Magnetorheological semi-active shock mitigation control. Part I: numerical analysis and preliminary tests

Abstract

Minimizing shock loads transmitted to sensitive structures or equipment is the objective of shock mitigation control. The core of shock mitigation control using magnetorheological (MR) energy absorber (EA) to minimize impact load is to make full use of the piston stroke of MREA so as to achieve “soft-landing.” The key lies in the precise description of the hysteresis of MREA and the shock mitigation control method. In this part, a single-degree-of-freedom (SDOF) shock mitigation control system using MREA is established, and the corresponding dynamic model and drop-induced shock mitigation test system are established. Based on the optimal Bi number control method and constant force control theory, feedforward controllers featuring resistor-capacitor (RC) operator-based hysteresis model and Bingham model are established to realize “soft-landing” in sequence. Specific performance evaluation indexes for shock mitigation control systems, that is, average velocity change rate (AVCR) and velocity-acceleration conversion ratio (V-ACR), are proposed. The effectiveness of shock mitigation control methods with different models on the MREA-based shock mitigation control system in profiles of simulation and tests are compared and analyzed.

Keywords

Shock mitigation semi-active control magnetorheological soft-landing RC operator hysteresis model

1. Introduction

Realizing adaptive mitigation of shock excitation shows great engineering value, such as helicopter seat suspensions (Choi and Wereley, 2005), spacecraft landers (Wang et al., 2019), vehicle longitudinal impact energy absorbers (Woo et al., 2007), and gun recoil systems (Singh and Wereley, 2014). Minimizing shock loads transmitted to force-sensitive structures and reducing damage to structures and injuries to the human body caused by shock excitation (Ayari et al., 2009; Bohman et al., 2011) are the most important research topics in the field of shock mitigation control. Magnetorheological (MR) energy absorber (EA) with the characteristics of rapid response, wide range of controllable damping force, and continuous controllability via electromagnetic field through the MR valve has been applied in vibration control cases (Phu et al., 2020; Woo et al., 2007; Zhu et al., 2019). In recent years, researchers have explored the possibilities of MREA application in shock mitigation control systems (Feng et al., 2022a, 2022b). Whether it is vibration control or shock mitigation control, the premise to realizing the efficient control of the system based on MREA is the precise control of MREA. The accurate description of the hysteretic nonlinear mechanical characteristics of MREA is the “soul.”

A few models have been proposed to characterize the hysteresis of MREA. For the shock mitigation control system, when the piston of MREA moves at high velocity, the off-state damping shows nonlinear characteristics with velocity and cannot be ignored. Considering minor loss based on Bingham-plastic model (Mao et al., 2005), Mao et al. (2013b) proposed Bingham-plastic nonlinear flow with minor losses (BPM) model and proved that the mechanical properties of MREA under shock conditions were more accurately predicted by the new model. Mao et al. (2013a) also proposed a hydromechanical model based on fluid dynamics, in which the dynamic response of MREA at high velocity is more accurately described by the hydromechanical model than unsteady-Bingham-plastic model and unsteady-BPM model. Bingham model (Stanway et al., 1987), bi-viscous model (Stanway et al., 1996), Bouc-Wen model (Ismail et al., 2009), polynomial model (Choi et al., 2001), resistor-capacitor (RC) operator-based model (Chen et al., 2018), and dynamic RC operator model (Bai and Tang, 2021) are commonly used to characterize the mechanical properties of MREA.

Compared with vibration excitation, shock excitation possesses the characteristics of short time and high transfer energy. Shock mitigation systems that rely on their passive mechanical properties to reduce impact loads are often controlled only for specific conditions. Passive systems often fail to decrease shock loads effectively when shock excitation is not in the optimal range. The semi-active shock mitigation control system using MREA adjusts the damping force adaptively according to the different shock excitations. It is possible to achieve optimal shock mitigation under different shock excitations. Shock mitigation control methods applied in different scenarios are being proposed. Maeda et al. (2016, 2019) replaced the conventional plastic deformation lander with a MREA on the lunar probe and proposed a semi-active landing control algorithm. The simulation and test results proved that the detector’s ability to land on complex terrain and prevent overturning is effectively enhanced by landers with MREA. Han et al. (2019) designed a hybrid controller for MR aircraft landing gear to improve landing efficiency. In order to decrease the discomfort and health risk caused by the shock excitation on passengers, Du et al. (2020) proposed an adaptive Skyhook control method based on an improved genetic algorithm, in which the vibration dose value and peak acceleration of the chassis are significantly reduced.

The key to minimizing the shock load is the precise “soft-landing” control, that is, maximizing the use of the piston stroke with zero terminal velocity. Wereley et al. (2011) proposed an optimal Bi number control method, under which “soft-landing” was realized by adjusting the controllable damping force of MREA. A “soft-landing” control method of a single-degree-of-freedom (SDOF) system based on MREA coupled with coil spring is further investigated by Singh and Wereley (2013). Later, Choi and Wereley (2015) proposed an optimal Bi number control method incorporating a time lag, so as to solve the issue that the controller is too sensitive to the response time. To prevent the dropping rate of the payload from exceeding the critical sink rate during the “soft-landing” shock mitigation control, Wang et al. (2021b) proposed a new control method based on the duration deceleration exposure control method and the optimal Bi number control method. Considering the quadratic damping in MREA shock mitigation process, Wang et al. (2021a) further proposed a generalized Bingham number control method and combined it with the duration deceleration exposure control method to achieve more precise “soft-landing” control. Combining the Skyhook control strategy and constant force control method, Bai and Yang (2019) proposed a switching control method to achieve the “soft-landing” under mixed shock-vibration excitations.

Currently, the majority of the research on MR shock mitigation remains in simulation. It is rare to find literature on the influence of MREA mechanical model and control method on the performance of shock mitigation. Based on the drop-induced shock mitigation control system, the influence of MREA mechanical model and control method on the “soft-landing” shock mitigation control result are thoroughly analyzed by simulation and tests in this study. The main technical contributions are as follows. (i) Based on the framework of the constant force control method and the optimal Bi number control method, combining the RC operator-based hysteresis model and Bingham model, the drop-induced “soft-landing” control is realized; (ii) the influence of mechanical models and control methods on the “soft-landing” shock mitigation control is analyzed from mechanical properties of MREA and payload motion state; (iii) two specific indexes, that is, average velocity change rate (AVCR) and velocity-acceleration conversion ratio (V-ACR), are proposed to evaluate the validity and stability of the control method.

2. “Soft-landing” shock mitigation control based on MREA

2.1. Optimal Bi number control method

Optimal Bi number control method is to calculate and output the controllable damping force of MREA using the initial excitation to achieve “soft-landing.” The control schematic is shown in Figure 1. When the shock excitation is transmitted to the MREA, the initial excitation is received and transmitted by the sensor of the optimal Bi number controller. By calculating, storing, and tracking the optimal Bi number, $B i_{o}$ , the shock mitigation controlled damping force is accurately controlled. In the figure, F_desired and F_tracking are the desired force from the upper controller and the tracking force output from MREA, respectively.

The drop-induced shock mitigation control system based on MREA is shown in Figure 2. As seen from the figure, d is the height of the drop distance. The dynamic equation of the shock mitigation control system is given by:

m \overset{\cdot\cdot}{x} (t) = F - m g

(1)

where F is the damping force of MREA; m is the payload mass; t is the time; $\overset{\cdot\cdot}{x} (t)$ is the piston acceleration of MREA; g is the acceleration of gravity.

The MREA damping force is composed of uncontrollable passive force and controllable yield force:

F = - c \overset{\cdot}{x} (t) - f_{y} sgn {\overset{\cdot}{x} (t)}

(2)

where c is the damping coefficient; $\overset{\cdot}{x} (t)$ is the piston velocity of MREA; f_y is the yield force; sgn() is the signum function.

The initial conditions for shock mitigation are given by:

x (0) = D; \overset{\cdot}{x} (0) = - v_{0}

(3)

where D is the maximum available stroke of MREA; $- v_{0}$ is the initial excitation, the excitation is specified as positive upward and negative downward.

From equations (1) and (2), the acceleration of the payload is obtained as:

\overset{\cdot\cdot}{x} (t) = - \frac{c}{m} \overset{\cdot}{x} (t) - \frac{f_{y}}{m} sgn {\overset{\cdot}{x} (t)} - g

(4)

Since the initial direction of the payload is downward, $sgn {\overset{\cdot}{x} (t)} = - 1$ , equation (4) can be rewritten as:

\overset{\cdot\cdot}{x} (t) = - \frac{c}{m} \overset{\cdot}{x} (t) + \frac{f_{y}}{m} - g

(5)

According to the initial conditions in equation (3), the payload velocity can be expressed as:

\overset{\cdot}{x} (t) = - v_{0} {[1 + Bi - R_{v}] e^{- \frac{t}{τ_{v}}} - Bi + R_{v}}

(6)

Bi = \frac{f_{y}}{c v_{0}}; τ_{v} = \frac{m}{c}; R_{v} = \frac{τ_{v} g}{v_{0}}

(7)

where Bi is the Bi number, defined as the ratio of the yield force to the product of the passive damping coefficient and the initial excitation; $τ_{v}$ is defined as the ratio of payload to the passive damping coefficient.

Combining displacement initial conditions given by equation (3), the displacement equation of the payload is obtained as:

x (t) = D [R_{s} (1 + Bi - R_{v}) (e^{- \frac{t}{τ_{v}}} - 1) + \frac{(Bi - 1) t}{τ_{v}} + 1]

(8)

R_{s} = \frac{R_{v} v_{0}}{D}

(9)

where x(t) is the piston travel displacement of MREA.

The piston acceleration of MREA is obtained by differentiating equation (6):

\overset{\cdot\cdot}{x} (t) = \frac{v_{0}}{τ_{v}} [1 + Bi - R_{v}] e^{- \frac{t}{τ_{v}}}

(10)

The velocity Bi number, ${Bi}_{o}^{v}$ , is obtained with $\overset{\cdot}{x} (t) = 0$ :

{Bi}_{o}^{v} = R_{v} - {(1 - e^{\frac{t}{τ_{v}}})}^{- 1}

(11)

The displacement Bi number, ${Bi}_{o}^{d}$ , is obtained with $x (t) = 0$ :

{Bi}_{o}^{d} = R_{v} - R_{s} \frac{1 + e^{\frac{t}{τ_{v}}} (\frac{1}{R_{s}} - 1)}{1 + e^{\frac{t}{τ_{v}}} (\frac{t}{τ_{v}} - 1)}

(12)

The “soft-landing” time, t_o, is obtained by setting:

{Bi}_{o}^{v} (t_{o}) = {Bi}_{o}^{d} (t_{o})

(13)

Then, the “soft-landing” time, t_o, is given by:

t_{o} = {\frac{1}{R_{s}} - 1 - W [e^{(\frac{1}{R_{s}} - 1)} (\frac{1}{R_{s}} - 1)]} τ_{v}

(14)

where W[] is lambert W function.

The optimal Bi number, Bi_o, is obtained by combining equations (11) and (14):

B i_{o} = R_{v} - \frac{1}{1 - e^{{\frac{1}{R_{s}} - 1 - W [e^{(\frac{1}{R_{s}} - 1)} (\frac{1}{R_{s}} - 1)]} τ_{v}}}

(15)

2.2. Constant force control method

The constant force control schematic is shown in Figure 3. According to Figure 3, after the desired acceleration is calculated by the constant force controller, the desired force is adjusted in real-time by comparing the error between the actual acceleration and the desired acceleration at the current moment. Finally, the desired current obtained from the force tracking module is transmitted to MREA.

The expression of the constant force control method is given by:

2 {\overset{\cdot\cdot}{x}}_{d} (t) x (t) = \overset{\cdot}{x} {(t)}^{2}

(16)

Δ + m {\overset{\cdot\cdot}{x}}_{d} (t) = m \overset{\cdot\cdot}{x} (t - 1)

(17)

F_{desired} (t) = F (t - 1) + Δ

(18)

where $\underset{d}{\overset{\cdot\cdot}{x}} (t)$ is the desired piston acceleration of MREA; $Δ$ is the external force error calculated by the desire piston acceleration and actual piston acceleration of MREA; $F_{desired}$ is the desired force from MREA.

3. Simulation and analysis

The sinusoidal displacement excitation with a frequency of 1 Hz and amplitude of 30 mm is set. The parameters in the RC operator-based hysteresis model and the Bingham model are identified by genetic algorithm, and the two mathematical models performances are obtained in Appendix A. The available stroke for MREA is 120 mm. Initial excitations are set at 1.83 and 3.65 m/s (Choi and Wereley, 2015). The parameters used in the simulation are listed in Table 1 and Table A.1 in Appendix A.

Table 1.

Parameters of the MREA system.

Parameters	Values
The maximum available stroke of MREA: D (m)	0.09
The damping coefficient: c (Ns/m)	1200
The payload mass: m (kg)	93
Low-velocity initial excitation: $v_{0}$ (m/s)	1.83
High-velocity initial excitation: $v_{0}$ (m/s)	3.65

Figure 4 presents the comparison of the desired forces of the constant force control method and the optimal Bi number control method under different initial excitations. According to Figure 4, both the constant force control method and the optimal Bi number control method adjust the desired force according to the initial excitations. The constant force method keeps the desired force almost unchanged during the shock mitigation, only a sudden change occurs at the end of the process due to force tracking delay. The optimal Bi number control method keeps the yield force constant, and the desired force decreases with time.

3.1. Force tracking of MREA

Based on the schematics of the optimal Bi number control and constant force control, as shown in Figures 1 and 3, after the desired force is obtained by the control methods using velocity and displacement, the desired current needs to be calculated through the force tracking module to control the MREA output damping force. A force tracking principle is given in Appendix A.

The force tracking results of different control methods under the initial excitation of 1.83 m/s are shown in Figure 5. Based on the constant force control method, as shown in Figure 5(a) and (b), the RC operator-based hysteresis model shows similar tracking capability with the Bingham model, and the tracking errors occur around 0.1 s. The constant force control method is a feedback control method. The occurrence of tracking error causes the desired force to be continuously raised to compensate for the previous moment force tracking error. However, the force tracking capability of the model is limited by the mechanical properties of the MREA. Near the end of the control process, the decreasing MREA velocity results in a narrow range of the MREA output force range, and the output damping force is difficult to maintain even with the increasing desired force of the constant force control method. To keep the output damping force constant, the desired currents of the RC operator-based hysteresis model and the Bingham model increase with time, and finally remain at the peak set current of 1.8 A. The decrease in the output force range of the MREA mechanical characteristic model is also indicated by the trend of current variation. Although both desired currents are increasing, the increasing trend in the RC operator-based hysteresis model tends to increase, while the increasing trend in the Bingham model tends to be flat.

The force tracking results based on the optimal Bi number control method are shown in Figure 5(c) and (d). From the figures, based on the optimal Bi number control method, both the RC operator-based hysteresis model and the Bingham model precisely track the desired force without force tracking error. Comparing the tracking current, under the optimal Bi number control, the RC operator-based hysteresis model current continues to increase over time, but does not increase to the maximum current of 1.8 A. The increasing trend of current is significantly lower than that of the constant force control method, even the tracking current of the Bingham model continues to decrease with time.

When the initial excitation increases to 3.65 m/s, as shown in Figure 6, the force tracking capability of the RC operator-based hysteresis model and the Bingham model is significantly weakened as compared with that at low-velocity initial excitation. With the constant force control method, the RC operator-based hysteresis model still achieves accurate force tracking over a period. However, as compared with the force tracking results at low velocity, the occurrence time of the force tracking error and peak current is apparently advanced. The desired force cannot be tracked by the tracking force of the Bingham model during the whole control process, as shown in Figure 6(b), where the tracking current is maintained at the peak set current of 1.8 A at all times.

In the case of the optimal Bi number control method, the force tracking error moment and the peak current occurrence moment of the RC operator-based hysteresis model are also apparently advanced as compared with the tracking results under low-velocity initial excitation. Similar degradation of the force tracking capability of the RC operator-based hysteresis model is demonstrated in the constant force control method. Because of the open-loop characteristics of the optimal Bi number control system, the desired force is not changed by the force tracking error, as compared with the constant force control method. The RC operator-based hysteresis model shows better force tracking capability than the Bingham model under high-velocity initial excitation, because of the coupling relationship between current and velocity in the hysteresis model.

According to Figures 5 and 6, both the RC operator-based hysteresis model and the Bingham model show similar force tracking capability under the excitation of 1.83 m/s. In this case, the optimal Bi number control method possesses better force tracking results than the constant force control method. However, with high-velocity initial excitation, the force tracking capability of the model declines. Compared with the tracking results of the Bingham model which fails to track the desired force, the force tracking capability of the RC operator-based hysteresis model is much better.

3.2. Motion state of MREA

Figure 7 presents the piston travel displacement variations of MREA. The optimal Bi number control method and the constant force control method take 0.22 and 0.12 s respectively to exhaust the maximum available stroke of MREA under the initial excitation of 1.83 m/s. In such excitation, due to the similar force tracking results between the RC operator-based hysteresis model and the Bingham model, the displacement variations of the MREA by different control methods are almost identical. According to Figure 7(b), when the initial excitation is 3.65 m/s, the desired current of the control method based on the Bingham model maintains the maximum current of 1.8 A with the same tracking force. Based on the RC operator-based hysteresis model, it takes 0.068 and 0.055 s for the maximum stroke of MREA to be exhausted by the optimal Bi number control method and the constant force control method, respectively.

Consuming the available stroke of MREA, the external force equivalent to the weight of the payload is applied to the payload to maintain a constant velocity of MREA. The control results under different control methods are compared by the end stroke velocity of the MREA. The MREA velocities are decreasing continuously under the influence of the MREA damping force. According to Figure 8(a), similar force tracking results under low-velocity initial excitation also result in almost identical piston velocity variations using the same control method. Compared with the linearly decreasing variation trend of velocity using the constant force control method, the velocity decrease trend of the optimal Bi number control method is gradually flattening. Due to the slight force tracking error of the constant force control method, the velocity of the optimal Bi number control method is closer to zero than that of the constant force control method, when the maximum available stroke is exhausted.

When the initial excitation increases to 3.65 m/s, as shown in Figure 8(b), different degrees of increase in MREA terminal velocity under control methods are observed, as compared with low-velocity initial excitation. Due to the less effectiveness of the Bingham model to track the desired forces, MREA velocity variations and maximum MREA terminal velocity are induced not as expected. Based on the RC operator-based hysteresis model, because of the better desired force tracking performance, the MREA terminal velocity of the optimal Bi number is closer to zero as compared with the constant force control method.

According to Figure 9(a), when the initial excitation is 1.83 m/s, the acceleration variations of MREA using optimal Bi number control method are almost identical, and the acceleration decreases with time and gradually tends to be flat. Based on constant force control method, the acceleration of MREA is generally maintained constant by the RC operator-based hysteresis model and the Bingham model. Only a short time before the end, the acceleration inevitably decreases due to the reduced tracking force. Besides, an earlier time and a lower amplitude of acceleration attenuation are observed compared to the RC operator-based hysteresis model. By comparing the acceleration variations of different control methods, the acceleration of the constant force control method is supposed to remain stable, while that of the optimal Bi number control method is supposed to decrease with time. Higher peak acceleration and lower minimum acceleration are achieved by the optimal Bi number control method compared with constant force control method.

When the initial excitation is 3.65 m/s, the acceleration of the Bingham model is consistent and much lower than that of the RC operator-based hysteresis model. Based on the RC operator-based hysteresis model, the optimal Bi number control method still provides higher peak acceleration and lower minimum acceleration than the constant force control method. However, the difference between the peak acceleration and the minimum acceleration is narrowed compared with difference under the low-velocity initial excitation of 1.83 m/s. Both the acceleration deviation moment of the optimal Bi number control method and the constant force control method are advanced.

From the motion state of the MREA under the different control methods, “soft-landing” shock mitigation control can be achieved by both the constant force control method and optimal Bi number control method under low-velocity initial excitation. In this case, the control results are not significantly influenced by the hysteresis model. When the initial excitation is high, the control capabilities of different control methods decrease to some extent. Based on the Bingham model, the motion states of the MREA are always consistent under different control methods, therefore the control method is invalidated.

4. Evaluation indexes

To evaluate the control performances, two specific evaluation indexes, that is, AVCR and V-ACR, are proposed to characterize the stability and validity of the shock mitigation process. AVCR and V-ACR are respectively defined as:

AVCR = \frac{| v_{\max} - v_{\min} |}{v_{0} S_{u}}

(19)

V - ACR = \frac{| \overset{\cdot\cdot}{x} (t) |}{v_{0}}

(20)

where $S_{u}$ is the available stroke of the MREA; $v_{\max}$ and $v_{\min}$ is the maximum velocity and minimum velocity of MREA.

When the initial excitation is 1.83 m/s, AVCR and V-ACR under the different control methods are shown in Figure 10(a) and (b), respectively. According to Figure 10(a), similar AVCRs are presented by different control methods. AVCR of the optimal Bi number control method is higher than that of the constant force control method due to the less force tracking error. The optimal Bi number control method based on the RC operator-based hysteresis model possesses the highest AVCR of 8.21, while the constant force control method based on the Bingham model has the lowest AVCR of 7.51. According to Figure 10(b), the optimal Bi number control shows apparently higher peak V-ACR and lower minimum V-ACR than the constant force control. Based on constant force control method, the peak AVCRs of the RC operator-based hysteresis model and the Bingham model are almost the same. However, lower minimum V-ACR and higher V-ACR range of the RC operator-based hysteresis model are observed. Among the different control methods, the optimal Bi number control method based on the RC operator-based hysteresis model possesses the highest range of the V-ACR. The maximum V-ACR is 15.5 times the minimum V-ACR. The constant force control method based on the Bingham model possesses the lowest range of V-ACR, and the maximum V-ACR is only 1.4 times of the minimum V-ACR.

When the initial excitation is 3.65 m/s, AVCR and V-ACR under different control methods are presented in Figure 11(a) and (b), respectively. According to Figure 11(a), higher AVCR is presented by the RC operator-based hysteresis model compared with the Bingham model. The optimal Bi number control method based on the RC operator-based hysteresis model possesses the highest AVCR of 6.93. Due to less performance of the force tracking of the Bingham model, AVCR of the Bingham model-based control method is the lowest at 3.72, which is less than half of AVCR of the RC operator-based hysteresis model. Based on the RC operator-based hysteresis model, according to Figure 11(b), the characteristics of higher peak V-ACR, lower minimum V-ACR and higher V-ACR range are still possessed by the optimal Bi number control method. However, the V-ACR range is apparently narrowed, and the maximum V-ACR is only 3.62 times of the minimum V-ACR. The ratio of the maximum V-ACR to the minimum V-ACR based on the constant force control method is exceeded to 1.98.

According to Figures 10 and 11, compared with constant force control method, the optimal Bi number control method possesses an enormous range of V-ACR and worse stability during the shock mitigation process. When the initial excitation is high, the stability during the shock mitigation process of the optimal Bi number control method is improved, while that of the constant force control method deteriorates. In the case of low-velocity initial excitation, AVCR and V-ACR under the control methods are not significantly influenced by the hysteresis model. With high-velocity initial excitation, the AVCRs of the Bingham model are lower than that of the RC operator-based hysteresis model.

5. Experimental tests

The drop-induced shock mitigation test system is shown in Figure 12. During the test, NI-cRIO collects the velocity and displacement of MREA and outputs the control signal calculated by PC to the controllable current driver. The damping force of the MREA is controlled by the current from the controllable current driver according to the initial excitations. The payload mass is 93 kg. The maximum available stroke for the MREA is set to be 90 mm.

5.1. Force tracking under shock excitation

Different from the numerical simulation, the desired current is output to the MREA by the force tracking module in the tests. The force tracking results are shown in Figure 13. According to Figure 13(a) and (b), the MREA accelerates instantaneously after shock excitation from the payload. The sudden increase in velocity results in a much higher desired force calculated by the constant force control method than the output of MREA. Desired force remains stable after two control steps. The MREA desired force of constant force control methods is accurately tracked by the tracking force after a response time of about 0.02 s. The force tracking results of the optimal Bi number control method are shown in Figure 13(c) and (d). Because the MREA is not accelerated beyond the velocity threshold, the optimal Bi number control method is not able to iterate out the convergent solution for a period after the MREA is accelerated. During this period, the desired force is still zero, and the MREA only outputs the passive damping force of 1 kN. The control method assumes that the energy transmitted to the MREA can be dissipated by the passive damping force. After 0.04 s, the yield force and the desired force are obtained by the optimal Bi number control method. The yield force remains constant after a short fluctuation. The desired force decreases with time after reaching the peak, and finally coincides with the controllable force. Similar variations of the MREA controllable and desired forces are obtained by comparing simulation and experimental results. Due to the response time delay of MREA (Bai et al., 2019) and velocity threshold, different from the simulation results, the tracking force of MREA is not changed until 0.03 s of desired force is obtained.

5.2. Motion state under shock excitation

Figure 14 shows the piston travel displacement of the MREA. Because the optimal Bi number control method does not work until the MREA reaches the velocity threshold, the optimal Bi number control method achieves faster displacement reduction than constant force control, and both models under the optimal Bi number control method exceed the set maximum available stroke. The constant force control method adjusts the desired force in real-time after the acceleration of MREA with accurate force tracking. Based on the constant force control method, the MREA displacement of the RC operator-based hysteresis model and the Bingham model basically remains unchanged after reaching 83.3 and 72.0 mm, that is, 92.6% and 80% of the maximum available stroke, respectively. The constant force control method based on the RC operator-based hysteresis model is verified to achieve the “soft-landing.” For the constant force control method based on the Bingham model, although the MREA is also decelerated to zero within the maximum available stroke, the MREA stroke is not fully exhausted.

Figure 15 presents the piston velocity in the shock mitigation test. After the payload contacts the MREA, the velocity of the MREA increases rapidly to about 1.2 m/s. Then, under the damping force and the limit belt, the MREA velocity gradually decreases to zero. Based on the constant force control method, the deceleration process of the RC operator-based hysteresis model is closer to constant deceleration than the Bingham model. The deceleration of the Bingham model suddenly decreases at 0.1 s. Based on the optimal Bi number control method, when reaching the peak velocity, the MREA maintains a constant velocity for a period before decelerating under the controlled damping force. The MREA decelerates to zero with 0.04 s and the trend of velocity reduction increases further after 0.1 s. Since the optimal Bi number control method works, the available stroke of MREA has been consumed more than half. There is no apparently difference in the piston velocity variation between the two different hysteresis models. Compared with the constant force control method, high peak velocity and shorter deceleration time are obtained by the optimal Bi number control method.

The piston acceleration of the MREA is shown in Figure 16. Due to the impact of the payload, the MREA produces enormous downward acceleration at 0.03 s under different control methods. Based on the constant force control method, the MREA produces an upward acceleration of about 0.04 s. The deceleration of the MREA is effectively controlled at about 10 m/s² by both the RC operator-based hysteresis model and the Bingham model. Based on the optimal Bi number control method, the acceleration of the RC operator-based hysteresis model and the Bingham model is kept near zero from 0.04 to 0.09 s. When the optimal Bi number control method works, the available stroke of MREA is nearly exhausted, and the remaining stroke is not sufficient to slow down MREA to stop. Restricted by the limit belt, the MREA generates a peak acceleration of about 40 m/s² before stopping. The RC operator-based hysteresis model produces a higher limit acceleration than the Bingham model. From the acceleration variations, under the current test conditions, the optimal Bi number control method fails to completely dissipate the energy transferred from the shock excitation to the MREA within the set maximum stroke.

5.3. Shock mitigation evaluation

As shown in Figure 17, the AVCR of the constant force control method is much higher than that of the optimal Bi number control method in the test. The constant force control method based on the Bingham model achieves the highest AVCR of 13.4. Compared with the simulation results, the AVCR of the optimal Bi number control method obtained from the test is significantly decreased.

Comparing AVCR obtained from the test with the MREA motion state, the control method with a higher AVCR achieves faster deceleration within the available stroke of the MREA. This verifies that the efficiency of the shock mitigation control method is accurately evaluated by AVCR.

To accurately evaluate the stability of the shock mitigation test process, the abnormal acceleration caused by payload impact and limit belt limitation is removed when analyzing the V-ACR of the shock mitigation test process. According to Figure 18, based on the constant force control method, the RC operator-based hysteresis model achieves higher peak V-ACR, lower minimum V-ACR, and higher V-ACR range than the Bingham model. Based on the optimal Bi number control method, compared with the simulation results, the minimum V-ACRs of the RC operator-based hysteresis model and the Bingham model are significantly increased, and the V-ACR ranges are sharply reduced. Compared with the Bingham model, the RC operator-based hysteresis model possesses lower V-ACR and worse V-ACR range in the constant force control method and the optimal Bi number control method.

6. Conclusions

The constant force control method and the optimal Bi number control method for the MREA shock mitigation control system were presented and established in this study. Then, based on the RC operator-based hysteresis model and the Bingham model, the control methods were simulated and preliminarily tested for the adaptive shock mitigation. Finally, the evaluation indexes of AVCR and V-ACR were proposed to evaluate the effectiveness and stability of shock mitigation control methods.

The main conclusions are summarized as follows.

The constant force control method is desired to keep the MREA damping force constant throughout the shock mitigation process. The desired force of the optimal Bi number control method decreases with velocity because only the controllable force remains constant.

From the simulation results, similar force tracking results are observed for different models under low-velocity initial excitation, and shock mitigation controls are not apparently influenced by the hysteresis models. AVCRs under different control methods are similar. The constant force control method shows better control stability than the optimal Bi number control method. When the initial excitation is high, the control method based on the Bingham model is invalidated, while the shock excitations can still be apparently reduced by the control method based on RC operator-based hysteresis model.

In the tests, “soft-landing” is basically achieved by the constant force control method based on the RC operator-based hysteresis model using 92.6% of the maximum available stroke. The constant force control method based on the Bingham model also reduces the MREA velocity to zero within the set maximum available stroke, but the maximum available stroke of MREA is not fully utilized. Due to velocity threshold conditions and MREA response delay, the optimal Bi number control method is not as good as expected to achieve the “soft-landing.”

In the tests, the constant force control method possesses a higher AVCR than the optimal Bi number control method. Based on the constant force control method and the optimal Bi number control method, the RC operator-based hysteresis model possesses lower peak V-ACR, higher minimum V-ACR, minor V-ACR range, and stable shock mitigation process, compared with the Bingham model.

Comparing the motion state with evaluation indexes, the validity and stability of the shock mitigation control method are accurately reflected by AVCR and V-ACR. During the test, the control results of the optimal Bi number control method are inconsistent with the simulation. Apparent differences in the optimal Bi number control method between the test and the simulation are compared, and the evaluation results conforming to the MREA motion state are still presented by the evaluation indexes.

Based on the drop-induced shock mitigation system, the control results of the optimal shock mitigation control methods were simulated and tested. MREA-based shock mitigation control method is difficult to be directly applied in practical shock mitigation systems with complex configurations. The optimal shock mitigation method for SDOF and 2DOF systems will be investigated in Part II of this study.

Footnotes

Appendix A

The Bingham model is given by:

(A.1)

{\begin{matrix} F = c \overset{\cdot}{x} (t) + f_{y} sgn (\overset{\cdot}{x} (t)) \\ f_{y} = a_{1} I^{2} + a_{2} I + a_{3} \end{matrix}

where I is the applied current; $a_{1}$ , $a_{2}$ and $a_{3}$ are the constant coefficients to be identified for f_y.

The RC operator-based hysteresis model is given by:

(A.2)

F = c \overset{\cdot}{x} (t) + α z + k_{r} x (t)

(A.3)

z (t) = 1 - 2 e^{- \frac{g_{1} (S)}{p}}, when \overset{\cdot}{x} (t) > 0

(A.4)

z (t) = - 1 + 2 e^{\frac{g_{2} (S)}{p}}, when \overset{\cdot}{x} (t) < 0

(A.5)

\overset{\cdot}{z} (t) = 0, when \overset{\cdot}{x} (t) = 0

(A.6)

S_{0} = {g_{1}}^{- 1} (- \ln (\frac{(1 - z (t^{*}))}{2}) p), x_{0} = x (t^{*}), when \overset{\cdot}{x} (t) > 0

(A.7)

S_{0} = {g_{2}}^{- 1} (\ln (\frac{(1 + z (t^{*}))}{2}) p), x_{0} = x (t^{*}), when \overset{\cdot}{x} (t) < 0

(A.8)

S (t) = S_{0} + x (t) - x_{0}

(A.9)

g_{1} (S) = {| S |}^{q}

(A.10)

g_{2} (S) = - {| S |}^{q}

where $α$ , $k_{r}$ are the stiffness coefficient and hysteresis coefficient; $z$ is the hysteresis operator of MREA; the range of the hysteresis operator $z$ is (−1,1) expressed by equations (2)–(5), and $\overset{\cdot}{z} (t)$ is its derivative; p is the tuning parameter, and p > 0; $S$ is the virtual displacement; tuning function $g_{1}$ and $g_{2}$ are the monotonically function of virtual displacement S as shown in equations (A.9)–(A.10); $q$ is a hysteresis factor; ${g_{1}}^{- 1}$ and ${g_{2}}^{- 1}$ are the inverse functions of $g_{1}$ and $g_{2}$ ; $S_{0}$ and $x_{0}$ are the references of the virtual displacement and displacement at the moment $t^{*}$ that the direction of excitation changes.

According to equations (A.1)–(A.10), four parameters in the Bingham model are to be determined, which are $c, a_{1}, a_{2}, a_{3}$ . Five parameters in the RC operator-based hysteresis model are to be determined. By analyzing the experimental data, the undetermined parameters $c$ and $α$ of RC operator-based hysteresis model are fitted by a quadratic function with current as variable, as shown in equations (A.11) and (A.12). Therefore, there are nine parameters in the RC operator hysteresis model to be determined, which are $p, q, c_{1}, c_{2}, c_{3}, α_{1}, α_{2}, α_{3}, k$ .

(A.11)

c = c_{1} I^{2} + c_{2} I + c_{3}

(A.12)

α = α_{1} I^{2} + α_{2} I + α_{3}

The mechanical properties of MREA are characterized by the model after parameter identification. Parameter identification is to determine the undetermined parameters in the model by fitting the test data with the model. The test data used for parameter identification are sinusoidal displacement excitation with an amplitude of 30 mm and a frequency of 1.5 Hz. The applied current is 0–1.8 A.

Undetermined parameters in the RC operator-based hysteresis model and the Bingham model are solved by genetic algorithm with an objective function Obj:

(A.13)

Obj = \sum_{I = 0}^{1.8} (\frac{1}{n} \sum_{N = 1}^{N = n} (F_{test} (I) - F_{model} (I)))

where $F_{test}$ and $F_{model}$ are the damping forces of the test and model, respectively; $n$ is the number of test data.

The parameters $c, a_{1}, a_{2}, a_{3}$ in the Bingham model and the parameters $p, q, c_{1}, c_{2}, c_{3}, α_{1}, α_{2}, α_{3}, k$ in the RC operator-based hysteresis model are shown in Table A.1.

Figure A.1 presents the identification results of the RC operator-based hysteresis model and the Bingham model. As seen from the figure, compared with the Bingham model, smaller fitting errors are observed for the RC operator-based hysteresis models when MREA velocity is near zero.

The damping force prediction results of the RC operator-based hysteresis model and the Bingham model are shown in Figure A.2. Under excitations, the hysteresis model accurately characterizes the mechanical properties of MREA. MREA hysteresis nonlinearity is accurately predicted by the RC operator-based hysteresis model.

To achieve the objective of continuously controllable damping force, force tracking is also required. Force tracking is the calculation of the desired current matching the desired force under a given displacement and velocity excitation. According to equations (A.1), (A.2), (A.11), and (A.12), the Bingham model and the RC operator-based hysteresis model are quadratic functions of applied current. The optimal force tracking calculation is equivalent to the minimum error calculation between the model force and the desired force, and the desired current matching the desired force is obtained by solving the optimal solution of the quadratic function. The expressions are given by:

(A.14)

\min (| F_{model} (I) - F_{desired} |), 0 \leq I \leq I_{\max}

(A.15)

min (| A I^{2} + BI + C |), 0 \leq I \leq I_{\max}

where A, B, and C are represented as $(c_{1} \overset{\cdot}{x} + α_{1} z)$ , $(c_{2} \overset{\cdot}{x} + α_{2} z)$ , and $(c_{3} \overset{\cdot}{x} + α_{3} z + k_{r} x - F_{desired})$ ; $I_{\max}$ is the maximum applied current.

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

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors wish to acknowledge the National Natural Science Foundation of China (Grant No. 52272392), the Innovation Project of New Energy Vehicle and Intelligent Connected Vehicle of Anhui Province, and Key Research and Development Projects in Anhui Province (202104a05020002), for their support of this research.

ORCID iDs

Weihan Li

Xianxu “Frank” Bai

Pu Gao

Jiabao Pan

Data availability statements

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

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