The magnetorheological elastomer (MRE) is an intelligent material whose mechanical properties can be rapidly adjusted under a magnetic flux density. This material’s mechanical properties change due to the interaction between the iron particles inside the material. Understanding the influence of magnetic flux on iron particles in MRE materials is essential. Studies have proven that the distance and angle of inclination between iron particles significantly affect the magnetic flux density and the interaction force between the particles. Therefore, the distribution of iron particles substantially affects the material’s properties. However, understanding magnetic flux through magnetic particles is necessary to improve the material’s mechanical properties and to design magnetic field systems in systems using the materials. This study maps three problems affecting magnetic flux density to the properties of MRE. First, the mechanical characteristics of the MRE were presented in the frequency, amplitude, magnetic flux density, and magnetic flux inclination domains relative to the particle chain. Next, the influence of the magnetic flux on the particle chain was investigated based on the dipole interaction model and the magnetic force on iron particles. The finite element method also explored the magnetic flux distribution in the MRE material. Finally, the response of the single-degree-of-freedom damping system is tested experimentally. The results show that the influence of the magnetic flux on the iron particles in the MRE material is significant. The research results aim to improve the mechanical properties of MRE materials.

Magnetorheological elastomers (MREs) are usually composed of micron-sized iron particles (typically iron, nickel, or cobalt) dispersed in the elastomer. The change in the physical properties of the MRE is due to the magnetic force of the magnetic particles and the change in the magnetic flux density. Magnetic particles with high magnetic permeability and low residual magnetization are commonly used, and natural rubber or silicone rubber is often used as the substrate. In MREs, the magnetic particles are fixed in the matrix. Isotropic and anisotropic MREs are formed in the presence or absence of magnetic fields during vulcanization. The iron particles in the isotropic MRE are randomly distributed, while the iron particles in the anisotropic MRE are distributed in series.

The properties of MREs are greatly affected by several factors, such as size, geometry, spatial distribution, and volume of particles. Among these, the volume fraction of iron particles has the most remarkable influence on the MRE properties. Furthermore, the formation of particle chains during vulcanization is shown to be highly dependent on the particle volume for anisotropic MREs.1 Many studies suggest that the mechanical properties of MRE materials are highly effective when the iron particles are completely magnetized. The distance between particles, as well as the defects of particle chains, is also taken into consideration. It has been observed that a high percentage of iron particles can significantly enhance the mechanical properties of MRE materials.2,3 A high particle volume ratio is desirable as the distance between particles reduces, making them more sensitive and responsive to magnetic fields.4 However, the particle density should not be too high as this can cause the particles to become too large, thereby leading to defects in the material. The large volume (above 50 vol. %) does not increase the MR effect but makes the material have a high zero-field modulus. In the case of a diluted particle concentration in the MRE, such as a concentration less than 20 vol. %, the concentration of magnetic particles is so low that individual chains and aggregates do not form. The volume fraction also depends on the base material and the particle diameter, with volume percentages claimed to be effective ranging from 27% to 40%.5–8 

The dynamic modulus can describe the frequency-dependent property of the MRE. The dynamic modulus is usually characterized by its storage and loss modulus, representing the material’s elastic and dissipation components. The frequency-dependent behavior of the MRE can be further illustrated by the loss factor, which is the ratio of the loss modulus to the storage modulus. The loss factor is used to quantify a material’s ability to dissipate energy and can be used to optimize the performance of an MRE for specific applications. The properties of a magnetic rheological elastomer (MRE) are highly dependent on the flux density of the applied magnetic field. MREs respond to the strength of magnetic fields by changing their mechanical and rheological properties. The flux density significantly depends on the MRE material’s stiffness (modulus) and hysteresis.9,10 At low flux densities, MREs exhibit soft properties, while at higher flux densities, they become stiffer and more solid. The magnitude of this effect depends on the strength of the applied magnetic field. At a specific frequency, the hysteresis of the MRE is nonlinear and increases significantly as the magnetic flux increases. Hysteresis is the rate of energy loss in one cycle. It is the result of friction occurring within the material. Many models have been used for the hysteresis of MRE materials, such as Bingham, Dahn, Bouc–Wen, and smooth Coulomb friction models.11–18 The property of an amplitude-dependent rheological material is the ability of a material to change its properties in response to different amplitude magnitudes. At a given frequency and magnetic field strength, the elastic modulus (or stiffness) becomes large or small when performed at small and large amplitudes. It is the Fletcher–Gent or Payne effect property of elastic materials.19,20 This property is also highly dependent on the strength of the magnetic field. MRE properties also depend on iron particle size and iron particle volume ratio. The structure of particles in the MRE is also attracting the interest of many researchers. The tilt of the magnetic field relative to the particle chain is thought to significantly affect the stiffness (module) and hysteresis, which needs to be elucidated.21–23 

The iron particles in the stored magnetic material interact under a magnetic field strength. The particles are magnetized when exposed to an external magnetic flux and are polarized, so these iron particles interact with each other in the MRE material. This interaction force and the magnetic force cause a change in the properties of a material. Many studies are interested in the influence of the magnetic field and the distribution of iron particles inside the material to improve the material’s mechanical properties during the fabrication process.24–27 Flux distribution and flux intensity in rheological materials were determined by using finite element method magnetic (FEMM) software to analyze the flux distribution inside the material under different bead structures.28–30 Studies have shown that understanding the influence of magnetic flux on iron particles in MRE materials is of great importance due to the interaction between iron particles. The structure of the iron particles in the MRE material significantly affects the properties of the material. The extraction of the compressed-mode MRE with the tilt angle has received little interest. A magneto-mechanical model for anisotropic MREs, using magnetic dipole theory and energy methods, was developed for the compressive strain on the magnetic particle chain microstructure.31 Furthermore, the distance and inclination angle between iron particles greatly influence the magnetic flux strength and the inter-particle interaction force. However, studying the influence of magnetic fields on particle interactions in magnetic rheological materials is a difficult task and is of interest to many scientists.

Vibration systems using MRE-based devices require a semi-active controller to improve efficiency. The simplest and most common controller is the on-off skyhook controller, where the stiffness of the MRE material is set to the maximum and minimum values.32,33 Proportional integral differential (PID) controllers are also widely used in many vibration applications.34,35 Fuzzy logic control is a form of artificial intelligence widely used in semi-active controllers.36–38 The error value is continuously calculated by a PID so that the system reaches the desired value. Linear quadratic modulation (LQ) is one of the most widely used methods for semi-active controllers.39 Semi-active adaptive controllers are suitable for vibration control systems with uncertain parameters. The basis of the adaptive controller is parameter estimation. Lyapunov stabilization is used to determine the updated laws and to ensure the system is stable.40 The fuzzy logic can be used to adjust the high or low stiffness of the MRE corresponding to the high or low value of the applied current smoothly. The determination of the output values is based on fuzzy rules. The use of artificial neural networks (ANNs) for semi-active control is one of the promising techniques in the field of vibration control, which has been shown to provide better performance than conventional controllers.41–44 An ANN is used to model the system’s dynamic behavior and generate the appropriate control signal based on the input. The controller can be used to predict the system’s future behavior and adjust the controller accordingly. The h-infinity controller is also preferred for the semi-active controller. The potent feedback regulator can minimize the effects of uncertainty and disturbance. The h-infinity control method can achieve high-precision control, suitable for many control applications.45 It is a simple structured approach, making it easier to design and implement. There are also sliding mode controllers and composited controls for semi-active controllers.46,47 The application of vibration systems not only with semi-active control of MRE absorbers and isolators but also with tunable periodic local resonance metamaterial structures was developed using MREs. A compact design for a novel metamaterial MRE sandwich beam was developed.48 The primary objective of this design is to create a real-time tunable local resonance bandgap for effective vibration isolation. A groundbreaking metamaterial sheet was proposed, in which MREs and multiple gradient resonators are combined.49 This innovative design offers low-frequency capability, wide-band coverage, and adjustable bandgap properties.

This study investigated the influence of magnetic particles on the magnetic flux. Mechanical properties of MREs are presented first with different tilt angles between the magnetic field and particle chain. Then, the effect of the magnetic flux density between the iron particles was investigated by the finite element method. The dipole force model between iron particles in the material is determined. The analysis aims to evaluate the agreement between the dipole model between iron particles in the MRE material and the experimental results. Finally, the response of the single-degree-of-freedom damping system is tested experimentally.

In this study, three MRE samples with chain angles to the direction of the magnetic field, including 0°, 30°, and 60°, were fabricated. The sample fabrication process is as shown in Fig. 1. Materials include iron particles (20 µm size, 30% volume, BASF SG-BH), room temperature vulcanized silicone (Shinetsu KE1416, 68% volume), and silicon oil (2% volume). These materials are mixed to form a homogeneous mixture. The mixture was put into a centrifugal mixer to create a homogeneous mixture within 10 min. The mixture was then put into a vacuum chamber for about 20 min to remove the air bubbles inside the material. The mixture is then poured into plastic molds. Vulcanization was carried out under a magnetic field of 0.5 T for 24 h. The mold cavities have different inclination angles so that the normal to the mold face creates an inclination angle of 0°, 30°, and 60° corresponding to the three MRE samples. Samples S0, S30, and S60 correspond to particle chains with inclination angles of 0°, 30°, and 60°, respectively.

FIG. 1.

Process of making MRE samples consists of three main steps: mixing to a homogeneous mixture, removing air bubbles, and curing in a magnetic field with different mold inclination angles.

FIG. 1.

Process of making MRE samples consists of three main steps: mixing to a homogeneous mixture, removing air bubbles, and curing in a magnetic field with different mold inclination angles.

Close modal

The values of the factors that affect the formation of particle chains, such as the particle volume, applied magnetic field, time, and silicon oil (as a surfactant), were 30%, 0.5 mT, 24 h, and 2%, respectively. Assume that the chains have been formed in the direction of the magnetic field during vulcanization and the response of iron particles under a magnetic field strength follows a particle-magnetic dipole theory. The mechanical properties of the MRE samples (S0, S30, and S60) were determined in the absence of magnetic flux density and the presence of a magnetic flux density. MRE samples, with the same particle volume fraction and different particle chain inclination in the magnetic field direction, were tested for mechanical properties. Experimental results show that the MRE samples have the same values of mechanical properties in the absence of an applied magnetic field. When the magnetic flux density is applied, the MRE samples have significantly different values of mechanical properties. This demonstrates that the tilt angle of the particle chain relative to the direction of the magnetic field significantly affects the properties of the MRE material. The mechanical properties of MRE materials, depending on the frequency, amplitude, magnetic flux density, and chain inclination angle with the magnetic field direction, will be analyzed in the following section.

The optimal volume fraction of iron particles in a magnetorheological elastomer (MRE) can significantly impact its mechanical properties. Increasing the volume fraction of iron particles generally leads to an increase in the stiffness and elastic modulus of the MRE. On the other hand, increasing the volume fraction of iron particles can reduce the flexibility and resilience of the MRE. The volume fraction of iron particles can also affect the fatigue resistance and durability of the MRE. Optimal volume fractions should be chosen to balance the mechanical properties and long-term durability. Therefore, careful optimization is required to find the volume fraction that achieves the desired balance between the magnetic and mechanical properties for a specific application. Experimental testing and characterization are essential to evaluate the mechanical behavior of MREs with different volume fractions and validate their suitability for intended applications.

The mechanical properties of the MRE were determined based on the dynamic properties of an experimental system, as shown in Fig. 2. Two MRE samples were placed between the fixed iron cores and the movable part to induce shearing motion. The fixed core is fitted with a force sensor (PCB PIEZOTRONICS LOAD CELL FORCE SENSOR 208B02). The movable core is wound by a copper coil with a diameter of 1 mm and wound 600 turns when fixed on an oscillating table. The amount of displacement of the table is the shear strain of the MRE. The excitation table performs displacement [x=x0sin(ωt) based on an eccentric wheel. The deflection of the eccentric wheel adjusts the displacement amplitude. The eccentric wheel is driven by a servo motor whose rotational speed controls the frequency. A DC power supply (KIKUSUI PMC-5AS) regulates the magnetic flux passing through the core. A laser sensor (KEYECE LB-02) detects table vibrations. An oscilloscope (TEKTRONIX MDO3022) was used to analyze the force-displacement response. Varying frequency, amplitude, magnetic field strength, and particle chain angle were identified to determine the MRE characteristics.

FIG. 2.

Testing system for measuring the mechanical properties of materials.

FIG. 2.

Testing system for measuring the mechanical properties of materials.

Close modal

1. General properties

In this section, the properties of the MRE are elucidated. Many studies have shown that the MRE properties depend on frequency, amplitude, and magnetic flux density. MRE properties depending on the particle size and ratio have been studied by many researchers. However, there are few studies on the structure-dependent MRE properties of the particle chain. The MRE mechanical properties under different angles of the chain are also investigated as the interaction of particles in the magnetic field.

The force-displacement hysteresis loops of the MRE under different conditions are analyzed. Besides elastic properties, hysteresis properties of materials are also essential properties of MRE materials, including frequency-independent hysteresis and frequency-dependent hysteresis. Experiments were performed at low frequency (f ≤ 1 Hz) to investigate frequency-independent hysteresis, as shown in Fig. 3. The hysteresis loop area still has a significant value even though the frequency value is very small. It is the cause of the Fletcher–Gent phenomenon, where the stiffness/elastic modulus decreases as the strain amplitude increases.19,20 This property is similar to the property of frictional behavior. We use friction to represent this property of the material. As the strain frequency increases, the loop area increases, which is the fundamental viscosity of the material (frequency-dependent hysteresis). Therefore, the mechanical properties of the MRE including elastic force, frictional force, and viscous force is
F=Fe+Ff+Fv,
(1)
where Fe is the elasticity of the material, Ff is the frictional property of the material representing the frequency-independent hysteresis property, and Fv is the viscosity representing the hysteresis depends on the frequency.
FIG. 3.

Force-displacement response at low frequency performance (f = 1 Hz); the viscosity is very small, and elasticity and friction are the main properties: (a) absence of an applied magnetic field and (b) a magnetic flux strength is applied.

FIG. 3.

Force-displacement response at low frequency performance (f = 1 Hz); the viscosity is very small, and elasticity and friction are the main properties: (a) absence of an applied magnetic field and (b) a magnetic flux strength is applied.

Close modal
The elastic (Fe) and frictional properties (Ff) increase significantly when a magnetic flux density is applied. The particle chain angle also influences the MRE mechanical properties. Assuming that the excitation frequency is small (f = 1 Hz), the viscosity effect (frequency-dependent hysteresis) is negligible (Fv ≈ 0). Equation (1) is rewritten as F = Fe + Ff. The force-displacement response is shown in Fig. 3. The force amplitude is determined as
F0=Fe0+Ff0,
(2)
where F0 is the force amplitude, Fe0 is the elastic force amplitude, and Ff0 is the friction force amplitude.

2. Frequency dependence

MRE properties under different frequencies were analyzed for sample S0. The amplitude is set to 0.8 mm, and the frequency is changed with varying values of 1, 4, 8, and 20 Hz. The experiment was performed under different magnetic flux densities of 0 mT (0 A), 112 mT (2 A), and 226 mT (4 A). The force-displacement response is shown in Fig. 4. The experiment is performed in the absence and the presence of an applied magnetic field. The response loop area is still significant even at 1 Hz as the property of a friction force; the area and amplitude of the force increase slightly with the increasing frequency, as shown in Fig. 4(a). It is the basic viscoelasticity of MRE materials.

FIG. 4.

Force response–displacement under different frequency levels in the case of with and without an applied magnetic field performed sample S0: (a) no magnetic field; (b) in the presence of 265 mT (3 A).

FIG. 4.

Force response–displacement under different frequency levels in the case of with and without an applied magnetic field performed sample S0: (a) no magnetic field; (b) in the presence of 265 mT (3 A).

Close modal

As the applied flux strength increases, the loop area and force amplitude increase significantly at 1 Hz, as shown in Fig. 4(b). However, the area and amplitude increased slightly with increasing excitation frequency. Many studies have suggested that in elastic materials, the material’s hysteresis includes nominal and frequency-dependent hysteresis. The friction is evident when performing at low frequencies (1 Hz) and large amplitudes. It is the Fletcher–Gent property of elastomers that the stiffness increases with performance at low amplitudes. This problem will be evident when observing the response loop at different amplitudes, which will be presented in the next section. This nominal hysteresis is strongly dependent on the magnetic field strength. Many models have been proposed to represent this nominal hysteresis, such as the Bouc–Wen, smooth Coulomb friction, and Dah model. Meanwhile, frequency-dependent hysteresis means the fundamental viscoelasticity of an elastic material.

3. Magnetic flux density dependence

The MRE properties strongly depend on the magnetic flux density, including hysteresis and stiffness. The frequency is initially set to 1 Hz to limit frequency-dependent hysteresis. The frequency is then increased to 12 Hz to observe frequency-dependent hysteresis. The amplitude is set to medium (x0 = 0.8 mm). The magnetic field strength is adjustable at different levels from 0–226 mT, corresponding to the adjusted current of 0–4 A.

Experimental results are shown in Fig. 5. When the low frequency is used to reduce the effect of viscosity, as shown in Fig. 5(a), the force amplitude and the slope of the response loop increase significantly with increasing magnetic flux density. The value of the amplitude increases by about 100% when the flux strength is increased to 227 mT. Moreover, the area of the loop is also significantly increased. The elastic and frictional properties depend significantly on the strength of the applied magnetic flux. Figure 5(b) shows the force-displacement response performed at a frequency of 12 Hz. The loops become more elliptical, and the amplitude of the loop increases slightly. The rate of change of increase with the frequency increasing to 12 Hz is almost the same as the corresponding loops at 1 Hz; the trend is the same with magnetic flux density levels. It is the fundamental viscous property of the material.

FIG. 5.

The force-displacement response under different current levels with the amplitudes set: (a) excitation frequency of 1 Hz; (b) excitation frequency of 12 Hz.

FIG. 5.

The force-displacement response under different current levels with the amplitudes set: (a) excitation frequency of 1 Hz; (b) excitation frequency of 12 Hz.

Close modal

4. Amplitude dependence

The MRE property is significantly dependent on the excitation amplitude. The amplitude is set to small, medium, and large values corresponding to x0 = 0.5, 0.85, and 1.4 mm, respectively; the frequency is set to 1 Hz. The experiment was performed without a magnetic flux and in the presence of an applied magnetic flux density of 167 mT (3 A).

The force-displacement response under different amplitudes is shown in Fig. 6. Due to the low-frequency performance, the effect of viscosity is low. The force response still exists in a hysteresis loop; this property is similar to friction. This property explains the Fletcher–Gent phenomenon when the low amplitude of the equivalent hardness of the material increases. When performed at large amplitudes, the friction is clearly shown. As the amplitude increases, the slope of the response loops decreases moderately. The loop slope is slightly reduced and significantly reduced without a magnetic flux density and with an applied magnetic flux density. The friction amplitude (Ff0) is evident when performing large amplitudes. The (Ff0) value increased significantly by 100% at 167 mT. The elasticity and frictional properties are significantly affected when a magnetic field strength is applied.

FIG. 6.

Force-displacement responses under different amplitude levels for sample S0; frequency is set to f = 1 Hz: (a) in the absence of magnetic field 0 mT (0 A); (b) the presence of magnetic field 167 mT (3 A).

FIG. 6.

Force-displacement responses under different amplitude levels for sample S0; frequency is set to f = 1 Hz: (a) in the absence of magnetic field 0 mT (0 A); (b) the presence of magnetic field 167 mT (3 A).

Close modal

5. Dependence on the tilt angle of the particle chain

The influence of the tilt angle relative to the direction of the magnetic flux on the MRE properties was also investigated. The three material samples, S0, S30, and S60, have different bead inclination angles of 0°, 30°, and 60°, respectively. The experiment was performed at 1 Hz frequency, 0.85 mm amplitude, and different levels of magnetic flux density. The force-displacement response is shown in Fig. 7. In the absence of magnetic field, the force amplitudes and response loop areas are the same, which indicates that the substrate material is the same for all samples. However, when applying the magnetic flux density, the response loops are significantly different in both force amplitude and response loop area. The force amplitude and the response loop area decrease as the tilt angle increases. The elastic force amplitude Fe0 and the maximum friction force (Ff0) reached the maximum and minimum values for samples S0 and S60, respectively.

FIG. 7.

Force-displacement loop response when performed with samples with different tilt angles 30°, 60° and 90° and amplitude and frequency of x0 = 0.85 and f = 1 Hz, respectively; experiments were performed under varying levels of electric flux density: (a) 0 mT (0 A); (b) 112 mT (2 A); (c) 224 mT (4 A); (d) 336 mT (6 A).

FIG. 7.

Force-displacement loop response when performed with samples with different tilt angles 30°, 60° and 90° and amplitude and frequency of x0 = 0.85 and f = 1 Hz, respectively; experiments were performed under varying levels of electric flux density: (a) 0 mT (0 A); (b) 112 mT (2 A); (c) 224 mT (4 A); (d) 336 mT (6 A).

Close modal

6. Discussion

MRE materials possess viscosity, elasticity, and frictional properties. Viscosity exhibits an MRE force response in the frequency domain and is less susceptible to magnetic fields. While the elasticity and frictional properties vary significantly in the magnetic field strength domain, the frictional property explains the Fletcher–Gent phenomenon, where the modulus of elasticity decreases with increasing amplitude. The structure of the chain, such as the angle of inclination of the chain, also significantly affects the elasticity and frictional properties of the material. Table I presents the values of force amplitude (F0), friction amplitude (Ff0), loop area (A, and loop slope (s) under the strain amplitude, magnetic flux density, and strain frequency levels for sample S0. The force amplitude values increase due to increasing the displacement amplitude. However, the loop slope decreases by 20% and 50% in the absence of magnetic flux density and the presence of applied magnetic flux density, respectively. Fletcher–Gent properties change significantly with changing magnetic flux density. The table shows that the above-mentioned values increase considerably in the magnetic flux density domain as the magnetic field strength increases from 0 to 330 mT. The force amplitude, friction amplitude, and slope of the loop increased by 130%, 370%, and 140% with increasing current, respectively. The loop slope and loop area represent equivalent stiffness and energy absorption after each cycle. As the strain frequency increases to 12 Hz, the values increase slightly due to the influence of the viscosity of the material. The values increase slightly with increasing frequency. It is the viscosity of the material, depending on the frequency. When the magnetic flux density is applied, these values are raised to a significant value.

TABLE I.

Force amplitude (F0), friction force amplitude (Ff0), loop area (A), and loop slope (s = F0/x0) were determined under different levels of amplitude, magnetic flux density (applied current), and frequency for sample S0.

F0 (N)Ff0 (N)A (mm2)S (N/mm)F0 (N)Ff0 (N)A (mm2)S (N/mm)
Strain amplitude (mm)f = 1 Hz, 0 mT (0 A)f = 1 Hz, 170 mT (3 A)
0.25 4.7 0.8 0.6 19.9 14.5 4.2 2.9 60.3 
0.50 8.6 1.3 1.8 17.9 21.8 6.2 8.6 45.5 
0.75 12.3 1.8 3.8 17.1 28.2 7.4 15.8 39.2 
1.0 15.9 2.2 6.2 16.6 33.5 8.2 23.8 34.8 
1.2 19.6 2.7 9.2 16.5 39.3 8.8 32.9 32.8 
1.4 23.8 3.1 13.2 16.3 43.4 9.4 43.4 30.2 
1.70 27.1 3.4 17.4 16.1 48.0 10 54.5 28.6 
F0 (N)Ff0 (N)A (mm2)S (N/mm)F0 (N)Ff0 (N)A (mm2)S (N/mm)
Strain amplitude (mm)f = 1 Hz, 0 mT (0 A)f = 1 Hz, 170 mT (3 A)
0.25 4.7 0.8 0.6 19.9 14.5 4.2 2.9 60.3 
0.50 8.6 1.3 1.8 17.9 21.8 6.2 8.6 45.5 
0.75 12.3 1.8 3.8 17.1 28.2 7.4 15.8 39.2 
1.0 15.9 2.2 6.2 16.6 33.5 8.2 23.8 34.8 
1.2 19.6 2.7 9.2 16.5 39.3 8.8 32.9 32.8 
1.4 23.8 3.1 13.2 16.3 43.4 9.4 43.4 30.2 
1.70 27.1 3.4 17.4 16.1 48.0 10 54.5 28.6 
Magnetic flux density (mT)f = 1 Hz, x0 = 0.85 mmf = 12 Hz, x0 = 0.85 mm
0 mT (0 A) 14.7 2.2 5.6 17.6 15.3 2.3 6.2 18.3 
57 mT (1 A) 20.2 4.1 9.9 24.0 22.3 4.3 12.1 26.5 
113 mT (2 A) 25.7 6.7 16.6 30.6 29.5 6.9 18.1 35.1 
170 mT (3 A) 30.4 8.2 20.4 36.2 34.4 8.1 22.8 40.9 
227 mT (4 A) 33.2 9.2 23.1 39.5 36.8 9.0 24.6 43.8 
270 mT (5 A) 34.5 10.0 24.1 41.1 38.3 9.8 26.3 45.6 
330 mT (6 A) 35.4 10.5 25.0 42.1 38.9 10.0 26.2 46.4 
Magnetic flux density (mT)f = 1 Hz, x0 = 0.85 mmf = 12 Hz, x0 = 0.85 mm
0 mT (0 A) 14.7 2.2 5.6 17.6 15.3 2.3 6.2 18.3 
57 mT (1 A) 20.2 4.1 9.9 24.0 22.3 4.3 12.1 26.5 
113 mT (2 A) 25.7 6.7 16.6 30.6 29.5 6.9 18.1 35.1 
170 mT (3 A) 30.4 8.2 20.4 36.2 34.4 8.1 22.8 40.9 
227 mT (4 A) 33.2 9.2 23.1 39.5 36.8 9.0 24.6 43.8 
270 mT (5 A) 34.5 10.0 24.1 41.1 38.3 9.8 26.3 45.6 
330 mT (6 A) 35.4 10.5 25.0 42.1 38.9 10.0 26.2 46.4 
Strain frequency (Hz)x0 = 0.85 mm, 0 mT (0 A)x0 = 0.85 mm, 170 mT (3 A)
1 Hz 14.7  5.6 17.7 28.2  18.3 33.6 
4 Hz 15.5  5.8 18.5 30.1  19.3 35.7 
8 Hz 16.0  6.0 19.3 33.4  20.8 39.7 
12 Hz 16.2  6.5 19.5 33.8  21.1 40.3 
20 Hz 17.0  7.4 20.8 35.6  22.7 42.3 
25 Hz 17.1  7.5 20.6 35.7  23.9 42.6 
31 Hz 17.6  8.0 21.5 37.7  23.3 44.8 
Strain frequency (Hz)x0 = 0.85 mm, 0 mT (0 A)x0 = 0.85 mm, 170 mT (3 A)
1 Hz 14.7  5.6 17.7 28.2  18.3 33.6 
4 Hz 15.5  5.8 18.5 30.1  19.3 35.7 
8 Hz 16.0  6.0 19.3 33.4  20.8 39.7 
12 Hz 16.2  6.5 19.5 33.8  21.1 40.3 
20 Hz 17.0  7.4 20.8 35.6  22.7 42.3 
25 Hz 17.1  7.5 20.6 35.7  23.9 42.6 
31 Hz 17.6  8.0 21.5 37.7  23.3 44.8 
When performed at 1 Hz, the frequency-dependent hysteresis is negligible, Fv ≈ 0. Assume the response force includes elastic force and friction force, F = Fe + Ff. The response force amplitude is determined (F0 = Fe0 + Ff0), as shown in Fig. 3. When a magnetic flux is applied, the force amplitude values are analyzed,
Fe0=Fe00+ΔFe0;Ff0=Ff0(0)+ΔFf0,
(3)
where Fe0(0) and Ff0(0) are the elastic and frictional force amplitudes in the absence of an applied magnetic field, respectively. ΔFe0 and ΔFf0 are the increments in the elastic and friction force amplitudes, respectively, when applying a magnetic field.

The experimental results have been presented in Table II and Fig. 8. From Fig. 8, it can be seen that the largest and smallest increases in elastic force happen for samples S30 and S60, respectively. Meanwhile, the maximum and minimum increases in friction are for the S0 and S60 models. In the next part of this study, we use the dipole interaction model to explain the influence of the magnetic flux density on elastic and frictional forces. This model also explains the effect of the chain tilt angle on the elasticity and frictional properties of the material.

TABLE II.

The elastic force and the friction force amplitude (Fe0, Ff0) under the magnetic flux strength levels for samples S0, S30, and S60. The part in parentheses shows the increase in the amplitudes (ΔFe0 and ΔFf0).

Elastic force amplitude Fe0Friction force amplitude Ff0
(increase of elastic force amplitude ΔFe0)(increase of friction force amplitude ΔFf0)
Sample0 A1 A2 A3 A4 A0 A1 A2 A3 A4 A
S0 12.5 15.8 18.7 21.9 23.3 2.2 4.2 6.8 8.3 9.5 
(0) (3.3) (6.2) (9.4) (10.8) (0) (2) (4.6) (6.1) (7.3) 
S30 11.9 15.25 18.9 21.9 23.5 2.0 3.8 5.9 7.2 8.2 
(0) (3.35) (7.0) (10) (11.6) (0) (1.8) (3.9) (5.2) (6.2) 
S60 12.8 14.5 16.5 18.8 19.3 2.4 3.8 5.0 6.5 6.6 
(0) (1.7) (3.7) (6) (6.45) (0) (1.4) (2.6) (3.2) (4.2) 
Elastic force amplitude Fe0Friction force amplitude Ff0
(increase of elastic force amplitude ΔFe0)(increase of friction force amplitude ΔFf0)
Sample0 A1 A2 A3 A4 A0 A1 A2 A3 A4 A
S0 12.5 15.8 18.7 21.9 23.3 2.2 4.2 6.8 8.3 9.5 
(0) (3.3) (6.2) (9.4) (10.8) (0) (2) (4.6) (6.1) (7.3) 
S30 11.9 15.25 18.9 21.9 23.5 2.0 3.8 5.9 7.2 8.2 
(0) (3.35) (7.0) (10) (11.6) (0) (1.8) (3.9) (5.2) (6.2) 
S60 12.8 14.5 16.5 18.8 19.3 2.4 3.8 5.0 6.5 6.6 
(0) (1.7) (3.7) (6) (6.45) (0) (1.4) (2.6) (3.2) (4.2) 
FIG. 8.

Increase in elastic force amplitude and friction force amplitude when tested at different applied current values for the samples (S0, S30, and S60); the values are calculated as shown in Fig. 3(b): (a) the increase in elastic force amplitude ΔFe0; (b) the friction force amplitude ΔFf0.

FIG. 8.

Increase in elastic force amplitude and friction force amplitude when tested at different applied current values for the samples (S0, S30, and S60); the values are calculated as shown in Fig. 3(b): (a) the increase in elastic force amplitude ΔFe0; (b) the friction force amplitude ΔFf0.

Close modal
The force between the magnetic dipoles is shown in Fig. 9. When two magnetic particles are placed close together in a magnetic field, the magnetic moments of the particles interact, creating a magnetic force between them. The strength of the dipole–dipole interaction force depends on the magnitude of the dipoles and the distance between them. The force is stronger for larger dipoles and closer distances. In addition, the orientation of the dipoles concerning each other also plays a role in determining the strength of the force. The interaction energy is expressed as16 
Um=m2(13cos2α)4πμ1μ0l3,
(4)
where α is the angle of inclination between the two particles, μ0 = 4π × 10−7 N/A2 is the vacuum permeability, μ1 is the magnetic permeability of the medium, and m and l are a dipole moment and the distance between two magnetic dipoles, respectively.
FIG. 9.

Schematic diagram of magnetization of two magnetic particles under external magnetic flux: (a) two magnetic dipoles; (b) shear deformation.

FIG. 9.

Schematic diagram of magnetization of two magnetic particles under external magnetic flux: (a) two magnetic dipoles; (b) shear deformation.

Close modal
Equation (4) is rewritten in terms of x and y as
Um=m2(12y2/x2+y2)4πμ1μ0x2+y23=m2(x22y2)4πμ1μ0x2+y25/2.
(5)
The force of interaction between two iron particles in the x and y directions is
Fmx=dUmdx=m24πμ1μ0x(3x2+12r2)x2+r27/2
=dUmdx=m24πμ1μ0sinα[3(sinα)2+12cosα2]l2,
(6)
Fmy=dUmdy=m24πμ1μ0r(x2+14r2)x2+r27/2
=dUmdx=m24πμ1μ0cosα[(sinα)2+14cosα2]l2.
(7)
The dipole moment of a particle is determined as
m=JpVi,
(8)
where Jp = μ0Mp is the dipole moment per unit particle volume and Vi=43πd3 is the volume of an iron particle.
According to the Frohlich–Kennelly law, the magnetic field strength Mp is defined as
Mp=(μp1)MsHMs+μp1H,
(9)
where Ms denotes the saturated magnetization, μ0Ms = 2.1 T for ferromagnetic materials, and μp = 1000 represents the permeability of the ferromagnetic material at low magnetic fields. H = kI/lc (A/m) is the external magnetic flux; k, I, and lc are the number of turns, the applied current, and the length of the coil, respectively.
The dipole moment m̄ for all particles in the sample volume MRE is
m̄=nm=nJpVi=VJp,
(10)
where n = ∅V/Vi is the number of particles in the sample MRE, ∅ = 40% is the volume ratio of iron particles in the material sample, V is the sample volume, and Vi is the volume of an iron particle.
The magnetic shear force amplitudes of the MRE sample are determined as
Fx0=m̄24πμ1μ0sinα+δ[3(sinα+δ)2+12cosα+δ2]l2,
(11)
Fy0=m̄24πμ1μ0cosα+δ[(sinα+δ)2+14cosα+δ2]l2,
(12)
where δ = x0/h is the inclined chain strain angle, x0 is the shear displacement amplitude, and h is the thickness of the MRE sample.

Assume the iron particles in the anisotropic MRE sample are arranged in an ideal series. The tilt angle (α) is the angle between the direction of the magnetic flux and the particle chain. The total magnetic forces (Fx0, Fy0) are simulated in the domain of applied current and the tilt angle, as shown in Figs. 10 and 11. Both forces increase significantly as the applied current increases. However, the increased magnitude depends on the tilt angle between the magnetic flux and the particle chain. Figure 10 shows the magnetic forces in the current domain for the three tilt angles 0°, 30°, and 60°. The magnetic force Fx0 increases the most for the tilt angle of 30° and the slowest for the angle of addiction 60° in the x direction. Meanwhile, the magnetic force Fy0 increases the most when the angle of inclination is 0° and increases insignificantly when the angle of inclination is 60°. Figure 11 examines the magnetic force in the angular domain between the magnetic field and the particle chain. It can be seen that the force Fx0 reaches the maximum value in about 30°. In comparison, the force (Fy0) reaches its maximum value at 0° and is negligible when the tilt angle is large (over 60°).

FIG. 10.

Force amplitudes of the whole particle Fx0 and Fy0) are calculated in the magnetic field strength domain with different inclination angles: (a) Fx0; (b) Fy0.

FIG. 10.

Force amplitudes of the whole particle Fx0 and Fy0) are calculated in the magnetic field strength domain with different inclination angles: (a) Fx0; (b) Fy0.

Close modal
FIG. 11.

Forces of the whole particle Fx0 and Fy0) are calculated in the tilt angle domain: (a) Fx0; (b) Fy0.

FIG. 11.

Forces of the whole particle Fx0 and Fy0) are calculated in the tilt angle domain: (a) Fx0; (b) Fy0.

Close modal

This section compares the experimental results (Fig. 8) and the simulation results (Fig. 10). Assume that the increase in the elastic force amplitude (ΔFe0) is the cause of the magnetic force Fx0. Meanwhile, the increase in friction force (ΔFf0) is due to the influence of the magnetic force Fy0. The force Fy0 is similar to the normal force load and significantly affects the friction force through the coefficient of friction. The results show a forced agreement between simulation and experiment when the applied magnetic flux and different inclination angles exist. Considering the relationship between two forces (ΔFe0, Fx0) from Fig. 8(a) and Fig. 10(a) shows that they have similar properties when examined at three different angle values. The forces increase significantly with increasing amperage, and the rate-dependent increase in the inclination angle between the magnetic flux and the particle chain is in descending order of 30°, 0°, and 60°. Besides, the relationship between two forces (ΔFf0, Fy0) also has the same properties. Comparing the results between Figs. 8(b) and 10(b), the forces increase with increasing the amperage. The degree of increase depends on the angle of inclination, reaching the maximum value at 0° inclination and decreasing with an increasing tilt angle. However, the force value obtained experimentally is not the same as the calculated one because the particles in the sample have not been arranged in an ideal sequence and there is an interaction between the particles. Both simulation and experimentation can lead to discoveries and insights into the behavior of MRE materials.

This study used finite element method magnetic (FEMM) software to analyze the magnetic flux distribution in MRE materials with different particle structures. The influence of magnetic flux on iron particles considerably influences the mechanical properties of the MRE. Figure 12 shows the magnetic model in an MRE consisting of three parts, the elastomer, the magnetic particle, and the coil. The first part corresponds to the coil that generates the magnetic field inside the material. The coil has ten turns to produce an external magnetic flux strength (point A) of the same value as that of the experiment. The influence of magnetic flux on iron particles dramatically influences the mechanical properties of the MRE. The first part corresponds to the coil that generates the magnetic field inside the material. The coil has ten turns to produce an external magnetic flux strength (point A) of the same value as that of the experiment. Two pure iron particles with a diameter of 20 μm and a distance between the particles of 10 μm were established. The third part deals with the elastomer matrix (natural silicone rubber) to which an elastomer is designated as a non-magnetic material. After establishing the boundary conditions, the two-dimensional geometry and material properties of the magnetic particles in the MRE were modeled with a fine triangular mesh in the critical region of the magnetic particle and a coarse triangular mesh in the outer part, as shown in Fig. 12(b). Magnetic particles with angles from 0° to 90° were included to assess the effect of magnetic flux on the MRE. In addition, two pure iron particles with a diameter of 20 μm and a distance of 10 μm between them were included. The elastomer matrix (natural silicone rubber) was modeled as a non-magnetic material. The fine and coarse triangular meshes were used to accurately depict the two-dimensional geometry and material properties of the magnetic particles in the MRE.

FIG. 12.

Finite element model of iron particles in MRE materials placed in the magnetic field: (a) model; (b) meshing.

FIG. 12.

Finite element model of iron particles in MRE materials placed in the magnetic field: (a) model; (b) meshing.

Close modal

The magnetic flux density passing through the iron particle and the elastomer is shown in Fig. 13 and Table III. The magnetic field lines flow smoothly to form a closed loop at the iron-free region. In the area near the iron particles, the field lines are attracted to the iron particles, causing the field lines to be deflected, and the magnetic flux in this area is large. When a current (2 A) is applied, the magnetic flux outside the material, as observed at point B, is the same and reaches a value of 112 mT for the four different bead structure cases. The value is equal to the experimental flux value. Observed at the midpoint of the two iron particles (point A), the flux value differs significantly between the different grain structures. The magnetic flux is dense when the tilt angle of the two particles is in the same direction as the magnetic field [Fig. 13(a)]. The flux density decreases as the tilt angle increases. When the bead chain's inclination angle is in the same direction as the magnetic flux, the number of closed field-line circles between the two particles is large. This proves that the dipole between the two particles is large, as shown in Fig. 13(a). The magnetic flux circles between two particles, the dipole between the particles decreases as the angle between the inclination and the direction of the magnetic flux density increases, and the dipole is almost negligible when the tilt angle reaches 90°. The value of the flux density at points A and C is shown in Fig. 14. From Table III, it can be seen that the flux value is the same at point A and decreases significantly at C with the change in the bead inclination angle. The inclination angle of the particles and the value of the magnetic flux between the particles are used to calculate the interaction force between the particles, thereby determining the interaction force in a volume of the MRE material. This interaction directly affects the elastic and friction troops when the MRE material is subjected to shear deformation.

FIG. 13.

Magnetic flux passing through the iron particles corresponds to the external flux (point A) of 2 A (112 mT) under the angles of two particles: (a) 0°; (b) 30°; (c) 60°; (d) 90°.

FIG. 13.

Magnetic flux passing through the iron particles corresponds to the external flux (point A) of 2 A (112 mT) under the angles of two particles: (a) 0°; (b) 30°; (c) 60°; (d) 90°.

Close modal
TABLE III.

Magnetic flux values at points A and C when changing the bead inclination angle at 2 and 4 A amperage.

The tilt angles of two particles
Applied current (A)Observation point15°30°45°60°75°90°
2 A (112 mT) At point A (mT) 113 112 110 108 106.5 105.4 104 
At point C (mT) 311 300 270 221 160 89.4 36 
4 A (227 mT) At point A (mT) 227 225 220 217 213 211 212 
At point C (mT) 620 590 540 444 320 179 73 
The tilt angles of two particles
Applied current (A)Observation point15°30°45°60°75°90°
2 A (112 mT) At point A (mT) 113 112 110 108 106.5 105.4 104 
At point C (mT) 311 300 270 221 160 89.4 36 
4 A (227 mT) At point A (mT) 227 225 220 217 213 211 212 
At point C (mT) 620 590 540 444 320 179 73 
FIG. 14.

Flux response at points A and C as the tilt angle between two particles increases with two levels of amperage.

FIG. 14.

Flux response at points A and C as the tilt angle between two particles increases with two levels of amperage.

Close modal

In this section, the responses of a semi-active vibration system using an MRE are investigated with the tilt angles between the magnetic field direction and the particle chains. The electric current was varied from 0 to 4 A (0–227 mT), and the tilt angle was varied using different MRE samples. The experiments showed that the tilt angle significantly affects the response of the system.

The SDOF vibration system incorporating the aforementioned MREs as stiffness elements is shown in Fig. 15(a). The dynamic equation of the system is described as follows:
mẍ+cẋ+kx=cu̇+ku,
(13)
where m is the mass, u is the displacement of the base, x the displacement of the mass, and c is the damping coefficient of the MRE. In addition, k = k0 + km, where k is the variable stiffness parameter of the MRE, k0 is the nominal stiffness, and km is the magnetic-induced stiffness.
FIG. 15.

Single-degree-of-freedom vibration system incorporating MREs. (a) Mathematical model and (b) schematic of the experimental system.

FIG. 15.

Single-degree-of-freedom vibration system incorporating MREs. (a) Mathematical model and (b) schematic of the experimental system.

Close modal
The transfer function H(s), which represents the ratio of the output to input in the Laplace domain, is defined as follows:
Hs=X(s)U(s)=cks+1mks2+cks+1,
(14)
where U(s) is the Laplace transform function of u(t) and X(s) is the transform of x(t).
By replacing S with in the function H(s), the displacement transmissibility TR(ω) of system H() is further defined in the frequency domain as follows:
TRω=H(jω)=1+2ζγ21γ22+2ζγ2,
(15)
where γ = ω0/ω, ω=k/m, ζ=c/(2km), ω is the tunable natural frequency of the system, ω0 is the excitation frequency, and ϑ and γ are the damping ratio and dimensionless frequency, respectively.

Experiments were conducted to investigate the response of the sDOF system. A schematic of the experimental apparatus is shown in Fig. 15(b). Two MREs were placed between the upper and lower core. The lower core was fixed in a shaker to induce the stimulation. The upper core and coil work as the system mass. Thus, the magnetic field forms a closed loop that passes through the MREs. A DC generator was used to supply current to the electromagnet. A swept sinusoidal signal, whose frequency varied from 1 to 40 Hz, was used to excite the base. Two laser displacement sensors measured the displacement of the mass and base excitation. The signals were saved and analyzed using a fast Fourier transform (FFT) analyzer. Under the same excitation, three cases were tested for all samples: passive-on (0 A), passive-off (4 A), and switching control scheme.

The switching control algorithm was used to optimize the response in the frequency domain,
I=I=0ifff0,I=6iff>f0,
(16)
where f is the base excitation frequency and f0 is the switching frequency value.

The experimental results are shown in Figs. 16 and 17. Figure 16 shows the transmissibility response of the mass displacement in the frequency domain with fixed current values of 0 and 4 A for all samples. When no current was applied, the responses were the same, and the natural frequency of the system was 13 Hz in all cases. The natural frequencies were changed to 26, 28, and 30 Hz for the corresponding samples S60, S30, and S0, respectively, at an applied current of 4 A. The frequency tuning range was 12 Hz for S60 and 17 Hz for S0. The displacement responses under different control strategies are shown in Fig. 17 for S0. The response was remarkably suppressed when the on-off control algorithm was applied. The system using samples S0 and S30 achieved better performance than the other samples. The response when using S30 was slightly better than that when using S0. The tuning stiffness was higher, but the effect of hysteresis was significant when using S0. The results are in good agreement with the properties analyzed in Sec. III.

FIG. 16.

Transmissibility responses of the SDOF system in frequency domain use different samples. Executed when there is no current (0 A) and an applied current value (4 A).

FIG. 16.

Transmissibility responses of the SDOF system in frequency domain use different samples. Executed when there is no current (0 A) and an applied current value (4 A).

Close modal
FIG. 17.

Response of the SDOF using the MRE in the frequency domain for S0. (a) Transmissibility; (b) displacement response.

FIG. 17.

Response of the SDOF using the MRE in the frequency domain for S0. (a) Transmissibility; (b) displacement response.

Close modal

This study details the properties of the MRE and the interactions between the particles in the MRE. The analysis is based on experimental results, FEMM, and the principles of magnetic dipoles. The physical attributes of MREs are influenced not solely by the particle volume ratio, matrix material, particle diameter, and magnetic flux density; this study finds that the tilt angle between the particle chain and the direction of the magnetic field also plays a significant role. The force-displacement responses were conducted on three samples—S0, S30, and S60—with tilt angles of 0°, 30°, and 60°, respectively. The elastic force amplitude reached the maximum and minimum values in samples S30 and S60, respectively. Meanwhile, the friction force amplitude (frequency-independent hysteresis) reached the maximum and minimum values in samples S0 and S60, respectively. The magnetic dipole model was also presented in agreement with the experimental results.

A two-dimensional model was developed using FEMM software to investigate magnetic properties in MRE materials. The simulation involves two magnetic particles, specified as pure iron, surrounded by an elastic polymer matrix. When the external magnetic flux density is constant, the magnetic flux density at the midpoint of the two particles is significantly affected by the tilt angle between the two particles compared to the magnetic field direction. The larger the angle, the lower the magnetic field strength between the two particles, which affects the interaction between iron particles in the magnetic field. The interaction involves the mechanical properties of the material. The sDOF system’s response to different samples is consistent with the characteristics of the samples. The frequency tuning range was 12 Hz for S60, 15 Hz for S30, and 17 Hz for S0. The system using samples S0 and S30 performed better than using other samples in reducing the displacement response when applying the on-off control algorithm. In MRE vibration systems, magnetic field-dependent elastic properties need to be enhanced to enable rapid response to external forces. Conversely, magnetic field-dependent hysteresis impedes the response, and it is necessary to minimize this characteristic.

In conclusion, the arrangement of iron particles in the MRE plays an essential role in influencing the elasticity and hysteresis of the material. Choosing the angle of inclination between the magnetic flux and the chain will increase the effectiveness of the MRE.

This research is funded by University of Technology and Education – The University of Danang under project number T2022-06-02.

The authors have no conflicts to disclose.

Quang Du Nguyen: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal). Hoa Thi Truong: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Xuan Bao Nguyen: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Cung Le: Conceptualization (equal); Data curation (equal); Resources (equal); Writing – review & editing (equal). Minh Tien Nguyen: Data curation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available on request from the corresponding authors. The data are not publicly available due to privacy or ethical restrictions.

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