Modeling of the Dynamic Characteristics for a High-Load Magnetorheological Fluid-Elastomer Isolator

Abstract

To meet the vibration isolation requirements of engines under diverse operating conditions, this paper proposes a novel magnetorheological fluid-elastomer isolator with high load and tunable parameters. The mechanical and magnetic circuit structures of the isolator were designed and optimized through theoretical calculations and finite element simulations, achieving effective vibration isolation within confined spaces. The dynamic performance of the isolator was experimentally evaluated using a hydraulic testing system under varying excitation amplitudes, frequencies, initial positions, and magnetic fields. Experimental results indicate that the isolator achieves a static stiffness of 3 × 10⁶ N/m and a maximum adjustable compression load range of 105.4%. In light of the asymmetric nonlinear dynamic behavior of the isolator, an improved nine-parameter Bouc–Wen model is proposed. Parameter identification performed via a genetic algorithm demonstrates a model accuracy of 95.0%, with a minimum error reduction of 28.8% compared to the conventional Bouc–Wen model.

1. Introduction

The engine serves is a major source of vibration during vehicle operation, with its complex framework, cyclic fuel ignition, and sporadic energy output generate considerable oscillations. These vibrations not only damage vehicle components, shortening the service life of the body, but also reduce passenger comfort. Engine mounts, which connect the vehicle frame to the engine, are designed to suppress the bidirectional transmission of vibrations between the engine and the body while also supporting the engine [1]. An ideal engine mount should exhibit high stiffness and damping at low frequencies, and low stiffness and damping at high frequencies [2].

Typically, mounts are categorized as passive, semi-active, and active types [3]. Passive mounts are mainly divided into rubber mounts and hydraulic mounts, both of which have fixed damping and stiffness once they are set, limiting their adaptability [4]. Active mounts work by using actuators to apply dynamic forces opposite to the engine’s vibration sources, theoretically providing superior isolation [5]. However, active mounts have a relatively complex structure, high production costs, and consume significant energy, limiting their practical applications. In contrast, semi-active mounts offer controllable stiffness and damping. Compared to passive mounts, they better meet the demands for engine vibration reduction and ride comfort while consuming significantly less energy than active mounts and providing excellent isolation performance across a wide frequency range.

Magnetorheological (MR) mounts, as a type of semi-active mount with rapidly reversible and controllable stiffness and damping, theoretically provide broad-frequency vibration isolation for engines. Yang et al. [6] designed and optimized a step-by-step bypass MR damper for heavy-duty vehicles. The damping channel length of this damper reached 46 mm, with a maximum output damping force of approximately 4.5 kN and a damping force adjustability range of up to 5.4 times. One year later, Yang et al. [7] developed a labyrinth-style dual-channel valve MR damper for seat suspensions. Experimental results indicated that, under a 2 A current, the damping force could reach 21 kN, with a damping force range that was adjustable by 5.61 times. Jiang et al. [8] designed a six-degrees-of-freedom MR damper with a 42 mm damping channel. Experimental results showed that the maximum damping force of this damper was about 27 kN, with an adjustable range of up to 15 times. Chen et al. [9] proposed an energy-fed step-type MR damper, with experiments showing that at a 7 mm amplitude, the damper output damping force could reach 4.5 kN, exhibiting asymmetric tension-compression characteristics. Liang et al. [10] introduced a novel MR damper with an asymmetric conical flow channel, where the damping channel length was 38.4 mm. The conical flow channel allowed for asymmetric force output. Hu et al. [11] proposed a new MR damper for vibration energy harvesting with a 13 mm damping channel, achieving a maximum damping force of 1.62 kN, with an adjustable damping range exceeding 10 times. Xi Jun et al. [12] proposed a design methodology for MR dampers with variable inertia and damping for torsional vibrations, investigating the influence of excitation current and rotational speed on the damper’s output inertia and damping force, achieving a damping force adjustability range of up to 2.8 times. Jiang et al. [13] developed a dual-tube MR damper with a bending channel and a diameter of 114 mm. At an 8 mm amplitude and 0.5 A current, the maximum damping force reached 4.5 kN, with a range that was adjustable by 10.76 times. To achieve better damping performance within a constrained volume, Hu et al. [14] designed and optimized a novel MR damper with a diameter of 79 mm. Without any external current, the output damping force was 0.116 kN, and with a 2 A excitation current, the output damping force reached 7.139 kN.

To accurately describe the dynamic characteristics of MR dampers, mathematical models must be established, typically classified into parametric and non-parametric models [15]. Among these, non-parametric models are data-driven and do not assume a predefined mechanical structure. And parametric models primarily focus on the viscoelastic properties of MR dampers, with simple forms of expression. In these models, the damper is considered as a combination of several damping and elastic elements, with model parameters determined based on experimental data. These models offer high interpretability and facilitate direct correlation between magnetic input and mechanical response, which is advantageous for control-oriented applications. Yan et al. [16] designed an MR damper with a double-ring damping gap and conducted magnetic circuit simulations to validate its feasibility. A theoretical model for damping force under high-speed conditions was derived, and a mathematical model for damping force at high speeds was established. Comparison with experimental results showed a good fit between the theoretical model and the experimental data. One year later, Yan et al. [17] obtained an analytical solution for the pressure drop gradient across the damping gap based on the Bingham model and proposed a four-parameter theoretical model that more effectively reflects the mechanical characteristics of the damping gap at both low and high speeds. Simulation results of the proposed theoretical model, compared with experimental data, showed that the model effectively described the experimental results. Jiang et al. [18] introduced a new phenomenological model to describe the nonlinear dynamic behavior of MR dampers. They tested a multi-channel bypass MR damper with three operational modes, and the new phenomenological model demonstrated higher accuracy compared to the commonly used Bouc–Wen and Bingham models. Deng et al. [19] proposed a new hysteresis characteristic model for MR dampers based on gray theory. At different frequencies and currents, the fitting results were consistent with the measured data, showing good smoothness and no oscillation in the fitting curves, with an average relative error within 2.04%. Lv et al. [20] applied neural networks and classification algorithms to model MR damper systems. By predicting the desired damping force and using decision tree classification algorithms, the expected current for the MR damper was adjusted to meet the immediate damping force requirements of the structural system. Experimental results showed that the damping force predicted by this method had an error of 5.28%. Qian et al. [21] proposed a modified Bouc–Wen model that can be adapted to various currents, excitation frequencies, and amplitudes. The parameters in the Bouc–Wen model were identified by combining the harmonic balance method and the genetic algorithm. The agreement between the modified prediction model and the test results was verified under a sinusoidal excitation of 80 mm and 1 Hz.

The studies primarily focus on individual MR dampers. Jiang et al. [22] proposed an MR fluid-elastomer isolator for a helicopter rotor with a maximum stroke of 5 mm, emphasizing the design and optimization of the magnetic circuit structure. By modeling the elastomer in the isolator as two nonlinear springs and a damper, they introduced an improved eight-parameter Bouc–Wen model. Comparison with experimental data showed that the model effectively fitted the force–displacement and force–velocity curves. Yin [23] proposed a flow-type MR isolator with a circular damping channel, based on traditional hydraulic mounts. By serially connecting a spring and an MR damper, isolation was achieved. An improved Bingham model was then proposed, which accurately predicted the dynamic characteristics of the isolator. Zhu et al. [24] introduced an isolator consisting of a series connection of an MR damper and an MR elastomer. The dynamic characteristics of the isolator were described using the Bouc–Wen model, and experimental verification confirmed the accuracy of model. Wang et al. [25] proposed an MR isolator consisting of magnetorheological fluid (MRF) encapsulated in polybutadiene polyol-based polyurethane, with a diameter of approximately 40 mm. They developed a phenomenological model consisting of a variable friction damper and a nonlinear spring, which successfully described the dynamic characteristics of the isolator.

Therefore, to achieve adaptive vibration reduction for engines under different working conditions, this paper presents a novel high-load MRF-elastomer isolator as the engine mount. The mechanical configuration and magnetic circuit of the isolator were developed and refined using theoretical computations and finite element analysis techniques, considering the actual installation conditions and performance criteria of the isolator. The dynamic behavior of the MR isolator was tested under different operating conditions, and the effects of excitation frequency, displacement amplitude, initial position, excitation current, and other factors on the load were analyzed. Based on the dynamic performance characteristics of the isolator, an improved Bouc–Wen model was proposed. Parameter identification was then performed using a genetic algorithm. The accuracy and correctness of the improved model were verified by comparing with experimental data and the complex modulus method.

The main contributions and innovations of this work can be summarized as follows:

(1) The proposed MR isolator achieves a high static stiffness and a compressive load adjustment range. It provides a broader variable range compared to those with high initial stiffness, and higher initial stiffness compared to designs with similar tunability.

(2) By integrating a rubber elastic element and an MRF damping unit in a parallel configuration, the design effectively balances high load support and frequency adaptability within a constrained space.

(3) A modified nine-parameter Bouc–Wen model is introduced that incorporates the influence of initial position, which is an important factor neglected in eight-parameter models. This improvement reduces modeling error and provides a more accurate characterization of the nonlinear hysteresis of the system.

We believe these contributions provide meaningful advances in the design of MR-based isolators, particularly for applications requiring a high load capacity, a compact form factor, and improved dynamic performance.

2. MRF-Elastomer Isolator Structure Design and Optimization

2.1. Analysis of Static Stiffness

To meet the high-load, wide-frequency, variable-parameter damping requirements, this paper proposes an MRF-elastomer isolator, as illustrated in Figure 1. Figure 2 shows a sketch of the engine–vehicle body isolation system.

The shell and the inner sleeve are linked by rubber, with MRF filling the inner shell. The shell is connected to the engine by an install bracket, and the transition seat is connected to the vehicle body. When the isolator shell is exposed to force, the fluid injection end cap is fastened by the transition base, resulting in a synchronous displacement of the piston and the rubber. In the study of the isolator’s working principle, its effect is regarded as the combination of the elastomeric damping part and the MRF damping part.

When the isolator is operating, the elastomeric damping element is subjected to an axial load $P$ as shown in Figure 3. The stiffness of rubber can be regarded as the parallel connection of the upper and lower elastomeric parts, as shown in Equation (1). Therefore, the total axial stiffness $k_{sum}$ of the isolator is as follows:

$$k_{sum}=k_1+k_2$$

(1)

The upper part is conical in shape, with the rubber in a combined shear and compression-tension working mode. The stiffness $k_1$ of the upper part can be approximated as follows [26]:

$$k_1={\displaystyle \frac{\pi (R_2^2-R_1^2)}{R_2-R_3}}\left[\left(1+{\displaystyle \frac{m(h-h_1)}{2(R_2-R_3){\cos }^2\alpha }}\right)E\tan \alpha +G\cot \alpha \right],$$

(2)

The lower part is cylindrical in shape, with the rubber in a pure shear working mode. According to Hooke’s law, the stiffness of the lower part can be obtained as follows:

$$k_2={\displaystyle \frac{2\pi R_2{h}_{1}G}{R_2-R_3}},$$

(3)

in the formula, $R_1$ denotes the internal radius at the top section of the outer shell, $R_2$ represents the internal radius at the bottom portion of the outer shell, and $R_3$ corresponds to the external diameter at the base region of the inner sleeve. The parameter h indicates the total height of the isolator, with $h_1$ specifying the vertical measurement of its lower segment. The angle α characterizes the inclined angle of the external protective layer. Material properties include $G$ as the shear modulus and $E$ as Young’s modulus, complemented by the m coefficient associated with the material’s hardness where $m$ is constant at 1.05.

Natural rubber, selected for its high elasticity, outstanding abrasion resistance, and low compression set, is utilized as the material. The relationship between the shear modulus $G$, elastic modulus $E$, and Shore hardness $HA$ of natural rubber satisfies the following equation [27]:

$$E={\displaystyle \frac{15.75+2.15HA}{100-HA}},G\doteq E/3,$$

(4)

To make full use of the limited installation space, the outer shell radius $R_2$ and total height h are selected based on their maximum allowable dimensions. The inner radius $R_1$ at the upper end of the outer shell can be regarded as a dependent variable of $h_1$ and $\alpha$.

$$R_1=R_2-(h-h_1)\tan \alpha$$

(5)

From Equation (5), it can be seen that $R_1$ increases with a larger $h_1$ and smaller $\alpha$. Substituting Equations (2), (3), and (5) into Equation (1) yields the total stiffness of the elastomeric component.

$$k_{sum}={\displaystyle \frac{\pi (R_2^2-{\left(R_2-(h-h_1)\tan \alpha \right)}^2)}{R_2-R_3}}\left[\left(1+{\displaystyle \frac{m(h-h_1)}{2(R_2-R_3){\cos }^2\alpha }}\right)E\tan \alpha +G\cot \alpha \right]+{\displaystyle \frac{2\pi R_2{h}_{1}G}{R_2-R_3}}$$

(6)

as indicated in Equation (6), enhancing the tilt angle $\alpha$ contributes to higher system stiffness. At a 7.5° inclination configuration, Figure 4 visually presents the correlation between the isolator’s total stiffness $k_{sum}$ and critical design variables $R_3$ (inner sleeve base radius) and $h_1$ (lower segment height). The results demonstrate an inverse proportionality: system rigidity diminishes as both the vertical dimension $h_1$ of the shear-dominant elastomer section and the material thickness increase. Moreover, the larger $h_1$ is, the smaller the effect of the rubber thickness on the total stiffness. The larger the rubber thickness, the smaller the effect of $h_1$ on the total stiffness.

When $h_1=43.5\mathrm{mm}$, the relationship between the total stiffness $k_{sum}$ of the isolator and the structural parameters $\alpha$ and $R_3$ is shown in Figure 5. It can be observed that the total stiffness increases with the inclination angle $\alpha$ of the rubber, while it decreases with an increase in the thickness of rubber. Moreover, the smaller the thickness, the greater the effect of the inclination angle on the total stiffness. The larger the inclination angle, the greater the effect of the thickness on the total stiffness.

When $R_3=31 \mathrm{mm}$, the relationship between the total stiffness $k_{sum}$ of the isolator and the structural parameters $\alpha$ and $h_1$ is shown in Figure 6. It can be observed that the smaller the height $h_1$, the greater the effect of the inclination angle on the total stiffness. The larger the inclination angle, the greater the effect of $h_1$.

Since the inner sleeve of the isolator also serves as the magnetic shell for the damper, the height $h_1$ must be designed to accommodate the maximum stroke of the piston while meeting the static stiffness requirements. The internal radius $R_3$ at the base of the inner sleeve must satisfy performance criteria for vibration reduction and excitation requirements. Meanwhile, the inner radius $R_1$ constrains the allowable dimension of the piston rod.

Consequently, $h_1$, $R_1$, and $R_3$ should be maximized, and α should be minimized. However, enhancing the total stiffness of the isolator requires a reduction in $h_1$ and an increase in $\alpha$. Thus, the structure of the isolator needs to be optimized to meet both the static stiffness and damping force requirements.

2.2. Analysis of Damping Force

In the MR damper, the pressure differential induced by the reciprocating motion of the piston causes the MRF to transfer from the high-pressure chamber to the low-pressure chamber through the damping conduit. By controlling the magnetic field intensity at the damping channel, the rheological properties of the MRF are altered, thereby achieving the purpose of controlling the damping force. The operating mode of the MR damper is of the shear valve type, and the damping force is as follows [28]:

$$F={\displaystyle \frac{12\eta l{A}^2}{2\pi r_1{w}^3}}\nu +{\displaystyle \frac{3lA\tau \left(B\right)}{w\pi }}\mathrm{sgn}(\nu )$$

(7)

$$A=\pi (r_2^2-r_1^2)$$

(8)

where $\eta$ describes the viscosity of MRF, $l$ corresponds to the axial length of the damping channel, $r_2$ defines the outer radius of the piston, $r_1$ reflects the inner radius of the piston rod, $w$ characterizes the width of the damping channel, and $\tau$ represents the shear yield stress of MRF. Equation (7) consists of two distinct components: the initial term quantifies the velocity-dependent viscous damping force proportional to motion speed v, while the subsequent term embodies the magnetic field-regulated Coulomb damping force. Notably, when $B=0$, the yield stress $\tau (B)$ diminishes entirely, eliminating the Coulomb damping contribution.

From Equation (7), it can be seen that the smaller the piston rod radius $r_1$ and the gap $w$ of the damping channel are, the larger the two damping forces will be. The piston rod radius $r_1$ is limited by the inner radius $R_1$ of the upper end of the outer shell and must meet the tensile and compressive strength requirements of the piston.

$$r_1\ge \sqrt{{\displaystyle \frac{F_{\max }}{\pi [\sigma ]}}}$$

(9)

in the equation, $F_{\max }$ is the maximum output damping force of the damper, and $\left[\sigma \right]$ is the allowable stress of the material.

Modifications in the excitation current induce corresponding variations in magnetic flux density within the damping channel, directly modulating the Coulomb damping force magnitude. The width of the damping channel critically determines the magnetic flux intensity distribution. Under constant excitation current and coil configurations, narrowing the channel gap reduces magnetic reluctance, amplifying flux density. To satisfy engine vibration control requirements, the damping force adjustment range is defined as $F\in [-8050N,8050N]{\displaystyle \frac{\delta y}{\delta x}}$, with an initial channel clearance of 1 mm. Both viscous and Coulomb damping forces exhibit positive correlations with piston radius $r_2$ and cumulative channel length l. The maximum feasible value of $r_2$ is constrained by the outer radius $R_3$ of the inner sleeve and the width of damping channel $w$. Lengthening the controllable damping channel improves magnetic field efficiency and increases Coulomb damping capacity. However, this parameter is physically limited by the piston travel range and the height h₁ of the inner sleeve, which is initially set to 40 mm.

2.3. Magnetic Circuit Analysis

In the magnetic circuit design of the MR damper, lower reluctance corresponds to higher operational efficiency. As illustrated in Figure 7, the magnetic circuit consists of five critical components. $R_{m1}$ represents the equivalent reluctance of the central region within the piston core. $R_{m2}$ corresponds to the reluctance of the damping gap. $R_{m3}$ and $R_{m4}$ denote the reluctances of the upper and lower sections of the piston assembly, respectively. And, $R_{m5}$ quantifies the equivalent reluctance contributed by the inner sleeve. When a magnetic field is applied, the ferromagnetic particles in the MRF align into chain-like structures along the direction of the magnetic field, altering its rheological properties.

Different materials exhibit different saturation magnetic flux densities. As the magnetic field intensity increases, when a component reaches magnetic saturation, its magnetic flux density no longer increases with further field strength enhancement. To improve the utilization of the magnetic field, it is necessary to ensure that magnetic saturation is reached first at the damping channel. Magnetic circuit optimization involves reverse calculation to determine the magnetic flux cross-sectional area in different regions. The larger the magnetic flux cross-section, the smaller the magnetic reluctance. Therefore, based on Gauss’s law for magnetism and Ampère’s circuital law, the cross-sectional areas in other regions must satisfy the following equation:

$$S_s\ge S_{s,\min }={\displaystyle \frac{B_{f,\max }{S}_f}{B_s}}$$

(10)

where $B_{f,\max }$ and $S_f$ denote the maximum flux density and the cross-sectional area of magnetic flux within the adjustable damping channel, respectively. The parameters $S_{s,\min }$ and $B_s$ represent the minimal flux cross-section and peak flux density across distinct zones, while $S_s$ indicates the effective flux area in auxiliary regions. Due to the significantly higher magnetic permeability of ferromagnetic materials compared to that of MRF and air, flux leakage is neglected in the magnetic circuit optimization. Consequently, the total magnetic reluctance can be approximated by that of the adjustable damping channel. To guarantee a minimum flux density of 600 mT across the controllable damping channel, the required magnetomotive force can be derived through Equation (11).

$$NI\doteq B_f{S}_f\cdot R_{m2}={\displaystyle \frac{l{B}_f}{{\mu }_m}}$$

(11)

in the equation, $B_f$ is the magnetic flux density in the damping channel, $S_f$ is the magnetic flux area at the damping channel, and $\mu$ is the magnetic permeability of the material. To prevent excessive heat generation, the peak current must not surpass 2 A. Furthermore, considering effects such as flux leakage in the real system, the coil turns are fixed at 500, which determines the required coil space.

2.4. Structural Optimization and Simulation Verification

First, the magnetic circuit of the MR damper is modeled and optimized using Ansys software. By adjusting $r_1\in \left[2 \mathrm{mm}, 8 \mathrm{mm}\right]$, $r_2\in \left[20 \mathrm{mm}, 30 \mathrm{mm}\right]$, and $l\in \left[10 \mathrm{mm}, 30 \mathrm{mm}\right]$, the controllable damping channel is designed to reach saturation magnetic flux density first, ensuring that the damping force variation range of the damper is maximized while maintaining the required strength. After optimization, the values of $r_1=4 \mathrm{mm}$, $r_2=25 \mathrm{mm}$, and $l=20 \mathrm{mm}$ are determined. The distribution of the device’s magnetic circuit is shown in Figure 8, where the magnetic flux density at the controllable damping channel reaches saturation first and can achieve 600 mT.

The static displacement under the applied load was obtained through finite element analysis, as shown in Figure 9. Under a 3 kN load, the elastomeric deformation is approximately 1 mm, satisfying the constraints and functional requirements for supporting the vehicle-mounted engine.

This paper first determines the damping scheme and structure of the isolator by considering the installation space, static load, and vibration reduction requirements, with the damper and elastomeric isolator connected and linked in parallel. Then, by balancing the total stiffness of the rubber isolator, R₃, and h₁, the parameters of the outer shell and inner sleeve are optimized. Simulation confirms that the static stiffness meets requirements. Based on the damping force requirements, the piston structure is initially designed. The dimensions of the damper working section are optimized according to the structural layout and magnetic circuit design specifications. Finally, the dimensions of the isolator piston are listed in

Table 1. Main parameters of the magnetorheological fluid-elastomer isolator.

Name	Symbols	Size
Radius of the Piston Rod	$r_1$	4 [mm]
Radius of the Piston	$r_2$	25 [mm]
Width of the Damping Channel	$w$	1 [mm]
Length of the Damping Channel	$l$	20 [mm]
Inner Radius of the Upper End of the Shell	$R_1$	32.5 [mm]
Inner Radius of the Lower End of the Shell	$R_2$	43.5 [mm]
Outer Radius of the Lower End of the Inner Sleeve	$R_3$	31 [mm]
Total Height of the Isolator	$h$	139 [mm]
Height of the Lower Part of the Isolator	$h_1$	58 [mm]
Inclination Angle of the Shell	$\alpha$	7.5 [deg]

, and the physical prototype of the isolator is shown in Figure 10.

3. Fatigue Testing and Parameter Identification

3.1. Experimental System

A testing system, as shown in Figure 11, was established to investigate the dynamic mechanical properties of the MRF-elastomer isolator under different conditions of excitation frequency, initial positions, excitation currents, and displacement amplitudes. The isolator was mounted between the upper and lower hydraulic grips of a fatigue testing machine (MTS Landmark 370.25, MTS Systems Corporation, Eden Prairie, MN, USA). The displacement of the lower clamp was controlled via programmed steps set on the control panel. Force and displacement sensors were used to measure the load and relative displacement at both ends of the isolator, respectively. A DC power supply provided the excitation current. When the initial position was set to 0 mm, the MRF-elastomer isolator experienced zero force, allowing for a clear distinction between its tension and compression characteristics. Considering the operating mode of the MRF-elastomer isolator under real-world loading conditions, the initial position was set to −3.5 mm, with the displacement amplitude adjusted to not exceed 3 mm, ensuring that the isolator remained in a compressed state throughout. The experimental scheme is summarized in

Table 2. Fatigue testing experimental plan for magnetorheological fluid-elastomer isolator.

Scheme	Amplitude (mm)	Vibration Frequency (Hz)	Initial Position (mm)	Current (A)
1	1	0.1	0	0/2
2		0.5		0/0.5/1/1.5/2/2.5
3		1		0/2
4		5
5		10
6	3	0.1
7		0.5
8		1
9	5	0.1
10		0.5
11		1
12	1	0.1	−3.5
13		0.5
14		1
15		5
16		10
17	3	0.1
18		0.5
19		1

. A total of 19 groups of tests were conducted under different excitation currents, with each group running for multiple cycles.

3.2. Experiment Result

At the zero initial position with a 0.5 Hz excitation frequency and 1 mm of amplitude, the load–displacement curves at different excitation currents are shown in Figure 12. Figure 13, Figure 14 and Figure 15, respectively, display the load–displacement curves of the MRF-elastomer isolator at different excitation currents under various amplitudes, vibration frequencies, and initial positions.

The load amplitude test results of the MRF-elastomer isolator under different operating conditions are shown in Figure 16.

3.3. Result Analysis

As shown in Figure 12, Figure 13, Figure 14 and Figure 15, the load of the MRF-elastomer isolator increases with the excitation current, displacement amplitude, and frequency. However, as the current continues to increase, the rate of increase becomes smaller. In Figure 13, the major axis in hysteresis elliptical shapes rotates clockwise by increasing displacement amplitude. This indicates that the stiffness of the isolator decreases as the displacement amplitude increases. From Figure 14, there are no significant changes in hysteresis loops by increasing frequency when the isolator is compressed. This may be because the compressive force of the isolator is larger than the tensile force. And according to Equation (7), the damping force is related to the displacement and velocity. When the displacement amplitude of the isolator is only 1 mm, the damping force generated by the isolator is quite small and even less noticeable compared to the larger compressive force. From Figure 16, frequency variation has the smallest effect on load amplitude. When the frequency changes, the absolute increment of the maximum load caused by the excitation current remains nearly unchanged. When the displacement amplitude changes, the absolute increment of the maximum load caused by the excitation current slightly increases. When the initial position changes, the absolute increment of the maximum load caused by the excitation current slightly decreases. However, under the same operating conditions, the increase in the maximum compression load caused by the excitation current always exceeds the increase in the maximum tensile load.

Under experimental conditions with a displacement amplitude of 1 mm and a frequency of 0.1 Hz (equilibrium position fixed at 0 mm), the compressive load amplitude increases from 3.083 kN to 6.331 kN as the excitation current intensifies, achieving a 105.4% enhancement. When the displacement amplitude is 5 mm, the frequency is 0.5 Hz, and the excitation current ranges from 0 to 2 A (equilibrium position at 0 mm), the initial compressive load amplitude is 16.533 kN, with a 19.98% increase under maximum current excitation. For tensile load characterization under a displacement amplitude of 1 mm, a frequency of 0.5 Hz, and an excitation current range of 0–2 A (equilibrium position at 0 mm), the initial tensile load amplitude measures 3.115 kN, showing an 83.7% increase at peak current. Under dynamic loading conditions (5 mm displacement amplitude, 1 Hz frequency, 0 A baseline current, equilibrium position at 0 mm), the tensile load amplitude is 13.177 kN, rising by 22.3% upon current activation.

The complex modulus approach for calculating equivalent stiffness and damping is a typical method for describing the linear behavior of dampers [21], where the complex stiffness k* is viewed as the combination of storage stiffness k′ and loss stiffness k″.

$$k^{*}=k\prime +k″$$

(12)

The position data and measured loads from experiments can be approximated using the dominant Fourier harmonic components (sine and cosine) corresponding to the excitation frequency, as outlined below:

$$X=X_c\cos (2\pi ft)+X_s\sin (2\pi ft)$$

(13)

$$F=F_c\cos (2\pi ft)+F_s\sin (2\pi ft)$$

(14)

The force can be expressed as the combination of the synchronous elastic force and the perpendicular resistive force as follows:

$$F=k\prime X(t)+(k″/2\pi ft)\stackrel{·}{X}(t)$$

(15)

Based on Equations (12)–(15), the in-phase stiffness k′ and the loss stiffness k″ are given by the following equations:

$$k\prime ={\displaystyle \frac{F_c{X}_c+F_s{X}_s}{X_c^2+X_s^2}}$$

(16)

$$k″={\displaystyle \frac{F_c{X}_s-F_s{X}_c}{X_c^2+X_s^2}}$$

(17)

Therefore, the equivalent damping C_MR can be approximated as follows:

$$C_{MR}\approx {\displaystyle \frac{k″}{2\pi f}}={\displaystyle \frac{F_c{X}_s-F_s{X}_c}{2\pi f(X_c^2+X_s^2)}}$$

(18)

Based on the measured data, the equivalent damping C_MR and in-phase stiffness k′ of the MRF-elastomer isolator are listed in

Table 3. Calculation of stiffness and damping using the complex modulus method.

Scheme	Stiffness/N/m			Damping/Nsm-1
	0 A	2 A	Variation	0 A	2 A	Variation
1	3.115 × 106	5.468 × 106	75.6%	6.09 × 105	4.73 × 106	676.0%
2	3.121 × 106	3.019 × 106	−3.3%	1.43 × 105	2.24 × 106	1472.9%
3	3.167 × 106	5.327 × 106	68.2%	7.13 × 104	4.46 × 105	525.9%
4	3.272 × 106	5.622 × 106	71.8%	1.76 × 104	9.44 × 104	435.4%
5	3.338 × 106	5.623 × 106	68.5%	9.29 × 103	4.81 × 104	417.8%
6	2.874 × 106	3.332 × 106	15.9%	3.50 × 105	2.14 × 106	510.0%
7	2.897 × 106	2.406 × 106	−16.9%	7.79 × 104	1.19 × 106	1434.6%
8	2.917 × 106	3.383 × 106	16.0%	4.31 × 104	2.20 × 105	409.7%
9	2.811 × 106	3.005 × 106	6.9%	2.82 × 105	1.43 × 106	407.8%
10	2.864 × 106	2.384 × 106	−16.7%	6.36 × 104	1.05 × 106	1545.1%
11	2.888 × 106	3.129 × 106	8.3%	3.36 × 104	1.55 × 105	362.0%
12	3.839 × 106	6.368 × 106	65.9%	7.27 × 105	3.81 × 106	424.4%
13	3.815 × 106	3.849 × 106	0.9%	1.80 × 105	2.15 × 106	1100.3%
14	3.846 × 106	5.849 × 106	52.1%	9.00 × 104	3.74 × 105	315.5%
15	4.081 × 106	6.481 × 106	58.8%	1.93 × 104	7.55 × 104	290.1%
16	4.215 × 106	6.598 × 106	56.5%	1.08 × 104	4.26 × 104	296.3%
17	3.432 × 106	3.944 × 106	14.9%	4.63 × 105	2.06 × 106	346.1%
18	3.481 × 106	2.803 × 106	−19.5%	1.03 × 105	1.45 × 106	1308.7%
19	3.500 × 106	4.006 × 106	14.4%	5.41 × 104	2.10 × 105	288.2%

.

As shown in Table 3, the stiffness of the MRF-elastomer isolator increases with increasing current and frequency, exhibiting a maximum amplitude of 75.6%. Conversely, stiffness decreases with increasing displacement amplitude. Among these factors, frequency has the least absolute effect on the isolator’s stiffness, while current has the greatest effect. The damping of the isolator only increases with increasing current, reaching a maximum amplitude of 1545.1%, and decreases with increasing frequency and amplitude. Frequency has the largest absolute effect on the damping of the isolator, whereas amplitude has the smallest effect. When the amplitude and frequency increase or the initial position changes, the relative proportion of stiffness and damping growth with current decreases.

3.4. Dynamic Modeling and Parameter Identification

From the experimental curves, the MRF-elastomer isolator’s dynamic performance shows asymmetric hysteresis loops. The complex modulus method can quickly determine the general trends of stiffness and damping variations with current, but it cannot accurately represent the isolator’s real-time dynamic characteristics. Therefore, it remains necessary to establish a mathematical model to precisely describe the dynamic characteristics of the isolator. The Bouc–Wen model is currently the most widely used model [29], but it has deficiencies in describing low-speed motions. In this model, the isolator’s elastic and damping elements are connected in parallel, and the hysteresis loop operator can describe the isolator’s hysteresis behavior. Its expression is shown as follows:

$$\begin{array}{c}F=kx+c\dot{x}+\theta z\\ \dot{z}=\alpha \dot{x}-\beta \left|\dot{x}\right|{\left|z\right|}^{n-1}z-\gamma \dot{x}{\left|z\right|}^n\end{array}$$

(19)

in the equations, $\alpha ,\beta ,\theta$, and $\gamma$ are dimensionless parameters that influence the shape and size of the load–displacement hysteresis loop. $\alpha$ represents the linearity degree of the hysteresis loop. The parameter $\theta$ is related to the inclination of the model and affects the magnitude of the output force. The smoothness coefficient n is typically set to 2 [30]. $\gamma$ is the width adjustment coefficient of the hysteresis model, $\beta$ is the height adjustment coefficient of the hysteresis model, $k$ is the stiffness of the hysteresis model, and c is the damping of the hysteresis model [31].

To better describe the asymmetry in the compression and extension operating modes of the MRF-elastomer isolator, a step function $\epsilon (x)$ and parameter d are introduced in Equation (19), allowing the stiffness to change based on determining the operating mode of the damper. Additionally, within one cycle, the force–displacement curve of the damper exhibits steps when the displacement decreases from the maximum and is at a distance W from the equilibrium position, and when the displacement increases from the minimum and is at a distance W^′. In Figure 17, $\left[1-\epsilon \left(x\right)\right]k$ represents the stiffness of the damper, $f(x,\dot{x})$ represents the damping of the damper, x represents displacement, and F represents load.

Through observation, the positions $W$ and $W\prime$, as shown in Figure 18, have an approximate relationship with the differences in the isolator’s compression and tension stiffness.

$$W\prime =\left(1-d\right)W$$

(20)

The improved model is shown as follows:

$$\begin{array}{c}F=\left\{1-\epsilon (x)\right\}kx+f(x,\dot{x})c\dot{x}+\theta z\\ \dot{z}=\alpha \dot{x}-\beta \left|\dot{x}\right|{\left|z\right|}^{n-1}z-\gamma \dot{x}{\left|z\right|}^n\end{array}$$

(21)

$$f(x,\dot{x})=\left\{\begin{array}{cc}0,& x\ge W,\dot{x}<0\cup x<-W\prime ,\dot{x}\ge 0\\ O(W-x),& W-1/M\le x<W,\dot{x}<0\\ O\left(W\prime +x\right),& -W\prime \le x<-W\prime +1/O,\dot{x}\ge 0\\ 1,& others\end{array}\right.$$

(22)

$$\epsilon (x)=\left\{\begin{array}{ll}0,& x<0\\ d,& x\ge 0\end{array}\right.$$

(23)

in the equations, d mainly changes the slope of the hysteresis curve during extension, $O$ primarily affects the slope of the hysteresis curve at the step, and $W$ causes the position of the step to change. When the displacement is subjected to a sinusoidal excitation with an amplitude of 1 mm and a frequency of 0.5 Hz, the force–displacement response is shown in Figure 19.

To evaluate the accuracy of the proposed model, the fitness function is defined as follows:

$$J=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left[\frac{\left|F_{\exp i}-F_{simi}\right|}{(F_{\max \mathrm{e}}-F_{\min \mathrm{e}})/2}\right]}^2}}$$

(24)

in the equations, $F_{\exp i}$ and $F_{simi}$ are the test and simulation load values, $F_{\max e}$ and $F_{\min e}$ are the maximum and minimum load experimental data, respectively, and $J$ is the root mean square error between the simulation and test values. The parameters are identified using a genetic algorithm [32]. The simulation results of the hysteresis curves of the MRF-elastomer isolator under different currents for Scheme 2 are shown in Figure 20, and the parameter identification results are shown in

Table 4. Improved model parameter identification results.

I/A	a	β	γ	θ	k/N/m	c/Nsm−1	d	W/mm	O/N/m
0	6301	9254	56,694	309	3.24 × 106	1.05 × 105	0.098	0.890	8116
0.5	727	447	66,829	8881	3.75 × 106	3.10 × 105	0.074	0.694	8952
1	2408	4634	11,571	3484	3.96 × 106	3.96 × 105	0.120	0.643	9567
1.5	943	1928	28,303	8827	4.14 × 106	4.26 × 105	0.158	0.608	11,822
2	2025	23,390	15,213	5872	4.51 × 106	4.83 × 105	0.134	0.625	6473
2.5	1520	2187	41,086	6554	4.73 × 106	5.86 × 105	0.139	0.601	8461

. The parameter d slightly increases with increasing current, W decreases with increasing current, and O fluctuates between 6000 and 12,000.

The trends of stiffness and damping with the current for different models are shown in Figure 21, where the compression stiffness is given in Table 4, and the tensile stiffness is obtained according to Equation (25).

$$k_c=k\times \left(1-d\right)$$

(25)

As shown in Figure 21, all three methods exhibit consistent trends with stiffness and damping increasing with excitation current. At 0 A, the stiffness and damping of all three models are similar, and as the current increases, the differences between the models and the differences in compression and tension stiffness in the improved Bouc–Wen model also increase. At 2 A, the complex modulus method yields the largest parameters whereas the improved Bouc–Wen model gives the smallest. Figure 22 presents a comparison of the simulation results under the same operating condition using the complex modulus method, the traditional Bouc–Wen model, and the improved Bouc–Wen model. The corresponding normalized root mean square errors between simulation and experimental data are illustrated in Figure 23, and those of the improved Bouc–Wen model under varying operating conditions are summarized in

Table 5. Error of the improved Bouc–Wen model under different operating conditions.

Scheme	Amplitude (mm)	Vibration Frequency (Hz)	Initial Position (mm)	Current (A)	Error
2	1	0.5	0	0	2.0%
				2	4.9%
5		10		0	2.4%
				2	5.9%
7	3	0.5	0	0	1.3%
				2	3.2%
13	1	0.5	−3.5	0	2.7%
				2	6.6%

. From Figure 23, the error of all three models increases with the increase in current. At 0 A, the error of the improved Bouc–Wen model is only 2.0%, which is a 52.4% reduction compared to the complex modulus method and the traditional Bouc–Wen model. At 2.5 A, the error of the improved Bouc–Wen model is 5.0%, which is a 49.0% and 39.8% reduction compared to the complex modulus method and the traditional Bouc–Wen model, respectively. As shown in Table 5, with the decrease in amplitude and increase in vibration frequency, the error of the model increases. The error is generally lower at 0 A than at 2 A, and the error is higher at the preloaded position than at the zero position.

4. Conclusions

This paper presents a novel high-load MRF-elastomer isolator for engine vibration reduction from both theoretical and experimental perspectives. Through theoretical calculations and finite element simulations, the mechanical and magnetic circuit structures of the MRF-elastomer isolator were simulated and optimized. The dynamic mechanical performance of the isolator was tested using a fatigue testing machine under various conditions, with displacement amplitudes ranging from 1 to 5 mm, frequencies ranging from 0.1 to 10 Hz, excitation currents ranging from 0 to 2 A, and equilibrium positions at 0 mm and −3.5 mm. When the displacement amplitude was 1 mm, the frequency was 0.5 Hz, the excitation current was between 0 and 2 A, and the equilibrium position was at 0 mm, the tensile load amplitude of the isolator varied from 3.115 kN to 5.721 kN, a relative change of up to 83.7%. When the displacement amplitude was 1 mm, the frequency was 0.1 Hz, the excitation current was between 0 and 2 A, and the equilibrium position was at 0 mm, the maximum compression load amplitude varied from 3.083 kN to 6.331 kN, a relative change of up to 105.4%. After parameter identification using the complex modulus method, it was found that the stiffness could vary by up to 75.6%, and the damping could vary by up to 1545.1%. Finally, based on the hysteresis characteristics of the isolator, an improved Bouc–Wen model was proposed to describe the nonlinear characteristics of the isolator, and the parameters were identified using a genetic algorithm. Comparing with experimental data, the proposed improved Bouc–Wen model has a maximum error of only 5.0% at a 0.5 Hz vibration frequency, 1 mm of amplitude, and the zero initial position under different currents, which is a reduction of 49.0% and 39.8% compared to the complex modulus method and the traditional Bouc–Wen model, respectively, with the model error increasing as the excitation current increased. Therefore, the proposed MRF-elastomer isolator has great potential in the vibration control field for heavy-duty equipment, and the proposed improved Bouc–Wen model provides a theoretical basis for the precise control of MR damping devices.