According to the configurations of inerter, nonlinear damping, and linear springs, four single-degree-of-freedom suspension system models with different IDES (Inerter-based Dynamic Energy Suspension) configurations were designed. The nonlinear dynamic equations of these systems were established using Newton’s second law. Subsequently, the HBM-PALM was employed to solve the dynamic responses of these four suspension systems under harmonic excitation, thereby analyzing their dynamic performance.

The distinctions among Coupled Hybrid Type I, II, and III IDES primarily derive from the configuration of the linear spring, nonlinear damping, and inerter within the system, as illustrated in Figure 1.

Type I: Vertically installed on the suspension by connecting a linear spring and a nonlinear damper in parallel, which are then connected in series with an inerter.

Type II: Vertically installed on the suspension by connecting an inerter with a parallel combination of a linear spring and a nonlinear damper.

Type III: Vertically installed on the suspension by connecting an inerter and a nonlinear damper in parallel, which are then connected in series with a linear spring.

Type IV: Vertically installed on the suspension by connecting a linear spring in series with a parallel combination of a nonlinear damper and an inerter.

A harmonic excitation (Acos(ωt)) is applied to the base to investigate the vibration attenuation performance of the suspension system, where A denotes the excitation amplitude, ω the excitation frequency, z₂ the vertical displacement of the suspension, z₁ the vertical displacement of the base, and z_b the vertical displacement of the IDES inerter. The specific structural parameters are listed in Table 2.

Using Newton’s second law, the displacement dynamic equations for the suspension system with the coupled hybrid type I IDES, and the IDES, can be derived as Equation 1.

The displacement dynamics equations for the suspension system of the coupled hybrid II IDES and the displacement dynamics equations for the IDES are m z ″   2 + c 1 z ′   2 − z ′   1 + k 1 z 2 − z 1 + b z ″   2 − z ″   b = 0 b z ″   2 − z ″   b − k 2 z b − z 1 − c 2 z ′   b − z ′   1 z b − z 1 2 = 0 (2)

The displacement dynamics equations for the suspension system of the coupled hybrid Ⅲ IDES and the displacement dynamics equations for the IDES are m z ″   2 + c 1 z ′   2 − z ′   1 + c 2 z ′   2 − z ′   b z 2 − z b 2 + k 1 z 2 − z 1 + b z 2 − z b = 0 b z ″   2 − z ″   b − k 2 z b − z 1 − c 2 z ′   2 − z ′   b z 2 − z b 2 = 0 (3)

The displacement dynamics equations for the suspension system of the coupled hybrid type Ⅳ IDES and the displacement dynamics equations for the IDES are m z ″   2 + c 1 z ′   2 − z ′   1 + k 1 z 2 − z 1 + k 2 z 2 − z b = 0 b z ″   b − z ″   1 + k 2 z b − z 2 + c 2 z ′   b − z ′   1 z b − z 1 2 = 0 (4)

In the dynamic Equations 1–4,the mass term m z 2 ″ is expressed using absolute acceleration because Newton’s second law must be applied in an inertial reference frame. The effects of base motion (such as z 1 and z b ) are transmitted to the mass through the relative displacements or velocities of the springs and dampers, ensuring that the mass response in the inertial frame is correctly coupled with the base motion and avoiding any physical inconsistency.

To realize the displacement-modulated nonlinear damping mechanism described in Equations 1–4, c x 1 − x 2 2 v 1 − v 2 , a passive hydraulic damper with specially designed flow channels can be employed as the actuating element. In this model, the squared relative displacement term x 1 − x 2 2 acts as a modulation function and, together with the coefficient c, constitutes a transient effective damping coefficient, causing the damping force to increase quadratically with the deformation amplitude. In such a passive hydraulic damper, the flow area of the throttle valve can be mechanically designed to vary quadratically with the piston displacement, thereby directly achieving the displacement-dependent nonlinear damping behavior without the need for external control.

The dynamic response of the four IDES suspension systems is obtained using the HBM-PALM method. The Harmonic Balance Method (HBM) is a powerful tool for analyzing the steady-state response of nonlinear vibration systems. It represents the system’s dynamic response as a series of harmonic components, transforming the problem into solving a set of algebraic equations. Due to the nonlinear nature of the system, only low-order harmonics (such as the first-order harmonic) are typically retained as an approximate solution, and higher-order harmonic terms are ignored. This method provides significant advantages in simplifying the analysis of nonlinear vibrations, especially for vibration control problems such as suspension systems. The Pseudo-Arc Length Method (PALM) is a numerical method commonly used to solve the steady-state solutions of nonlinear systems. By introducing a pseudo-arc length parameter, it avoids the branch switching failure that may occur in conventional arc length methods, effectively tracking multiple solutions of the nonlinear system. In the analysis presented here, PALM is combined with HBM to precisely track the dynamic response of the complex nonlinear system under different operating conditions.

Since Equations 1–3 all contain cubic nonlinear terms, higher-order harmonic terms can be omitted, retaining only the first-order harmonic as an approximate solution. The solutions for the suspension displacement z 1 and IDES displacement z b are assumed to be Z = z 2 z b = a 1 ⁡ cos ω t + a 2 ⁡ sin ω t b 1 ⁡ cos ω t + b 2 ⁡ sin ω t (5)

Substituting Equation 5 into Equations 1–4 and balancing the parameters for the oscillatory terms cos(ωt) and sin(ωt) leads to a set of nonlinear algebraic equations, which are given by Equation 6. Q a 1 , a 2 , b 1 , b 2 , ω = 0 (6)

Equation 5 includes four nonlinear equations governing the dynamic behavior of the coupled IDES half-car system. To resolve the dimensional deficiency of the system, an arc-length parameter is introduced, extending the problem into a 5-dimensional space, formulated as Equation 7. Y = a 1 , a 2 , b 1 , b 2 , ω T (7)

At this point the system of equations can be expressed as Equation 8. D Y , s = 0 (8)

At this point the system of equations has 5 equations, and constraints should be imposed in order to avoid the folding point of the solution ε · Δ Y − Δ s = 0 (9)

Equation 9 can be transformed into the Cauchy problem d Y d s = ε Y 0 , Y 0 = Y 0 (10)

Equation 10 can be used to calculate the corresponding predictive solution using the modified Eulerian method Y j = Y j − 1 + ε Y j − 1 s j − s j − 1 + s j − s j − 1 2 2 · ε Y j − 1 − ε Y j − 2 s j − 1 − s j − 2 , j = 1 , 2 , … (11)

Newton-type iterative corrections defined in Equation 12 can be used to control the accuracy of the solution. Y j 0 = Y j , Y j n = Y j n − 1 − D Q Y j P Y j − 1 Q Y j n − 1 0 , n = 1 , 2 , … (12)

Through a limited series of iterations, variable Y * (derived from solving Equations 10, 11 asymptotically approaches a state fulfilling condition Q Y * = 0 . The resulting equilibrium magnitudes induced by harmonic roadway vibrations are Z m = z 1 z b = a 1 2 + a 2 2 b 1 2 + b 2 2 (13)

As shown in Figure 2, the suspension dynamic displacements of the coupled suspension systems with four different IDES configurations under harmonic excitation are illustrated. It can be observed that the displacement curves of the hybrid type I and type II overlap. This is because, although the mechanical connection forms differ between hybrid type I and hybrid type II, the linear spring and nonlinear damper always form an equivalent parallel branch with the inerter, and the other end of this branch is fixed to the base. Therefore, under harmonic excitation, the motion equations of the suspension mass can be transformed into the same form in the frequency domain, exhibiting equivalent dynamic effects and resulting in identical suspension responses.

In contrast, the connection patterns of hybrid type III and type IV alter the force transmission paths: in hybrid type III, the nonlinear damper and inerter are connected in parallel and directly coupled with the suspension, whereas in hybrid type IV, a linear spring serves as an intermediary, making the inerter and damper indirectly coupled to the suspension. This structural difference changes the acting points of the inertial and damping forces relative to the base, thereby leading to distinct frequency response characteristics. Consequently, the suspension dynamic responses of hybrid type III and type IV under harmonic excitation are not identical.

In summary, compared to the other two systems, the coupled suspension systems with hybrid type I and hybrid type II IDES exhibit smaller resonance peaks in suspension displacement. Since hybrid type I and type II are dynamically equivalent, the hybrid type I IDES suspension system is selected for subsequent coupling with the half-car system in the next section.

The road surface simple harmonic excitation is the periodic vibration excitation caused by the uneven road surface when the vehicle is traveling on the road. Subjecting fore and aft tires to harmonic road excitations, the frequency-domain dynamics of the IDES-integrated half-car system are analyzed. Applied to the leading wheels, periodic pavement excitations are mathematically represented by Equation 25. z r f = A ⁡ cos ω t (25)

The rear tires experience identical harmonic road excitation to the front tires in amplitude (A) and angular frequency (ω), but with a phase delay of (l_{_}f + l_{_}r)/v, where v denotes the longitudinal vehicle velocity, as described in Equation 26. z r r = A ⁡ cos ω t − l f + l r v (26)

Featuring cubic nonlinearities per Equations 14–19, the IDES-enhanced half-car model derives dynamic responses using HBM-PALM (Wang et al., 2014; Wu and Tang, 2022). Partitioned co-simulation (Hollari et al., 2014; Rahikainen et al., 2020) offers subsystem-decoupling advantages for real-time multi-physics systems. The spectral solution converges to first-harmonic approximations Z = z s ϕ z u f z u r z b f z b r = a 1 ⁡ cos ω t + a 2 ⁡ sin ω t b 1 ⁡ cos ω t + b 2 ⁡ sin ω t c 1 ⁡ cos ω t + c 2 ⁡ sin ω t d 1 ⁡ cos ω t + d 2 ⁡ sin ω t e 1 ⁡ cos ω t + e 2 ⁡ sin ω t g 1 ⁡ cos ω t + g 2 ⁡ sin ω t (27)

Substituting Equation 27 into Equations 13–18 and balancing the coefficients of harmonic terms cos ω t and sin ω t yields F 1 a 1 , · · · g 1 , a 2 , · · · g 2 , ω cos ω t + F 2 a 1 , · · · g 1 , a 2 , · · · g 2 , ω sin ω t = 0 F 3 a 1 , · · · g 1 , a 2 , · · · g 2 , ω cos ω t + F 4 a 1 , · · · g 1 , a 2 , · · · g 2 , ω sin ω t = 0 F 5 a 1 , · · · g 1 , a 2 , · · · g 2 , ω cos ω t + F 6 a 1 , · · · g 1 , a 2 , · · · g 2 , ω sin ω t = 0 F 7 a 1 , · · · g 1 , a 2 , · · · g 2 , ω cos ω t + F 8 a 1 , · · · g 1 , a 2 , · · · g 2 , ω sin ω t = 0 F 9 a 1 , · · · g 1 , a 2 , · · · g 2 , ω cos ω t + F 10 a 1 , · · · g 1 , a 2 , · · · g 2 , ω sin ω t = 0 F 11 a 1 , · · · g 1 , a 2 , · · · g 2 , ω cos ω t + F 12 a 1 , · · · g 1 , a 2 , · · · g 2 , ω sin ω t = 0 (28)

Nullification of harmonic coefficients in Equation 28 formulates the harmonic balance conditions F 1 a 1 , · · · g 1 , a 2 , · · · g 2 , ω · · · F i a 1 , · · · g 1 , a 2 , · · · g 2 , ω · · · F 12 a 1 , · · · g 1 , a 2 , · · · g 2 , ω = 0 (29)

The mathematical formulations governing these twelve nonlinear equations are documented in the Appendix.

Equation 28 constitutes a twelve-equation nonlinear system governing the dynamic characteristics of the half-vehicle chassis integrated with IDES. Equation 28 has 13 parameters a 1 , · · · g 1 , a 2 , · · · g 2 , ω , which may be expressed as Equation 30. Q X , ω = 0 (30) where Q = F 1 , · · · , F 12 T and X = a 1 , · · · g 1 , a 2 , · · · g 2 , ω , Equation 29 admits the reformulated representation as Equation 31. Q Y = 0 (31)

After a finite number of iteration steps, the solution of Equation 29 is obtained. The equilibrium vibrational magnitude induced by pavement periodic forcing converges to Z m = z s m ϕ m z u f m z u r m z b f m z b r m = a 1 2 + a 2 2 b 1 2 + b 2 2 c 1 2 + c 2 2 d 1 2 + d 2 2 e 1 2 + e 2 2 g 1 2 + g 2 2 (32)

The vehicle tires experience random excitation originating from the road surface, which functions as the primary input force. The dynamic response characteristics of the IDES half-car system are subsequently analyzed using time-domain simulation. Furthermore, six critical evaluation metrics are utilized to quantify the root-mean-square (RMS) value associated with the system’s dynamic response. The pavement-induced random excitation is inherently dependent on both the type of road surface and the vehicle’s traveling speed. In this investigation, a Class C road profile is adopted, and the excitation is mathematically modeled through the filtered white noise method. Consequently, the stochastic excitation exerted on the front tires can be expressed as Equation 33. z r f ′ t = − 2 π f 0 z r f t + 2 π n 0 G q n 0 v ω t (33)

Here, f 0 represents the frequency, and G q n 0 denotes the roughness coefficient of the pavement surface, c-level road profile to take 64 × 10 − 6 m 3 ; n 0 denotes the reference spatial frequency, set to 0.1 m − 1 ; vehicle speed v, set to 15 m / s ; ω t represents a Gaussian white noise with a mean of 0 and an intensity of 1. Analogous to the harmonic excitation from the road surface, the random excitation applied to the rear tires of the vehicle exhibits a temporal lag compared to that of the front tires, which is mathematically expressed as Equation 34. z r r t = z r f t − τ (34)

Based on Equation 32, the dynamic behavior of the half-car system incorporating IDES is evaluated using a set of performance metrics, which include the vertical acceleration of the vehicle body ( z s ″ , J V B V A ), pitch acceleration ( ϕ ″ , J V B P A ), front suspension deflection ( z s f − z u f , J S D f ), rear suspension deflection ( z s r − z u r , J S D r ), front tire load fluctuation ( k t f z s f − z u f , J D T L f ), and rear tire load fluctuation ( k t r z s r − z u r , J D T L r ). The detailed formulations of these dynamic performance indicators are provided as follows J V B V A = ω 2 a 1 2 + a 2 2 J V B P A = ω 2 b 1 2 + b 2 2 J S D f = a 1 − l f b 1 − c 1 2 + a 2 − l f b 2 − c 2 2 J S D r = a 1 + l r b 1 − d 1 2 + a 2 + l r b 2 − d 2 2 J D T L f = k t f c 1 − A 2 + c 2 2 J D T L r = k t r d 1 − A ⁡ cos ω l f + l r v 2 + d 2 − A ⁡ sin ω l f + l r v 2 (35)

Given that the semi-car system’s original model is based on linear dynamics, its response characteristics remain accessible through Laplace transform operations. Adopting a parallel algorithmic framework to the IDES-integrated configuration, vibrational outputs of both the original architecture and the INES-augmented system are resolved via the Harmonic Balance Method (HBM) and Pseudo-Arc Length Method (PALM) respectively, with all dynamic performance descriptors maintaining analytical consistency with Equation 35.

In this section, we have detailed the excitation boundaries of the half-car system and constructed relevant indicators for evaluating the system’s vibration damping performance. These boundary conditions provide the foundation for the dynamic response analysis in the subsequent chapters, while the evaluation indicators will be used to assess the specific vibration damping performance of the system.

Figure 5 illustrates the analytical dynamic behavior of the half-car system incorporating IDES through the combined HBM-PALM methodology. Assuming homogeneous tire configurations at both ends with equivalent structural properties, the vehicular velocity is maintained at 15 m/s as a moderate operational condition. Owing to the transient phase lag across axles, the displacement and pitch dynamics of the vehicle body manifest multi-band resonant behavior with periodic amplitude maxima. When the nonlinear damping of IDES is 1 × 106 N/m, the system’s dynamic response becomes single-valued, demonstrating linear characteristics. At approximately 10 Hz, a distinct dynamic phenomenon can be observed: the vertical displacement of the vehicle body z s and the pitch angle ϕ are nearly suppressed to zero, while the tire displacements ( z u f , z u r )and the inerter displacements within the front and rear suspension branches ( z b f , z b r )exhibit peak responses. This behavior indicates the occurrence of a modal decoupling effect, in which the vibration energy is predominantly confined to the tire–suspension–inerter subsystem rather than being transmitted to the vehicle body. Consequently, the body motion is effectively isolated through a mechanism analogous to anti-resonance, where the primary system response approaches a minimum while the auxiliary subsystem undergoes large oscillations. In Figure 5, the analytical results are

Figure 6 shows the frequency-domain responses of six important dynamic performance metrics for the baseline half-car system, the system combined with INES, and the system combined with IDES under three operational speeds: 5 m/s (low speed), 15 m/s (mid-range speed), and 35 m/s (high speed). The assessment metrics include vertical acceleration of the vehicle body, pitch angular acceleration, dynamic deflection of both front and rear suspensions, and dynamic loading fluctuations on the front and rear tires.

As shown in Figures a and b, the frequency-domain responses of the vehicle body vertical acceleration and pitch angular acceleration under harmonic excitation in the half-vehicle system exhibit multiple resonance peaks. Particularly at low vehicle speeds, the low-frequency resonance phenomena become more pronounced, leading to an increase in the number of resonance peaks. This is due to the higher sensitivity of the suspension system to low-frequency excitations at low speeds, where the natural frequencies of the system align with the excitation frequencies, thereby generating more resonance peaks. At high vehicle speeds, the system’s dynamic response shifts toward the high-frequency range, resulting in a reduction in the number of resonance peaks. Therefore, the appearance of multiple resonance peaks at low vehicle speeds is a natural consequence of the interaction between the system’s inherent frequencies and low-frequency excitations, which is a normal phenomenon.

At low speed, compared to the original half-car system, the IDES-equipped system significantly suppresses resonance peaks in both suspension deflections and tire dynamic loads at the front and rear axles. Demonstrating excellent vibration reduction capability. However, as system damping rises overall, the maximum vertical and pitch accelerations of the vehicle body exhibit a small increase. Compared with the INES-integrated system, although the suspension and tire indicators are similar, the IDES system markedly reduces the peak values of vertical and pitch accelerations, indicating superior performance in ride comfort.

At medium speed, the IDES-integrated system continues to effectively suppress the resonance peaks of suspension deflection and tire dynamic loads. Compared with the INES system, IDES significantly mitigates the deterioration of vertical and pitch accelerations caused by the inerter, further demonstrating its synergistic advantage in mid-speed vibration control.

At high speed, the IDES system still effectively reduces the resonance responses of suspension deflection and tire dynamic loads while maintaining good control over the vertical acceleration of the vehicle body. Although the pitch acceleration resonance peak increases slightly, it is still significantly lower than that of the INES-integrated system, indicating that IDES can still balance stability and comfort at high speeds.

Overall, the half-car system integrated with IDES exhibits favorable energy dissipation characteristics and coordinated dynamic responses across all speed conditions. Although the resonance peaks of suspension deflection and tire dynamic forces are somewhat higher than those of the INES system in certain cases, its suppression of vertical and pitch accelerations is significantly better, which is more conducive to achieving a balance between ride comfort and handling stability. In addition, compared to INES, IDES can effectively reduce the vertical acceleration and pitch angle acceleration of the vehicle body across a wide frequency range, significantly improving the vibration reduction performance of the vehicle body. Although the dynamic load on the front and rear tires and the dynamic deflection of the front and rear suspensions are slightly higher than those of INES in the same frequency band, the differences are minor and do not adversely affect the overall performance of the system. This shows that IDES achieves wide-band vibration reduction characteristics with relatively little cost through full-band optimization of the vehicle body’s response.

When incorporating an inerter into the traditional NES to meet lightweight design requirements, a 1 kg ball screw inerter can achieve an adjustable inertance ranging from 60 to 240 kg (Smith, 2002).

The comparison of the vertical and pitch angular accelerations of the vehicle body for the baseline half-car system, the system combined with IDES, and the system combined with INES under road random excitation is shown in Figure 7. Table 4 provides the accurate RMS values for six evaluation metrics. When driven on a C-class pavement, metrics RMS J V B V A , RMS J S D f , RMS J S D r , RMS J D T L f , RMS J D T L f and RMS J D T L r of the IDES-equipped system are reduced by 0.69%, 5.75%, 5.56%, 6.07%, and 5.66%, respectively, compared to the original system, while metric RMS J V B P A is slightly elevated by 1.94%, consistent with the road harmonic excitation scenario. Compared to the half-car system incorporating INES, IDES places greater emphasis on overall balance. In terms of “local” performance, it requires sacrificing some vibration suppression efficiency. For example, metrics RMS J V B V A and RMS J V B P A are reduced by 3.51% and 14.41%, respectively, significantly lowering the body pitch acceleration. However, metrics RMS J S D f , RMS J S D r , RMS J D T L f , and RMS J D T L f show slight increases, though the magnitudes remain small. Furthermore, from an engineering application perspective, this balanced performance of IDES is more attractive, as vehicle suspension prioritizes “overall comfort and stability” rather than extreme optimization of individual metrics.

The parametric study reveals that applying the IDES structural parameters listed in Table 3 to the original suspension slightly degrades metric RMS J V B P A . To further enhance overall driver comfort, this study employs a genetic algorithm to optimize the structural parameters of the IDES suspension system, specifically stiffness, damping, and inertance coefficients. The optimization objective is to improve metrics J V B V A and RMS J V B P A while ensuring metrics RMS J S D f , RMS J S D r , RMS J D T L f ,and RMS J D T L r remain within acceptable bounds. Consequently, the objective function formulated to optimize IDES suspension system dynamic performance is expressed as Equation 36. L = ρ 1 RMS J V B P A RMS J V B P A O + ρ 2 RMS J V B V A RMS J V B V A O (36)

Vehicle dynamic performance must simultaneously consider ride comfort (vertical vibration) and handling stability (pitch vibration). In the formulation, weighting coefficients   ρ 1 and   ρ 2   for the body vertical acceleration and pitch angular acceleration are both set to 0.5, as they hold equal priority in the optimization objectives to prevent excessive optimization of a single metric from degrading other performance aspects. Metrics RMS J V B V A and RMS J V B P A represent the RMS values of the vertical and pitch angular accelerations of the vehicle body for the IDES suspension system, while metrics RMS J V B V A O and RMS J V B P A O denote those of the conventional suspension system.

During the parameter optimization process using genetic algorithms, for collision risk prevention, the RMS metrics of suspension dynamic deflections and wheel dynamic loads (at both vehicle ends) must be constrained within acceptable thresholds. Specifically, front/rear suspension deflection RMS must be capped at one-third of maximum travel, while dynamic wheel load RMS (front/rear) should not exceed one-third of static load RMS values. The corresponding constraint equations are as follows RMS J S D f ≤ S max / 3 RMS J S D r ≤ S max / 3 RMS J D T L f ≤ m u f + m s f · g / 3 RMS J D T L r ≤ m u r + m s r · g / 3 (37)

To ensure that the optimized parameters of the IDES suspension system align with the actual vehicle conditions, it is necessary to define the optimization range for the structural parameters. 2000 N / m ≤ k b f ≤ 3000 N / m 2000 N / m ≤ k b r ≤ 3000 N / m 500 N s / m ≤ c b f ≤ 2000 N s / m 500 N s / m ≤ c b r ≤ 2000 N s / m 0 k g ≤ b f ≤ 200 k g 0 k g ≤ b r ≤ 200 k g (38)

The parameter optimization process using a genetic algorithm is shown in Figure 8. Fundamentally, genetic algorithms emulate biological evolution’s natural selection through operators like selection, crossover, and mutation, progressively generating individuals with enhanced environmental fitness. As a powerful and robust algorithm, the genetic algorithm effectively addresses the optimization problem of suspension system structural parameters. The specific optimization procedure is as follows:

Using MATLAB’s built-in Genetic Algorithm Toolbox, the population size was set to 50, with 30 iterations performed. Selection, crossover, and mutation parameters were maintained at their default values. A real-number coding scheme was employed, with upper and lower bounds defined for each structural parameter. Under these settings, optimized parameters were efficiently obtained.

Figure 9 shows the iteration steps of the genetic algorithm. The convergence curve stabilizes within 30 iterations, indicating that the algorithm has found a near-optimal solution. Minimal variation in the curve beyond 10–20 iterations confirms the rationality of algorithmic settings (e.g., population size of 50, default genetic operator parameters). Table 5 provides the optimized design parameters of the IDES mechanism at low (5 m/s), medium (15 m/s), and high speeds (35 m/s). These parameters were applied to IDES suspension system simulations to evaluate the dynamic performance of the genetically optimized mechanism. To visually demonstrate the superiority of the genetic algorithm, Figure 9 compares dynamic performance metrics (original suspension, unoptimized IDES, and optimized IDES systems) under low-, medium-, and high-speed conditions.

As illustrated in Figure 10, versus the original suspension system, the IDES suspension system optimized by genetic algorithm exhibits significantly consistent improvements in dynamic performance indicators across different vehicle speeds. Due to the physical coupling between vertical body acceleration and pitch angular acceleration, the latter exhibits a slight deterioration (still within the specified constraint limits), while all other five performance indicators show significant enhancement. To further explore the specific effects of parameter optimization using genetic algorithms, this study selects medium vehicle speed as a typical operating condition for optimization analysis to obtain the most representative performance optimization results. A comparison is then made with the baseline suspension system and the optimized INES suspension system. The comparison of vertical body acceleration and pitch angular acceleration for the optimized IDES and INES suspension systems, as well as the baseline system, is shown in Figure 11, with specific dynamic performance indicators provided in Table 6.