Research on MRD Parametric Model Based on Magic Formula

In order to get a better description to the nonlinear characteristics of magnetorheological dampers, the magic formula is introduced into the general method of parametric modelling of magnetorheological dampers to propose a new parameterizedmodel calledmagic formula-hysteresis loopmodel (MFM).,e newmodel is simple in structure, the physical meaning of each parameter is clear, and the parameter identification is convenient. ,e fitting and experimental data of MFM and the phenomenon model under different conditions are applied for error analysis and comparison. ,e results show that the errors of MFM are more accurate and have better fitting and experimental data under different working conditions, which also have better adaptability and versatility.


Introduction
Magnetorheological Fluid (MRF) is a new intelligent material composed of nonmagnetic liquid and tiny soft magnetic particles.e particles are featured by high magnetic flux and low hysteresis [1]. is material exhibits low viscosity Newtonian fluid properties at zero magnetic field, but under high magnetic field, it shows high viscosity and low fluidity Bingham fluid characteristics.Under applied magnetic field, MR fluid's rheological properties such as viscoelasticity, yield strength, and so on can change rapidly (milliseconds) and reversibly [2].Magnetorheological damper (MRD), which takes magnetorheological fluid as the actuating medium, has many advantages such as high yield stress, high temperature adaptability, short response time, strong stability, low requirement voltage, and convenient semiactive control, etc [3].It has drawn more and more attention in the fields of automobile [4], architecture [5], aviation [6], and other fields that require vibration control.
By adjusting the damper input current, MRD could change the strength of the magnetic field to control the size of the damping force.erefore, MRD is often applied in vibration semiactive control.
e premise of accurate semiactive control is to establish accurate and reliable MRD model.But the output damping force of MRD, which is influenced by the coupling of external excitation, applied magnetic field, and internal flow field, shows a strong nonlinear character.
is results in difficulties of precise MRD modelling.In order to accurately describe the properties of MRD, many MRD models have been established, which are divided into two types: physical model and phenomenological model [7].Among them, the physical model refers to the use of magnetorheological fluid constitutive model and damper geometry to do the modelling and determine the damping force [8].Because there are still many assumptions about the mechanism of magnetorheological fluid, the established physical model is not accurate enough.erefore, the physical model can often be used to guide the damper design but seldom be applied to the control.Phenomenological model is divided into nonparametric model and parametric model.In the nonparametric model, each parameter has no explicit physical meaning and is only used to identify the hysteresis loop that controls damping force, such as fuzzy logic model [9], neural network model [10], and so on.Besides, the parameters in parametric models have comparatively clear physical meaning.
e commonly used parametric models are Bingham model [11], nonlinear double-viscosity model [12], Dahl model [13], Bouc-Wen model [14], and phenomenon model (PM) etc [15].Among them, PM, which is an improved version based on the Bouc-Wen Model, is a parameterized model that conforms well to the experimental phenomena.It can accurately describe the MRD low-velocity attenuation, yield characteristic, and hysteresis characteristic, which owns higher precision and incentive adaptability, that is, the model parameters identi ed by a certain excitation can be applied to a wider range of excitation.erefore, PM is the most widely used MRD parameter model, and its model expression is listed in the following equation: However, PM is composed of two di erential equations and some absolute and power function terms, which is di cult to establish and to solve by general numerical method [16].
e model needs to identify up to 12 parameters, of which the existing intermediate variable has ambiguous physical meaning; the relationship between other parameters and external incentives is not clear, so it is di cult to determine the initial value or a xed range of values, and repeated trial and error and iterative calculations to complete the model parameter identi cation [17][18][19] are needed.All of these have brought great di culties to the establishment and use of MRD model.erefore, in order to describe the dynamic characteristics of MR uid damper better, it is necessary to establish an MRD parameterization model which is convenient for modelling and solving and easy to identify parameters with high accuracy and strong universality.
e magic formula [20] can get a good approximation to the nonlinear curve through the combination of several trigonometric functions, therefore, is usually used to establish tire mechanics model.e general form of the magic formula is shown in the following equation: Among them, B, C, D, and E are parameters which control the curve shape; α is independent variable, and based on the actual situation it can select speed, displacement, de ection angle, and so on.
e magic formula can well describe the mechanical characteristics of the tire within the scope of the experimental data.At the same time, the mechanical properties of the tire can still be accurately expressed in a certain range outside the experimental data.
at is, the magic formula can predict from the limited working conditions and ensure a high degree of con dence.At the same time, we can see that, as shown in Figure 1, the curve of tire force-de ection angle is very similar to the plastic uid curve of MR damper; however, the plastic uid curve cannot describe the hysteresis characteristics of the magnetorheological damper.Considering that the tire mechanics model has similar nonlinear characteristics with the MRD, the Magic Formula is introduced to the MRD modelling process, and a magic formula-hysteresis loop model (MFM) is established.e new model is of simple structure, de nite physical parameters and convenient parameters identi cation.According to the tting with the experimental data, MFM is compared with PM and error analysis is done.e results prove the accuracy and universality of the new model, which also provide a reference for the application of MRD in vibration control.

Magnetorheological Damper Experiment
e general structure of MRD is shown in Figure 2. e magnetorheological uid is lled in the cavity of MRD. e piston has some xed damper channels inside, and the coil is wound inside the piston.When an external force acts axially on the piston rod, the piston moves within the MRD cavity, and the magnetorheological uid ows along the xed damping channels.When the coil is energized, a magnetic eld can be generated, and the properties of the magnetorheological uid in the damping channels change as the magnetic eld changes.e gas compensation chamber at the end of the damper can provide an appropriate compensation force.
An Instron8802 performance testing machine (Figure 3) is used for a type of MRD data test.e main structure parameters of MRD are as follows: the maximum input current of the damper is 3 A, the maximum displacement of the piston is 0.055 m, the length of the damper cylinder is 0.212 m, the outer cylinder diameter is 0.042 m, piston rod's diameter is 0.01 m, and the weight is 2.3 kg. e sinusoidal signal is used as the excitation to do the test.By changing the amplitude of the sinusoidal signal Amp (0.005 m, 0.01 m, 0.015 m, and 0.02 m), the frequency f (0.5 Hz, 1 Hz, 1.5 Hz, and 2 Hz) and the control current I (0 A, 0.5 A, 1 A, 1.5 A, 2 A, 2.5 A, and 3 A), the MRD damping force data under di erent working conditions were obtained.ere are too many working conditions, therefore, the working condition Amp 0.01 m, f 1 Hz in di erent currents is selected, and the experimental data are collected as shown in Figure 4.It 2 Shock and Vibration can be seen from Figure 3 that MRD shows obvious yield characteristic and hysteresis characteristic during its working process.

The Establishment of the Magic Formula Model
By studying the existing MRD parametric model (such as Bouc-Wen Model), the basic principle of parametric modelling is shown in Figure 5. e MRD moves at a speed of _ x under the action of external force F at the moment the damping force generated by the damper mainly consisting of three parts: hysteresis element, damping element, and sti ness element.e hysteresis element mainly describes nonlinear characteristics such as yield characteristics and hysteresis characteristics; the damping element describes the viscous damping after yielding; the sti ness element describes the sti ness characteristics of the damper.Part of the   In order to describe the hysteresis characteristic, an absolute value term is introduced, and a complete MRD parametric model is established based on the principle of MRD parametric model modelling to generate the damping force F MFM as is shown in the following equation: In Equation ( 3), x indicates the displacement of damper piston, _ x indicates the speed of damper piston.e other parameters' de nition and the range of their values are in Table 1.

Parameter Identification
For the convenience of calculation, Amp 0.01 m, f 1 Hz is selected as the reference condition for tting.Curve tting was performed by a combination of genetic algorithm and nonlinear least square method.
e tting steps are as follows.Firstly, the range of each parameter obtained from the model establishment process is optimized by genetic algorithm, and approximated global optimum can be obtained quickly by genetic algorithm in this interval, and then the approximate global optimal is taken as the nonlinear least square's initial value of the input and the accurate global optimum of each parameter can be obtained.e data of tting are as listed in Table 2.
According to the data in Table 1, to identify each parameter: (1) f 0 represents the bias, as a xed value, taking the average of the current, we can obtain (2) C, D, E, k c , and k in this condition can all be tted as linear function of the current I, as shown in Figure 6.
e tting results are shown in the following equations: When the magnitude of the external excitation frequency changes, the values of C, D, E, k c , k show only minor uctuations.It can be considered that these parameters do not change with the amplitude of the external excitation frequency.
(3) A, B in this condition can be t into a quadratic function of the current I, shown in Figure 7.
By changing the amplitude or frequency, the relationship of A and amplitude is indicated in Figure 8(a), the relationship of A and frequency is indicated in Figure 8(b).
By changing the amplitude or frequency, the relationship of B and amplitude is indicated in Figure 9(a), and the relationship of B and frequency is indicated in Figure 9(b).
erefore, the nal tted A, B is as follows: A −0.205I Take Equations ( 4)-( 11) of the recognized parameters in Equation ( 3) to get a complete model of the MRD magic formula.Numerical simulation is taken when the complete MFM in MATLAB is under di erent conditions for di erent current, and it is compared with the experimental data.Because there are so many di erent working conditions, only a few of them are selected, as shown in Figure 10.
It can be seen from Figure 9 that the parameters of MFM under di erent conditions are in good agreement with the experimental data.

Shock and Vibration
PM. Taking into account the di erential equation in PM is more di cult to directly model, therefore, it is chosen to establish the PM in MATLAB/Simulink, as shown in Figure 11.
From previous studies, the parametersc 0 , c 1 , α, and A can be expressed as a polynomial function of the current I in PM, and other parameters are constants.e genetic algorithm is used to further identify the parameters in PM based on experimental data.Taking into account the unknown value range of each parameter of PM, the method of continuously narrowing the parameter value interval gradually improves the recognition accuracy of genetic algorithm.e speci c method is as follows: set the value range of each parameter as [−1 × 10 8 , 1 × 10 8 ], and identify each parameter by genetic algorithm, respectively, under basic condition (Amp 0.01 m, f 1 Hz) and di erent currents (0∼3 A). e minimum value of each parameter obtained by di erent current identi cation is used as the lower limit, and the maximum value is taken as the upper limit, which is reset as the parameter value range and identi ed by genetic algorithm.After narrowing the parameter interval several times, the nal parameter identi cation result can be obtained (Table 3).
Taking into account many more working conditions, the basic condition Amp 0.01 m, f 1 Hz is selected to compare their own tting accuracy; select the condition Amp 0.02 m, f 2 Hz, which is far away from the basic condition, to do the damping force prediction accuracy 5.2.Self-Fitting Accuracy Comparison.MFM and PM are identi ed under the basic condition Amp 0.01 m, f 1 Hz, and the average error for di erent currents under this condition is listed in Table 4.When I 3 A, the di erence in accuracy is the greatest, as shown in Figure 12.
It can be seen from the error data and Figure 12 that under the condition Amp 0.01 m, f 1 Hz, the average error of each equation of MFM is less than 2.5% under di erent currents, and the average error of MFM under each current is less than the average error of PM at each current, that is, the tting equation of MFM to the experimental data is higher.Shock and Vibration 7 listed in Table 5.When I 1 A, the di erence in accuracy is the greatest, as shown in Figure 13.

Prediction Accuracy
As can be seen from the error data and Figure 13, under the condition Amp 0.02 m, f 2 Hz, the average error of the equation formula of MFM at each current is increased compared with the condition Amp 0.01 m, f 1 Hz, and the largest error is only 2.6%, which is still able to meet the accuracy requirements.At this time, the average error of MFM at each current is still less than the average error of PM at each current, indicating that the prediction accuracy of MFM is higher than that of PM.
Based on the above error analysis, it can be seen that compared with PM, MFM has higher accuracy; when the amplitude of the external excitation frequency changes, the magnitude and standardization tendency of the damping force can still be predicted accurately, and the prediction accuracy is higher than the phenomenon model with better applicability.

Conclusions
In the modelling process of MRD, MFM is established after magic formula is introduced.Based on the experimental data of MRD, the parameters of the new model are identi ed, and the accuracy and the error analysis of the tted MFM are compared with the commonly used PM.Under the basic condition Amp 0.01 m, f 1 Hz, MFM is more accurate than PM at di erent currents, and the maximum precision error of MFM is only 2.5%; under the condition Amp 0.01 m, f 1 Hz, the tting accuracy of MFM at di erent currents is also higher than that of PM, and the maximum accuracy error of MFM is only 2.6%.e results show that MFM is simple in form and does not include  Data Availability e data used to support the ndings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no con icts of interest regarding the publication of this paper.

Figure 4 :
Figure 4: Experimental date of Amp 0.01 m, f 1 Hz under di erent currents.(a) Piston speed at di erent currents-damping force curve.(b) Piston displacement at di erent currents-damping force curve.

5. 1 .
PM Parameter Identi cation.To further investigate the accuracy and applicability of MFM, MFM is compared with

Figure 6 :
Figure 6: Parameter C, D, E, k c , and k and current t graph.e points represent the experimental values, and the line represents the tted value.(a) Parameter C and current t graph.(b) Parameter D and current t graph.(c) Parameter E and current t graph.(d) Parameter k c and current t graph.(e) Parameter k and current t graph.

Figure 7 :
Figure 7: Parameter A, B and current t graph.e points represent the experimental values, and the line represents the tted value.(a) Parameter A and current t graph.(b) Parameter B and current t graph.

Figure 8 :
Figure 8: e tting map of Parameter A and amplitude and frequency.e points represent the experimental values, and the line represents the tted value.(a) Parameter A and amplitude t graph.(b) Parameter A and frequency t graph.

Figure 9 :Figure 10 :
Figure 9: e tting map of Parameter B and amplitude and frequency.e points represent the experimental values, and the line represents the tted value.(a) Parameter B and amplitude t graph.(b) Parameter B and frequency t graph.

Figure 12 :
Figure 12: e comparison between MFM, PM, and experimental data when Amp 0.01 m, f 1 Hz, and I 3 A. (a) Piston speed-damping force curve.(b) Piston displacement-damping force curve.

Figure 13 :
Figure 13: e comparison between MFM, PM, and experimental data when Amp 0.02 m, f 2 Hz, and I 1 A (a) Piston speed-damping force curve.(b) Piston displacement-damping force curve.

Table 1 :
Parameters' de nition and the range of their values.

Table 2 :
e tting data of Amp 0.01 m, f 1 Hz in di erent current conditions.

Table 4 :
e average current error when Amp 0.01 m, f 1 Hz.physical meaning and range of values of each parameter are clear, and parameter identi cation and simulation calculations can be performed quickly and easily.e simulation results under di erent conditions are in good agreement with the experimental data.With high precision and good adaptability, it has great potential in the eld of MRD control. e

Table 5 :
e average current error when Amp 0.02 m, f 2 Hz.