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“Shape function + memory mechanism”: based hysteresis modeling of magnetorheological fluid actuators
Li-Jun Qian, Peng Chen, Fei-Long Cai, Xian-Xu Bai
Li-Jun Qian, Peng Chen, Fei-Long Cai, Xian-Xu Bai, "“Shape function + memory mechanism”: based hysteresis modeling of magnetorheological fluid actuators," Proc. SPIE 10596, Behavior and Mechanics of Multifunctional Materials and Composites XII, 105961H (22 March 2018); doi: 10.1117/12.2294504 Event: SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring, 2018, Denver, Colorado, United States Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 3/26/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
“Shape function + memory mechanism”-based hysteresis modeling of magnetorheological fluid actuators Li-Jun Qian, Peng Chen, Fei-Long Cai and Xian-Xu Bai* Laboratory for Adaptive Structures and Intelligent Systems (LASIS), Department of Vehicle Engineering, Hefei University of Technology, Hefei 230009, People’s Republic of China ABSTRACT A hysteresis model based on “shape function + memory mechanism” is presented and its feasibility is verified through modeling the hysteresis behavior of a magnetorheological (MR) damper. A hysteresis phenomenon in resistor-capacitor (RC) circuit is first presented and analyzed. In the hysteresis model, the “memory mechanism” originating from the charging and discharging processes of the RC circuit is constructed by adopting a virtual displacement variable and updating laws for the reference points. The “shape function” is achieved and generalized from analytical solutions of the simple semi-linear Duhem model. Using the approach, the memory mechanism reveals the essence of specific Duhem model and the general shape function provides a direct and clear means to fit the hysteresis loop. In the frame of the structure of a “Restructured phenomenological model”, the original hysteresis operator, i.e., the Bouc-Wen operator, is replaced with the new hysteresis operator. The comparative work with the Bouc-Wen operator based model demonstrates superior performances of high computational efficiency and comparable accuracy of the new hysteresis operator-based model. Keywords: Hysteresis, memory mechanism, shape function, RC circuit, computational efficiency, MR actuators.
1. INTRODUCTION Magnetorheological (MR) fluids, as a kind of smart materials, show tunable rheology with the presence of external magnetic field1, and the MR fluids based actuators (MR actuators), such as MR dampers2, MR brakes3 and MR clutches4, achieve controllable mechanical properties (damping or stiffness). They can be a promising candidate in semi-active vibration control systems, like the suspension systems in automobiles or trains5, seismic structural protection systems6 and human prosthetic knees4. However, the hysteresis property7, an intrinsic behavior of MR actuators, no matter whether they work in shear mode, flow mode, squeeze mode, or pinch mode, complicates the control of MR actuators-based systems, and the lack of an accurate hysteresis model restrains the control performance or further extensional applications. Hence, from the research beginning of MR applications, great efforts have been made on hysteresis modeling. The hysteresis behavior is presented when MR actuators are excited under a reciprocal displacement excitation, commonly under a sinusoidal excitation. Experimental tests show that a big change of the varying tendency occurs around the direction-change points2,7,8. Coincidentally, the associated velocity in this area is relatively small, and the varying tendency is displayed more distinctly in the force-velocity plot2,7, graphically like the magnetization curve (the B-H curve). Consequently, some hysteresis models based on force-velocity plot were formulated to fit the tested hysteresis loop under certain exciting conditions, such as nonlinear hysteretic bi-viscous model9, viscoelastic-plastic model10, hyperbolic tangent function-based model11, polynomial model12, division-based model13 and some no-parameter models14, such as the neural network or machine learning-based hysteresis models. However, only the identification result for the specific curve fitting are satisfying, while the prediction performance for behavior modeling is not comparable to that of the dynamic phenomenological models, such as the Bouc-Wen model15, Dahl model16 and LuGre model17. To more accurately characterize the hysteresis behavior of MR fluid actuators, their modified models18-20 were proposed for the damping force roll-off after the direction-change points. Apparently, these hysteresis models can well portray or predict the typical hysteresis behavior even under random excitation19. It is worthy to note that, compared to the force-velocity curve-based hysteresis models applying curve fitting, the BoucWen model, Dahl model and LuGre model, belonging to one category of Duhem model 21, are trying to formulate the *
Author to whom any correspondence should be addressed. E-mail:
[email protected] (Xian-Xu Bai); Behavior and Mechanics of Multifunctional Materials and Composites XII, edited by Hani E. Naguib, Proc. of SPIE Vol. 10596, 105961H · © 2018 SPIE CCC code: 0277-786X/18/$18 · doi: 10.1117/12.2294504 Proc. of SPIE Vol. 10596 105961H-1
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memory mechanism, which is actually the essence of hysteresis system, and the hysteresis in these models is defined between the generalized displacement input and generalized force output. Hence, the consistence between the hysteresis model and the hysteresis behavior in MR actuators promotes a clue to further understand the hysteresis mechanism in MR actuators. However, the above mentioned hysteresis models are expressed with differential equations, which makes it trivial to solve the equations and the specific format of the model limits the flexibility to model the variable hysteresis behavior of MR actuators, then a new hysteresis model with higher computational efficiency and more flexibility is needed. Subsequently, in this paper, a basic hysteresis operator is presented and generalized based on a hysteresis phenomenon in resistor-capacitor (RC) circuit. The “shape function + memory mechanism” strategy is applied, and the memory mechanism originating from the charging and discharging processes of the RC circuit is constructed. The shape function is achieved and generalized from analytical solutions of the simple semi-linear Duhem model. Eventually, within the structure of a “Restructured phenomenological model”, a parameter model for MR actuators will be constructed and evaluated.
2. “SHAPE FUNCTION + MEMORY MECHANISM”-BASED HYSTERESIS MODEL 2.1 Hysteresis phenomenon in RC circuit A normal RC circuit is presented in Figure 1, which can work in charging and discharging modes through the switching of contact points SA or SB. The zero-state response of the potential difference 𝑉𝑐+ between the ends of the capacitor in charging mode (when SA is switched on) and the zero-input response in discharging mode (when SB is switched on and assuming the initial potential difference 𝑉𝑐+ is equal to 𝑉𝑣 ) can be respectively expressed as 𝑉𝑐+ (𝑡) = 𝑉𝑣 (1 − 𝑒 −𝑡+⁄𝑅𝐶 ) 𝑉𝑐− (𝑡) = 𝑉𝑣 𝑒 𝑡−⁄𝑅𝐶
(1a) (1b)
where 𝑉𝑣 is the supply voltage; R and C are the resistance and capacitance, respectively. It should be noted that, the time in charging is marked as 𝑡+ ∈ [0,+∞) when taking the switching on point as reference time, while in discharging mode the time is labeled as 𝑡− (∈ (−∞,0]). The time response of the RC circuit in charging and discharging processes is presented in Figure 2. Mathematically, Eqs. (1a) and (1b) are the solutions of a simple semi-linear Duhem model22. It is worthy to note that the curves for charging and discharging processes reveal a memory of RC circuit and settle the basic shapes. As shown in Figure 2, in the discharging mode with an initial condition of 𝑉0 , the time history of potential difference will start from 𝑉0 and follow the shape of Eq. (1b). When the charging and discharging modes switch frequently, as Case A in Figure 2, the working mode changes before the convergence of potential difference, and following the continuity of 𝑉𝑐 , the discharging response will start from potential difference V0. Apparently, the time-potential difference relation follows the principle of hysteresis, and the change of working mode in RC circuit is equivalent to the direction change of displacement. In the hysteresis of RC circuit, we use the time labels 𝑡+ ∈ [0,+∞) and 𝑡− ∈ (−∞,0] to identify the working mode. As in Case A, the shifted point t0 can be obtained from its inverse function with V0, and then following the basic shape and the time interval from t0, the trajectory of potential difference can be fully predicted. Hence, we can find that the input-output trajectory follows the basic shape functions, and 𝑡+ and 𝑡− work as media between the input t and output 𝑉𝑐 and provide a strategy to realize the memory mechanism through calculation of t0. In this hysteresis phenomenon and Eq. (1), the different time labels 𝑡+ ∈ [0,+∞) and 𝑡− ∈ (−∞,0] for forward and backward processes are creatively used, and based on the absolute continuity of hysteresis output and the media of 𝑡+ and 𝑡− , the pattern from V0 to t0 reveals a mechanism to update the initial information as long as new monotonic excitation occurs. Moreover, as the solutions of a simple semi-linear Duhem model, instead of the differential expression, Eq. (1) provides an alternative to formulate the hysteresis loops. Furthermore, with the normalization concept and generality, Eqs. (1a) and (1b) can be rewritten as 𝑉̅𝑐+ (𝑡+ ) = 1 − 2𝑒 −𝑔1 (𝑡+)⁄𝑅𝐶 𝑉̅𝑐− (𝑡− ) = −1 + 2𝑒
𝑔2 (𝑡− )⁄𝑅𝐶
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(2a) (2b)
Switch
SA
SB
R
VV
VC C GND
Figure 1. Schematic of the RC circuit. Time (s) 0 -* +00
,,,-,,, Charge
V
v
- - - - Discharge Case A
>
vo
0V Time (s) -co
I
to
0
Figure 2. Hysteresis phenomenon in charging and discharging processes of RC circuit. where 𝑔1 (𝑡+ ) and 𝑔2 (𝑡− ) are monotonically increasing functions in terms of 𝑡+ ∈ [0,+∞) or 𝑡− ∈ (−∞,0], and satisfy g1(0) = g2(0) = 0. Through that, Eq. (1) provides more flexibility to model the hysteresis shape. Except the initial points of 𝑉̅𝑐+ = −1 and 𝑉̅𝑐− = 1, the normalized potential difference 𝑉̅𝑐 varies in the range of (-1,1), hence totally 𝑉̅𝑐 ∈ [−1,1]. 2.2 Principle of “shape function + memory mechanism”-based hysteresis model Inspired from the Duhem hysteresis characteristics in MR actuators, the memory mechanism in RC circuit and shape function of Eq. (2), a hysteresis model with “shape function + memory mechanism” strategy is expressed as 𝑧(𝑡) = 1 − 2𝑒 −𝑔1(𝑆)⁄𝑎 , when 𝑥̇ (𝑡) > 0
(3a)
𝑧(𝑡) = −1 + 2𝑒 𝑔2(𝑆)⁄𝑎 , when 𝑥̇ (𝑡) < 0
(3b)
𝑧̇ (𝑡) = 0, when 𝑥̇ (𝑡) = 0
(3c)
𝑆0 = 𝑔1 −1 (−ln ((1 − 𝑧(𝑡 ∗ ))/2) 𝑎) , 𝑥0 = 𝑥(𝑡 ∗ ), when change to forward 𝑆0 = 𝑔2 −1 (ln((1 + 𝑧(𝑡 ∗ ))⁄2)𝑎), 𝑥0 = 𝑥(𝑡 ∗ ), when change to backward 𝑆(𝑡) = 𝑆0 + 𝑥(𝑡) − 𝑥0
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(3d) (3e) (3f)
𝑧(0) = 𝑧0 ∈ (−1,1)
(3g)
where x and z are the general displacement input and force output, respectively; S is a virtual displacement variable, and S0 and x0 are the reference points of the virtual displacement and the displacement when a new monotonic excitation occurs at time 𝑡 ∗ (which means the direction changes at 𝑡 ∗ ); a is a positive hysteresis factor, and g1/g2 defined in Eq. (2) are employed here to fit the hysteresis loops; although not mathematically rigorous, it is practically reasonable to assume that the initial condition z0 is defined in (-1, 1) when considering the limited physical capacity of hysteresis systems.
2.3 Hysteresis model for MR actuators The macroscopic behavior of MR actuators is a coupling between hysteresis and viscous/elastic property. Hence in the modeling process, the basic mechanical elements, such as damping, spring, Coulomb and hysteresis, would be used and coupled to reveal the mechanism. Here, for MR actuators, “Restructured phenomenological model” shown in Figure 3 is used, which had shown high performance in hysteresis modeling of MR dampers20. The damping force F can be expressed as (4a)
𝐹 = 𝑐0 𝑥̇ + 𝛼𝑧 + 𝑘1 𝑥 + 𝑓0 𝑦̇ =
𝛼𝑧
(4b)
𝑐1
where F is the output damping force; c0 and k1 are the damping coefficient and stiffness coefficients, respectively; f0 represents the offset elastic force due to initial displacement; z is the hysteresis output as in Eq. (3), and 𝛼 is its coefficient. To fit the phenomenon of roll-off, a viscous element c0 is serially connected with hysteresis element. It should be noted that in the structure of “Restructured phenomenological model” the real input to the hysteresis element is (𝑥̇ − 𝑦̇ ), and for the generality, the monotonic functions g1 and g2 are redefined by a power law functions as 𝑔1 (𝑆) = |𝑆|𝑏
(5a)
𝑔2 (𝑆) = −|𝑆|𝑏
(5b)
where b is a positive factor to tuning the hysteresis shape. Moreover, the hysteresis behavior in MR actuators is currentdependent (or field-dependent), and as stated in the reference, the coefficients c0, c1 and 𝛼 are current-related, while the others are constants. Hence, in the new hysteresis operator based model, there are seven parameters to be identified ([𝑐0 , 𝑐1 , 𝑘1 , 𝑓0 , 𝛼, 𝑎, 𝑏]) under constant currents. While in the Bouc-Wen operator based model, the hysteresis output is given by 𝑧̇ = 𝜌(−𝜎|𝑥̇ − 𝑦̇ |z|𝑧|𝑛−1 − (1 − 𝜎)(𝑥̇ − 𝑦̇ )|𝑧|𝑛 + (𝑥̇ − 𝑦̇ ))
(6)
where 𝜌, 𝜎 and n are the hysteresis factors of Bouc-Wen operator based model, and in this case, there are eight parameters to be identified ([𝑐0 , 𝑐1 , 𝑘1 , 𝑓0 , 𝛼, 𝜌, 𝜎, 𝑛]).
y
x
Hysteresis Operator c1
c0
F
k1 Figure 3. Structure of the “Restructured phenomenological model”.
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3. VALIDATION OF THE “SHAPE FUNCTION + MEMORY MECHANISM”-BASED HYSTERESIS MODEL To validate the performance of the “shape function + memory mechanism”-based hysteresis model for MR actuators, the experimental tests of the MR damper is adopted for the parameters identification. As concerning the same hysteresis mechanism under different excitation, four sets of experimental tests are used: 2.0 Hz sinusoidal displacement excitation with amplitudes of 10 mm and 15 mm, and applied currents of 0.5 A and 1.0 A. The parameters c0, c1 and 𝛼 are currentdependent for MR actuators, so as presented in reference19, a current-related function is needed. For simplicity, a specific value would be optimized under a certain current, and the others would be constants in any case. For the optimization, genetic algorithm20 is used for the parameters identification to minimize the mean square error (MSE) between the simulated Fsim.i and experimental damping force Fexp.i, expressed as MSE =
1 𝑚
2
∑𝑚 𝑖=1(𝐹sim.𝑖 − 𝐹exp.𝑖 )
(7)
where m represents the number of all experimental data. IJVV
1500
2.0 A
1000
1000
. T,
1.0 A
Z o
500
500
Experimental Test New Hysteresis Model
o D.)
c E
0
0
- - - Bouc -Wen Model -500
-500
-1000
-1000
-1500
-1500
-10
-5
0
5
10
-15
Displacement (mm)
-10
-5 0 5 Displacement (mm)
10
15
(a) (b) Figure 4. Identification results for the MR damper under 2.0 Hz sinusoidal displacement excitations with different amplitudes: (a) 10 mm and (b) 15 mm.
Figure 5. The computational cost (Com cost) of the new hysteresis model and Bouc-Wen model through MSE.
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The identification results are presented in Figures 4(a) and 4(b). Apparently, within the structure of the “Restructured phenomenological model”, both the new hysteresis model and Bouc-Wen operator based model can well describe the hysteresis behavior of MR damper regardless of the applied currents or amplitudes. However, in the initial region after the direction-change point, the identification results show that neither of these two models can fully capture the variation tendency. It should be noted that an improved performance is possible using the presented model with more suitable functions g1 and g2, while Bouc-Wen operator based model does not have the option. As shown in Figure 5, the MSEs for different excitations are presented and compared between the two models. Although there are more hysteresis factors in Bouc-Wen operator based model, a comparable performance is achieved by the new hysteresis model. Moreover, in the profile of the computational cost that is defined by the consumed time for the parameters identification, the new hysteresis model shows a remarkably improved computational efficiency. Because instead of the differential expression, the algebraic form of the new hysteresis model would largely reduce the computational burden.
4. CONCLUSION With the theory of specific Duhem hysteresis model (such as Bouc-Wen, Dahl and LuGre models) and a hysteresis phenomenon in RC circuit, a hysteresis model was presented and verified with the “shape function + memory mechanism” strategy. In detail, a memory mechanism originating from the charging and discharging processes of the RC circuit was constructed by adopting a virtual displacement variable and updating laws for the reference points, and a generalized analytical solution of simple semi-linear Duhem model was adopted for the shape function. Due to the coupling between the hysteresis and viscous/elastic in macroscopic behavior of MR actuators, the “Restructured phenomenological model” is employed with the new hysteresis operator, and the performance of new hysteresis model for MR actuators is evaluated and compared with that of Bouc-Wen operator based model. It turns out that a comparable performance with Bouc-Wen operator based model could be achieved in term of the description accuracy, and the computational efficiency to identify the hysteresis of the new model is much better than Bouc-Wen operator based model.
ACKNOWLEDGEMENTS The authors wish to acknowledge the Equipment Pre-Research Foundation of China during the 13th Five-Year Plan Period (Grant No. 6140240040101), Key Research and Development Projects of Anhui Province (Grant No. 1704E1002211) and Fundamental Research Funds for the Central Universities (Grant No. JZ2017HGTB0202), for their support of this research.
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