Analysis and compensation methods for time delays in an impact buffer system based on magnetorheological dampers

Abstract

Magnetorheological dampers have been widely studied as versatile real-time actuators for solving vibration problems in various structures and systems. However, the inherent time delay problem of magnetorheological actuators may cause performance degradation of semi-active control systems. The primary purposes of this article are to provide a comprehensive analysis on the time delay of an impact buffer system based on a magnetorheological damper and to propose compensation methods to reduce the time delay. To this end, this study evaluated the electromagnetic circuit in the magnetorheological damper and designed an advanced correcting circuit to improve the response time. The simulation results show that the proposed circuit reduces the time delay by 5 ms. Using a magnetorheological buffer system setup, its force response times were tested. The experimental results show that the electromagnetic circuit plays a significant role in the time delay and it is highly dependent on the effectiveness of the electric parts of the control hardware. Furthermore, a proportional–integral–derivative controller is able to reduce the time delay and improve the dynamic performance of the magnetorheological impact buffer system.

Keywords

Magnetorheological buffer system time delay electromagnetic circuit

Introduction

Magnetorheological (MR) dampers have been effectively used for reducing seismic responses of civil engineering structures and controlling vibration responses of vehicles and other engineering systems where rapid responses are demanded (Lee and Kawashima, 2007; Ruangrassamee and Kawashima, 2001). Typical structural control systems consist of numerous components, such as sensors, amplifiers, filters, controllers, and control devices. Obviously, all these components require time to be properly activated, thus there exist inevitable time delays between input signals and output control forces. The time delay may cause performance degradation problems of control systems, particularly in semi-active systems where real-time controls are required. Hence, various methods have been proposed to compensate the time delay problems (Alkhatib and Golnaraghi, 2003; Bhardwaj and Datta, 2006).

Since Crosby and Karnopp (1973, 1974) first introduced semi-active dampers and semi-active controls, several investigations have been carried out to characterize the response time of the electrical and mechanical parts of MR damper based on semi-active systems (Goncalves et al., 2003, 2006; Koo et al., 2003; Zhu, 2005). Though the response time of MR fluid itself is 1–2 ms, the response time of MR devices is not as fast as anticipated because other factors (such as compliance of the system, control electronics) affect the total response time of MR devices (Goncalves et al., 2003, 2006). Christenson et al. (2008) measured the response time of a 200-kN commercially available MR damper from Lord Corporation, and it turned out to be approximately 30 ms, which is less than the response time of a 180-kN MR damper by Lord Corporation (i.e. 60 ms). Koo et al. (2006) experimentally studied the response time of an MR damper by considering the effects of the operating current, the piston velocity, and the system compliance. Occhiuzzi et al. (2003) defined three different time delays in a full-scale 50-kN MR damper system and carried out extensive semi-active tests and simulations to access the effects of time delays.

In an effort to address the time delay issues of MR damper systems, over the past decades, different control algorithms have been developed and investigated for MR damper systems. The clipped-optimal control (COC) was proposed by Dyke et al. (1996), and it was applied to a small-scale experimental building system equipped with MR dampers. Xu et al. (2003) proposed a semi-active control algorithm based on neural networks and applied it to a building structure. Cha and Agrawal (2011) developed the decentralized output feedback polynomial controllers based on a genetic algorithm, and they (Cha and Agrawal, 2012) applied it to a nonlinear highway bridge model and a large-scale three-story frame model. Besides, controllers based on fuzzy logic algorithms have been proposed and applied to civil structures and vehicle systems (Choi et al., 2004; Guclu and Yazici, 2008; Sun et al., 2005). Several advanced control algorithms have been proposed, such as the statistical process control (SPC) controller. Zhang (2012) implemented the SPC to civil structures without any controller or sensors.

Despite the fact that the response time of MR systems for civil engineering fields and suspension systems has been studied, the effect of time delay on the performance of semi-active control systems subjected to impact loadings has rarely been investigated. In their recent work, Wereley et al. (2011) and Singh HJ and Wereley NM (2013) studied an optimal control design of adaptive magnetorheological shock isolation (MRSI) mounts for drop-induced impacts and performed non-dimensional analysis of the system. The results showed that the time constant and the Bingham number, which is defined as the ratio of the yield force to the viscous force, are of utmost importance in effectively controlling the system response. Considering the extremely short time of the force control (less than 600 ms) in MR buffer systems under impact loadings, the time delay study would be critically important for the effect of the MR systems. Nonetheless, there exists little research on compensation methods for time delay in MR systems for such applications. Thus, this article focuses on the analysis and compensation methods against time delays of an MR buffer system under impact loadings. It evaluated the effects of the basic electromagnetic circuit as well as an advanced correcting circuit on time delays. In addition to the circuit analyses, the effect of driving electronics on the overall response time was also evaluated. The MR damper used in this article is a specially designed long-stroke MR damper, and the time delay of the MR buffer system was experimentally measured. Furthermore, a proportional–integral–derivative (PID) controller was designed to compensate the time delay in the real-time control system.

Experimental setup

Figure 1 shows an impact buffer system based on an MR damper, which is intended to control the damping characteristics continuously by varying the input power of the electromagnet. The experimental setup consists of an industrial control computer, a real-time simulation system (dSPACE), a current controller, and an experimental platform (Li et al., 2009). The damper is installed in shaft sleeves with a recoil mass, and the piston rod is connected to the ground mount.

Figure 1.

Impact test platform system.

During the experiment, the explosion of gunpowder in the close bomb produces a large impulsive force, which pushes the recoil mass along with the MR damper cylinder back along the guide rail. The elastomeric bumper is installed at the rear of the test setup to prevent the recoil from bumping into the test rig in a case that the recoil exceeds the maximum stroke length. The current controller is used to control the damping force according to the control input prescribed by the dSPACE control system, which is also used as the data acquisition system.

The MR damper in the buffer system works as a shock absorber, and it is used to dissipate energy of the impact loading. Thus, the impact loading within an extremely short time is transformed into small damping force acting on a long displacement. Such dynamics can be expressed as the following differential equation

m_{h} \frac{{dv}_{r}}{dt} = F_{pt} - F_{cv} - F_{f}

(1)

where $F_{pt}$ is the impact load, $m_{h}$ is the recoil mass, $v_{r}$ is the recoil velocity, $F_{cv}$ is the sum of Coulomb force ( $F_{τ}$ ) and viscous force ( $F_{η}$ ), and $F_{f}$ is the friction force between the contact surfaces.

Agrawal and Yang (1997) defined the time delay as a summation of the fixed time delay and the time taken by actuators to build up the required control force. The fixed time delay, which includes the on-line data acquisition from sensors at different points of the structure, filtering, processing of data, calculation of forces, and transmission of the control force signals from computer to the actuator, is almost exclusively dependent on the effectiveness of the electric part of the control hardware. Thus, this study focuses on the total “process” time that requires building up the desired control force, which depends on the specific dynamics of actuators as well as the fixed time of the control hardware.

Electromagnetic circuit analysis

In this section, the research on the time delay starts with a simplified electromagnetic circuit analysis. Based on the simplified circuit analysis, an electromagnetic drive circuit of the current controller is investigated. A feasibility of reducing the time delay by modifying the transfer function of electromagnetic circuit is discussed.

Basic electromagnetic circuit model

In order to illustrate the principle of the electromagnetic circuit for an MR damper, a basic model was first investigated. The electromagnetic circuit of an MR damper is generally simplified into an equivalent resistance and an equivalent inductance connected in series. The control current cannot make a sudden change because the inductance needs a time to store and release energy, which delays the “activation” of electromagnetic coil. Figure 2 illustrates an electromagnetic circuit of an MR damper by assuming $R_{m}$ to be an equivalent resistance and $L_{m}$ an equivalent inductance. According to the Kirchhoff laws, the differential equation of a control electric circuit in the zero state can be described as follows

L_{m} \frac{{dI}_{m} (t)}{dt} + R_{m} \cdot I_{m} (t) = V (t)

(2)

Figure 2.

Electromagnetic circuit of MR damper.

If the loading voltage is a constant, the solution of the above equation can be given as

I_{m} (t) = \frac{V_{o}}{R_{m}} (1 - e^{- \frac{R_{m}}{L_{m}} t})

(3)

where $V_{o}$ is the initial value of the loading voltage.

Figure 3 shows the simulated results of the relationship between the equivalent resistance $R_{m}$ and the response time of the coil current, and Figure 4 shows a relationship between the equivalent inductance $L_{m}$ and the response time of the coil current. To be consistent with the literature, the authors define the response time of the coil current as the time required to reach to 95% of the final state from the initial state (Yang, 2001). As shown in Figure 3, the response time of current triples when the resistance reduces from 8 to 2 Ω. However, it decreases 4 ms when the inductance reduces from 6 to 2 mH. The response time of the magnetic field and the response time of the current should be the same in the ideal case. However, due to the presence of eddy currents, the response of the magnetic field always lags the changing current (Guo et al., 2009). Besides, the magnetic loop of MR damper passes through steel and ferroalloy, which have the ability to keep magnetisms. In other words, the magnetic flux strength B is always lagging behind the magnetic field intensity H, which makes the time delay last longer.

Figure 3.

Resistance versus time.

Figure 4.

Inductance versus time.

Current controller

In order to control the output force of an MR damper, the input current must be continuously adjustable by a current controller. In this article, the pulse width modulation (PWM) current drive technology is adopted to control the current on the electromagnetic coil of the MR damper, which generates magnetic fields and controls the output damping forces. Figure 5 shows a current controller that includes a current driver and a digital signal processor (DSP) controller. The PWM signal $V_{c}$ is generated by the DSP controller, and the current driver is connected to the electromagnetic coil to produce the controllable current. Figure 6 shows a schematic of the control system for MR damper, in which the control algorithm is carried out by the DSP controller when the analog-to-digital (A/D)-converted force and velocity transducer signals together with the reference signal are transmitted as the inputs. The electromagnetic drive circuit is activated by the PWM signal with a certain duty cycle. Finally, the desired operating current is applied to the electromagnetic coil to achieve the desired damping force.

Figure 5.

Current controller.

Figure 6.

Schematic of MR control system.

The PWM signal $V_{c}$ with a certain duty cycle, which is produced by the DSP controller, switches the MOSFET Q to on or off state, transforming the input voltage $U_{i}$ to pulse signals. The coil current $I_{m}$ shows an upward trend when the tube voltage $U_{Q}$ is at low level, setting the MOSFET at off state, as shown in Figure 7. The inductor–capacitor (LC) filter circuit is designed to decrease the current fluctuation. The loop circuit is composed of the fast recovery diode ( $F$ ) and the inductance ( $L_{1}$ ), thus the current is continuously supplied when the input voltage ( $U_{i}$ ) becomes zero. Based on this current driver circuit, the steady current flowing through the MR damper coil can be achieved, and its amount can be controlled by the duty cycle of the input PWM signal ( $V_{c}$ ).

Figure 7.

Waveform diagram for the switch regulator circuit.

As shown in Figure 7, the duty cycle is defined as

D = \frac{T_{on}}{T}

(4)

where $T$ is the period of a pulse and $T_{on}$ is the working time for one cycle.

The response of the drive circuit can be described by the following transfer function

\frac{U_{0}}{U_{i}} = \frac{L_{m} s + R_{m}}{L_{1} L_{m} {Cs}^{3} + R_{m} {CL}_{1} s^{2} + (L_{1} + L_{m}) s + R_{m}}

(5)

\frac{U_{0}}{I_{m}} = L_{m} s + R_{m}

(6)

where $U_{i}$ is the input voltage and $U_{o}$ is the output voltage working on the electromagnetic coil. Dividing equation (5) by equation (6), the circuit transfer function can be achieved as

G_{I} = \frac{I_{m}}{U_{i}} = \frac{1}{\frac{L_{1} L_{m} C}{R_{m}} s^{3} + L_{1} {Cs}^{2} + \frac{(L_{1} + L_{m})}{R_{m}} s + 1} \times \frac{1}{R_{m}}

(7)

The input voltage $U_{i}$ is a pulse signal and its Laplace transform can be described as

U_{i} (s) = \frac{U (1 - e^{- T_{on}})}{s (1 - e^{- sT})}

(8)

where $U$ is the constant voltage (12 V) supplied by the direct current (DC) regulated power supply.

Since the coefficient of s ³ is small enough to be neglected, it can be simplified as

G_{I} = \frac{I_{m}}{U_{i}} \approx \frac{1}{L_{1} {Cs}^{2} + \frac{(L_{1} + L_{m})}{R_{m}} s + 1} \times \frac{1}{R_{m}}

(9)

Then, the Laplace transformation of the output current in the electromagnetic coil of MR damper can be expressed as

\begin{matrix} I_{m} (s) & = U_{i} (s) \cdot G_{I} (s) \\ = \frac{1}{L_{1} {Cs}^{2} + \frac{(L_{1} + L_{m})}{R_{m}} s + 1} \cdot \frac{U (1 - e^{- {sT}_{on}})}{s (1 - e^{- sT})} \times \frac{1}{R_{m}} \end{matrix}

(10)

According to the final value theorem of Laplace, the stable value of $I_{m}$ is achieved as

\begin{matrix} lim_{t \to \propto} I_{m} (t) & = lim_{s \to 0} {sI}_{m} (s) \\ = lim_{s \to 0} \frac{1}{L_{1} {Cs}^{2} + \frac{(L_{1} + L_{m})}{R_{m}} s + 1} \cdot lim_{s \to 0} \frac{U (1 - e^{- {sT}_{on}})}{(1 - e^{- sT})} \times \frac{1}{R_{m}} \\ = lim_{s \to 0} \frac{τ e^{- {sT}_{on}}}{{Te}^{- sT}} \times \frac{U}{R_{m}} = \frac{T_{on}}{T} \times \frac{U}{R_{m}} = D \times \frac{U}{R_{m}} \end{matrix}

(11)

The above analysis indicates that when the current drive circuit approaches a stable state, the current $I_{m}$ in the electromagnetic coil of the MR damper is proportional to the duty cycle of input PWM signal ( $V_{c}$ ). Besides, if the pulse period ( $T$ ) is much smaller than the time constant $τ$ ( $L_{m} / R_{m}$ ), that is $T < < τ$ , the current fluctuation can be neglected.

When a current controller is added in series with electromagnetic coils, the corresponding electromagnetic circuit can be simplified as a second-order linear system with the PWM signal ( $V_{c}$ ) as the input and the coil current ( $I_{m}$ ) as the output. This can be expressed as follows

G_{Ic} = \frac{I_{m}}{V_{c}} \approx \frac{1}{L_{1} {Cs}^{2} + \frac{(L_{1} + L_{m})}{R_{m}} s + 1} \times K

(12)

where $K$ is a proportional coefficient, $K = (D \times U) / (V_{c} \times R_{m})$ .

The model of equation (12) is used to design a simulation model, and the simulation results are compared with the experimental data in the subsequent sections. The transfer functions are obtained by a parameter identification technique to the second-order model of current responses.

Experimental results and analysis

Figure 8 illustrates the design concept of the long-stroke MR damper used in this study, and Table 1 shows the critical parameters of the damper including the number of the coils $N$ , the effective length $L_{f}$ , the diameter of piston rod $D_{r}$ , the diameter of the piston head $D_{h}$ , the thickness of the shear gap $g_{p}$ , and the stroke length $Z$ . The responses of the electromagnetic circuit under impact loadings were measured and analyzed based on the designed MR damper in the subsequent sections.

Figure 8.

Long-stroke MR damper.

Table 1.

MR damper design parameters.

Stroke length $Z$ (mm)	Thickness of shear gap $g_{p}$ (mm)	Diameter of piston head $D_{h}$ (mm)	Diameter of piston rod $D_{r}$ (mm)	Effective length $L_{f}$ (mm)	Number of coils $N$	Resistance of coils $R_{m}$ ( $Ω$ )	Inductance of coils $L_{m}$ ( $mH$ )
±380	1.5	47	28	84	3	10.68	37.7

MR: magnetorheological.

Response analysis of electromagnetic drive circuit

This part of the study evaluates the time delay of an electromagnetic drive circuit with four different PWM signals. The experiment was carried out by measuring the voltage $U_{s}$ upon the sampling resistance $R_{s}$ (0.5 Ω), when the electromagnetic drive circuit was applied with various PWM signals. The coil current $I_{m}$ was then obtained by the division of $U_{s} / R_{s}$ . In order to compare the resulting responses distinctively, the PWM signal $V_{c}$ was normalized by scaling down to the corresponding final values of current $I_{m}$ . Figure 9 shows the measured current responses of the electromagnetic coil by applying a normalized signal ${\bar{V}}_{c}$ with four different values from 0.5 to 2.0 with a step of 0.5.

Figure 9.

Response of coil current.

The current response time exhibits an approximated increase when the PWM signal $V_{c}$ rises. Furthermore, there is a large amount of overshoot which increases as the input voltage rises, and its dynamic characteristics perform undesirably. The response time varies approximately 40, 30, 40, and 65 ms when the final output currents $I_{m}$ changes 0.5, 1.0, 1.5, and 2 A, respectively. However, the magnitudes of the PWM signal $V_{c}$ and the output current $I_{m}$ are proportional, and the output current $I_{m}$ tends to be saturated when the PWM signal is applied greater than 5 V (see Figure 10).

Figure 10.

Current versus input voltage.

Response analysis of impact MR buffer system

To further evaluate the time delay of the MR buffer system, a series of tests were conducted under different PWM signals $V_{c}$ . The output damping force of the MR damper consists of viscous damping force ( $F_{η}$ ) and Coulomb damping force ( $F_{τ}$ ). The symbol $F_{η}$ is only relevant with the recoil velocity $v_{r}$ and reflects the characteristics of fluid viscous damping. The symbol $F_{τ}$ is determined by the magnetic field intensity in the working gap and controlled by the input current, which reflects the characteristics of adjustable damping force. In order to determine the response time of Coulomb damping force ( $F_{τ}$ ) with respect to output current $I_{m}$ response, the dynamic force model can be obtained as the following form which neglects the friction force $F_{T}$ (Zhang et al., 2008)

F_{τ} \tan h (a_{1} v_{r} + a_{2}) = F_{R} - α v_{r}^{n}

(13)

where $\tan h (a_{1} v_{r} + a_{2})$ represents the Coulomb force ( $F_{τ}$ ) when the recoil velocity $v_{r}$ approaches 0, $a_{1} = 10$ , $a_{2} = 0.5$ , $α = 1100$ , and $n = 1.6$ . The coefficients were obtained by the nonlinear least-squares fitting technique based on the experimental data, and the total damping force $F_{R}$ and the recoil velocity $v_{r}$ were measured real-time utilizing the force and velocity sensors.

Table 2 shows a relationship between the Coulomb force ( $F_{τ}$ ) and the current ( $I_{m}$ ). The Coulomb force increases when the coil current rises. However, it reaches the saturated point when the current $I_{m}$ exceeds 1.8 A as shown in Table 2.

Table 2.

Relationship between $I_{m}$ and $F_{τ}$ .

PWM signal $V_{c}$	1.18	1.60	2.21	2.66	3.32	3.75	4.37	5.0
Current $I_{m}$ (A)	0.3	0.5	0.8	1.0	1.3	1.5	1.8	2.0
Coulomb force $F_{τ}$ (N)	1550	2058	3220	3800	4550	5100	6150	6150

PWM: pulse width modulation.

In order to compare the response between the Coulomb damping force ( $F_{τ}$ ) and the electromagnetic coil current $I_{m}$ intuitively, all the PWM signals in Figure 11 are normalized to one. Similarly, the value of $F_{τ}$ is normalized through dividing by an integer $M$ which depends on the corresponding final value of the current $I_{m}$ . After calculation, the values of the integer $M$ are 1600, 3400, 4300, and 6500, when the current $I_{m}$ is applied with 0.5, 1.0, 1.5, and 2.0 A, respectively.

Figure 11.

Response of MR buffer system: (a) V_c = 1.6 V, (b) V_c = 2.66 V, (c) V_c = 3.75 V, and (d) V_c = 5 V.

As shown in Figure 11, the response of Coulomb force ( $F_{τ}$ ) is slower than that of the coil current $I_{m}$ . The total time delay $τ$ seems to be affected by (1) time delay $τ_{α}$ , due to the electronics parts of the current controller, and (2) time delay $τ_{β}$ , the response time of mechanical parts of the damper and the MR fluid. Thus, the total time delay can be expressed as

τ = τ_{α} + τ_{β}

(14)

When the PWM signal $V_{c}$ (current $I_{m}$ ) increases, the time delay $τ_{α}$ slightly increases, the time delay $τ_{β}$ decreases, and the total time delay stabilizes at about 90 ms. As shown in Figure 12, the time delay $τ_{β}$ is about 15 ms which contains the hysteresis of magnetic materials and the response time of MR fluid when the input current $I_{m}$ is 2 A. Hence, the response of the electromagnetic circuit has an increasing effect on the time delay of the impact MR buffer system along with the increment of PWM signal $V_{c}$ (current $I_{m}$ ).

Figure 12.

Time delay of MR buffer system.

The above results indicate that the response time of the MR buffer system to step PWM control signals is the summation of those coming from electromagnet circuit, magnetic hysteresis (BH), MR effect, etc. The electromagnetic circuit can be simplified to a linear second-order system response and the hysteresis and MR effect can be regarded as simple delays. Besides, compared with other factors, the response of electromagnetic circuit has the most significant influence on the response time of the entire MR buffer system.

Proposed time delay compensation methods

Advanced correcting circuit

In order to reduce the time delay in the electromagnetic circuit, this study proposes an advanced circuit for the MR damper. A new equation can be obtained by Laplace transformation of equation (3)

I (s) = \frac{1}{s} \times \frac{V_{o}}{L_{M} s + R_{m}}

(15)

It is noted that the electromagnetic model of an MR damper can be simplified as the transfer function of a first-order inertial system and the current regarded as step signal. Therefore, it is possible to modify the transfer function of electromagnetic circuit model by incorporating an advanced correcting transfer function.

Figure 13 shows the advanced correcting circuit with input $U_{a} (s)$ and output $U_{o} (s)$ , which is supposed to connect the electromagnetic circuit of an MR damper shown in Figure 6 in series. Generally, a simple transfer function of the advanced correcting circuit can be written as

G (s) = α \frac{Ts + 1}{α Ts + 1}

(16)

Figure 13.

Advanced correcting circuit diagram.

where the coefficients $α$ and $T$ should be determined by the parameters of capacitance and resistance in the advanced correcting circuit, $T = R_{1} C$ , $α = R_{2} / (R_{1} + R_{2})$ .

To figure out the parameters involved in the transfer function and validate the response model of an MR damper under a constant voltage, a testing experiment was setup as shown in Figure 14. During testing, an external resistance was connected in series in order to monitor the changes in the current by measuring the voltage signal at the ends for the wire. A hall probe was positioned at the magnetic pole to measure the magnetic field intensity by a Gaussmeter (450 Lake Shore), which was connected to an oscilloscope.

Figure 14.

Experimental test.

Since the time delay is unrelated with the values of the input voltage, the response of the magnetic field and the current is compared by normalizing their magnitudes to 1. As shown in Figure 15, the time delay of the magnetic field is much longer than that of the current due to the eddy current effects. Based on the experimental data, the responses of the current and the magnetic field were fitted into two first-order inertial transfer functions, $G_{I} (s)$ and $G_{M} (s)$ . Therefore, the numerical values of parameters $R_{1} 3 Ω$ , $R_{2} 1 Ω$ , and $C 2.17 mF$ were obtained by assuming $T 0.0065 s$ and $α 0.25$ in equation (16). By applying the obtained parameters into advanced correcting circuit, it takes 5 ms to achieve 95% of the final state, which increases 75% of the response time as compared with the original response time as shown in Figure 16.

Figure 15.

Current and magnetic field response.

Figure 16.

Magnetic field response time.

DSP PID controller

The transfer function of equation (12) can be obtained by a parameter identification technique on the second-order model based on the experimental data of the current response. Figure 17 shows the comparison of parameterized second-order models $G_{Ic} (s)$ and the experimental results. Due to nonlinear effect of inductance by the electromagnetic coil with the iron core, the natural frequency and the damping ratio of the electromagnetic loop control system keep changing within a certain range.

Figure 17.

Comparison of second-order model and experimental results.

A PID controller was designed to adjust the delayed response using the DSP controller. The typical PID equation can be written in Laplace transform form

G_{pid} (s) = K_{p} + \frac{K_{i}}{s} + K_{d} s

(17)

In order to reduce the overall time delay, the transfer function model $G_{Ic} (s)$ which had the longest response time of coil current (I_m = 2 A) was chosen for the PID controller. Figure 18 shows a block diagram of the PID controller with a unit feedback loop, in which PWM signal $V_{c}$ was treated as the input control. An error value, which is defined as the difference between the PWM signal $V_{c}$ and the output current $I_{m}$ , was minimized to obtain the optimal three constant parameters ( $K_{p}$ , $K_{i}$ , and $K_{d}$ ). In order to balance the performance and the robustness, the PID gains were obtained as 10, 0.5, and 0.08, respectively, by setting a rise time (the time required for the response to rise from 10% to 90% of its final value) t_r = 0.005 s, the phase margin >45°, the bandwidth w_c ≥ 628 rad/s.

Figure 18.

Block diagram of PID controller.

The results show that the step response of the coil current after the PID regulation is greatly improved as compared with original system (Figure 19). It is obvious that the current response ( $I_{m}$ ) decreases its overshoot and improves its performance under different PWM signals $V_{c}$ . The response time with the PID control is reduced to 5 and 15 ms as compared with that of original cases for 0.5 and 1 A, respectively. However, the effect of PID regulation becomes less significant due to the saturation characteristic of current controller for higher current considered in this study (i.e. 1.5 and 2.0 A). Overall, such designs (advanced correction circuits, PID regulations, etc.) effectively reduce the time delay of MR systems subjected to impact loadings.

Figure 19.

Effect of PID controller.

Conclusion

In this article, a comprehensive research on the time delay of an impact buffer system based on an MR damper was investigated. It is found that the total time delay of the impact buffer system is contributed by the electromagnetic circuit delay, the hysteresis delay (B vs H), and the MR fluid’s response time. It is also found that the time delay is highly dependent on the effectiveness of the electric part of the control hardware. Rheology characteristics of MR fluids seem to play a negligible role in the time delay of impact buffer systems, thus the focus should be placed on the analysis of the electronic parts for such systems.

An MR damper’s electromagnetic circuit can be regarded as a first-order inertial system, whose transfer function can be changed to reduce the time delay. Thus, an advanced correcting circuit was designed and augmented to the first-order circuit model of an MR damper. The simulation results show that the correcting circuit is effective in reducing the response time of magnetic fields. For the electromagnetic drive circuit of the MR buffer system, it can be regarded as a second-order linear system with PWM signals as the input. The experimental results show that the response of the electromagnetic circuit has the most significant influence on the whole MR buffer system. The results also showed that a PID controller is an effective compensation method of the time delay problem of MR systems under impact loadings.

Footnotes

Declaration of conflicting interests

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

Funding

This work was supported by a Natural Science Foundation of China (NSFC) grant funded by the Chinese Government (nos 51175265 and 51305207).

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