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Abstract

The magnetically tunable modulus of soft magnetorheological elastomer offers broad applications in vibration control. This material exhibits an obvious Payne effect that the dynamic modulus decreases as the strain amplitude increases. It is necessary to analyze the influence of amplitude-dependent modulus and nonlinear viscoelastic behavior for the application of soft magnetorheological elastomer in vibration control. A fractional viscoelastic constitutive model is developed in a nonlinear manner to depict the dynamic mechanical behaviour of soft magnetorheological elastomer accurately and provide guidance for applications in vibration control. Model parameters are identified by minimizing the standard error ratio of dynamic modulus. The numerical calculations and the experimental data are used for model validation. This nonlinear fractional viscoelastic model has superior ability to describe the stress-strain hysteresis curve compared with the revised Bouc-Wen model and the linear fractional viscoelastic model. The effects of nonlinear viscoelasticity on the dynamic behavior of vibration isolation systems are presented, and the amplitude dependence is found to be beneficial to the isolation performance. A control algorithm based on a switchable magnetic field that depends on load frequency is employed as an example to enhance the vibration isolation performance for the application of soft magnetorheological elastomers. This study reveals the inherent characteristics of soft magnetorheological elastomer and provides a theoretical basis for its vibration control design in practical applications.

Keywords

soft magnetorheological elastomer amplitude-dependent modulus nonlinear fractional viscoelastic model dynamic mechanical analysis vibration isolation

Introduction

Magnetorheological elastomer (MRE) is composed of magnetizable particles embedded in elastic matrices. The material is isotropic when particles are distributed randomly. In contrast, it is anisotropic as particles are distributed directionally.^1,2 The magnetizable particles are either magnetically soft or hard, and magnetizable particles in soft MRE (s-MRE) can be easily magnetized and demagnetized.^3,4 When magnetic fields are applied to s-MRE, there are enhanced modulus and obvious magnetostriction, which is known as the magnetorheological (MR) effect. The mechanical characteristic of magnetically controllable modulus offers broad application in vibration absorbers,^5,6 isolators^7–9 and vehicle suspensions,^10,11 and increasing efforts are also devoted to the application of the magneto-induced deformation in magnetic actuators and soft robotics.^12,13 In comparison, the s-MRE structure for magnetic actuation is slenderer than that for vibration control, and the required magnetic field strength of s-MRE for magnetic actuation is lower than that for vibration control.

For the practical applications of MRE in both vibration control and magnetic actuation, modelling the mechanical behaviour is an essential basis. Constitutive models focusing on the enhancement of dynamic modulus of MRE attracted significant attention, to investigate the influence of the magnetic field on the viscoelastic behaviour and the controllable modulus for the design of vibration control.^14,15 As Maxwell stress dominates the magnetic-induced deformation rather than the enhancement of modulus, the magneto-mechanical theory can be simplified to describe the macroscopic performance of magnetic actuation.^16,17

On–off control,¹⁸ H-infinity control,¹⁹ sliding mode control and fuzzy control have been applied to s-MRE based vibration systems.^20,21 Time compensation strategy and phase compensation strategy have been adopted for the time delay of s-MRE.^22,23 As the demand for control strategies increases in s-MRE based vibration systems, accurate models need to be developed to capture the nonlinear viscoelasticity of s-MRE. While most literature has focused on the quasi-static behavior of s-MRE, it is important to note that s-MRE is sensitive to both frequency and strain amplitude.^24,25 Classical viscoelastic models, such as the Kelvin-Voigt model, Maxwell model and Zener model, fail to accurately describe the dynamic behavior of s-MRE within a wide range of strain and frequency. To address this, Bouc-Wen,²⁶ Ramberg-Osgood,²⁷ Prandtl-Ishlinskii or other elements have been incorporated into viscoelastic models to better describe the dynamic mechanical behaviour of s-MRE.²⁸

Viscoelastic materials exhibit behavior intermediate between that of a viscous fluid and an elastic material, and the fractional derivative has been introduced to describe the stress-strain relation for viscoelastic materials.^29,30 However, to predict the magneto-mechanical coupling behavior of s-MRE, which is inherently soft, an assumption of finite strain range must be applied.³¹ Besides of the intrinsic viscoelasticity, the s-MRE material also exhibits pronounced Payne effect, Mullins effect and temperature dependence,^32,33 which are typical behaviors of filled rubber.^32,34 Experimental studies demonstrate that the dynamic modulus of s-MRE decreases when the strain amplitude increases,^1,2,24,25 and the transition behavior of s-MRE occurs at a certain temperature as the dynamic modulus exhibits different trends with the temperature variation.³⁵ Considering potential nonlinear oscillation in certain scenarios of s-MRE-based systems, constitutive models with the strain amplitude dependence are preferable to accurately depict the nonlinear viscoelastic behavior.^36,37 Taking into account the amplitude dependence of s-MRE, the constitutive model remains insufficiently developed, and the influence of nonlinear viscoelasticity on the dynamic behavior of s-MRE-based isolation systems needs to be explored to advance practical applications. This paper incorporates the amplitude dependence of s-MRE into its viscoelastic constitutive model and investigates the influence of nonlinear viscoelasticity on s-MRE-based vibration isolation systems.

In this paper, the experimental verification is conducted through the dynamic mechanical analysis (DMA) tests to investigate the amplitude dependent behavior and the nonlinear viscoelasticity of s-MRE. Subsequently, a nonlinear fractional viscoelastic constitutive model is developed to capture the amplitude-dependent behavior, incorporating a magnetic spring to account for the magnetic field-dependent behavior. Model parameters are identified by optimizing the standard error ratio of dynamic modulus, and the model validation is performed using the numerical calculation and experimental data of the stress-strain hysteresis curve. Compared with the Bouc-Wen model and the linear fractional viscoelastic model, this nonlinear fractional viscoelastic model demonstrates superior accuracy. Furthermore, the influence of nonlinear viscoelasticity on the dynamic behavior of s-MRE-based isolation systems is analyzed, and the isolation efficiency is improved through a control algorithm that employs a switchable magnetic field. In comparison with linear fractional-order models, the amplitude dependence of s-MRE is found to be beneficial to the vibration control effect. This work aims to provide theoretical basis and guidance for the design of s-MRE-based vibration control systems.

Nonlinearity on dynamic mechanical properties of s-MREs

S-MREs and DMA experiments

The investigated type of s-MRE was produced as follows. The carbonyl iron powder (Sigma-Aldrich, US) was chosen as the magnetizable particle because of its low remnant magnetization, and the silicone rubber (Wacker Chemie AG, Germany) was selected as the matrix due to its good elasticity. During the fabrication of particle-reinforced rubber, different particles and rubbers were selected to obtain good long-term stability, which is a necessary consideration for s-MRE. Compared to MR fluids, s-MREs have better stability. The shape and dimensions of samples folowed the BS ISO 4664-1:2011, as shown in Figure 1(a), and the magnetic field was produced by neodymium disc magnets (E-magnets, UK) of various dimensions. Other details of s-MRE samples can be found in the illustration of a previous experiment.³⁸ In Figure 1(b), the DMA tests for s-MREs in shear mode used Instron Electropuls E1000. The range of strain amplitude from 1 % to 5 % was selected to examine the amplitude dependence and nonlinear viscoelasticity of the mechanical properties of s-MREs. During the testing the displacement and load were tracked along the vibration directions which was vertical in Figure 1(b), and the direction of external magnetic field was parallel to magnetic particle chains which were oriented horizontally.

Figure 1.

DMA tests of s-MRE in shear mode (a) samples and (b) experimental setup.

Amplitude dependence of the modulus

As a kind of particle-reinforced rubber, s-MRE material responds to a sinusoidal load with a time delay. The dynamic shear modulus of s-MRE is mathematically considered as a complex modulus, comprising both a real part and an imaginary part.

G = G^{'} + i G^{″}

(1)

where G′ and G″ denote the storage modulus and the loss modulus, respectively. The storage modulus presents the ability of storing energy during deformations, and the loss modulus indicates the ability of dissipating energy with heat. During the experimental data processing, the dynamic modulus magnitude G^* = |G | is the directly obtained result. Both the storage modulus and the loss modulus are calculated based on the modulus magnitude G^*. According to experimental study, the modulus magnitude G^* is found to decrease as the strain amplitude increases. However, the material stiffness cannot decrease to zero, unless the material fails. Therefore, the dynamic shear modulus magnitude G^* can be expressed in a nonlinear form

G^{*} (θ) = G_{0} (1 - β θ)

(2)

where G₀ is a shear modulus when the influence of the deformation is ignored, G₀ > 0; β is a nonlinear coefficient, 0 < β < 2/π rad^-1; θ = arctanγ_A, θ and γ_A are the angle of shear deformation and the amplitude of shear strain, respectively. When the amplitude dependence of the modulus is ignored in the above expression, the nonlinear coefficient β = 0 and the modulus magnitude G^*(θ) = G₀, implying a constant modulus which was adopted in references;^30,39 when the amplitude dependence of s-MREs is enhanced, the resultant nonlinear coefficient β increases.

The ratio of the standard error of estimation S_e to the standard error of deviation S_d is defined as the standard error ratio S_e/S_d, which is selected as a dimensionless statistic to reflect the quality of curve-fitting in Figure 2, avoiding the influence of different sizes and orders of magnitude. The standard error ratio can be defined as

\frac{{S_{e}}^{2}}{{S_{d}}^{2}} = \frac{(n - 1) \sum {(y_{\exp} - y_{f i t})}^{2}}{n \sum {(y_{\exp} - {\bar{y}}_{\exp})}^{2}}

(3)

where n is the number of data samples, and each curve in Figures 2 and 3 was fitted by nine data samples, which were obtained by averaging the data for three pairs of s-MRE samples; ͞y_exp denotes the average of experimental data, y_exp and y_fit denote experimental data and fitted results, respectively. When the data samples y are the modulus magnitude G^*, the curve-fitting and the histogram of standard error ratio S_e/S_d can be obtained in Figures 2 and 3. For linear viscoelastic materials (β = 0), the modulus magnitude G^* remains constant with the amplitude variation, while the modulus magnitude G^* decreases with increasing amplitude. Because of their viscoelasticity nature, s-MRE materials do not respond to excitation instantaneously. Researchers found that the response time of s-MRE to a displacement excitation was on the order of several milliseconds, and the response time of s-MRE to a step current excitation was about 12 ms.²³ The experimental results in this piece of work was limited to 1 Hz to 50 Hz, as the tests failed at high frequencies when the s-MRE samples failed to respond to excitation.

Figure 2.

Experimental data and fitted results of shear modulus with different strain amplitudes at frequency of (a) 1 Hz and (b) 5 Hz.

Figure 3.

Experimental data and fitted results of shear modulus with different strain amplitudes at frequency of (a) 10 Hz and (b) 50 Hz.

Mathematical modelling for s-MREs

Mathematical models for s-MREs

For viscoelastic materials, the simplest models are the classical Maxwell and Kelvin-Voigt rheological models, in which there are only two components including an elastic spring and a Newtonian dashpot. It is hard to obtain quantitative agreement with experimental data using simple models. Meanwhile, more complex models are more time-consuming, due to the increased number of fitting parameters. The fractional Zener model has been found to be a good solution to balance complexity and accuracy.^30,39

As one of the most accurate rheological models, it is justified to select the fractional Zener model incorporated with a magnetic spring to present the magnetic field-dependent behavior as shown in Figure 4, where the relationship between the stress τ(t) and the strain γ(t) (referred to Appendix A) can be expressed as

τ (t) = (G_{e} + G_{m} + \frac{G_{v} τ_{0}^{α} D^{α}}{1 + τ_{0}^{α} D^{α}}) γ (t)

(4)

where G_e denotes the shear modulus of the elastic model, 0 < G_e < G₀; G_v denotes the shear modulus of the Fractional Maxwell model, 0 < G_v < G₀; G_m denotes the controllable modulus generated from the MR effect, 0 < G_m < G₀; τ₀ is a parameter related to the relaxation time, τ₀ > 0; D is a derivative operator; α is the order of a fractional derivative, 0 < α < 1. The fractional springpot in the Fractional Maxwell model can be turned into an a dashpot (α = 1) or a spring (α = 0) as an extreme case for viscous fluid or elastic materials, respectively.

Figure 4.

Fractional viscoelatic model of s-MRE.

When the nonlinear viscoelasticity is introduced to depict the amplitude dependence, the modulus can be expressed as

G (θ) = (G_{e} + G_{m} + \frac{G_{v} τ_{0}^{α} D^{α}}{1 + τ_{0}^{α} D^{α}}) (1 - β θ)

(5)

As a commonly chosen type of fractional derivatives, the Caputo derivative owns practically measurable initial conditions, whose physical meanings are well understood. Using the predictor-corrector approach to solve Caputo fractional differential equations,^40,41 the stress-strain relation in the fractional Zener model can be numerically calculated for equation (5).

Figure 5 shows the influence of fractional order α and nonlinear coefficient β on hysteresis loops of s-MREs. In Figure 5(a), when the fractional order α increases, the viscosity of the material increases, and the slope of ellipse increases, which means the shear modulus increases. The area enclosed by the hysteresis loop is calculated by the numerical integration as the dissipating energy. The area of the ellipse in Figure 5(a) increases as the fractional order α increases, which indicates the ability to dissipate energy increases. In Figure 5(b), the slope of ellipse decreases as the nonlinear coefficient β increases, so the shear modulus decreases when the amplitude dependence increases. But the area of ellipse barely changes when the nonlinear coefficient β varies.

Figure 5.

Viscoelastic stress response of different (a) fractional-orders and (b) nonlinear coefficients.

Parameter identification

When viscoelastic problems are converted into elastic problems through the Fourier transform, the storage modulus and loss modulus are expressed as functions od the frequency (referred to Appendix A). When nonlinear viscoelasticity is introduced to depict the amplitude dependence, the storage modulus G′ and the loss modulus G″ are expressed as follows

G^{'} (ω, θ) = {G_{e} + G_{m} + \frac{G_{v} τ_{0}^{α} ω^{α} [\cos (\frac{α π}{2}) + τ_{0}^{α} ω^{α}]}{1 + τ_{0}^{2 α} ω^{2 α} + 2 τ_{0}^{α} ω^{α} \cos (\frac{α π}{2})}}

(6)

G^{″} (ω, θ) = [\frac{G_{v} τ_{0}^{α} ω^{α} \sin (\frac{α π}{2})}{1 + τ_{0}^{2 α} ω^{2 α} + 2 τ_{0}^{α} ω^{α} \cos (\frac{α π}{2})}]

(7)

where ω = 2πf, ω and f are the angular frequency and the frequency, respectively.

The model parameters can be identified by minimizing the sum of standard error ratios as

s u m = \sum_{i = 1}^{2} {(S_{e} / S_{d})}_{i}

(8)

If i = 1, the data sample y is the experimental data of storage modulus G'; alternatively, the data sample y is the experimental data of loss modulus G″ if i = 2.

As shown in Figure 6, the curve-fitting exhibits a quantitative agreement with experimental data, and the dependence of modulus on the frequency can be also observed. Both the storage modulus G′ and the loss modulus G″ are constant with variable strain amplitude for linear viscoelastic materials (β = 0), while both the moduli vary with the strain amplitude for nonlinear viscoelasticity, as shown in Figure 6. The parameter identification was carried out for magnetic fields of 0 and 500 mT, and the results are listed in Table 1.

Figure 6.

Experimetnal data and fitted results on the frequency dependence of (a) storage modulus and (b) loss modulus.

Table 1.

The results of parameter identification.

Strain amplitude	Magnetic intensity	G_e (MPa)	G_v (MPa)	G_m (MPa)	τ (s)	a	β (rad^-1)
1%	0	1.0097	0.8595	0	0.0165	0.5145	0.2047
1%	500 mT	1.0551	0.9053	0.3665	0.0152	0.4612	0.1928
2%	0	0.9293	0.8677	0	0.0138	0.5072	0.1975
2%	500 mT	0.9319	0.8806	0.3639	0.0131	0.4377	0.1923
3%	0	0.8582	0.8415	0	0.0122	0.5053	0.1954
3%	500 mT	0.8610	0.8717	0.2894	0.0117	0.4243	0.1903
4%	0	0.8005	0.7765	0	0.0106	0.4757	0.1937
4%	500 mT	0.8484	0.8231	0.2676	0.0091	0.4124	0.1878
5%	0	0.7497	0.7638	0	0.0092	0.4697	0.1915
5%	500 mT	0.7596	0.7819	0.2084	0.0069	0.3981	0.1874

It has been found that when the magnetic field intensity is below 500 mT, the dynamic shear modulus magnitude of s-MRE can be increased linearly by strengthening the magnetic field.^1,2 Due to the saturation magnetization, above a certain value of magnetic field intensity, the modulus magnitude of s-MRE cannot be effectively increased any more by further strengthening the magnetic field.

Numerical solution and validation

The numerical results of equation (5) can be used to depict the stress-strain hysteresis curve for s-MREs. The numerical calculations of different models are compared with experimental data in Figures 7 and 8, where the dimensionless standard error ratio is calculated to validate the curve fitting. The nonlinear fractional-order model performs better in depicting the stress-strain relation of s-MRE than the revised Bouc-Wen model (referred to Appendix B) and the fractional-order model.^30,39

Figure 7.

Experimental data and predicted results of stress-strain hysteresis loops with a strain amplitude of 1% in a magnetic field of (a) 0 mT and (b) 500 mT.

Figure 8.

Experimental data and predicted results of stress-strain hysteresis loops with a strain amplitude of 5% in a magnetic field of (a) 0 mT and (b) 500 mT.

Dynamical analysis of s-MRE vibration systems

Vibration isolation against harmonic force

As a kind of magnetically sensitive rubber, s-MRE materials can be used for vibration isolation, and the nonlinear fractional viscoelastic model can be introduced to describe the force-displacement relation. When the vertical direction is the only considered direction, the schematic diagram of a vibration isolation system is shown in Figure 9, where the machine represents a rotating machine or sensitive equipment, and the foundation is either fixed or flexible. When the DMA tests of s-MREs were performed under harmonic load, vibration isolation systems herein also experience harmonic load. The influence of viscoelasticity was studied in s-MRE-based vibration isolation systems,⁴² and this piece of work focuses on the nonlinear system stiffness, which results from the amplitude dependence of the modulus of s-MREs. The dynamic analysis in this study is limited to single-degree-of-freedom systems, while multi-body systems and rigid-flexible coupled systems need to be considered in practice.

Figure 9.

Schematic configuration for the model of s-MRE-based isolation system.

When the s-MRE is used to protect the foundation against the exerted loading, dynamic differential equations can be written as follows

\begin{array}{l} m \ddot{x} (t) + K (θ) x (t) = F s i n ω t \\ K (θ) = (K_{e} + K_{m} + \frac{K_{v} τ_{0}^{α} D^{α}}{1 + τ_{0}^{α} D^{α}}) (1 - β θ) \end{array}

(9)

where m and x(t) are the mass and the displacement of the machine on the top, respectively; K_e and K_v denote the system stiffness in the elastic model and the fractional Maxwell model, respectively; K_m denotes the magnetically adjustable field-induced stiffness. ω = 2πf, ω and f are the angular frequency and the frequency, respectively; θ is the angular amplitude of the s-MRE-based isolator, θ = arctanx_A, where x_A is the amplitude of the displacement x(t). When the infinite foundation at the bottom is fixed, the displacements of the machine and the s-MRE-based isolatorare the same. The transmissibility λ_f can be defined as the ratio of the response amplitude F_p (the force transmitted to the fixed foundation) to the excitation amplitude F.

λ_{f} = \frac{F_{p}}{F}

(10)

In vibration isolation systems, s-MRE structures can be designed in parallel or in series for the desired stiffness, such as laminated structures.^15,35 Considering the results of parameter identification for s-MRE material, in this isolation system the parameters are set as m = 30 kg, K_e = 200 kN/m, K_v = 180 kN/m, τ₀ = 0.03 s and F = 20 N. When the response amplitude is smaller than the excitation amplitude, the transmissibility is below unity, and then the vibration isolation is considered to be effective. The transmissibility λ_f for the isolation system can be calculated numerically according to the definition in equation (10).

Figure 10 displays the influence of amplitude dependence on the transmissibility of this isolation system in the frequency domain. The nonlinearity of the system stiffness β = 0, regardless of the amplitude dependence; however, when the amplitude dependence is considered, the nonlinear coefficient β ≠ 0. As the nonlinear coefficient β increases from 0 to 0.6 rad^-1, the resonance frequency is reduced from 16.1 Hz to 8.0 Hz, which is in agreement with the observation that the material stiffness decreases when the nonlinear coefficient β increases in Figure 5(b). At the resonance frequency, the maximum peak can be observed on the transmissibility curves. As the amplitude dependence of s-MRE increases with the nonlinear coefficient β, the maximum peak transmissibility decreases from 12.76 to 11.91 (1.106 dB to 1.076 dB). The decrease in system stiffness due to increased amplitude dependence of s-MRE, as a result, the amplitude dependence is beneficial to the reduction of the force transmitted to the foundation. To improve the performance of vibration isolation, both the peak transmissibility and the resonance frequency can be lowered by increasing the nonlinear coefficient β, which can be achieved by enhancing the amplitude dependence of s-MRE during the fabrication.

Figure 10.

Influence of the nonlinear coefficient on the transmissibility against harmonic forces.

For the concern over magnetization efficiency, the ideal range of magnetic flux density for controllable properties of s-MRE is found to be below 500 mT, where the modulus of the material can be increased linearly with the increasing magnetic field intensity.^1,2 In comparison with traditional vibration isolation, the isolation effect can be further improved by switching the state of the external magnetic field at a certain frequency f_s, where the transmissibility without a magnetic field and in a magnetic field of 500 mT are the same. The control strategy can be illustrated as

B (f) = {\begin{cases} 0 f < f_{s} \\ 500 mT f > f_{s} \end{cases}

(11)

where B is the magnetic flux density of the switchable magnetic field, and f is the frequency.

When the performance of vibration isolation is studied in switchable magnetic fields, the maximum peak transmissibility occurs at the switch frequency f_s. If the amplitude dependence of s-MRE is neglected, the nonlinearity of the system stiffness β = 0; while the amplitude dependence is considered, the nonlinear coefficient β ≠ 0. In Figure 11, as the amplitude dependence of s-MRE increases with the nonlinear coefficient β, the switch frequency decreases from 17.9 Hz to 9.5 Hz, and the peak transmissibility decreases from 3.75 to 1.66 (0.574 dB to 0.220 dB). The nonlinear coefficient β can be increased by enhancing the amplitude dependence of s-MRE during fabrication, which is beneficial to the improvement of isolation performance from the perspective of lowering both the switch frequency and the peak transmissibility. In comparison with the traditional passive isolation system, the peak transmissibility is reduced by 48.0% to 78.7% in switchable magnetic fields.

Figure 11.

Influence of the nonlinear coefficient on the isolation effect against harmonic forces.

Vibration isolation against harmonic motion

In Figure 9, when the s-MRE is used to isolate the machine from ground motions, dynamic differential equations can be written as follows

\begin{array}{l} m \ddot{x} (t) + K (θ) δ (t) = 0 \\ x (t) = δ (t) + X s i n ω t \\ K (θ) = (K_{e} + K_{m} + \frac{K_{v} τ_{0}^{α} D^{α}}{1 + τ_{0}^{α} D^{α}}) (1 - β θ) \end{array}

(12)

where Xsinωt is the displacement of the flexible foundation at the bottom; δ(t) is the displacement of the s-MRE-based isolator; θ is the angular amplitude of the s-MRE-based isolator, θ = arctanδ_A, where δ_A is the amplitude of the displacement δ(t). The transmissibility in this isolation system λ_d is defined as the ratio of the response amplitude X_p (the displacement amplitude of the machine on the top) to the excitation amplitude X.

λ_{d} = \frac{X_{p}}{X}

(13)

In this isolation system, the parameters are set as m = 30 kg, K_e = 200 kN/m, K_v = 180 kN/m, τ = 0.03 s and X = 5 mm. The transmissibility λ_d can be calculated numerically according to the definition in equation (13).

In Figure 12, as the vibration frequency increases to higher than 40 Hz, the transmissibility in the s-MRE-based isolation system against harmonic motions increases and the efficiency of vibration control decreases (but the transmissibility is still below unity, 0 dB), which is different from the transmissibility in the s-MRE-based isolation system against harmonic force. The response time of s-MRE is reported to be at the several milliseconds level.¹⁹ As a result, the material deforms very slightly and even loses its elasticity at high frequencies, which then affects the performance of vibration isolation.

Figure 12.

Influence of the nonlinear coefficient on (a) the transmissibility and (b) the islation effect against ground motions.

The influence of the amplitude dependence on the transmissibility of s-MRE based isolation systems in the frequency domain is shown in Figure 12(a). The nonlinearity of the system stiffness β = 0, regardless of the amplitude dependence; when the amplitude dependence is considered, the nonlinear coefficient β ≠ 0. As the nonlinear coefficient β increases from 0 to 0.6 rad^-1, the resonance frequency is reduced from 16.2 Hz to 11.3 Hz. This is in agreement with the result of material stiffness shown in Figure 5(b), where the shear modulus of s-MRE decreases with increasing nonlinear coefficient β. When the amplitude dependence of s-MRE increases with the nonlinear coefficient β, the peak transmissibility decreases from13.7 to 6.01 (1.137 dB to 0.779 dB), but it barely changes until the nonlinear coefficient β is above 0.45 rad^-1. When the nonlinear coefficient β is higher than 0.45 rad^-1, the peak of the transmissibility curves is observed to become sharper, because the transmissibility increases faster to the maximum value as the frequency increases to the resonance frequency. It may result from the jump phenomenon and the stiffness softening, which can be enhanced by increasing the nonlinearity of system stiffness β. As the system stiffness decreases with increasing amplitude, the vibration transmitted to the machine is reduced at resonance frequencies. When low resonance frequency and low peak transmissibility are beneficial to the performance of vibration isolation, increasing the nonlinear coefficient β is an effective method, this can be realized by enhancing the amplitude dependence during the fabrication of s-MRE.

The control strategy in equation (11) is also adopted in this vibration isolation system. When the performance of vibration isolation is studied in switchable magnetic fields, the maximum peak of the transmissibility curves can be observed at the switch frequency f_s. In Figure (12), as the amplitude dependence of s-MRE increases with the nonlinear coefficient β, the switch frequency decreases from 15.6 Hz to 12.7 Hz, and the peak transmissibility decreases from 4.32 to 3.63 ( 0.588 dB to 0.370 dB). Considering the nonlinearity of the system stiffness, enhancing the amplitude dependence is beneficial for the reduction of the maximum peak transmissibility and the improvement of the isolation performance. Compared with the system in a zero magnetic field, the peak transmissibility is reduced by 39.6 % to 68.5 % in switchable magnetic fields.

Conclusion and outlook

In this work, the amplitude dependence of mechanical properties for s-MRE is experimentally characterized. The dynamic modulus magnitude of s-MRE decreases with increasing strain amplitude. To address the influence of amplitude on the viscoelasticity, a nonlinear fractional-order model is developed for s-MRE. The nonlinear fractional-order model is compared with the Bouc-Wen model and the linear fractional viscoelastic model using the numerical calculation and experimental data of the stress-strain hysteresis curve. This nonlinear fractional viscoelastic model demonstrates superior precision in depicting the dynamic behavior of s-MRE. Additionally, the impact of nonlinear viscoelasticity is analyzed for s-MRE-based isolation systems. Enhancing the amplitude dependence of s-MRE is expected to improve the isolation performance.

Other than the intrinsic advance in revealing the insights into how the amplitude dependence affects the dynamic behaviour of s-MRE, this research is expected to contribute to the theoretical basis and the optimal design of s-MRE-based vibration isolation systems. It is feasible to enhance the amplitude dependence during the fabrication of s-MRE, and the technique has potential for commercial applications. Furthermore, the influence of nonlinear viscoelasticity on the dynamic behaviour of s-MRE-based multi-body systems and rigid-flexible coupled systems needs to be explored to further promote practical applications.

Footnotes

Acknowledgement

Financial support from the National Natural Science Foundation of China (grant numbers 12172279 and 12472031) is gratefully acknowledged. Additionally, the proofreading by Xiaochong Zhang is highly appreciated.

ORCID iD

Guanghong Zhu

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: National Natural Science Foundation of China (grant numbers 12172279 and 12472031).

Declaration of conflicting interests

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

Data Availability Statement

Data will be made available on request.*

Fractional viscoelastic model

In the linear fractional viscoelastic model as shown in Figure 3, the elastic shear stress τ_e(t) can be expressed as (A1)

τ_{e} (t) = G_{e} γ (t)

The shear stress in fractional Maxwell model $τ_{v} (t)$ can be expressed as (A2)

G_{v} τ_{0}^{a} D^{a} [γ_{v} (t)] = G_{v} [γ (t) - γ_{v} (t)]

(A3)

γ (t) = [τ_{0}^{a} D^{a} + 1] γ_{v} (t)

The field-induced shear stress $τ_{m} (t)$ can be expressed as (A4)

τ_{m} (t) = G_{m} γ (t)

As shown in Figure 3, the relationship between the total stress $τ (t)$ and strain $γ (t)$ can be expressed as (A5)

τ (t) = (G_{e} + G_{m} + \frac{G_{v} τ_{0}^{α} D^{α}}{1 + τ_{0}^{α} D^{α}}) γ (t)

The relationship between the stress $τ (t)$ and strain $γ (t)$ can be transformed in the frequency domain by applying the Fourier transform. (A6)

τ (ω) = [G_{e} + G_{m} + \frac{G_{v} τ_{0}^{α} {(i ω)}^{α}}{1 + τ_{0}^{α} {(i ω)}^{α}}] γ (ω)

Substituting $i α = \cos (a π / 2) + i s i n (a π / 2)$ into equation (A6), the complex modulus of s-MRE G* can be obtained as (A7)

G^{*} (ω) = G_{e} + G_{m} + \frac{G_{v} τ_{0}^{α} ω^{α} [\cos (\frac{α π}{2}) + isin (\frac{α π}{2})]}{1 + τ_{0}^{α} ω^{α} [\cos (\frac{α π}{2}) + isin (\frac{α π}{2})]}

Rationalizing the complex denominator (A8)

G^{*} (ω) = G_{e} + G_{m} + \frac{G_{v} τ_{0}^{α} ω^{α} [\cos (\frac{α π}{2}) + isin (\frac{α π}{2})]}{1 + τ_{0}^{α} ω^{α} [\cos (\frac{α π}{2}) + isin (\frac{α π}{2})]} \times \frac{[1 + τ_{0}^{α} ω^{α} \cos (\frac{α π}{2})] - i τ_{0}^{α} ω^{α} \sin (\frac{α π}{2})}{[1 + τ_{0}^{α} ω^{α} \cos (\frac{α π}{2})] - i τ_{0}^{α} ω^{α} \sin (\frac{α π}{2})}

Separating the real and imaginary parts, (A9)

G^{*} (ω) = G_{e} + G_{m} + \frac{G_{v} τ_{0}^{α} ω^{α} [\cos (\frac{α π}{2}) + τ_{0}^{α} ω^{α}]}{1 + τ_{0}^{2 α} ω^{2 α} + 2 τ_{0}^{α} ω^{α} \sin (\frac{α π}{2})} + i \frac{G_{v} τ_{0}^{α} ω^{α} \sin (\frac{α π}{2})}{1 + τ_{0}^{2 α} ω^{2 α} + 2 τ_{0}^{α} ω^{α} \sin (\frac{α π}{2})}

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A nonlinear fractional-order model of soft magnetorheological elastomers for applications in vibration isolation

Abstract

Keywords

Introduction

Nonlinearity on dynamic mechanical properties of s-MREs

S-MREs and DMA experiments

Amplitude dependence of the modulus

Mathematical modelling for s-MREs

Mathematical models for s-MREs

Parameter identification

Numerical solution and validation

Dynamical analysis of s-MRE vibration systems

Vibration isolation against harmonic force

Vibration isolation against harmonic motion

Conclusion and outlook

Footnotes

Acknowledgement

ORCID iD

Funding

Declaration of conflicting interests

Data Availability Statement

Fractional viscoelastic model

References