Modeling,heat dissipation design,and force tracking control for temperature-dependent hysteresis of magnetorheological damper

Abstract

The temperature-dependence (T-dependence) characteristics of magnetorheological fluids (MRFs) cause the damping force of magnetorheological dampers (MRDs) to change with temperature. The rapid temperature rise can lead to performance degradation or even failure of MRFs, reduced damping force of MRDs, and decline in control performance. In this paper, numerical simulations and predictions of the temperature rise characteristics of the MRD are performed and heat sinks are designed and optimized. The experimental results verify the efficiency of the simulations and predictions, and the heat sinks can significantly reduce the rate of temperature increase and improve the ability of the damper to operate for long hours. In order to accurately compensate for T-dependence characteristics of the MRD, a T-dependence hysteresis model and a model-based feedforward force tracking control method with disturbance observation of the MRD are proposed and validated by experiments. The experimental results indicate that the proposed T-dependence model has better prediction accuracy than the general hysteresis model, and the feedforward control method achieves good force tracking performance even without expensive force sensors.

Keywords

Magnetorheological damper temperature-dependence hysteresis compensation heat sink

1. Introduction

Because of the fast response, reversible characteristics, and controllability, lower energy requirements, magnetorheological dampers (MRDs) are gaining popularity in many vibration control applications, such as vehicle and seat suspensions, and buildings, bridges (Bai and He, 2021; Bahiuddin et al., 2018; McKee et al., 2018; Ning et al., 2016; Raja et al., 2013). These dampers convert mechanical vibration energy into heat energy through, thereby reducing vibration, which will cause the dampers’ temperature to rise rapidly during long working periods.

The flow characteristics of MR fluid are closely related to temperature (Sidpara et al., 2009). Hemmatian et al. (2020) remind that the effect of temperature on the viscosity and shear stress of magnetorheological fluids (MRFs) diminished with increasing magnetic field. Sherman et al. (2015) discuss the temperature-dependent (T-dependent) characteristics of the material components of MR fluid. Chen et al. (2015) analyzed the influence of temperature on the rheological characteristics of MR fluid. Wang et al. (2017) measured the temperature related mechanical properties of MR fluids using a parallel disc shear stress testing device. It is found that the viscosity stress decreases much more than the yield stress with the increase of MR fluid temperature. In conclusion, changing the MR fluid temperature can influence the flow and mechanical characteristics. Further, control performances may be deteriorated with the variation of temperature (Choi et al., 2005 ; Marathe et al., 2004; Wang and Liao, 2011). For instance, the RMS of the vibration acceleration of the controlled system increased with increasing temperature (Yang et al., 2009; Yu et al., 2017).

In order to effectively control the damping force of the MRDs and improve MRDs-based vibration control performance, it is very meaningful to accurately model the T-dependent hysteresis and dynamic characteristics of the MRDs (Gamota and Filisko, 1991). Various efforts have been devoted to developing hysteresis models based on numerous concepts including the Bingham model (Piatkowski, 2014), the Dahl model (Johanastrom and Canudas-de-Wit, 2008), LuGre model (Ismail et al., 2009), and the Bouc-Wen model (Zhu and Rui, 2014). Most of the existing researches focus on the influence of structure and current on the damping force of the MRD, while there are few researches on the influence of T-temperature. It is of great significance for the future application of MRDs to systematically discuss the influence of T-temperature on the damping force of MRD (Xia et al., 2020). Tang and Yue (2011) tested the damping force characteristics of MRD at different temperatures and then point out that the viscosity stability of magnetic fluid is good in the range of 20°C–100°C. To capture the hysteretic characteristics of MR dampers accurately, Yu et al. (2019) developed a hysteretic model considering the temperature effects and asymmetric characteristics. However, it is still extremely difficult for these models to provide satisfactory estimation and control accuracy over a wide temperature range.

To suppress temperature disturbances, the closed-loop control method with force sensors is frequently used to achieve force tracking (Yu et al., 2019). In practical engineering applications, force sensors have disadvantages such as high price and difficulty of installation. Temperature sensors (e.g. thermocouples) have the advantages of being inexpensive and easy to install. The model-based feedforward compensation control method with temperature sensors is a more economical and practical method.

The primary goal of this paper is to build a modeling, heat dissipation design and force feed-forward tracking control method for the T-dependent hysteresis of MRDs. This paper is organized as follows: Section 2 shows the T-dependent and dynamic characteristics of MRFs and a typical MRD. The numerical simulations and predictions of the temperature rise characteristics of the MRD are carried out and the heat sinks are designed and optimized in Section 3. Section 4 introduces a hysteresis model and a force tracking control method to describe and compensate the T-dependence characteristics of the MRD. Finally, Section 5 concludes the work.

2. T-dependent characteristics of the MRD

2.1. Temperature rise characteristics of the MRD

The temperature rise properties of a double-ended MRD are tested using an electro-hydraulic servo fatigue tester (type LFVHH, manufactured by W+B Ltd, Switzerland), as shown in Figure 1. The magnetorheological fluid damper was subjected to a sinusoidal excitation with an amplitude of 15 mm and a frequency of 1 Hz for an operating time of 1800 s at an ambient temperature of 20°C. The temperature rise of the MRD’s piston is monitored using a thermocouple with a range of −50°C to 200°C. Figure 2 shows measured temperature rise curve of the MRD with and without input current.

As can be seen from Figure 2, the temperature of the damper under 2 A current rises to 140°C within 35 min, essentially reaching the maximum operating temperature of MRFs. Rapid temperature rise can lead to: performance degradation or even failure of MRFs; and reduced damping force, which will lead to poor vibration control performance if the model does not take T-characteristics into account.

2.2. T-dependent characteristics of MRF

The performance of MRFs (type: MRF-132D, Operating temperature range: −40°C to 130°C, LORD ltd., USA) is tested experimentally using Anton Paar’s MCR302 rheometer as shown in Figure 3. Test conditions: sample volume of MRFs is 0.314 ml; rotor-tray gap is 1 mm; shear rate is 10 rpm; oil bath temperature control with 1°C/min increase in temperature.

The relationship between viscosity and temperature in the absence of a magnetic field is shown in Figure 4. As can be seen from Figure 4, the viscosity of MRFs decreases as the temperature rises. The slope of decrease is large for the temperature range 20°C–60°C. The decreasing trend slows down for the temperature range 60°C–138°C. The viscosity drops sharply when it exceeds 138°C, which coincides with the fact that the maximum operating temperature of the MRF cannot exceed 130°C. The viscosity of MRFs decreased by 65% with increasing temperature from 20°C to 100°C, which shows that the viscosity is strongly influenced by temperature.

By applying magnetic fields of 100, 200, 400, and 600 mT to the MRF, the shear yield stress curves at different temperatures can be obtained, as shown in Figure 5. It can be seen from Figure 5 that the shear yield stress decreases linearly with increasing temperature, but is less influenced by temperature than viscosity.

The reason for these phenomena may be that temperature has a greater effect on the viscosity of the base fluid silicone oil of the magnetorheological fluid and less on the shear yield stress of the hydroxy iron powder.

2.3. T-dependent hysteresis of MRD

The mechanical properties of the double-ended MRD are tested at different temperatures using the electro-hydraulic servo fatigue device, as shown in Figure 6. The damper is temperature controlled using a silicone heating pad and kept the target temperature for 1.5 h before being tested.

A sinusoidal excitation with amplitude of 5 mm and frequency of 1 Hz is applied to the damper. The force versus displacement and force versus velocity curves for different temperatures under different currents are shown in Figure 7. As can be seen from Figure 7, the damping force of the MRD decreases as the temperature rises. When the current is higher, the percentage of decrease becomes smaller. This phenomenon indicates that temperature has a greater effect on the viscous damping force and a smaller effect on the Coulomb damping force. This conclusion is in perfect agreement with the results of the tests and analysis in section 2.2.

Thus, a sinusoidal excitation with amplitude of 3 mm and frequency of 2 Hz is applied to the damper. The measured force versus displacement hysteresis curves of the MRD under 2 A input current are shown in Figure 8. Figure 9 shows the force curves with different temperatures for the MRD with 2 A input current under a non-periodic displacement excitation shown in Figure 10. Comparing Figures 7, 8, and 9, it can be seen that the temperature characteristics of the magnetorheological dampers remain invariant as the frequency and amplitude of the excitation displacement changes.

3. Design and optimization of the heat sinks

In order to reduce the rate of temperature rise of the MRD and extend its continuous operation, this section attempts to design and optimize the heat dissipation scheme of the MRD.

3.1. Heat dissipation model

In the case of the MRD, the energy to overcome the damping force and the heat generated by the coil resistance is the thermal energy entering the system. The heat dissipated through exchange with the environment is the thermal energy leaving the system. When the energy entering the damper is greater than the leaving the damper, the temperature of the damper will rise.

Figure 11 shows the heat exchange model of the MRD. Assuming that: (1) the temperature of the fluid inside the MRD is a function of position x and time t; (2) in the transverse section of the MRD, the MRFs have the same temperature $T (x, t)$ . The energy balance equation of the MRD can be expressed as

\overset{\cdot}{W} - q = \frac{du}{dt}

(1)

where, $W$ is the external energy consumed by the MRD, $q$ is the change rate of heat dissipation, u is the internal energy.

Heat energy is transferred in three ways: heat conduction, heat convection, and heat radiation:

(1) heat conduction

The phenomenon of heat conduction can be described by Fourier’s law

Φ_{1} = - λ A \frac{\partial T}{\partial x}

where, λ is the thermal conductivity, the negative sign indicates that the direction of heat transfer is opposite to the direction of temperature increase, and x is the axis perpendicular to the area A.

(2) heat convection

Convective heat transfer can be achieved using Newton’s formula for heat transfer expressed as,

When the fluid is heated, $Φ_{2} = h_{2} (T_{w} - T_{f})$ ; when the fluid is cooled, $Φ_{2} = h_{2} (T_{f} - T_{w})$ .

where, $T_{w}$ is the temperature of the convective surface of the damper, $T_{f}$ is the fluid temperature. q is the heat flow density proportionality factor, h₂ is the surface heat transfer coefficient.

(3) heat radiation

The radiant heat flux of a real object can be expressed in the form of an empirical modification of the Theban-Boltzmann law

Φ_{2} = ε A σ T^{4}

where, T is the thermodynamic temperature of the blackbody, $σ = 5.67 \times 1 0^{- 8} W / (m^{2} \cdot K^{4})$ is the Stepan-Boltzmann constant, and $ε$ is the emissivity of the object.

Based on the theory of heat transfer, the change rate of heat dissipation $q$ can be given by

q = [\frac{4 {v_{0}}^{0.7}}{{D_{2}}^{0.3}} + ε σ_{b} (\frac{T {(t)}^{4} - {T_{0}}^{4}}{T (t) - T_{0}})] \cdot A [T (t) - T_{0}]

(2)

where, $ε$ is the blackness of the outer wall of the damper, $σ_{b}$ is the blackbody radiation constant, $A = \frac{π {D_{2}}^{2}}{2} + π D_{2} L$ is the total heat dissipation area of the damper, $[\frac{4 {v_{0}}^{0.7}}{{D_{2}}^{0.3}} + ε σ_{b} (\frac{T {(t)}^{4} - {T_{0}}^{4}}{T (t) - T_{0}})]$ denotes the total heat transfer coefficient h of the damper including the integrated heat dissipation coefficient considering the heat conduction, heat transfer, and heat radiation, which can be obtained from finite element simulation, $v_{0}$ is the relative air flow rate, $D_{2}$ is the outer diameter of the damper cylinder, L is the length of the damper cylinder, $T (t)$ is the average temperature of the MRD, $T_{0}$ is the ambient temperature.

The external energy consumed by the MRD includes the mechanical energy of external excitation and the electrical energy fed into the coil. Therefore, we can get

\overset{\cdot}{W} = F (t) \frac{dx (t)}{dt} + I^{2} (t) R

(3)

where, x is the piston position, I is the control current of the damper, R is the coil resistance of the MRD, $F (t)$ is the damping force of the MRD, which can obtain by the Bouc-Wen model proposed by Section 4.

The energy consumed by the average temperature rise of the damper is

\frac{du}{dt} = \frac{dT (t)}{dt} C_{Overall}

(4)

where, $C_{Overall}$ is the overall heat capacity of the MRD.

Combining the above equations, the temperature rise model of the MRD can be obtained as

\frac{dT (t)}{dt} = \frac{P_{d} + P_{I} - q}{C_{Overall}}

(5)

where, the coil heat production power under the maximum current is

P_{I} = I^{2} R = 30.4 W

(6)

The measured force-displacement curves of the MRD at each temperature can be obtained by the experimental test system shown in Figure 6. The heat production $P_{d}$ of the MRD in one cycle at each temperature is as shown in Table 1. Fitting the data in Table 1, we can obtain

P_{d} = 112.1 - 0.4 T + 7.0 \times 10^{- 4} T^{2} + 2.2 \times 10^{- 6} T^{3}

(7)

Table 1.

Heat production with different temperatures of the MRD.

Temperature (°C)	20	40	60	80	100	120
Heat production (J)	104.2	97.9	90.6	85.1	81.7	77.5

The measured overall heat capacity as a function of temperature can be measured using a constant pressure calorimeter, is shown in Figure 12. Fitting the data in Figure 12, we can obtain

C_{Overall} = 13.1 - 0 . T + 0.3 \times 10^{- 2} T^{2} - 1.1 \times 10^{- 5} T^{3}

(8)

Because the MRD is a double-ended type, the working process of the working cylinder cavity volume is constant. The density of MRFs is set to 2.3 $g / c m^{3}$ . The thermal conductivity is taken as 22.5 $W / (m \cdot K)$ with reference to silicone oil.

Fitting the data in Figure 4, a function of viscosity versus temperature is obtained as

η (T) = - 0.0011 T^{3} + 0.41 T^{2} - 52.51 T + 3160.63

(9)

The measured and predicted temperature rise curves for the MRD are shown in Figure 11. As can be seen from Figure 13, the predicted temperature rise characteristics are in good agreement with the actual characteristics, indicating that the proposed heat dissipation model can be used for design and optimization of the heat sink system.

3.2. Optimization of the heat sinks

The temperature gradient in the damper piston stroke is small and the heat is concentrated and distributed evenly. The heat dissipation ribs are installed at the center of the piston stroke of the damper and at the top and bottom of the stroke. The material of the ribs is aluminum alloy with high thermal conductivity, cheap material, easy processing, and light weight.

To optimize the parameters of the heat sinks, the heat dissipation efficiency is defined as

\begin{matrix} η_{h} = (1 - \frac{Temperature with heat sinks}{Temperature without heat sinks}) \\ \times \frac{1}{\begin{matrix} Ratio of Heat dissipation area \end{matrix}} \end{matrix}

(10)

The design variables are the width and number of heat sink, and the objective function is to maximize the heat dissipation efficiency given by equation (10).

The design diagram of the heat sinks with different outer diameters is shown in Figure 14. The outer diameter of the MRD is 23 mm, and therefore we set the initial design width of the heat sink to 23 mm. The predicted temperature rise curves with width (18, 23, 28, and 33 mm) by proposed heat dissipation model and the finite element software are shown in Figure 13. From Figure 15, as the outer diameter of the heat sink increases, the rate of the temperature rise is decreasing. The heat dissipation efficiency is listed in Table 2.

Table 2.

Heat dissipation efficiency of heat sink ribs with different outside diameters.

Outer diameter	0 mm	18 mm	23 mm	28 mm	33 mm	38 mm
Surface area (mm²)	83,994.9	107,254.5	116,510.4	125,840.9	136,113.9	147,329.4
Ratio of heat dissipation area	—	127.7%	138.7%	149.8%	162.1%	175.4%
Temperature (°C)	138.3	88.3	82.4	79.3	76.8	72.5
Heat dissipation efficiency $η_{h}$	—	0.2831	0.2914	0.2848	0.2743	0.2712

As can be seen from Table 2, when the width is greater than 23 mm, the heat dissipation efficiency is decreasing as the outer diameter increases. In contrast, the increase in the outer diameter leads to an increase in the space required for damper installation. The reason for this phenomenon is the low efficiency of heat conduction from the damper to the periphery of the heat sinks. Therefore, we choose the width of the heat sink to be 23 mm.

The design diagram of the heat sinks with different numbers (3, 4, and 5) is shown in Figure 16. The predicted temperature rise curves with different numbers of the heat sinks by proposed heat dissipation model are shown in Figure 17. From Figure 17, as the number of the heat sink increases, the rate of the temperature rise is decreasing. The heat dissipation efficiency with different numbers of the heat sinks is listed in Table 3.

Table 3.

Heat dissipation efficiency of heat sink ribs with different numbers of the heat sinks.

	0	3	4	5	6
Surface area (mm²)	83,994.878	116,510.362	127,348.857	138,187.351	149,025.847
Ratio of heat dissipation area	—	138.7%	151.6%	164.5%	177.4%
Temperature (°C)	138.3	82.4	76.4	67.4	64.8
Heat dissipation efficiency $η_{h}$	—	0.2914	0.2952	0.3116	0.2996

As can be seen from Table 3, the heat dissipation efficiency rises at first and then falls as the number of the heat sink increases. The reason for this phenomenon is that the air convection becomes worse as the distance between heat sinks becomes smaller, resulting in a decrease in heat dissipation efficiency. Therefore, we choose the number of the heat sink to be 5.

3.3. Experimental verification of the heat sink system

Based on design and optimization results, the photo of the MRD with heat sinks is shown in Figure 18. The number of heat sinks is 5, the outer diameter is 23 mm, and the pitch is 20 mm. The measured and predicted temperature rise curves for the MRD without and with the heat sinks are shown in Figure 19 under the sinusoidal excitation with amplitude of 5 mm and frequency of 1 Hz.

From Figure 19, the predicted results are in good agreement with the experimental results. After 30 min of operation, the maximum temperature of the MRD with heat sinks is 76.9°C, which is 48.23% lower than that without heat dissipation. The damping force decreases to 15.7% due to the increase in temperature, while the damping force of the MRD without heat sinks decreases to 25.9%. That is a good improvement.

The measured and predicted temperature rise curves for the MRD with the heat sinks are shown in Figure 20 under the sinusoidal excitation with amplitude of 3 mm and frequency of 2 Hz. Figures 19 and 20 indicate the temperature rise prediction curves are in good agreement with the experimental results under different excitation.

4. Modeling and control for T-dependent hysteresis of the MRD

In order to suppress the degradation of the damper control performance due to the temperature rise, we attempt to establish a temperature feed-forward compensation method in this section.

4.1. Modeling method and experimental results

The Bouc-Wen model with T-dependence parameters of the MRD is given by

F_{d} (I, T) = c (I, T) \overset{\cdot}{x} + k (I, T) (x - x_{0}) + α (I, T) z

(11)

\overset{\cdot}{z} = - γ (I, T) | \overset{\cdot}{x} | z {| z |}^{n - 1} - β (I, T) \overset{\cdot}{x} {| z |}^{n} + A (I, T) \overset{\cdot}{x}

(12)

where $F_{d}$ is the force of the MRD, $I$ is the control current, $T$ is the temperature, $x$ and $\overset{\cdot}{x}$ are the damper displacement and velocity; $z$ is the Bouc-Wen hysteresis operator; “.” at the top of variables represents the first order derivative of the variables with respect to time; $i$ is the current applied to the MRD; $k$ and $c$ are the stiffness and damping function of the efficient current, respectively; $α$ is function related to the MR material yield stress; $x_{0}$ is the initial displacement, $A$ , $β$ , $γ$ , and $n$ are the parameters of the Bouc-Wen hysteresis operator.

Based on the genetic algorithm identification method proposed in Jiang et al. (2021), the parameters can be obtained as

\begin{matrix} {\begin{matrix} C_{0} = 2883 - 40.83 T + 2540.5 I + 0.22444 T^{2} - 335 I^{2} + 4.34 T \times I \\ α = 97.18 + 10.06 T - 61.61 I - 0.068 T^{2} + 80 I^{2} + 3.68 T \times I \\ γ = - 201 + 215 T - 1648.78 I - 0.833 T^{2} + 864.36 I^{2} - 64.69 T \times I \\ β = 952 - 22 T - 904.2 I + 0.149 T^{2} + 163.44 I^{2} + 16.78 T \times I \\ A = 2396 - 11.7 T + 5177 I - 0.042 T^{2} - 1822.7 I^{2} - 4.41 T \times I \\ k_{0} = 80.5 \\ n = 2 \\ x_{0} = 0 \end{matrix} \end{matrix}

(13)

To verify the feasibility of the model shown in equations (11) and (12), the measured and modeling of the input-output relationships for the MRD with different temperature are shown in Figures 21 to 24. The root mean square value of the simulation error is defined as

RMSE = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} (f_{mearsured} - f_{modeling})}

(14)

The values of the RMSE under the temperatures of 20°C, 40°C, 80°C, and 120°C are 65.2, 64.8, 71.3, and 72.6 N, respectively. Therefore, it can be concluded that the proposed Bouc-Wen model can accurately predict the T-dependent nonlinearity hysteresis characteristics of the MRD. It is worth noting that there is an obvious difference between predict model and experimental data in low velocity range. The main reason is that the proposed model does not consider the friction force between the piston and the barrel wall of the MRD. The friction force is switching between dynamic friction and static friction at low speed. To describe that and improve the modeling accuracy, one idea is to introduce a friction force function into the model. Equation (11) can be rewritten as

F_{d} (I, T) = c (I, T) \overset{\cdot}{x} + F_{f} sign (\overset{\cdot}{x}) + k (I, T) (x - x_{0}) + α (I, T) z

(15)

where $sign (\overset{\cdot}{x}) = {\begin{matrix} 1 \overset{\cdot}{x} \geq 0 \\ - 1 \overset{\cdot}{x} < 0 \end{matrix}$ , $F_{f} = 43.52 N$ is the friction force of the MRD and its value can be obtained by fatigue testing at very slow loading speeds. Using equation (15), the measured and modeling of the forces for the MRD under the non-periodic displacement excitation shown in Figure 10 with different temperature are shown in Figure 25. The value of the RMSE under the temperatures of 20°C and 120°C are 32.4 and 35.5 N respectively. Therefore, the introduction of friction into the model can improve prediction accuracy. We will research that in detail in our future work.

4.2. Force tracking control method and experimental results

The process of solving the inverse Bouc-Wen model to compensate for hysteresis magnifies the modeling error. Therefore, we use a feed-forward compensation method for the T-dependent hysteresis that does not require an inverse hysteresis model. The principle diagram of the feed-forward force tracking control method is shown in Figure 26. The Bouc–Wen model of the MRD is computed in real-time for the constant currents $I = 0, 0.1, \dots, I A_{\max}$ for the actual displacement, velocity, and temperature measured by sensors. The corresponding estimated forces of the MRD can be obtained theoretically by calculation. Based on the estimated forces and the desired control force, the control current $I_{control}$ is derived by piecewise interpolation.

In order to experimentally validate the proposed control method, the schematic and photograph of the experimental system are shown in Figure 27. The rapid control prototype uses Mobile platform with an integrated IO-135 board of the Speedgoat Ltd. from Switzerland. In the test, the ADC module in the IO-135 board is used to collect the sensor signals and the DAC module is used to output the control current signal to the current source (type: ILD-810, linearity: 99.3%).

Figure 28 shows the tracking results of the control system for a sinusoidal target with amplitude of 900 N and frequency of 1 Hz using the generalized Bouc-Wen model with considering T-dependent characteristics. The Maximum control error of the generalized Bouc-Wen model with considering T-dependent characteristics is 62.5 N. As a baseline performance, the control results using the traditional Bouc-Wen model without considering temperature are plotted as shown in Figure 29. The Maximum control error of the traditional Bouc-Wen model is 289.7 N. Compared to the control results based on the traditional model, the control system using the generalized Bouc-Wen model can improve accuracy in the force tracking control of the MRD with different temperatures. Figure 30 shows the tracking results of the control system for a sinusoidal target with amplitude of 1100 N and frequency of 2 Hz using the generalized Bouc-Wen model with considering T-dependent characteristics. Figure 31 shows the tracking control results for a non-periodic tracking target under the excitation shown in Figure 10. Note that better force control has been achieved up to 123°C with the based-model controller using the generalized Bouc-Wen model with considering temperature.

5. Conclusions

In order to solve the impact of rapid temperature rise during the operation of the damper, the heat sink system are designed and optimized, and a dynamic model and a force control method considering temperature effects is proposed and analyzed. The details of the conclusion are as follows.

(1) The predicted temperature rise curve is in good agreement with the measured curve, indicating that the proposed heat dissipation model can be used for design and optimization of the heat sink system.

(2) The heat dissipation efficiency is decreasing as the outer diameter increases. The reason for this phenomenon is the low efficiency of heat conduction from the damper to the periphery of the heat sinks.

(3) The heat sinks can effectively reduce temperature and improve operating time of the MRD

(4) The control system using the generalized Bouc-Wen model can improve accuracy in the force tracking control of the MRD with different.

The research in this paper provides a new idea to suppress the adverse effects of temperature rise on the MRD.

Footnotes

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.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors wish to acknowledge the financial support by National Natural Science Foundation of China (Grant No. 51975298), Natural Science Foundation of Jiangsu Province, China (Grant No. BK20181301), and Fundamental Research Funds for the Central Universities (Grant No. 30921011105).

Ethical compliance

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards.

ORCID iD

Wei Zhu

Data availability statement

The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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(2014) Semiactive vibration control using a magnetorheological damper and a magnetorheological elastomer based on the Bouc-Wen model. Shock and Vibration 2014: 1–10.