Fuzzy control of MRE-based precision measurement platform under random vibration

Abstract

To suppress vibrations on a precision measurement platform challenged by random vibration, a semi-active fuzzy control method utilizing magnetorheological elastomer (MRE) vibration isolator is proposed in this article. Firstly, based on stochastic process theory, Adams software is employed to establish an accurate model and analyze the random vibration response of the isolation platform. Secondly, based on the relative speed and displacement signal feedback, a robust fuzzy control algorithm which is independent of the system model is designed to suppress multiple spectral peaks. Finally, the results of Adams and Simulink co-simulation and semi-physical experiment based on dSPACE show that the designed fuzzy controller can effectively suppress the wide-frequency random vibrations of 15–2000 Hz. Compared with the passive system, the acceleration RMS, displacement RMS and peak acceleration power spectral density of the system are attenuated by 5.56%, 34.86%, and 48.42%, respectively. Especially, the two response peaks in the acceleration power spectral density attenuated by 48.42% and 73.74% compared with the natural resonance peak of the passive system and the excitation peak respectively, indicating the effectiveness of the designed control algorithm for vibration reduction of complex excitation spectrum.

Keywords

magnetorheological elastomer isolator variable stiffness and damping random vibration fuzzy control

Introduction

Random vibration suppression poses a challenge in precision machining and measurement, as ground motion, jet noise, and flow turbulence produce broadband excitations that easily induce resonance and reduce system accuracy and stability. For instance, airborne instruments, space cameras (Bian et al., 2023; Li et al., 2023), and lithography tools (Al-Rawashdeh et al., 2022) are highly sensitive to micro-vibrations. Conventional passive isolators (rubber, springs) have fixed parameters (Fu et al., 2022; Jurevicius et al., 2019; Wu et al., 2016) and fail against low-frequency broadband excitation, whereas active methods (piezoelectric, shape memory alloys, magnetostrictive actuators) consume high power and risk instability at high frequencies (He et al., 2021; Wang et al., 2016; Zhang et al., 2018). A semi-active device (Li et al., 2025; Wang et al., 2023; Zhu et al., 2025), such as an MRE isolator, utilizes the phase transition characteristics of magnetorheological materials under a magnetic field to continuously adjust stiffness and damping. This enables frequency shifting and attenuation of resonance peaks through external magnetic field control, offering an alternative approach to vibration isolation. Their low power use, fast response, wide tunability, and fail-safe protection highlight strong potential in civil engineering (Gu et al., 2019; Yang et al., 2016; Yarra et al., 2018; Zhu et al., 2025), precision platforms (Fu et al., 2019, 2023), seat suspension (Li et al., 2012) and so on (Gao et al., 2024; Liu et al., 2021; Wu et al., 2024; Yu et al., 2016).

The tunable range of the vibration isolation performance of an MRE isolator depends on its structural configuration. Magnetorheological elastomer (MRE) isolators function in two primary modes: compression (Zhu et al., 2025) and shear (Liao et al., 2012), determined by the orientation of the magnetic field relative to vibration. Compression mode supports higher loads but limited stiffness adjustability due to lateral material expansion. Shear mode provides a wider adjustable range despite lower load capacity, making it suitable for applications with limited weight requirements. Given the need for greater tunability rather than high load-bearing in this study, the shear mode was adopted.

For the MRE vibration isolation system with definite structure, the performance of vibration reduction is determined by the control algorithm. Typically, it is integrated with On-Off control (Li et al., 2012; Liao et al., 2012), Fuzzy control (Fu et al., 2019, 2023), Lyapunov-based seismic controller (Behrooz et al., 2014; Liao et al., 2012), clipped optimal controller (Wang et al., 2010), and H-infinity controller (Du et al., 2011) to guarantee the effective vibration suppression. However, On-Off control may introduce significant oscillations in the system response, leading to unsatisfactory vibration suppression. Control methods such as Lyapunov-based control, clipped-optimal control, and H-infinity control often require an accurate system model. In practice, it is challenging to establish a precise mathematical model for MRE isolation systems due to their nonlinearities and uncertainties. The fuzzy control algorithm, which does not rely on complex mathematical models and has strong robustness (Fu et al., 2023), has been proven to effectively control the strong nonlinearity and uncertainty of MRE vibration isolation systems. Fu et al. (2019, 2023) earlier applied fuzzy control to nonlinear MRE vibration isolation system, successfully suppressed single-frequency, time-varying and multi-frequency line spectrum vibration, and verified the effectiveness of the designed fuzzy controller. Li et al. (2012, Yang et al., 2016) and Nguyen et al. (2018) all adopted a fuzzy controller based on MRE vibration isolation system for El Centro seismic waves, which effectively alleviated the vibration in the two-story building. The above research verifies the effectiveness of the fuzzy controller for the vibration isolation system of magnetorheological elastomer (MRE). However, most of the excitation in the above studies are sinusoidal excitation, seismic excitation or narrowband random excitation, and there is little research on the control of wide-band random vibration excitation.

In this article, the power spectral density of random excitation is a narrow-band peak of white noise spectral superposition with limited bandwidth, which has the characteristics of wide frequency band, complex spectrum characteristics and random uncertainty. To address broadband vibration characteristics, the MRE vibration isolation system increases the vibration isolation bandwidth by reducing the zero-field stiffness (resonance frequency), but this method still cannot avoid the system resonance under the wideband random excitation, which needs to be suppressed by the controller. The traditional fuzzy controller controls the stiffness by applying current to avoid resonance based on the variable stiffness principle of displacement feedback. Nevertheless, the shifted resonance frequency still lie within the broadband excitation range, making it impossible to avoid system resonance. Moreover, this method may make the frequency of the resonant peak close to the peak of the excitation spectrum, magnify the peak response in the excitation spectrum, and introduce a new peak. Therefore, the traditional control method cannot be applied to this object, which brings challenges to the design of the controller. So, this article innovatively proposes a fuzzy control strategy that relies on the feedback of relative velocity and displacement. By introducing the damping feedback mechanism related to velocity, the peak value of the response power spectral density can be effectively reduced.

Based on the analysis above, this article proposes a fuzzy control algorithm based on MRE vibration isolation platform to address significant issues such as the serious impact of random vibration interference on the measurement accuracy of precision equipment. Firstly, the dynamic response of the platform under various excitation currents is simulated using Adams software, providing a foundation for designing the control strategy. Secondly, a fuzzy control algorithm independent of the system model is designed. Finally, Adams and Smulink co-simulation and semi-experimental verification based on dSPACE are carried out. The results demonstrate that the fuzzy control algorithm achieves superior vibration suppression performance compared to both passive and On-Off control strategies. In conclusion, the fuzzy control strategy proposed in this study can provide excellent performance in random vibration control.

Random response analysis of MRE vibration isolation platform based on Adams

The MRE vibration isolation system and its dynamic response under random excitation are shown in Figure 1. Figure 1(a) illustrates the MRE vibration isolation platform, comprising a base, four symmetrically arranged MRE vibration isolators, and the main platform. The core component of the vibration isolation platform is the shear MRE vibration isolator with high magnetorheological effect designed by our research group. The vibration of the platform base is attenuated by the MRE isolator, which features a small volume, lightweight design and compact structure, as depicted in Figure 1(b). The primary parts consist of the top connector, bottom connector, excitation coil, top plate, bottom plate, outer sleeve, inner shaft, and two pieces of shear MRE. The MRE material is made of carbonyl iron powder and silicone rubber mixed with a mass ratio of 7:3. The inner shaft, top plate and outer sleeve are all made of magnetic conductive material to form a closed magnetic loop, so that the shear MRE is positioned within the magnetic circuit to achieve the maximum magnetic field. The main parameters of the vibration isolator are listed in Table 1.

Table 1.

Parameters of MRE vibration isolation system.

Models	Parameters	Values
Isolator	Dimension	φ 56 mm × 50 mm
	Outside diameter of the shear MRE	28 mm
	Inner diameter of the shear MRE	22 mm
	Shear MRE thickness	9 mm
	Coil turns	500
	Coil wire diameter	0.45 mm
	Resistance	4.8 Ω
	Maximum current	2.0 A
Base/host platform	Length (a)	250 mm
	Width (b)	250 mm
	Height (c)	14 mm
	Mass	5.4 kg

The magnetorheological elastomer (MRE) isolator is an intelligent vibration isolation device that operates based on the controllable modulus of the MRE material under a magnetic field. The material consists of micron-sized magnetic particles dispersed in a rubber matrix. When an external magnetic field is applied, the interactions between these particles reversibly alter the material’s viscoelastic properties, enabling rapid and reversible adjustment of key mechanical characteristics such as the isolator’s stiffness and damping. Stiffness and damping are critical design parameters for the MRE isolator; the detailed calculation process is provided in Appendix A, and the resulting values are summarized in Table 2.

Table 2.

Calculation of MRE isolator leveling stiffness and damping under different currents.

Current (A)	Mass(kg)	Z-directionstiffness (kN/m)	Z-directiondamping (N s/m)
0	1.35	30.41	99.08
0.5	1.35	40.12	101.44
1.0	1.35	65.12	120.60
1.5	1.35	89.34	133.23
2.0	1.35	121.79	144.04

The vibration coupling of a platform with six degrees of freedom is extremely complex, so it need decoupled based on the following premise:

The coordinate axis of the fixed coordinate system overlaps with the inertial main axis of the vibration isolation body center at rest;

Make the inertia main axis of the vibration isolator and the elastic main axis coincide;

The MRE vibration isolator is symmetrically set at four corners as a support, and the vibration isolator parameters are standard.

Therefore, the vibration isolation platform should consider the vertical degrees of freedom and two transverse degrees of freedom (X- and Y-rotations), utilizing three generalized coordinates $(z, θ_{X}, θ_{Y})$ to define the rigid body’s spatial position. The six-degree-of-freedom dynamic model of the motion of the MRE vibration isolation platform is shown in Figure 1(c). Because the vibration studied in this article only occurs in the vertical direction, meaning the platform is only excited by the translational motion along the Z-axis. So the vibration of the platform in the Z-axis direction is completely decoupled from the vibration in other directions.

Then the dynamic equation of the decoupled MRE vibration isolation platform system is as follows:

m x_{2}^{″} = k_{0} (x_{2} - x_{1}) + c_{0} (x_{2}^{'} - x_{1}^{'})

(1)

Where $x_{1}$ , $x_{2}$ are the displacements of the base and the main platform in the Z-direction, respectively, $k_{0}$ and $c_{0}$ represent the equivalent stiffness and equivalent damping in the Z-direction.

Based on the Laplace transform, the transfer function is obtained by sorting out the equation:

H (ω) = \frac{X_{2}}{X_{1}} = \frac{(k_{0} + j c_{0} ω)}{(- m ω^{2} + j c_{0} ω + k_{0})}

(2)

According to the theory of stochastic processes, the spectral density of the random response of all steady linear vibration systems excited by stationary stochastic processes is equal to the product of the square of the corresponding amplitude-frequency characteristics of the system and the spectral density of the excitation, as shown below:

S_{y} (ω) = H^{2} (ω) S_{x} (ω)

(3)

Where $S_{y} (ω)$ is the spectrum density of the system response, $H (ω)$ is the frequency response function of the system, and $S_{x} (ω)$ is the spectrum density of the system excitation.

The random response variance of the vibration system $E {X^{2} (t)}$ is:

E {X^{2} (t)} = \frac{1}{2 π} \int_{- \infty}^{\infty} S_{y} (ω) d ω

(4)

Based on the above dynamic analysis of the MRE isolation platform under random excitation, Adams/vibration module was used to simulate its random vibration response. As shown in Figure 1(d). Firstly, the 3D model of MRE vibration isolation platform built in Solidwork software is imported into Adams. Then, the stiffness and damping parameters of the MRE vibration isolation platform under different currents were configured in Adams, based on the data provided in Table 2. Finally, random acceleration power spectral density signals in Figure 1(e) were imported into the base platform, and the acceleration power spectral density curves of random vibration response of the MRE vibration isolation platform under different excitation currents were obtained, as shown in Figure 1(f). The simulation analysis results of the random vibration response are shown in Table 3.

S_{x} (ω) = {\begin{matrix} 0.001, 15 \leq | ω | \leq 64.5 \\ 0.01, 64.5 < | ω | < 71.5 \\ 0.001, 71.5 \leq | ω | \leq 129.2 \\ 0.002512, 129.2 < | ω | < 142.8 \\ 0.001, 142.8 \leq | ω | \leq 193.8 \\ 0.00112, 193.8 < | ω | < 241.2 \\ 0.001, 241.2 \leq | ω | \leq 2000 \\ 0, | ω | < 15, | ω | > 2000 \end{matrix}

(5)

Table 3.

Random vibration response analysis results.

Parameters	Peakfrequency (Hz)	Peak power spectraldensity (g²/Hz)	RMS (g)
0 A	22.54	0.53 × 10	0.28
0.5 A	26.91	0.65 × 10	0.32
1.0 A	33.66	0.73 × 10	0.39
1.5 A	39.61	0.80 × 10	0.45
2.0 A	65.02	1.35 × 10	0.54

When the current increases from 0 to 2A, the peak frequency of platform vibration response acceleration increases from 22.54 to 65.02 Hz, the peak power spectral density increases from 0.53 × 10⁻²g²/Hz to 1.35 × 10⁻²g²/Hz, and the root mean square (RMS) value of acceleration increases from 0.28 to 0.54 g. This indicates that the random vibration response characteristics of the system—including the peak frequency, peak power spectral density, and time-domain RMS—can be modulated by applying different currents to adjust the stiffness and damping parameters of the MRE vibration isolation platform. However, the direct application of a constant current does not improve the vibration control performance of the MRE-based isolation platform.

Therefore, in order to give full play to the vibration isolation performance of the MRE isolator, it is necessary to design a suitable vibration control strategy to further improve the vibration control effect of the MRE isolation platform.

Design of semi-active fuzzy controller

In order to suppress the multiple peaks of random signal spectral density, absolute velocity and displacement are selected as feedback. In order to fully utilize the magnetic control capabilities of the MRE isolation system, the control strategy is extracted by experience and semi-active conditions, and its expression is as follows:

F = {\begin{matrix} F_{max} & x_{r} {x^{'}}_{r} \geq 0, F_{c} \geq F_{max} \\ F_{c} & x_{r} {x^{'}}_{r} \geq 0, F_{c} < F_{max} \\ 0 & x_{r} {x^{'}}_{r} < 0 \end{matrix}

(6)

Where $F$ is the control force of the output of the MRE isolator, $F_{c}$ is the expected output force calculated by the controller, and $F_{max}$ is the maximum control force that the MRE isolator can output.

Based on the semi-active control condition for the MRE vibration isolation platform specified in equation (6), the relative displacement $x_{r}$ and relative velocity $v_{r}$ are used as the state inputs of the fuzzy controller, while the control force $F_{MRE}$ serves as the output. By incorporating the semi-active control condition into the fuzzy rule table, a fuzzy controller is systematically designed. Its structure is shown in Figure 2.

The controller designed in this article describes the input and output with seven fuzzy language variables: ${NB, NM, NS, ZO, PS, PM, PB}$ , corresponding to {Negative large, negative medium, negative small, zero, positive small, positive middle, positive large}. The specific realization of fuzzification is accomplished by input membership function, which can represent the degree to which the precise numerical quantity in the theoretic domain belongs to a certain fuzzy language variable. In this article, the “NB” and “PB” at both ends of the discourse domain use Gauss type 2 membership function to ensure the smoothness and stability of the boundary, and the triangle membership function is used in the middle to improve the sensitivity of the controller.

Before fuzzification, it is usually necessary to shrink the input signal into the membership function’s domain; [−7,7] by quantizing factors. The value of the quantization factor is shown in equation (7):

k_{RV} = \frac{7}{A_{0}}, k_{RD} = \frac{7}{A_{1}}

(7)

$k_{RV}$ , $k_{RD}$ is the quantification factors corresponding to the relative displacement $x_{r}$ and relative velocity $v_{r}$ of the MRE vibration isolation platform, and $A_{0}$ , $A_{1}$ is the peak value of the relative displacement and relative velocity in the theoretical domain of matter respectively.

The fuzzy reasoning method adopts Mamdani method, and the fuzzy rules is designed based on the equation (6), which are shown in Table 4.

Table 4.

Semi-active fuzzy control rules for MRE vibration isolation platform.

U		RD
U		PB	PM	PS	ZO	NS	NM	NB
RV	PB	NB	NB	NM	NS	ZO	ZO	ZO
	PM	NB	NM	NM	NS	ZO	ZO	ZO
	PS	NM	NM	NS	NS	ZO	ZO	ZO
	ZO	NS	NS	NS	ZO	PS	PS	PS
	NS	ZO	ZO	ZO	PS	PS	PM	PM
	NM	ZO	ZO	ZO	PS	PM	PM	PB
	NB	ZO	ZO	ZO	PS	PM	PB	PB

Finally, the precise output value of the controller is scaled by the scale factor to match the actual control quantity. That is:

U^{'} = kU

(8)

Where $k$ is the scaling factor of the controller, $U$ is the output of the fuzzy controller, $U^{'}$ is the precise output of the controller.

MRE vibration isolation platform control Adams–Simulink co-simulation verification

The general block diagram of Adams–Simulink co-simulation is shown in Figure 3. The main steps of the co-simulation are as follows: First, the dynamics model of the MRE vibration isolation platform is established in Adams. The isolator’s stiffness, damping coefficient, and control force are defined as input signals to the model. The velocity and acceleration of the main platform and the base are defined as output signals. Secondly, the control algorithm is developed in Simulink, in this step, the physical quantity output by Adams is taken as input, and the controlling force is returned to Adams as input. Finally, through the data exchange of Adams and Simulink with utilizing their respective solvers, the simulation verification of the control algorithm is completed.

A fuzzy control model for the MRE vibration isolation platform was developed using co-simulation technology, and its performance was compared with that of On-Off control. The random vibration test data is applied in Adams as the excitation input to the system. At the same time, the vibration excitation and passive control state (MRE isolator works in zero field state) are taken as references, and the vibration acceleration and displacement response of the MRE isolation platform are obtained. Among them, the vibration acceleration power spectral density of the platform is shown in Figure 4. The acceleration time domain signal and the control force output of the controller are shown in Figure 5.

In order to quantify the vibration control effect of MRE vibration isolation platform, the vibration isolation efficiency for applying vibration control is defined respectively.

The vibration isolation efficiency $η$ is defined as the relative change between the RMS value of the platform’s response acceleration and the RMS value of the vibration excitation acceleration after the control algorithm is applied. The expression is as follows:

η = \frac{a_{control_RMS} - a_{exc_RMS}}{a_{exc_RMS}}

(9)

Where, $a_{control_RMS}$ is the RMS value of the platform response acceleration after the control is applied, and $a_{exc_RMS}$ is the RMS value of the vibration excitation acceleration.

The peak attenuation of the power spectral density of the vibration acceleration before and after the control is calculated according to equation (9), and the vibration isolation efficiency is shown in Table 5. The attenuation of absolute displacement of vibration before and after control is shown in Table 6.

Table 5.

Acceleration PSD peak attenuation and vibration isolation efficiency.

Test condition	Peak frequency(Hz)	Peak power spectraldensity (g²/Hz)	Attenuationrate	RMS (g)	Vibration isolationefficiency
Excitation	68	9.82 × 10⁻³	—	1.22	—
Passive control	30.4	8.20 × 10⁻³	16.50%	0.24	80.33%
On-Off control	34.9	5.05 × 10⁻³	48.57%	0.25	79.51%
Fuzzy control	40.4	3.98 × 10⁻³	59.47%	0.22	81.97%

Table 6.

Absolute displacement response signal attenuation table (mm).

Control method	RMS	Attenuation rate
Passive control	0.086	—
On-Off control	0.074	13.95%
Fuzzy control	0.059	31.40%

Results from combining the figures and tables:

Compared with the excitation signal, the peak value of the acceleration power spectrum density under the fuzzy control of the MRE vibration isolation platform can decrease by 59.47%, and the peak value is increased by 10.90% and 42.97% compared with the On-Off control and the passive control, respectively. Especially, the two response peaks in the acceleration power spectral density attenuated by 51.46% and 80.14% compared with the natural resonance peak of the passive system and the excitation peak respectively. The reason for the improvement is that the variable stiffness and variable damping characteristics of the system can effectively suppress the resonance peak and the excitation peak at the same time by the fuzzy control.

Compared with the excitation signal, the maximum RMS value of the response acceleration under the fuzzy control of the MRE vibration isolation platform decreases by 81.97%, and it is increased by 2.46% and 1.64% compared with On-Off control and the passive control, respectively. The reason is that the fuzzy control will not increase the response in the high frequency range while effectively suppressing the resonance peak and the excitation peak. In contrast, On-Off control amplifies the response in the high frequency range.

Compared with the response of the passive system, the RMS value of the response displacement in the time domain under the fuzzy control of the MRE vibration isolation platform can be attenuated by a maximum of 31.40% and increased by 17.45% compared with the On-Off control.

Compared with On-Off control, fuzzy control can obtain better control effect with lower energy consumption. The RMS value of the output force of fuzzy controller is only 1.95 N, while that of the output force of On-Off controller is 14.13 N.

Experiment on semi-physical vibration control of MRE vibration isolation platform

To verify the vibration control effect of the MRE vibration isolation platform, a vibration control experimental system is constructed, with its physical diagram shown in Figure 6. The electromagnetic vibration table provides vertical vibration excitation to the platform, with the base of the MRE vibration isolation platform fastened to the table surface using bolts. A vibration controller (VT-9008, Hangzhou Ehang, China) is used to generate a random acceleration excitation signal that matches the power spectral density shown in Figure 1(e). Two identical acceleration sensors (PCB333B52, Piezotronics, USA) are installed respectively at the base of the platform and at the center of mass in the main platform. The measured acceleration signals are converted into voltage signals by the signal conditioner, and then transmitted to the dSPACE/MicroBox semi-physical experimental platform (DS1202, dSPACE, Germany) for recording. The laser displacement sensor (KEYENCE LK-H025 model, Japan) is clamped to the switching magnetic seat on the bench to measure the displacement signal. The control algorithm established by Simulink is downloaded to dSPACE, and the required control force is calculated by dSPACE in real time according to the input signal, and converted into the corresponding control signal (voltage) through the inverse model of MRE vibration isolator. The control voltage signal output by the controller is converted by the current driver to the control current acting on the excitation coil in the vibration isolator in equal proportion. The 12 V switching power supply supplies power to the current driver. PC and dSPACE are connected through the network cable with TCP/IP protocol. The signal acquisition and control model is established in Simulink. The control system structure of MRE vibration isolation platform is shown in Figure 7.

The response of vibration acceleration and displacement of the MRE isolation platform is obtained by using the different control algorithms designed above and taking vibration excitation and passive control state (MRE isolator works in zero field state) as reference. Among them, the vibration acceleration power spectral density of the platform is shown in Figure 8 and the time domain signals of acceleration, displacement and control current output by the controller are shown in Figure 9. Consistent with the simulation calculation method, the peak attenuation of vibration acceleration power spectral density and vibration isolation efficiency before and after the control are shown in Table 7. The attenuation of the absolute displacement response of vibration before and after control is shown in Table 8.

Table 7.

Acceleration PSD peak attenuation and vibration isolation efficiency.

Test condition	Peak frequency (Hz)	Peak power spectral density (g²/Hz)	Attenuation rate	RMS (g)	Vibration isolation efficiency
Excitation	68	9.94 × 10⁻³	—	1.40	—
Passive control	32.93	7.93 × 10⁻³	20.22%	0.36	74.29%
On-Off controls	42.02	5.38 × 10⁻³	45.88%	0.38	72.86%
Fuzzy control	40.16	4.09 × 10⁻³	58.85%	0.34	75.71%

Table 8.

Vibration absolute displacement time range signal (mm).

Control method	RMS	Attenuation rate
Passive control	0.109	—
On-Off control	0.096	11.93%
Fuzzy control	0.071	34.86%

Results from combining the figures and tables:

Compared with the excitation signal, the peak value of the acceleration power spectrum density under the fuzzy control of the MRE vibration isolation platform can decrease by 58.85%, and the peak value is increased by 12.97% and 38.63% compared with the On-Off control and the passive control, respectively. Especially, the two response peaks in the acceleration power spectral density attenuated by 48.42% and 73.74% compared with the natural resonance peak of the passive system and the excitation peak respectively.

Compared with the excitation signal, The maximum RMS value of the response acceleration under the fuzzy control of the MRE vibration isolation platform decreases by 75.71%, and it is increased by 2.85% and 1.42% compared with On-Off control and the passive control, respectively.

Compared with the response of the passive system, the RMS value of the response displacement in the time domain under the fuzzy control of the MRE vibration isolation platform can be attenuated by a maximum of 34.86% and increased by 22.93% compared with the On-Off control.

Compared with On-Off control, fuzzy control can obtain better control effect with lower energy consumption. The RMS value of the output force of fuzzy controller is only 0.56 A, while that of the output force of On-Off controller is 1.36 A.

The trends observed in the simulation and experimental results are generally consistent. However, the overall control performance in experiments is inferior to that in simulations, primarily due to various uncertainties present in the actual control system. These include fluctuations in the random vibration excitation signal, inherent hysteresis of the MRE isolator, acquisition noise from sensors, and computational delays in the control process, all of which adversely affect the control effectiveness.

Conclusion

In this study, based on the MRE vibration isolation system, a model-independent fuzzy control algorithm is designed to solve the problem that the broadband random vibration excitation is difficult to control. The designed fuzzy control algorithm can give full play to the variable stiffness and variable damping characteristics of MRE vibration isolation system through fuzzy control strategy, and can effectively reduce the amplitude of power spectral density without amplifying high frequency vibration when the excitation spectrum is not uniform in the wide frequency range. Then the control effect is verified by Adams and Simulink co-simulation and semi-physical experiment based on dSPACE. The experimental results show that when the fuzzy control is adopted, the peak power spectral density of the vibration acceleration of the platform attenuates from 9.94 × 10⁻³g²/Hz to 4.09 × 10⁻³g²/Hz, and the attenuation is 58.85%, which is 38.63% higher than that of the passive control, and the vibration isolation efficiency is 75.71%. Furthermore, the maximum attenuation of the absolute displacement RMS value of the platform can be 34.86%, which is improved by 22.93% compared with the On-Off control.

The fuzzy control algorithm can effectively improve the vibration control effect of the vibration isolation platform, and provide an important reference for the future high-precision vibration control system design.

Footnotes

Appendix A

Since MRE materials are particle-reinforced rubber composites, the stiffness and damping parameters of MRE isolators can be estimated using methods similar to those for conventional rubber isolators. The static vertical stiffness of an MRE isolator is approximated by the equation:

(A.1)

k = 2 π lG / (\ln (D / d))

where $l$ is the length of the elastomer bushing, $G$ is the shear storage modulus of the MRE, $D$ is the outer diameter, $d$ is the inner diameter, and $k$ represents the vertical stiffness.

The structural damping parameters are estimated using the following expressions:

(A.2)

c = 2 m ξ ω

(A.3)

ξ = \frac{G^{'}}{2 G}

(A.4)

ω = \sqrt{\frac{k}{m}}

where $m$ denotes the supported mass, $ξ$ is the structural damping ratio, $G^{'}$ is the loss modulus of the MRE, $G$ is the shear storage modulus of the MRE, $ω$ is the natural frequency of the isolation system, and $k$ is the stiffness of the isolator.

To numerically compute the stiffness and damping parameters of the isolator, the shear storage modulus $G$ and loss modulus $G^{'}$ of the MRE material must first be obtained. These properties were measured using a rotational rheometer (Anton Paar, MCR301, Austria). The test system, illustrated in Figure A1, was operated under a magnetic flux density ranging from 0 to 0.8 T, at a frequency of 30 Hz and a temperature of 25°C. The variations of $G$ and $G^{'}$ with respect to the applied magnetic field are presented in Figure A2.

By substituting the values of $G$ and $G^{'}$ under different magnetic fields into equations (A.1)–(A.4), the vertical static stiffness and structural damping parameters of the MRE isolator were calculated for various currents (converted from magnetic flux density to excitation coil current). The numerical results are summarized in Table 2.

Acknowledgements

The authors are grateful for the helpful comments from the reviewers and editor to improve the article.

ORCID iDs

Jie Fu

Miao Yu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (Grant Nos. 12372146, 52575100) and Beijing Key Laboratory of Advanced Optical Remote Sensing Technology (Grant No. AORS202314).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

The data supporting the findings of this study are simulation-generated and are available from the corresponding author on reasonable request.

References

Al-Rawashdeh

Janaideh

Heertjes

(2022) On characterization of a generic lithography machine in a multi-directional space. Mechanism and Machine Theory 170: 104638.

Behrooz

Wang

Gordaninejad

(2014) Performance of a new magnetorheological elastomer isolation system. Smart Materials and Structures 23: 045014.

Bian

Zhou

(2023) Jitter-caused clutter and drift-caused clutter of staring infrared sensor in geostationary orbit. Sensors 23(11): 5278.

Zhang

(2011) Semi-active variable stiffness vibration control of vehicle seat suspension using an MR elastomer isolator. Smart Materials and Structures 20: 105003.

Bai

Lai

, et al. (2019) Adaptive fuzzy control of a magnetorheological elastomer vibration isolation system with time-varying sinusoidal excitations. Journal of Sound and Vibration 456: 386–406.

Fan

, et al. (2022) Vibration reduction design and test of UAV load radar. International Journal of Aerospace Engineering 2: 4186303.

Liu

Lai

, et al. (2023) Robustness analysis of magnetorheological elastomer-based vibration isolation system with optimal fuzzy controller. Smart Materials and Structures 32(3): 035018.

Gao

Han

, et al. (2024) Improving the performance of magnetorheological elastomer-based adaptive isolator through integrated compression-torsion structure. Smart Materials and Structures 33: 075022.

, et al. (2019) Experimental study of semi-active magnetorheological elastomer base isolation system using optimal neuro fuzzy logic control. Mechanical Systems and Signal Processing 119: 380–398.

10.

Liu

, et al. (2021) Active vibration isolation of ultra-stable optical reference cavity of space optical clock. Aerospace Science and Technology 112: 106633.

11.

Jurevicius

Vekteris

Turla

, et al. (2019) Investigation of the dynamic efficiency of complex passive low-frequency vibration isolation systems. Journal of Low Frequency Noise Vibration and Active Control 38(2): 608–614.

12.

Yuan

Wang

, et al. (2023) Image motion and experimental study of a 0.100 space pointing measuring instrument for micro-vibration conditions. Chinese Journal of Aeronautics 36(2): 191–200.

13.

Zhang

(2012) Development and simulation evaluation of a magnetorheological elastomer isolator for seat vibration control. Journal of Intelligent Material Systems and Structures 23(9): 1041–1048.

14.

Ding

Dzwairo

, et al. (2025) A magnetorheological damping-based quasi-zero-stiffness isolator for large-amplitude vibration isolation. International Journal of Mechanical Sciences 304: 110702.

15.

Liao

Gong

Xuan

, et al. (2012) Development of a real-time tunable stiffness and damping vibration isolator based on magnetorheological elastomer. Journal of Intelligent Material Systems and Structures 23(1): 25–33.

16.

Liu

Sedaghati

Peng

(2021) A novel semi-active switching control scheme for magnetorheological elastomer-based vibration isolator under dynamic input saturation. Smart Materials and Structures 30(9): 095008.

17.

Nguyen

Komatsuzaki

Iwata

, et al. (2018) Modeling and semi-active fuzzy control of magnetorheological elastomer-based isolator for seismic response reduction. Mechanical Systems and Signal Processing 101: 449–466.

18.

Wang

Xie

Chen

, et al. (2016) Investigation on active vibration isolation of a Stewart platform with piezoelectric actuators. Journal of Sound and Vibration 383: 1–19.

19.

Wang

Yang

(2010) Study on a semi-active shock isolation technology with magnetorheological devices. Applied Mechanics and Materials 26–28: 770–775.

20.

Wang

Zhou

Cao

, et al. (2023) Design and control of helicopter main reducer semi-active vibration isolation system based on MR damper. Journal of Intelligent Material Systems and Structures 34(17): 12.

21.

Gong

Yang

, et al. (2024) Investigation of a semi-active suspension system for high-speed trains based on magnetorheological isolator with negative stiffness characteristics. Mechanical Systems and Signal Processing 208: 111085.

22.

Jing

Sun

, et al. (2016) A 6DOF passive vibration isolator using X-shape supporting structures. Journal of Sound and Vibration 380: 90–111.

23.

Yang

Sun

Tian

, et al. (2016) Development of a novel multilayer MRE isolator for suppression of building vibrations under seismice events. Mechanical Systems and Signal Processing 70–71: 811–820.

24.

Yarra

Gordaninejad

Behrooz

, et al. (2018) Performance of a large-scale magnetorheological elastomer–based vibration isolator for highway bridges. Journal of Intelligent Material Systems and Structures 29(20): 1–12.

25.

Royel

, et al. (2016) Magnetorheological elastomer base isolator for earthquake response mitigation on building structures: Modeling and second-order sliding mode control. Earthquakes and Structures 11(6): 943–966.

26.

Zhang

Sheng

, et al. (2018) Dynamic analysis and control application of vibration isolation system with magnetic suspension on satellites. Aerospace Science and Technology 75: 99–114.

27.

Zhu

Yao

, et al. (2025) CBi-AM: A fusion model for predicting inverse nonlinear dynamics of magnetorheological elastomer isolator. Mechanical Systems and Signal Processing 234: 112815.

U		RD
U		PB	PM	PS	ZO	NS	NM	NB
RV	PB	NB	NB	NM	NS	ZO	ZO	ZO
	PM	NB	NM	NM	NS	ZO	ZO	ZO
	PS	NM	NM	NS	NS	ZO	ZO	ZO
	ZO	NS	NS	NS	ZO	PS	PS	PS
	NS	ZO	ZO	ZO	PS	PS	PM	PM
	NM	ZO	ZO	ZO	PS	PM	PM	PB
	NB	ZO	ZO	ZO	PS	PM	PB	PB

U		RD
U		PB	PM	PS	ZO	NS	NM	NB
RV	PB	NB	NB	NM	NS	ZO	ZO	ZO
	PM	NB	NM	NM	NS	ZO	ZO	ZO
	PS	NM	NM	NS	NS	ZO	ZO	ZO
	ZO	NS	NS	NS	ZO	PS	PS	PS
	NS	ZO	ZO	ZO	PS	PS	PM	PM
	NM	ZO	ZO	ZO	PS	PM	PM	PB
	NB	ZO	ZO	ZO	PS	PM	PB	PB

U		RD
U		PB	PM	PS	ZO	NS	NM	NB
RV	PB	NB	NB	NM	NS	ZO	ZO	ZO
	PM	NB	NM	NM	NS	ZO	ZO	ZO
	PS	NM	NM	NS	NS	ZO	ZO	ZO
	ZO	NS	NS	NS	ZO	PS	PS	PS
	NS	ZO	ZO	ZO	PS	PS	PM	PM
	NM	ZO	ZO	ZO	PS	PM	PM	PB
	NB	ZO	ZO	ZO	PS	PM	PB	PB