2)\\ J\ddot{\theta }=a(k_{21}(Z_s-Z_u)+u_1)-b(k_{22}({Z_s}^{\prime }-{Z_u}^{\prime })+u_2)\\ u_1=b_1({\ddot{Z}}_s-{\ddot{Z}}_s)=c_2({\dot{Z}}_s-{\dot{Z}}_u)\\ u_2=b_2({{\ddot{Z}}_s}^{\prime }-{\ddot{Z}}_b^{\prime })=c_2({{\dot{Z}}_b}^{\prime }-{\dot{Z}}_u)\end{array}\right.$$

(22)

where u₁ and u₂ are the forces between the inerter and the damping.

When the pitch angle is small, the following relationship can be approximated:

$$\left\{\begin{array}{l}{Z}_s=Z_3-a\ddot{\theta }\\ Z_s^{\prime }=Z_3+b\ddot{\theta }\end{array}\right.$$

(23)

3.2. Selection of Road Excitation

The random road input is selected as the road excitation model to study the advantages of the two structures of the HPIS relative to the traditional suspension [34]. In this paper, a road roughness model is built in Simulink (The MathWorks, Inc., Natick, Massachusetts, United States). The constructed road surface is a B-level road surface, and the speed u is set to 10 m/s. Then, the road input model is as follows: When S is S2, the suspension structure is series HPIS, and Z_b and Z_b’ are the vertical displacement of the front/rear suspension inerter. Assuming that the vehicle is travelling at a speed of 10 m/s on a class B road, the road input model used in the simulation is shown in Figure 4:

$${\dot{Z}}_r(t)=-0.111[u{Z}_r(t)+40\sqrt{G_q(n_0)u}\omega (t)]$$

(24)

where Z_r (t) is the vertical displacement of the random road input; ω (t) is the Gaussian white noise; and G_q (n₀) is the roughness of the road surface, which is 64 × 10⁻⁶ m³·cycle.

The time domain simulation of the model is as follows:

4. Suspension Optimization Analysis

Figure 5 shows the specific steps of the simulated annealing algorithm:

In this research, the simulated annealing algorithm is employed to optimize the design of the suspension system, and several improvements are proposed [35]. It has undergone critical enhancements to augment its efficiency and precision in complex optimization challenges. The key enhancements encompass the introduction of an adaptive temperature adjustment mechanism, which dynamically alters the cooling rate based on ongoing optimization feedback. Secondly, a multi-start strategy is employed to improve the likelihood of discovering global optima, and the integration of local search techniques aims to fine-tune solutions within promising regions. These improvements aim to enhance the effectiveness and accuracy of optimization algorithms. Thirdly, a memory feature has been incorporated to capitalize on previously found high-quality solutions, preventing redundant searches. Furthermore, parallel processing capabilities have been implemented, substantially enhancing the algorithm’s computational speed and efficacy, particularly in large-scale problem contexts. These collective enhancements significantly elevate the algorithm’s capability to navigate and solve intricate and variable optimization problems with greater effectiveness and accuracy. It is widely used in various constrained optimal problems. In addition, the initial temperature is set to 1000 degrees, the temperature drop rate is 0.98, and the initial value is set to the data as follows: $X=[980\times i,980\times j,2800\times p,2800\times q]$, where i, j, p, and q are integers of 1–3, respectively.

5. Optimization Results

To evaluate ride comfort and handling stability in a vehicle, the key factors include the dynamic load on the wheels, suspension deflection, pitch angular acceleration, and body acceleration [18]. The traditional suspension is selected as the evaluation benchmark, and the optimal objective function is established as follows:

$$\min f=\frac{qdtl(X)}{qdt{l}_{BD}}+\frac{hdtl(X)}{hdt{l}_{BD}}+\frac{qdnd(X)}{qdn{d}_{BD}}+\frac{hdnd(X)}{hdn{d}_{BD}}+\frac{\ddot{\theta }(X)}{{\ddot{\theta }}_{BD}}+\frac{BA(X)}{B{A}_{BD}}+pu$$

(25)

$$X=[\begin{array}{cccc}{c}_1& c_2& b_1& b_2\end{array}]$$

(26)

$$lb<X_i<ub i=1,2,3,4$$

(27)

In the formula, f is the optimal objective function, and qdtl(X), hdtl(X), qdnd(X), hdnd(X), $\ddot{\theta }$(X), and BA(X) are the root mean square values of the dynamic load of the front and rear wheels of the HPIS, the dynamic deflection of the front and rear suspensions, the pitch angular acceleration, and the body acceleration, respectively. qdtl_BD, hdtl_BD, qdnd_BD, hdnd_BD, ${\ddot{\theta }}_{BD}$, and $B{A}_{BD}$ are the root mean square values of the dynamic load of the front and rear wheels of the traditional suspension, the dynamic deflection, the pitch angle acceleration and the body acceleration of the front and rear suspensions, respectively. pu is the penalty principle, which stipulates the dynamic load of the front and rear wheels of the HPIS, the root mean square values of the dynamic deflection, and the pitch angular acceleration and body acceleration of the front and rear suspensions, as long as there is a corresponding value lower than the traditional suspension; the penalty value is set to 200. X is expressed as a set of parameters to be optimized. Considering the actual structural requirements, lb and ub are the upper and lower limits of the parameters to be optimized, so lb = [0, 0, 0, 0] and ub = [3000, 3000, 10,000, 10,000]. The suspension parameter optimization is shown in

Table 2. Comparison of optimization variables for parallel and series HPIS.

Suspension Layout	b1	b2	c1	c2
S1	203.0	212.4	7687.7	7316.1
S2	2785.9	2940.0	2930.5	2800.0

:

6. Simulation Analysis

The six performance indicators of the suspension devices were compared and are shown in

Table 3. Comparison of RMS values of various suspension structure performance indicators.

Performance Index		Body Acceleration (m/s2)	Pitch Acceleration (rad/s2)	Front Suspension Working Space (m)	Rear Suspension Working Space (m)	Front Wheel Dynamic Tire Load (N)	Rear Wheel Dynamic Tire Load (N)
S0	RMS	0.3945	0.0479	0.0172	0.0130	3634.8	3431.9
S1	RMS	0.4067	0.0783	0.0102	0.0090	3169.9	3240.2
S2	RMS	0.3589	0.0447	0.0141	0.0123	2844.7	2948.7

. The Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 shows comparisons of the gain of vehicle body acceleration, pitch acceleration, front/rear suspension working space and front/rear wheel dynamic tire load among the S0, S1 and S2.

From

Table 4. Series-parallel optimal degrees.

Performance Index Decrease	Body Acceleration (%)	Pitch Acceleration (%)	Front Suspension Working Space (%)	Rear Suspension Working Space (%)	Front Wheel Dynamic Tire Load (%)	Rear Wheel Dynamic Tire Load (%)
S1	−3.1	−63.40	40.60	30.7	12.7	5.6
S2	9.0	6.70	18.0	5.40	21.7	14.1

and Figure 6, it can be seen that at the speed of 10 m/s, compared with the traditional suspension, the HPIS has a significant improvement in the six aspects of front and rear wheel dynamic load, front and rear suspension dynamic deflection, pitch angle acceleration and body acceleration, and the comprehensive suspension performance is improved. This can be caused by several reasons. The first point is the inherent characteristics of a three-element hydro-pneumatic spring when added to an inerter. Furthermore, some indicators may be poor while others may be good, which is the result obtained through the optimization process. The decrease in the results of front and rear suspension working space and the deterioration of body and pitch acceleration in S1 is due to the sub-optimal solution found, while the overall improvement of the six indicators in S2 is due to the optimal solution found.

It is noteworthy that in terms of the peak time-domain response of the front and rear suspension working space, the series HPIS exhibits a larger peak time-domain response compared to the traditional suspension model and the parallel HPIS model. However, in the remaining four evaluation indexes, the peak time-domain response of the HPIS is significantly smaller than that of the parallel HPIS with passive suspension. These findings suggest that while the series HPIS may exhibit a larger response in certain aspects, it still offers advantages in terms of ride comfort and handling stability. Therefore, further research and development on the series HPIS suspension is warranted to explore its full potential and optimize its performance.

From the above diagram, compared to the S0, it can be seen that S1 and S2 have a significant positive impact on the suspension working space and dynamic tire load of both the front and rear wheels at a vehicle speed of 10 m/s. The decrease of front and rear suspension working space of S1 is 40% and 30.7%, while S2 reduces by 18% and 5.4%, respectively. S1 significantly improves driving safety. In terms of the front and rear wheels dynamic tire load, S1 reduces by 12.7% and 5.6%, while S2 reduces by 21.7% and 14.1%, respectively. S2 plays a role of improving handling stability. On the contrary, the performances of S1 and S2 in the two indexes of body acceleration and pitch acceleration were very different, with the body acceleration and pitch acceleration deteriorating by 3.1% and 63.4%, respectively. However, the body acceleration and pitch acceleration of S2 were reduced by 9% and 6.7%, respectively, thus improving the ride comfort.

Figure 13 shows a comparison of the performance indices at different speeds. For the body acceleration, the RMS values of the S1 deteriorate by 3.1%, 3.2%, and 3.3% at speeds of 10 m/s, 15 m/s, and 20 m/s, respectively, while the RMS values of the S2 decrease by 9.0%, 9.2%, and 9.3%. For the pitch acceleration, the RMS values of the S1 deteriorate by 63.4%, 63.8%, and 64.0% at speeds of 10 m/s, 15 m/s, and 20 m/s, respectively, while the RMS values of the S2 decrease by 6.7%, 6.8%, and 6.9%. The results indicate that S1 shows varying degrees of deterioration at different vehicle speeds, which causes a bad experience for passenger ride comfort. The decrease of front and rear suspension working space of S1 is 40%, 40.1%, 40.2% and 30.7%, 30.8%, 30.9%, while S2 reduces by 18%, 18.2%, 18.3% and 5.4%, 5.4%, 5.5%, respectively. The decrease of front and rear wheel dynamic tire load of S1 is 40%, 40.1%, 40.2% and 30.7%, 30.8%, 30.9%, while S2 reduces by 18%, 18.2%, 18.3% and 5.4%, 5.4%, 5.5%, respectively. In terms of the front and rear wheels dynamic tire load, S1 reduces by 12.7%, 13.0%, 13.0% and 5.6%, 5.8%, 5.9%, while S2 reduces by 21.7%, 21.8%, 21.9% and 14.1%, 14.1%, 14.2%, respectively. In summary, only two of the six performance indicators for the S1 show a positive trend towards optimization. The structure S2 of HPIS provides enhanced ride comfort and handling stability.

7. Conclusions

This study investigates the working principles of HPIS, focusing on the single-chamber structure and motion parameters. Through an exploration of the analytical expressions for stiffness and damping, we propose a novel type of HPIS suspension. To further examine the performance of this suspension, the half-vehicle models for the HPIS in both series and parallel configurations are established. Using the simulated annealing algorithm, the key variables of these two HPIS structures are obtained. The research results demonstrate that the S1 structure has a deterioration in the two indexes of body acceleration and pitch acceleration. In contrast, compared to the S0 and S1 structures, the S2 structure of HPIS excels in ride comfort and handling stability, offering new insights for suspension design. For the three-element HPIS suspension model proposed in this article, there are serious deteriorations in both low-frequency and high-frequency amplitudes in practical applications, which is not conducive to practical engineering applications. In future research, two-stage series-connected HPIS suspensions can be considered to improve the practical applicability of the model.