Comfort-Oriented Optimization of Speed-Dependent Variable Inertance for Intelligent Vehicle Suspension Systems

Abstract

This paper investigates the performance of a speed-dependent variable inerter in improving vehicle suspension performance. Unlike conventional and passive inerter suspensions with fixed mechanical properties, the proposed speed-dependent variable inerter allows continuous adjustment of inertance according to the relative acceleration between the sprung and unsprung masses, enabling variable inertance under changing driving speeds and road conditions. A quarter-vehicle model is employed to evaluate a conventional passive inerter and both a linearly and non-linearly increasing variable inerter system in series and parallel layouts. A multi-objective genetic algorithm simultaneously optimizes the suspension damping and variable inertance range with respect to ride comfort and road-holding ability. To further validate the simulations, the optimized systems are evaluated under step, random and sinusoidal road profiles. The results showed that a linearly increasing variable inerter, particularly in parallel configuration, offers the best compromise between ride comfort and road holding, achieving up to 4.94% improvement in ride comfort under a random road profile, outperforming conventional passive inerter and non-linearly increasing inerter suspensions, while maintaining acceptable tire–road contact. Performance improvements under step and sinusoidal road profiles were moderate, while more significant performance gains were observed under a random road profile due to the larger acceleration change induced, which led to larger inertance variation. These findings confirmed the potential of variable inerters as an alternative approach to vehicle suspension systems, due to their passive implementation, absence of control requirement and compatibility with compact suspension architectures.

1. Introduction

The vehicle suspension system plays a crucial role in maintaining passenger comfort, vehicle stability and handling by isolating the vehicle chassis from road-induced vibrations and ensuring adequate tire–road contact [1]. Conventional suspension systems have relied solely on suspension springs and dampers with fixed suspension characteristics, which lack adaptability to changing road conditions and driving maneuvers. As a result, they often struggle to satisfy conflicting performance demands related to ride comfort and road-holding ability. To address this limitation, Smith (2002) [2] introduced the inerter—a mechanical element that replaces the mass element in the force–current analogy and has since emerged as a significant device for enhancing vibration control [3].

Analogous to a capacitor in an electrical circuit, the inerter is a two-terminal mechanical device that generates a force proportional to the relative acceleration across its two terminals by means of a flywheel [2], as represented in (1).

$$F_{inerter}=b(a_2-a_1)$$

(1)

where $F_{inerter}$ indicates the inerter’s force acting on its two terminals, b indicates the inerter’s characteristic, called inertance, and $a_1,a_2$ indicate the acceleration at the two terminals. The inerters are commonly realized through ball–screw or rack-and-pinion mechanisms [4,5,6,7], and their inclusion significantly expands the dynamic configurations of suspension systems, leading to superior performance characteristics that are unattainable with the combination of a spring and damper only.

Substantial research has demonstrated that integrating an inerter into vehicle suspension can improve ride comfort and road holding by tuning the force transmission and system dynamics [1,8,9,10,11,12,13,14]. Early studies primarily focused on theoretical modeling and passive inerter configurations, leading to improved dynamic response and reduced vibration across various suspension layouts [15,16,17,18]. Experimental investigations further validated these benefits [19,20,21]. Beyond automotive use, inerters have also been effectively applied in building structures [22,23,24,25,26], railway or train systems [27,28,29,30] and some other vibration mitigation systems [31,32,33,34], demonstrating their versatility.

Despite these promising developments, passive inerters inherently provide constant inertance only, which limits their adaptability under dynamic driving conditions, particularly in passenger vehicles, where performance demands change continuously with driving speed and road disturbances. To address this limitation, researchers have explored switchable, semi-active and active inerter mechanisms capable of modifying inertance in real time through mechanical actuation [35,36,37,38,39,40]. While these approaches have shown improved performance, a key limitation remains in their reliance on an external control mechanism, which may increase system complexity, cost, implementation challenges and involve additional energy input. Furthermore, most of them rely on discrete switching behavior, which can lead to abrupt force transitions. Therefore, the concept of a variable inerter is introduced.

Recent advances in vehicle suspension have explored alternative approaches to vibration isolation, particularly through the use of a fractional damping model [41]. The paper demonstrated that fractional damping can improve the vibration isolation effect across a broader frequency range compared to the classical damping model. However, a trade-off to achieve optimal ride comfort and vibration settling is necessary. Higher fractional order shortens the vehicle’s settling time but increase its peak amplitude, which may degrade its suspension performance. Hence, a variable inerter can be considered for inclusion and act as an additional tuning dimension to balance the ride comfort, stability and resonance control. Other than that, the variable inerter can also be considered in micromobility application, which has emerged as a trend in urban transportation [42]. The reports demonstrated that micromobility riders such as electric kick-scooters are highly exposed to significant vibration levels, especially within frequency ranges associated with human discomfort, due to the limited suspension capability of such vehicles [43,44]. This excessive vibration level can cause structural failure and present a severe stability risk to the rider [45,46,47]. Therefore, these cases highlight the need for compact and effective vibration isolation solutions. An in-wheel suspension concept for micromobility vehicles has been introduced to improve ride stability, comfort and safety [48]. The results showed that the ride dynamics were significantly enhanced by reducing vertical acceleration peaks, improving tire–road contact consistency and shortening stabilization time after impacts. However, the paper also found out that introducing unnecessary mass and stiffness lowered the overall efficiency through investigating the performance improvements of different numbers of introduced shock absorbers. Hence, a variable inerter can be implemented to enhance vibration isolation without substantially increasing physical mass, while enabling adaptive control of suspension system dynamics, responding to varying road profiles, speeds and maneuvering conditions.

A key research gap remains due to the lack of investigation into purely passive mechanisms that enable continuous and smooth variation in inertance without relying on an external control mechanism. Such a capability is particularly advantageous for automotive applications where performance, reliability and simplicity are critical design considerations. Existing studies have not fully addressed how the inertance should vary with relative acceleration or speed to minimize both ride comfort and road holding simultaneously in a systematic and multi-objective manner. Moreover, the optimal combination of suspension damping and inertance profile for such variable inerter suspension systems has not been clearly quantified. To bridge this research gap, this study proposes a speed-dependent variable inerter that allows real-time adaptation of inertial forces. A quarter-vehicle model (QVM) is implemented to evaluate the performance of a conventional, passive inerter and two forms of variable inerter systems, namely linearly increasing and non-linearly increasing inertance profiles under step, random and sinusoidal road excitation. A multi-objective genetic algorithm is then used to simultaneously optimize the suspension damping and variable inertance range, addressing the conflicting objectives of ride comfort and road-holding ability.

Hence, the main contributions of this paper are to introduce a speed-dependent variable inertance profile which enables continuous adaptation of inertance depending on relative acceleration into suspension systems. A multi-objective optimization framework is developed to determine the optimal suspension damping and inertance variation for a variable inerter. Then, a comprehensive performance comparison was conducted between conventional, passive, linearly increasing and non-linearly increasing variable inerter suspension configurations in both series and parallel layouts. The results demonstrate that a parallel configuration consistently outperforms the series configuration and that a linearly increasing variable inerter provides the best comfort-handling balance. It provides the largest performance improvements in ride comfort under a random road profile, followed by step and sinusoidal road profiles, since a random road profile leads to greater changes in acceleration and hence inertance variation. These findings highlighted the potential of integrating a variable inerter with intelligent optimization strategies for future adaptive vehicle suspension systems.

2. Modeling of Suspension System

2.1. The Inerter

This study aims to optimize the performance of a vehicle suspension system using a variable inerter, examining both series and parallel configurations. Figure 1 showed three types of suspension layout that were used in this study, which are conventional suspension consisting of a spring and damper only and suspension with the addition of an inerter in series and parallel configurations, respectively. In the series inerter suspension layout, the inerter is placed in series with a damper, but in parallel with a spring element. An intermediate node mass, as in Figure 1b, is introduced between the spring and damper elements. This node serves as a point of force transmission and displacement and is set to an insignificant value to simplify modeling without significantly affecting the system dynamics. Meanwhile, in the parallel suspension layout, the inerter is installed directly in parallel with both the spring and damper elements.

Table 1. Types of inerter with their mathematical models.

Types of Inerter	Mathematical Model
Passive inerter	$F_{inerter}=b(a_2-a_1)$
Linearly increasing variable inerter	$F_{inerter}=b(a_2-a_1)$ $b=m\omega +c; \omega ={\displaystyle \frac{2\pi }{p}}(v_2-v_1)$
Non-linearly increasing variable inerter (exponential)	$F_{inerter}=b(a_2-a_1)$ $b=c{e}^{m\omega }; \omega ={\displaystyle \frac{2\pi }{p}}(v_2-v_1)$

$F_{inerter}$ indicates the inerter’s force, b indicates the inertance, $v_1,v_2$ and $a_1,a_2$ indicate the velocity and acceleration at the two terminals, $\omega$ is the rotational speed of the flywheel and $p$ is the ball–screw pitch. The equations $b=m\omega +c$ and $b=c{e}^{m\omega }$ are the general equations for the linear equation and exponential equation, where m is the slope or growth rate and c is the initial condition or the y-intercept.

summarized the three types of inerter implemented in the model, namely the passive inerter and the linearly increasing and non-linearly increasing variable inerter, along with their corresponding mathematical models or force–acceleration relationships. For the passive inerter, the inertance is treated as a constant value. In contrast, the variable inerters with either a linearly increasing or non-linearly increasing profile allow the inertance to vary with relative acceleration between the sprung and unsprung masses. All inerter-equipped suspensions are then compared against the baseline conventional suspension.

2.2. Vehicle Suspension Configurations

The vehicle model used for optimizing the suspension system involves a two-degree-of-freedom QVM, which captures the essential dynamics of a single wheel, the suspension and a portion of the vehicle body [8,9,50,51] under road excitation. The QVM model includes sprung mass, unsprung mass, suspension stiffness, suspension damping and tire stiffness. The sprung and unsprung masses represent the quarter-vehicle body and the wheel, respectively. Meanwhile, suspension stiffness and damping indicate the spring constant and damping coefficient of the suspension. Lastly, tire stiffness captures the ability of the tire to resist deformation when subjected to forces. The quarter-vehicle parameter values were chosen to replicate real-world conditions for a mid-size passenger car [8], as shown in

Table 2. QVM parameters of a typical passenger car.

Vehicle Parameter	Passenger Car
Sprung mass, $m_s$(kg)	317.5
Unsprung mass, $m_u$(kg)	45.4
Node mass, $m_n$(kg)	0.001
Suspension stiffness, $k$(N/m)	22,000
Suspension damping, $c$(N·s/m)	1500
Tire stiffness, $k_t$(N/m)	192,000
Inertance, b (kg)	6

, and an inertance of 6 kg was selected to optimize the vehicle performance, as evidenced in past research on passive inerters [1]. A small node mass of 0.001 kg is also introduced in series inerter suspension to ensure numerical stability and prevent singularity in system equations. Its magnitude is sufficiently small to ensure negligible influence on system dynamics.

Equations of Motion of QVM

Figure 2 illustrates three types of QVM and the governing equations of motions are determined as in (2) to (3) for conventional suspension, (4) to (6) for series inerter suspension and (7) to (8) for parallel inerter suspension, as in [1,8,52].

Conventional suspension system,

$$m_s{\ddot{z}}_s=k\left(z_u-z_s\right)+c\left({\dot{z}}_u-{\dot{z}}_s\right)$$

(2)

$$m_u{\ddot{z}}_u=k_t\left(z_g-z_u\right)-k\left(z_u-z_s\right)-c\left({\dot{z}}_u-{\dot{z}}_s\right)$$

(3)

Series inerter suspension system,

$$m_s{\ddot{z}}_s=k\left(z_u-z_s\right)+c\left({\dot{z}}_n-{\dot{z}}_s\right)$$

(4)

$$m_n{\ddot{z}}_n=b\left({\ddot{z}}_u-{\ddot{z}}_n\right)-c\left({\dot{z}}_n-{\dot{z}}_s\right)$$

(5)

$$m_u{\ddot{z}}_u=k_t\left(z_g-z_u\right)-k\left(z_u-z_s\right)-b\left({\ddot{z}}_u-{\ddot{z}}_n\right)$$

(6)

Parallel inerter suspension system,

$$m_s{\ddot{z}}_s=k\left(z_u-z_s\right)+c\left({\dot{z}}_u-{\dot{z}}_s\right)+b\left({\ddot{z}}_u-{\ddot{z}}_s\right)$$

(7)

$$m_u{\ddot{z}}_u=k_t\left(z_g-z_u\right)-k\left(z_u-z_s\right)-c\left({\dot{z}}_u-{\dot{z}}_s\right)-b\left({\ddot{z}}_u-{\ddot{z}}_s\right)$$

(8)

in which $m_s$, $m_u$ and $m_n$ represent sprung mass, unsprung mass and nodal mass, $z_s$, $z_u$ and $z_n$ represent their displacements, ${\dot{z}}_s,{\dot{z}}_u,{\dot{z}}_n \mathrm{a}\mathrm{n}\mathrm{d} {\ddot{z}}_s, {\ddot{z}}_u,{\ddot{z}}_n$ are the corresponding velocities and accelerations, $z_g$ is the vertical road displacement input, k represents suspension stiffness, $k_t$ represents tire stiffness, c represents suspension damping and b represents inertance.

A total of seven suspension configurations were simulated in the MATLAB^® Simulink^® (R2019a) environment, each subjected to a step, random and sinusoidal road profiles. A step road profile simulates a sudden, discrete change in road elevation, such as driving over a bump or uneven pavement, and is commonly used to evaluate the transient response of a vehicle suspension system, as displayed in Figure 3a. A random road profile is a stochastic model that simulates the irregularities of real-world road surfaces, typically classified according to ISO 8608:1995 [53]. A class A random road profile (smooth road classification) was used in this study [52], as shown in Figure 3b. Meanwhile, a sinusoidal road profile represents a smooth, periodic and deterministic vertical oscillations of road surfaces, as shown in Figure 3c.

While the investigated suspension configurations are:

• Conventional suspension;
• Series passive inerter suspension;
• Series linearly increasing variable inerter;
• Series non-linearly increasing variable inerter;
• Parallel passive inerter suspension;
• Parallel linearly increasing variable inerter;
• Parallel non-linearly increasing variable inerter.

3. Optimization of Suspension Parameters Using Multi-Objective Genetic Algorithm

In this study, the suspension design problem was framed as a multi-objective optimization problem, where the aim is to simultaneously minimize two conflicting vehicle performance criteria, which are the ride comfort, represented by the root-mean-squared (RMS) of vehicle body acceleration, and road-holding ability, represented by the RMS of dynamic tire load [54,55,56]. Reducing the vehicle body acceleration could improve ride comfort by limiting the vertical motion of the vehicle body. However, this typically increases the dynamic tire load because the tire must accommodate larger variations in contact forces. Conversely, improving road holding tends to deteriorate the ride comfort. Due to this trade-off, both objectives must be optimized simultaneously to identify a balanced suspension design.

Single-objective optimization and aggregated multi-objective optimization often struggle to address these competing goals simultaneously. They lack the capability to handle multiple objectives with trade-offs. Hence, the use of a multi-objective genetic algorithm (MOGA) is particularly well-suited to this problem due to the complexity and non-linearity of the system. The genetic algorithm is a global search method inspired by the process of natural selection. It does not rely on gradient information and is effective in exploring large, multimodal and non-differentiable search spaces. Therefore, MOGA is adopted to optimize the variable inertance range and suspension damping.

The optimization seeks the optimal combination of variable inertance b and suspension damping c, where b may follow either linearly or non-linearly increasing profiles. The problem is expressed as in (9), where the objective functions to be minimized are in (10) and (11).

$${}_{x=\{b,c\}}{}^{min}\{f_1\left(x\right),f_2\left(x\right)\}$$

(9)

$$f_1\left(x\right)=\sqrt{{\displaystyle \frac{1}{T}}{\int }_0^T{{(\ddot{z}}_s(t))}^{2}dt}$$

(10)

$$f_2\left(x\right)=\sqrt{{\displaystyle \frac{1}{T}}{\int }_0^T{{[k}_t\left(z_g(t)-z_u(t)\right)]}^{2}dt}$$

(11)

The algorithm began by randomly initializing a population of candidate solutions within the specified bounds of the design parameters. The boundaries for inertance and damping were from 0 kg to 320 kg and 0 N·s/m to 6000 N·s/m, respectively. These are in line with previous studies which often considered the damping value to be a fraction of the critical damping coefficient $\left(c_{cr}=2\sqrt{mk}\right)$ [57] and the ratio of inertance to sprung mass to fall between zero and one [58]. For each candidate solution that represents a unique configuration of inertance and damping, the QVM is simulated to evaluate both objective functions.

MOGA then evolved the population using selection, crossover and mutation. Selection favors individuals with superior performance, while crossover combines characteristics from two parent candidates to produce offspring. Mutation introduces random variations, ensuring genetic diversity and preventing premature convergence. Through these processes, the algorithm gradually converged to a set of non-dominated solutions, known as Pareto-optimal solutions, in which improving one objective necessarily worsens at least one other objective [26,59,60,61]. These solutions represent the best possible trade-offs and form a boundary known as the Pareto front. Each point on the Pareto front represents a different compromise among the objectives [62]. The overall optimization technique can be summarized in a flowchart, as illustrated in Figure 4.

There are two key decision points, namely knee point and utopia point, used for selecting the best candidate solution from the Pareto solutions. The point of intersection of minimum value of the two objective function is known as the utopia point, $z^{\ast }$ [61]. It can be used as the ideal reference point to find the best solution from a set of Pareto-optimal solutions where both objectives are minimized to their theoretical minimum values for all objectives, as in (12).

$$z^{\ast }=(\min f_1(x_i),\min f_2(x_i))$$

(12)

The Pareto-optimal solution which has the shortest distance from the utopia point among other non-dominated solutions is known as the knee point, $x_{knee}$ [59], as shown in Figure 5. It has a balanced optimal design point between two conflicting objectives and is often chosen as the best compromise solution, since it provides balanced performance in both objectives without prioritizing one excessively over the other. To identify the knee point, the Pythagorean distance of each Pareto solution, $x_i$ to the utopia point was determined using (13).

$$d_i=\sqrt{{(f_1\left(x_i\right)-{z_1}^{\ast })}^2+{(f_2\left(x_i\right)-{z_2}^{\ast })}^2}$$

(13)

where $f_1\left(x_i\right)$ is the value of the first objective (e.g., ride comfort) at solution $x_i$, $f_2\left(x_i\right)$ is the value of the second objective (e.g., road-holding ability) at solution $x_i$, and ${z_1}^{\ast }$ and ${z_2}^{\ast }$ are the utopia point coordinates for objectives $f_1\left(x\right)$ and $f_2\left(x\right),$ respectively. The knee point $x_{knee}$ can be determined as in (14).

$$x_{knee}=\arg \genfrac{}{}{0ex}{}{\min }{x_i\in Pareto front}\hspace{0.5em}{d}_i$$

(14)

4. Simulation-Based Performance Analysis of Variable Inertance Profile

A detailed performance comparison of the seven suspension configurations, namely conventional suspension, series and parallel passive inerter, linearly increasing and non-linearly increasing variable inerter suspensions was conducted under step input, random input and sinusoidal input. The comparison focuses on two key objectives, which are RMS sprung mass acceleration (ride comfort) and RMS dynamic tire load (road holding). A MOGA is used to identify the optimal suspension damping and inertance bounds for each configuration.

4.1. Performance Analysis Under Step Road Profile

4.1.1. Pareto Front Characteristics

The optimized Pareto fronts, as shown in Figure 6 and Figure 7, indicate that both passive inerter and linearly increasing variable inerter configurations shift the Pareto fronts leftward compared with the conventional suspension, particularly in the region of low-RMS vehicle body acceleration. This confirmed that regardless of whether the inertance is constant or speed-dependent, inerter-equipped systems are able to provide measurable improvement in the ride comfort performance indices, while maintaining acceptable dynamic tire load. A closer inspection revealed that the Pareto fronts of the passive inerter and linearly increasing variable inerter largely overlap, indicating that their overall performance levels were comparable under the simulated step road excitation. This similarity can be attributed to the relatively moderate relative velocity and acceleration levels induced by the step excitation, which limit the extent to which the variable inerter’s inertance deviates from its nominal value. Consequently, the benefits of inertance variability were not fully activated across the entire operating range.

Nevertheless, the parallel linearly increasing variable inerter consistently produced a wider Pareto front, particularly evident in Figure 7, indicating a broader spectrum of non-dominated solutions. This implies greater design flexibility, allowing designers to select solutions that have balanced ride comfort and road holding according to desired performance priorities. In contrast, the non-linearly increasing variable inerter performed poorly, as its rapidly growing inertance generated excessive inertial force at higher acceleration change, leading to deterioration in both ride comfort and road-holding behavior.

Compared with the established literature, where passive inerters typically yield ride comfort improvement of approximately 2% [1,49,64,65,66]; the current results were consistent in magnitude while demonstrating that a speed-dependent variable inerter can achieve a comparable or slightly improved benefit without requiring an external control input or additional energy consumption. Although the absolute performance gains were modest, the ability of the variable inerter to operate as a purely mechanical, passive component while offering enhanced adaptability highlighted its practical value. Overall, the findings suggested that variable inerters, particularly those with linearly increasing inertance profile, represent a feasible intermediate solution between passive simplicity and fully active suspension systems.

4.1.2. Utopia and Knee Point Evaluations

Table 3. Utopia point of conventional, passive inerter, linearly increasing and non-linearly increasing variable inerter under step road profile.

Types of Suspension System		Utopia Point
		RMS BA (m/s2)	RMS DTL (N)	Difference in BA (%)	Difference in DTL (%)
Standard (without inerter)		1.9086	1087.16	Reference
Series Suspension	Passive inerter	1.8438	1140.25	−3.40	4.88
	Linearly increasing variable inerter	1.8444	1144.09	−3.37	5.24
	Non-linearly increasing variable inerter	2.6053	1378.54	36.50	26.80
Parallel Suspension	Passive inerter	1.8527	1087.91	−2.93	0.07
	Linearly increasing variable inerter	1.8360	1096.71	−3.80	0.88
	Non-linearly increasing variable inerter	1.9595	1097.32	2.66	0.93

and

Table 4. Knee point of conventional, passive inerter, linearly increasing and non-linearly increasing variable inerter under step road profile.

Types of Suspension System		Knee Point
		RMS BA (m/s2)	RMS DTL (N)	Difference in BA (%)	Difference in DTL (%)
Standard (without inerter)		2.0185	1141.81	Reference
Series Suspension	Passive inerter	1.9303	1178.70	−4.37	3.23
	Linearly increasing variable inerter	1.9325	1183.34	−4.26	3.64
	Non-linearly increasing variable inerter	2.6093	1386.27	29.27	21.41
Parallel Suspension	Passive inerter	2.0010	1152.82	−0.87	0.96
	Linearly increasing variable inerter	1.9856	1163.35	−1.63	1.89
	Non-linearly increasing variable inerter	2.0254	1146.27	0.34	0.39

summarize the utopia and knee point comparisons for each Pareto front. These representative points allow clearer assessment of suspension performances under extreme and practically relevant trade-off conditions, respectively. The utopia point is found by the extremities of Pareto front which defines the values that simultaneously minimize both objectives, and it often acts as the ideal reference point. Meanwhile, the knee point is found by the optimal set of variables which represents the best trade-off between the two objectives, having the shortest distance from the utopia point, and it is often chosen as the design solution.

At the utopia point, both passive and linearly increasing variable inerters consistently outperform the conventional suspension in terms of ride comfort. In series suspension layout, the passive inerter and the linearly increasing variable inerter provide nearly identical reductions in RMS vehicle body acceleration, which are 3.40% and 3.37%, respectively, accompanied by moderate increases in RMS dynamic tire load. This indicates that the benefits of inertance variability are limited when the relative acceleration across the inerter remains small. In contrast, the non-linearly increasing variable inerter exhibited a substantial deterioration in both ride comfort and road holding, confirming that excessively growing inertance variation led to unfavorable inertial force. In the parallel suspension layout, the linearly increasing variable inerter achieved the largest reduction in ride comfort among all configurations, approximately 3.80%, while maintaining a minimal increase in RMS dynamic tire load (below 1%). Compared with the passive inerter which achieved a 2.93% ride comfort improvement with negligible dynamic tire load, the linearly increasing variable inerter offered a modest but measurable additional benefit at the utopia point.

The knee point analysis provides insight into performance under more realistic compromise conditions. For the series layout, the passive and linearly increasing variable inerter again exhibited very similar behavior, with ride comfort improvement of 4.37% and 4.26%, respectively, and comparable gains in RMS dynamic tire load. This further supported the observation that, under step road excitation, the inertance variability is not strongly activated due to limited relative acceleration across the suspension. However, both inerter types outperform the conventional suspension, confirmed the fundamental benefit of introducing inertial elements into vehicle suspension applications. Meanwhile in the parallel layout, the linearly increasing variable inerter delivered a 1.63% reduction in RMS vehicle body acceleration, which is higher compared to 0.87% for the passive inerter, with only a slightly higher increase in RMS dynamic tire load.

Overall, the utopia point and knee point evaluations confirmed that the passive and linearly increasing variable inerters provide a comparable baseline performance under the simulated environment, while the linearly increasing variable inerter offered slightly higher ride comfort improvement and a broader Pareto-optimal solution space, particularly in parallel configuration, indicating increased flexibility in achieving different comfort-handling trade-offs. Importantly, these benefits are achieved without active control input or external energy execution, positioning the variable inerter as a promising intermediate solution between purely passive and purely active suspension systems.

4.1.3. Analysis of Optimized Design Variables

The optimal suspension parameters associated with the knee point for each configuration are listed in

Table 5. Design variables traced back from knee point in Figure 5 under step road profile.

Types of Suspension System		Suspension Damping, c (N·s/m)	Passive Inertance, b (kg)	Variable Inertance
				Minimum Inertance, bmin (kg)	Maximum Inertance, bmax (kg)	Inertance Variation (%)
Standard (without inerter)		1549	-	-	-	-
Series Suspension	Passive inerter	1587	316.91	-	-	-
	Linearly increasing variable inerter	1586	-	283.15	285.15	0.71
	Non-linearly increasing variable inerter	1325	-	73.69	96.89	31.48
Parallel Suspension	Passive inerter	1503	0.33	-	-	-
	Linearly increasing variable inerter	1458	-	0.46	0.64	39.13
	Non-linearly increasing variable inerter	1628	-	1.13	2.26	100

. These parameters provide insight into how the optimization process balanced ride comfort and road-holding ability under practical trade-off conditions. For conventional suspension, the knee point solution corresponds to a suspension damping of 1549 N·s/m, serving as the baseline for comparison. When the inerters were introduced in series configuration, both passive inerter and linearly increasing variable inerter reduced to similar suspension damping values of 1587 N·s/m and 1586 N·s/m, respectively, showing that a considerable level of dissipation in suspension damping was required to achieve balanced suspension performance. The optimized inertance range for the linearly increasing variable inerter spans a narrow range, from 283.15 kg to 285.15 kg only, suggested that only limited inertance variation was activated at the knee point. This observation was also consistent with the earlier Pareto analysis, where the performance of the passive inerter and linearly increasing variable inerter were shown to be similar due to the relatively small change in relative acceleration across the suspension. In contrast, the non-linearly increasing variable inerter adopted a significantly lower inertance range with a wider variation, which explained its weaker performance resulting from excessive sensitivity to acceleration change.

In the parallel suspension configuration, the optimized parameters exhibited clearer distinction. The inclusion of the inerter is to improve the ride comfort without deteriorating the dynamic tire load, while maintaining the vehicle dynamics. Hence, the linearly increasing variable inerter operated within a modest inertance range, from 0.46 kg to 0.64 kg, combined with a reduced suspension damping compared to the passive inerter. Conversely, the non-linearly increasing variable inerter had a much larger inertance range, from 1.13 kg to 2.26 kg, and higher suspension damping, reflecting a reduced balance between inertial and dissipative mechanisms, which aligned with its poor performance.

In short, the knee point design variables confirmed that better suspension performance was achieved through linear and moderate inertance changes, rather than large or non-linear inertance changes. The optimized parameters further supported the results that the linearly increasing variable inerter behaves similarly to a passive inerter under step excitation, while offering additional flexibility and improved performance benefits without increasing system complexity and energy demands.

4.2. Performance Analysis Under Multiple Road Profiles

To further evaluate the performance of the proposed speed-dependent variable inerter, additional simulations were conducted under multiple road profiles beyond step road input, including a sinusoidal road profile and random road profile based on standard road roughness classifications, ISO 8608:1995 class A [53]. The simulations focused exclusively on parallel suspension configuration, as it consistently demonstrated superior performance over series configuration in terms of ride comfort and overall trade-off characteristics in the preceding step road profile study. Specifically, the parallel linearly increasing variable inerter exhibited a reduction in RMS vehicle body acceleration while maintaining a comparable dynamic tire load. In contrast, the series configuration shows limited improvements, with greater compromise in road holding.

The parallel configuration also offered a stable and well-defined dynamic model under continuously varying inputs, making it suitable for simulations under harmonic and stochastic excitations. Meanwhile, the series configuration involved an intermediate node mass with negligible magnitude, which led to strong coupling between acceleration and inerter force under dynamic excitation. This resulted in a direct feedthrough path without sufficient dynamic delay. As a result, the solver encountered algebraic loop conditions and numerical instability.

Lastly, the comparison among conventional suspension, passive inerter, and variable inerter systems with both linearly and non-linearly increasing inertance profiles within the parallel configuration was carried out. This approach allowed a clearer evaluation of the performance improvements of the speed-dependent variable inerter under more realistic operating conditions.

4.2.1. Performance Under Random Road Profile

The Pareto front obtained under a random road profile is shown in Figure 8, with the corresponding utopia and knee point comparisons summarized in

Table 6. Utopia point of conventional, parallel passive inerter, parallel linearly increasing and non-linearly increasing variable inerter suspension systems under random road profile.

Types of Suspension System	Utopia Point
	RMS BA (m/s2)	RMS DTL (N)	Difference in BA (%)	Difference in DTL (%)
Standard (without inerter)	0.1544	96.63	Reference
Passive inerter	0.1539	96.63	−0.31	0
Linearly increasing variable inerter	0.1468	96.63	−4.94	0
Non-linearly increasing variable inerter	0.1624	96.69	5.20	0.07

and

Table 7. Knee point of conventional, parallel passive inerter, parallel linearly increasing and non-linearly increasing variable inerter suspension systems under random road profile.

Types of Suspension System	Knee Point
	RMS BA (m/s2)	RMS DTL (N)	Difference in BA (%)	Difference in DTL (%)
Standard (without inerter)	0.1692	103.15	Reference
Passive inerter	0.1656	105.56	−2.11	2.34
Linearly increasing variable inerter	0.1651	106.27	−2.39	3.03
Non-linearly increasing variable inerter	0.1741	100.91	2.90	−2.17

, respectively. The simulation results exhibited a consistent trade-off relationship between ride comfort and road-holding ability, similar to the trend observed under the step road profile.

Referring to Figure 8, it can be seen that the parallel linearly increasing variable inerter shifted the Pareto front towards lower-RMS vehicle body acceleration regions and showed a better improvement in ride comfort. Comparing between utopia points, the linearly increasing variable inerter also demonstrated the most significant improvement in ride comfort, achieving an improvement of approximately 4.94% compared to conventional suspension, while maintaining a similar dynamic tire load. In contrast, the passive inerter provided a marginal improvement of 0.31% only, indicating limited effectiveness under random road excitation. The non-linearly increasing variable inerter, however, resulted in a deterioration of ride comfort by 5.20%, proving that excessive inertance variation leads to unfavorable dynamic responses.

The knee point analysis further highlighted the trade-off between ride comfort and road holding under practical, real-world operating conditions. The linearly increasing variable inerter achieved up to a 2.39% reduction in RMS vehicle body acceleration and slightly outperformed the passive inerter, which improved ride comfort by 2.11%, despite a moderate increase in RMS dynamic tire load. This observation suggested that the linearly increasing variable inertance is able to enhance vibration reduction under a random road profile, in which the variation in road input leads to higher relative acceleration across the suspension. As a result, the vehicle ride performance improvement became more pronounced compared to step road excitation.

Overall, the results confirmed that the linearly increasing variable inerter provided consistent and better improvement in ride comfort compared to conventional, passive inerter and non-linearly increasing variable inerter suspensions under both step road and random road profiles, while preserving acceptable road-holding ability. These findings highlighted the applicability of smooth inertance variation for practical vehicle suspension applications.

4.2.2. Performance Under Sinusoidal Road Profile

The Pareto front under a sinusoidal road profile is presented in Figure 9, whereas its corresponding utopia point and knee point are tabulated in

Table 8. Utopia point of conventional, parallel passive inerter, parallel linearly increasing and non-linearly increasing variable inerter suspension systems under sinusoidal road profile.

Types of Suspension System	Utopia Point
	RMS BA (m/s2)	RMS DTL (N)	Difference in BA (%)	Difference in DTL (%)
Standard (without inerter)	0.1577	55.64	Reference
Passive inerter	0.1563	55.68	−0.84	0.08
Linearly increasing variable inerter	0.1562	55.64	−0.93	−0.01
Non-linearly increasing variable inerter	0.1563	55.64	−0.90	−0.01

and

Table 9. Knee point of conventional, parallel passive inerter, parallel linearly increasing and non-linearly increasing variable inerter suspension systems under sinusoidal road profile.

Types of Suspension System	Knee Point
	RMS BA (m/s2)	RMS DTL (N)	Difference in BA (%)	Difference in DTL (%)
Standard (without inerter)	0.1577	55.66	Reference
Passive inerter	0.1563	55.74	−0.86	0.15
Linearly increasing variable inerter	0.1562	55.67	−0.93	0.03
Non-linearly increasing variable inerter	0.1563	55.67	−0.90	0.03

. Similar to the step and random road profiles, the linearly increasing variable inerter provided the best trade-off between ride comfort and road-holding ability, since it significantly shifted the Pareto front towards the origin.

Based on Table 8 and Table 9, it can be observed that all inerter-based suspensions showed relatively small improvements in ride comfort compared to the conventional suspension under a sinusoidal road profile. The linearly increasing variable inerter achieved the largest ride comfort improvement, with approximately 0.93% at both the utopia point and knee point, slightly outperforming the non-linearly increasing variable inerter which offered 0.90% of ride comfort improvement, followed by the passive inerter which improved by 0.84% and 0.86% at the utopia point and knee point, respectively. Overall, these improvements were modest, and all configurations exhibited minor variations only in dynamic tire load.

The limited performance improvements under a sinusoidal road profile can be attributed to the limited variation in relative acceleration across the suspension system. Unlike random road input, which introduced continuously varying excitation over a broad frequency range, sinusoidal excitation was characterized by a single dominant frequency. As a result, the variation in inertance was constrained and led to a smaller performance gain.

Comparing all of the simulation results across multiple road profiles, it is proved that the performance of the variable inerter depends on the nature of road excitation. Under step input, the performance improvement was moderate due to limited variation in dynamic variation. Under the sinusoidal road profile, the improvement remained relatively small as the single frequency input restricted inertance variation. Meanwhile, under the random road profile, the continuously varying input induced higher relative acceleration, allowing the variable inerter to operate effectively and deliver more significant performance improvements. Overall, the linearly increasing variable inerter consistently demonstrated the largest performance improvement across multiple road conditions over all suspension types, providing improved ride comfort while maintaining acceptable road holding. The findings confirmed that smooth inertance variation was the most effective, while it does not require a complex control strategy or additional energy input, thereby highlighting the applicability of the variable inerter in vehicle suspension under realistic driving conditions.

4.3. Physical Realization of Speed-Dependent Variable Inerter

The practical implementation of the proposed speed-dependent variable inerter can be realized using mechanical mechanisms, such as ball–screw and rack-and-pinion systems coupled with rotating flywheels. In conventional passive inerter design, the inertance is constant and pre-defined based on the flywheel’s moment of inertia and gear transmission. In this study, the variable inertance can be interpreted as a speed-dependent moment of inertia of the flywheel within the inerter, which can be achieved through passive mechanical means using smart materials or control strategies.

One of the physical realizations of variable inertance is through a mechanism in which the moment of inertia of the flywheel varies with its rotational speed [9,67,68,69]. For example, movable masses or sliders attached with a returning spring can be introduced into the inerter’s flywheel [9,67,69,70]. As the flywheel rotates, the slotted sliders experience centrifugal force and push the sliders radially outwards, leading to a greater moment of inertia and inertance. The returning spring pulls the sliders back to the origin when the flywheel slows down, offering a fully passive yet adaptive solution. The use of smart materials such as magnetorheological or electrorheological fluids can also contribute to variable inertance [37,71,72,73]. The inertance of a magnetorheological inerter is varied by changing its viscosity in the presence of a magnetic field to control damping [74]. Meanwhile, the electrorheological inerter uses electrorheological fluid to change its viscosity in response to an electric field [36,74]. Therefore, the proposed speed-dependent variable inerter provided a practically feasible and mechanical realizable approach for achieving a variable inertial response in vehicle suspension systems without increasing system complexity or energy consumption.

5. Conclusions

This study presents a comprehensive investigation of a speed-dependent variable inerter for vehicle suspension applications, with particular focus on whether variable inertance can provide observable benefits over conventional and passive inerter suspensions. Using a QVM, seven suspension configurations including the conventional, passive inerter, linearly increasing variable inerter and non-linearly increasing variable inerter in both series and parallel configurations were evaluated. A multi-objective genetic algorithm was used to optimize both suspension damping and inertance variation with respect to ride comfort and road-holding ability under step, random and sinusoidal road excitations. The result showed that the parallel linearly increasing variable inerter provided the most favorable balance between the two conflicting objectives. Its performance improvements were modest under a sinusoidal road profile and become more significant under a step road profile, followed by a random road profile, where continuous varying dynamics allow greater inertance variation. Specifically, it can deliver up to 3.80%, 4.94% and 0.93% improvements in ride comfort relative to conventional suspension at the utopia point under step, random and sinusoidal road profiles, respectively. When compared to the more practical knee point, it is still able to contribute ride improvements of 1.63%, 2.39% and 0.93%, with only a marginal impact on road holding. Overall, the findings demonstrated that integrating a smooth, linearly varying inertance profile can enhance ride comfort while preserving road-holding ability, even if performance gains were modest. Importantly, these benefits required no active control input or additional energy consumption, positioning the variable inerter as a promising solution between passive simplicity and active adaptability in vehicle suspension design.