A method for establishing a new parametric model to describe the hysteretic behavior of a magnetorheological damper

Abstract

The present invention relates to the field of intelligent materials, and particularly to a method for establishing a new parameter model for describing the hysteretic behavior of magnetorheological fluid in a magnetorheological damper. The proposed parameter model is obtained based on the hysteretic response of the magnetorheological fluid. First, according to the LAOS experiment, the hysteresis curve of the magnetorheological fluid is obtained, and multiple sets of data are obtained by changing the amplitude, frequency, and current. The genetic algorithm is used to determine the model parameters, and the root mean square error is used as the evaluation criterion, and the modeling results are compared with the Bouc-Wen model and the Dahl model. The results show that the proposed parameter model can accurately predict the hysteresis phenomenon of the magnetorheological fluid and has a good effect on capturing the strain softening / hardening characteristics in the hysteresis curve. Finally, the parameter identification is realized by fitting the parameter values under different currents. The entire process from proposal to verification in the model construction is completed, forming a complete set of methodologies. The parameter model proposed by the present invention will also provide theoretical guidance for the semi-active control of magnetorheological dampers in practical applications.

Description

Technical Field

[0001] The present invention relates to the fields of intelligent materials and automotive parts, and in particular to a method for establishing a new parameter model for describing the hysteresis behavior of a magnetorheological damper. Background Art

[0002] As a new type of intelligent material, magnetorheological fluid is a new type of fluid with controllable fluidity and is an active branch in the research of intelligent materials. It exhibits the characteristics of a Newtonian fluid with low viscosity in the absence of an external magnetic field. When an external magnetic field is applied, it presents as a Bingham fluid with high viscosity and low fluidity. There is a corresponding relationship between the viscosity of the liquid and the magnetic flux. This conversion has low energy consumption, is easy to control, and has a rapid response. This provides very favorable conditions for the engineering application of magnetorheological fluid in dampers.

[0003] However, under different conditions of current, amplitude, and frequency, the hysteretic response of magnetorheological fluid will exhibit significant nonlinearity, and under certain conditions, it will have the characteristics of strain hardening and strain softening. Although magnetorheological fluid can play a role in the engineering application of dampers, it is necessary to model the dynamic nonlinear hysteretic behavior of magnetorheological fluid before actual engineering applications to guide production. Therefore, to fully exert the engineering application value of magnetorheological fluid and simplify the corresponding controller, a simple, reliable, and effective mechanical property model of magnetorheological fluid should be developed first.

[0004] The traditional mechanical property models of magnetorheological fluid can be divided into parameter models and non-parameter models. For the construction of non-parameter models, most of them adopt artificial intelligence algorithms for solution at the present stage. Yu et al. proposed a non-parameter model based on artificial neural network to describe the force-displacement hysteretic response in an MRE isolator. For parameter models, the parameter model proposed by Bouc in 1971 and improved by Wen in 1976 is the most classical model, and the model describes the strain hardening characteristics and the nonlinear relationship between force and displacement in the form of a differential equation. Spencer et al. proposed an improved Bouc-Wen model to describe the hysteretic response in a wide range of inputs. Kwok et al. proposed an asymmetric Bouc-Wen model to thoroughly capture the asymmetric phenomenon of the hysteretic response at the zero velocity position. Dahl proposed the Dahl model using a differential equation to describe the hysteresis curve with friction effects.

[0005] In the parametric model, for the parameter identification process, it is necessary to select an appropriate algorithm. Kwok et al. used the genetic algorithm to identify the parameters of the asymmetric Bouc-Wen model. The results show that the parameters obtained by the genetic algorithm can well describe the hysteretic characteristics of the magnetorheological damper. Boukhtache et al. used the simulated annealing algorithm to determine the parameters of the J-A hysteresis model of silicon iron sheets, and the prediction results are in good agreement with the experimental data. Yu et al. used the particle swarm optimization algorithm for parameter identification, and the results are in good agreement.

[0006] The object of the present invention is to provide a method for establishing a new parametric model to describe the hysteretic behavior of a magnetorheological damper. By obtaining the hysteresis curves under different conditions of current, frequency, and amplitude, and decomposing the characteristics shown in the curves, a new parametric model is established, and the genetic algorithm is used to optimize the parameters, with the root mean square error as the evaluation criterion. Then, the relationship between the applied current and the parameters is studied and generalized to meet the needs of engineering applications. Summary of the Invention

[0007] Aiming at the defects in the existing modeling technology, the object of the present invention is to provide a method for establishing a new parametric model to describe the hysteretic behavior of a magnetorheological damper, providing a theoretical basis for the application of magnetorheological fluids in semi-active control.

[0008] In the new parametric model for describing the hysteretic behavior of a magnetorheological damper, since the magnetorheological fluid exhibits different strain hardening / softening characteristics under the influence of high and low frequencies in the presence of a magnetic field, for this hysteretic response behavior, in order to meet the requirements of semi-active control in the engineering application of magnetorheological fluids, a new model needs to be constructed to predict its complex hysteretic behavior. Therefore, the present invention improves the traditional parametric model to improve the model prediction accuracy, provides a theoretical guidance basis for the engineering application field of magnetorheological fluids, and will propose a new parametric model to describe the hysteretic behavior of magnetorheological fluids in the application of magnetorheological dampers, especially for capturing the possible strain hardening phenomenon at high frequencies in the presence of a magnetic field.

[0009] To achieve the above object, the technical solutions provided by the embodiments of the present invention are as follows:

[0010] A method for establishing a new parametric model to describe the hysteretic behavior of a magnetorheological damper, comprising the following steps:

[0011] First, the hysteresis curve of the magnetorheological fluid can be divided into four main characteristics, namely viscous, elastic, strain softening, and strain hardening characteristics. And they are respectively represented by expressions and differential operators with respect to the shear rate and shear strain. Its components are represented as a parallel combination of a spring, a buffer, and a differential operator. Through the above analysis, the expression of the new model is obtained:

[0012]

[0013] where τ, γ, and are the shear stress, shear strain, and shear rate, respectively; c and k represent the viscosity and elastic parameter, respectively; ε is the coefficient of the hysteresis operator; z is the hysteretic differential operator, which is described by the following expression:

[0014]

[0015] where is the time derivative of z; α, β, and δ are parameters that control the general shape and scale of the hysteresis curve. The intermediate variable z will be solved using the Euler method, and its expression is described in detail as follows:

[0016]

[0017] where Δt is the time elapsed between sampling two data points.

[0018] Therefore, the proposed model contains six parameters, as follows:

[0019] ∩ = [α, β, δ, ε, c, k]

[0020] After constructing the new parameter model, it is necessary to identify the parameters of the proposed model according to the stress-strain response of the magnetorheological fluid. The present invention uses the root mean square error (RMSE) between the test data and the model modeling results as the fitness function for parameter optimization, and its objective function is as follows:

[0021]

[0022] where ε is the parameter in the model; N is the number of test data in the hysteresis curve; is the experimental test value. The smaller the value of the fitness function obj(ε), the better the obtained parameter results. In this case, minfit(∩) is used as the objective function of the optimization problem. And the robustness of the genetic algorithm will be used to solve this optimization problem, and its steps are as follows:

[0023] 1) Read data: shear stress, shear strain, and shear rate;

[0024] 2) Define the problem and genetic algorithm parameters: Determine the fitness function and set the number of chromosomes N = 50, the crossover rate pc = 0.75, the mutation probability pm = 0.02, and the finite number of iterations X = 500;

[0025] 3) Determine the model parameter ∩ and initialize the model parameter value ∩ = [α, β, δ, ε, c, k];

[0026] 4) Initialize a set of chromosomes for each model parameter;

[0027] 5) Evaluate each chromosome using the fitness function value and compare it with the previous one.

[0028] 6) Sort the chromosomes according to their fitness and select a new population using the roulette wheel method.

[0029] 7) Update the chromosomes through crossover and mutation operations.

[0030] 8) Evaluate the new chromosomes and check if the stopping criteria are met. If the stopping condition is satisfied or the number of iterations is higher than the limited number of iterations, report the best chromosome as the solution and output the optimal model parameters; otherwise, return to step 5).

[0031] After parameter identification, the rationality and accuracy of the new model need to be verified. In the present invention, the model prediction data is compared with the experimental test data, and at the same time, the parameter model is compared with two classical parameter models, namely the Bouc-Wen model and the Dahl model, to verify the accuracy and rationality of the proposed model.

[0032] The influence of the six parameters on the hysteresis curve will be obtained by testing by adjusting the parameter sizes (using equally spaced sampling points). Among them, the larger the parameters α and ε are, the larger the area enclosed by their hysteresis curves is, and these two parameters will also affect the range of strain softening behavior. The lower the parameter β is, the larger the obtained stress peak will be, which indicates that the equivalent stiffness decreases with the increase of the parameter value. For the parameter δ, it can be seen that both its equivalent stiffness and the area of the closed hysteresis loop decrease with the increase of the parameter. For the parameter c, the larger its value is, the larger the area of the obtained hysteresis curve is. Especially when the parameter c = -1000, strain hardening will occur; and higher equivalent stiffness can be obtained by increasing the value of k; and all hysteresis curves intersect at two positions where the shear strain is 0. On both sides of these positions, the stiffness and the value of k are different: in the strain ranges from 0 to 10% and 0 to -10%, the stiffness in the clockwise path increases with the increase of k. However, in the strain ranges from -10% to 0 and 10% to 0, the stiffness decreases with the decrease of k.

[0033] After analyzing the influence of each parameter on the hysteresis loop, parameter generalization needs to be carried out for the parameters, that is, to propose the relationship between the model parameters and the applied current. Under the conditions of the same strain amplitude and frequency, the parameter values under different current inputs are respectively tested, and it is obtained that the parameters α and δ decrease monotonically with the increase of the applied current, the parameters ε and k increase monotonically with the increase of the applied current, and the parameters β and c have a quadratic relationship with the current.

[0034] Under the conditions of a strain amplitude of 10% and a frequency of 0.1 Hz, the parameter values were tested under three different current inputs of 0 A, 1 A, and 2 A. It can be observed that the change trends of each parameter when the applied current ranges from 0 A to 2 A are as follows:

[0035] α = -94.5563I + 10.9272

[0036] β = -141.5086I 2 + 334.5158I + 29.9828

[0037] δ = -3613.8599I - 583.9001

[0038] ε = 30.7102I - 0.517

[0039] c = 4140.1628I 2 -4242.4883I + 82.3255

[0040] k = 17926.9296I - 1605.5493

[0041] By adopting the above technologies, compared with the prior art, the present invention has the following beneficial effects:

[0042] 1) The proposed parameter model has a high degree of agreement with the magnetorheological hysteresis effect: By comparing the parameter model with the experimental values and two traditional models, the RMSE index of the proposed parameter model is smaller and has better accuracy.

[0043] 2) The proposed parameter model can predict the magnetorheological hysteresis effect: Through the technical means of parameter generalization, it can be known that the constructed parameter generalization model can better capture the hysteresis phenomenon of magnetorheological fluids.

[0044] 3) Compared with the traditional parameter model, the proposed parameter model has fewer parameters and higher accuracy, especially under the working conditions of strain hardening at high current and high frequency.

[0045] 4) The invention principle of this parameter model is simple and the calculation is convenient, and it can well capture the hysteresis phenomenon of magnetorheological fluids. Therefore, the parameter model can provide a theoretical basis for the semi-active control of magnetorheological fluids in mechanical applications.

[0046] 5) The method for establishing the proposed parameter model forms a complete set of methodologies from establishment to verification and can be used to guide theoretical learning. Description of the Drawings

[0047] Figure 1 Schematic diagram of the hysteresis response curve of magnetorheological fluid;

[0048] Figure 2 Schematic diagram of the decomposition of the hysteretic response curve behavior

[0049] Figure 3 Schematic diagram of the representation of the new model components

[0050] Figure 4 Schematic diagram of the comparison between the model calculated values and the test values: the currents from top to bottom are 0A, 1A, 2A, and the frequencies from left to right are 0.1Hz, 5Hz, 15Hz

[0051] Figure 5 Schematic diagram of the comparison between the model calculated values and the test values when the frequency is fixed at 15Hz: the currents from top to bottom are 1A, 2A, and the models from left to right are the Bonc-Wen model, the Dahl model, and the proposed new model

[0052] Figure 6 Schematic diagram of the influence of different parameters on the hysteresis curve: the parameters from a to f are α, β, δ, ε, c, k in turn

[0053] Figure 7 Schematic diagram of the relationship between the applied current and the model parameters

[0054] Figure 8 Schematic diagram of the comparison between the measured data after parameter generalization and the model prediction results Detailed implementation manners

[0055] The present invention will be further described below in conjunction with the accompanying drawings and the detailed implementation manners

[0056] Figure 1 Schematic diagram of the hysteretic response curves of the used magnetorheological fluid at three current levels of 0A, 1A, 2A and three different frequencies of 0.1Hz, 5Hz, 15Hz under a 10% strain amplitude Figure 2 Schematic diagram of the decomposition of the hysteretic response curve behavior Figure 3 Schematic diagram of the representation of the new model components Figure 4 Schematic diagram of the comparison between the model calculated values and the test values: the currents from top to bottom are 0A, 1A, 2A, and the frequencies from left to right are 0.1Hz, 5Hz, 15Hz Figure 5 Schematic diagram of the comparison between the model calculated values and the test values when the frequency is fixed at 15Hz and the strain amplitude is fixed at 10%: the currents from top to bottom are 1A, 2A, and the models from left to right are the Bonc-Wen model, the Dahl model, and the proposed new model Figure 6 Schematic diagram of the influence of different parameters on the hysteresis curve: the parameters from a to f are α, β, δ, ε, c, k in turn Figure 7 Schematic diagram of fitting the relationship between the applied current and the model parameters using the least squares principle Figure 8Schematic diagram of the comparison between the measured data after parameter generalization and the model prediction results.

[0057] In this example, a new parameter model for describing the hysteretic behavior of a magnetorheological damper is proposed. The component composition of the parameter model can be expressed as the parallel connection of a spring, a buffer, and a hysteresis operator. The expression of the new model is described in detail as follows:

[0058]

[0059] where τ, γ, and are the shear stress, shear strain, and shear rate, respectively; c and k represent the viscosity and elastic parameters, respectively; ε is the coefficient of the hysteresis operator; z is the hysteretic differential operator, which is described by the following expression:

[0060]

[0061] where is the time derivative of z; α, β, and δ are parameters that control the general shape and scale of the hysteresis curve. To accurately capture the hysteretic response of the magnetorheological fluid through the proposed model, six parameters should be determined as follows:

[0062] ∩ = [α, β, δ, ε, c, k] (3)

[0063] The specific process of proposing the parameter model is as follows:

[0064] First, the dynamic hysteretic response of the magnetorheological fluid is measured by a commercial MRC302 rheometer of MRD 180. The hysteretic characteristics of the magnetorheological fluid are tested at 300k using a vibrating sample magnetometer (VSM). The magnetorheological fluid is subjected to strain harmonic loading under the conditions of a strain amplitude of 10%, current inputs of 0A, 1A, and 2A, and three different frequencies of 0.1Hz, 5Hz, and 15Hz, and a series of hysteretic responses of the magnetorheological fluid are obtained. Its hysteretic response curve is as Figure 1 shown. It can be observed that under the action of the magnetic field, the magnetorheological fluid exhibits obvious hysteretic characteristics in the stress-strain curve. However, in the absence of a magnetic field, the hysteresis loop of the magnetorheological fluid is almost elliptical, indicating that only viscous characteristics appear under this condition. In addition, under the action of a magnetic field, only strain softening occurs at a frequency of 0.1Hz (low frequency). However, after further increasing the frequency to 5Hz and 15Hz (high frequency), both strain hardening and strain softening occur. Therefore, the hysteretic response of the magnetorheological fluid is very complex. To meet the requirements of semi-active control in the engineering application of the magnetorheological fluid, a new model needs to be constructed to predict its complex hysteretic behavior.

[0065] According to Figure 1The hysteresis characteristics described in can divide the hysteresis curve of magnetorheological fluid into four main characteristics, as Figure 2 shown, namely viscosity, elasticity, strain softening and strain hardening. Therefore, the stress-strain hysteresis curve can be decomposed into four components: The first two characteristics: viscosity and elasticity, can be expressed by expressions relative to shear rate and shear strain respectively, while the other two components can be represented by operators. The component representation method is as Figure 3 shown. In this case, viscosity and elasticity can be represented as the parallel connection of a spring and a buffer. The phenomena of strain softening and strain hardening can be simulated simultaneously with respect to strain and rate through differential operators. Therefore, the proposed parameter model is a parallel combination of a spring, a buffer and a differential operator, and the expression of the new model is described in detail as follows:

[0066]

[0067] where τ, γ and are shear stress, shear strain and shear rate respectively; c and k represent viscosity and elasticity parameters respectively; ε is the coefficient of the hysteresis operator; z is the hysteresis differential operator, which is described by the following expression:

[0068]

[0069] where is the time derivative of z; α, β and δ are parameters that control the general shape and scale of the hysteresis curve. In order to accurately capture the hysteresis response of magnetorheological fluid through the proposed model, six parameters should be determined as follows:

[0070] ∩=[α,β,δ,ε,c,k] (3)

[0071] After constructing the new parameter model, it is necessary to identify the parameters of the proposed model according to the stress-strain response of magnetorheological fluid. Since the hysteresis response is significantly non-linear, it is difficult to identify this parameter through mechanical tests. Therefore, a minimization optimization method needs to be adopted to achieve optimal parameter identification. During the process of parameter identification, a suitable fitness function needs to be constructed. The present invention uses the root mean square error (RMSE) between the test data and the model modeling result as the fitness function for parameter optimization, and its objective function is as follows:

[0072]

[0073] where ε is the parameter in the model; N is the number of test data in the hysteresis curve; These are experimental test values. The smaller the value of the fitness function obj(ε), the better the resulting parameter values. In this case, minfit(∩) is used as the objective function for the optimization problem. Additionally, in the present invention, the Euler method is used to solve for the intermediate variable, z, in Equation (1). Its expression is described in detail as follows:

[0074]

[0075] where Δt is the time elapsed between sampling two data points.

[0076] Since the genetic algorithm has strong robustness, this patent will use the genetic algorithm method to solve this optimization problem, and the steps are as follows:

[0077] (1) Read data: shear stress, shear strain, and shear rate;

[0078] (2) Define the problem and genetic algorithm parameters: Determine the fitness function and set the number of chromosomes N = 50, the crossover rate pc = 0.75, the mutation probability pm = 0.02, and the finite number of iterations X = 500;

[0079] (3) Determine the model parameter ∩ and initialize the model parameter values ∩ = [α, β, δ, ε, c, k];

[0080] (4) Initialize a set of chromosomes for each model parameter;

[0081] (5) Evaluate each chromosome using the fitness function value and compare it with the previous one;

[0082] (6) Sort the chromosomes according to their fitness and select a new population using the roulette wheel method;

[0083] (7) Update the chromosomes through crossover and mutation operations;

[0084] (8) Evaluate the new chromosomes and check if the stopping criterion is met. If the stopping condition is satisfied or the number of iterations is higher than the limit number of iterations, report the best chromosome as the solution and output the optimal model parameters; otherwise, return to step (5).

[0085] After parameter identification, it is necessary to verify the rationality and accuracy of the new model. As Figure 4As shown, the model prediction data is compared with the experimental test data. From the comparison results, it can be seen that the constructed parametric model can effectively capture the hysteretic behavior of the magnetorheological fluid, especially under the action of currents of 1 A and 2 A and high frequencies of 5 Hz and 15 Hz, when obvious strain softening and strain hardening phenomena occur in the hysteretic response. Secondly, in order to further evaluate the ability of the constructed parametric model to capture the hysteretic characteristics of the magnetorheological fluid, the parametric model is compared with two classical parametric models, the Bouc-Wen model and the Dahl model. As Figure 5 shown, the RMSE is used as the evaluation criterion, the strain amplitude is fixed at 10%, and the accuracy is compared and analyzed under the conditions of current inputs of 1 A and 2 A and frequencies of 0.1 Hz, 5 Hz, and 15 Hz. As Figure 5 shown, the parametric model has the highest accuracy among these three models. In particular, compared with the Bouc-Wen model and the Dahl model under high-frequency conditions with current present, the constructed model can accurately capture the strain softening and strain hardening characteristics. In addition, the proposed model can well describe the peak stress region, while there are obvious errors in the Bouc-Wen model and the Dahl model in this region.

[0086] In order to meet the requirements of actual production applications and give full play to the advantages of semi-active control of magnetorheological fluids, it is necessary to further study the influence of the parameter values in the parametric model on the model performance. As mentioned above, the constructed parametric model has six parameters, namely ∩ = [α, β, δ, ε, c, k]. To ensure consistency, all parameters are studied under the same test conditions (1 A - 0.1 Hz - 10%). As Figure 6 shown, the influence of the six parameters on the stress-strain hysteresis loop is depicted. Among them, the larger the parameters α and ε, the larger the area enclosed by the hysteresis curve. And the hysteresis loop is more sensitive to the parameter α, and these two parameters will also affect the range of strain softening behavior. While the lower the parameter β, the larger the obtained stress peak, which indicates that the equivalent stiffness decreases with the increase of the parameter value. For the parameter δ, it can be seen that both its equivalent stiffness and the area of the closed hysteresis loop decrease with the increase of the parameter. For the parameter c, the larger its value, the larger the area of the obtained hysteresis curve. In particular, strain hardening occurs when the parameter c = -1000. And higher equivalent stiffness can be obtained by increasing the value of k. Another phenomenon worth considering is that all the hysteresis curves intersect at two positions where the shear strain is 0. The results show that on both sides of these positions, the stiffness and the value of k are different: in the strain ranges from 0 to 10% and 0 to -10%, the stiffness in the clockwise path increases with the increase of k. However, in the strain ranges from -10% to 0 and 10% to 0, the stiffness decreases with the decrease of k.

[0087] After analyzing the influence of each parameter on the hysteresis loop, the present invention also performs parameter generalization. That is, the relationship between the model parameters and the applied current is proposed. As Figure 7 shown, under the conditions of a strain amplitude of 10% and a frequency of 0.1 Hz, the parameter values under three different current inputs of 0 A, 1 A, and 2 A are respectively tested. It can be observed that the parameters α and δ decrease monotonically with the increase of the applied current, while the parameters ε and k increase monotonically within the range of 0 A to 2 A of the current. Different from this, the parameters β and c have a quadratic relationship with the current. The variation trends of each parameter when the applied current ranges from 0 A to 2 A are as follows:

[0088] α = -94.5563I + 10.9272 (16)

[0089] β = -141.5086I 2 + 334.5158I + 29.9828 (17)

[0090] δ = -3613.8599I - 583.9001 (18)

[0091] ε = 30.7102I - 0.517 (19)

[0092] c = 4140.1628I 2 - 4242.4883I + 82.3255 (20)

[0093] k = 17926.9296I - 1605.5493 (21)

[0094] In order to verify the reliability of the model parameter generalization, the present invention uses the equation with parameter generalization to predict the test data, and tests its hysteresis behavior under the conditions of a strain amplitude of 10%, a frequency of 0.1 Hz, and an applied current of 0 A and 2 A. As Figure 8 shown: The constructed parameter generalization model can well capture the hysteresis phenomenon of the magnetorheological fluid. Therefore, the parameter model can provide a theoretical basis for the semi-active control of the magnetorheological fluid in mechanical applications.

Claims

1. A method for establishing a new parametric model to describe the hysteretic behavior of a magnetorheological damper, characterized in that, It includes the following steps: 1) Divide the hysteresis curve of the magnetorheological fluid into four main characteristics, namely viscosity, elasticity, strain softening and strain hardening characteristics; among which the first two characteristics: viscosity and elasticity, are respectively expressed by expressions relative to the shear rate and shear strain, while the other two components are represented by differential operators; so its components are expressed as follows: viscosity and elasticity are represented as the parallel connection of a spring and a buffer; the strain softening and strain hardening phenomena are simulated simultaneously with respect to strain and rate through differential operators; therefore, the proposed parameter model is a parallel combination of a spring, a buffer and a differential operator; 1.1) The expression of the parameter model is described in detail as follows: where τ, γ, and are the shear stress, shear strain, and shear rate, respectively; c and k represent the viscosity and elastic parameter, respectively; ε is the coefficient of the hysteresis operator; z is the hysteresis differential operator; 1.2) Its hysteresis differential operator is described by the following expression: wherein is the time derivative of z; α, β, and δ are parameters that control the general shape and scale of the hysteresis curve; 1.3) In order to accurately capture the hysteresis response of the magnetorheological fluid through the proposed model, six parameters should be determined as follows: ∩ = [α, β, δ, ε, c, k] 1.4) Use the Euler method to solve the intermediate variable z; its expression is described in detail as follows: where Δt is the time elapsed between sampling two data points; o(Δt 2 ) is a higher-order infinitesimal quantity; 2) Perform parameter identification on the proposed model according to the stress-strain response of the magnetorheological fluid; 2.1) Use the root mean square error RMSE between the test data and the model modeling result as the fitness function to optimize the parameters, and its objective function is as follows: where ε is a parameter in the model; N is the number of test data in the hysteresis curve; is the experimental test value; the smaller the value of the fitness function obj(ε), the better the obtained parameter results; minfit(∩) is used as the objective function of the optimization problem; 2.2) Use the robustness of the genetic algorithm to solve the optimization problem, and its steps are as follows: 2.2.1) Read the data: shear stress, shear strain and shear rate; 2.2.2) Define the problem and genetic algorithm parameters: determine the fitness function and set the number of chromosomes N, crossover rate pc, mutation probability pm, and finite iteration times X; 2.2.3) Determine the model parameter ∩ and initialize the model parameter value ∩ = [α, β, δ, ε, c, k]; 2.2.4) Initialize a set of chromosomes for each model parameter; 2.2.5) Evaluate each chromosome using the fitness function value and compare it with the previous one; 2.2.6) Sort the chromosomes according to the fitness of the chromosomes and select a new population using the roulette wheel method; 2.2.7) Update the chromosomes through crossover and mutation behaviors; 2.2.8) Evaluate the new chromosomes and check whether the stop criterion is met. If the stop condition is met or the number of iterations is higher than the limit number of iterations, report the best chromosome as the solution and output the optimal model parameters; otherwise, return to step 2.2.5); 3) After parameter identification, compare the model prediction data with the experimental test data, and at the same time compare the parameter model with the classical parameter model to verify the accuracy and rationality of the proposed model; 4) The influence of six parameters on the hysteresis curve was tested by adjusting the parameter sizes. Among them, the larger the parameters α and ε, the larger the area enclosed by their hysteresis curves; the lower the parameter β, the larger the obtained stress peak, indicating that the equivalent stiffness decreases with the increase of the parameter value; for the parameter δ, both its equivalent stiffness and the area of the closed hysteresis loop decrease with the increase of the parameter; for the parameter c, the larger its value, the larger the area of the obtained hysteresis curve, and strain hardening occurs when c = -1000; and higher equivalent stiffness can be obtained by increasing the value of k; moreover, all hysteresis curves intersect at two positions where the shear strain is 0. On both sides of these positions, the stiffness and the value of k are different: in the strain ranges from 0 to 10% and 0 to -10%, the stiffness in the clockwise path increases with the increase of k; however, in the strain ranges from -10% to 0 and 10% to 0, the stiffness decreases with the decrease of k. 5) Parameter generalization was carried out on the parameters, that is, the relationship between the model parameters and the applied current was proposed; under the same strain amplitude and frequency conditions, the parameter values under different current inputs were tested respectively, and it was obtained that the parameters α and δ decrease monotonically with the increase of the applied current, the parameters ε and k increase monotonically with the increase of the applied current, and the parameters β and c have a quadratic relationship with the current.

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