Method for modeling the hysteretic dynamics of metal rubber materials based on high-order nonlinear friction

Abstract

The application relates to a high-order nonlinear friction-based metal rubber material hysteresis dynamics modeling method, which comprises the following steps: constructing a high-order nonlinear friction force model according to the contact form and spatial angle distribution of metal wires in the metal rubber; constructing a traditional dynamics model; substituting the high-order nonlinear friction force model into the traditional dynamics model to construct a high-order nonlinear friction-based metal rubber hysteresis dynamics model; performing parameter identification on the metal rubber hysteresis dynamics model; constructing an error model to represent the error of the test results affected by the dynamics model; performing parameter identification on the error model to determine the optimal number of model terms that should be included in the model, and completing the determination of the error model; completing error model parameter optimization, and completing model construction according to the model correlation coefficient of the high-order nonlinear metal rubber hysteresis dynamics model based on the error model. The method is beneficial to the establishment of a metal rubber dynamics model with high precision and universality.

Description

Technical Field

[0001] This invention belongs to the field of metal rubber, specifically relating to a hysteresis dynamics modeling method for metal rubber materials based on high-order nonlinear friction. Background Technology

[0002] Metal rubber is a type of highly elastic pure metal vibration damping material produced by cold stamping through blanking processes such as spiral winding, weaving, or laying of metal filaments. Its internal structure forms a disordered, mesh-like, porous structure, and it has found some applications in special service environments. However, due to its complex structure, metal rubber exhibits strong nonlinear hysteresis characteristics, making it difficult to accurately construct a dynamic model, ultimately limiting its widespread application. Current research attempts to equate the metal filaments within the metal rubber structure to micro-element models to construct constitutive models of the material. However, these models neglect the damping characteristics of metal rubber, resulting in insufficient model accuracy. To address the accuracy issue, researchers have adopted phenomenological theory and introduced damping forces to model the dynamics of metal rubber. However, these models still cannot study the variation of damping forces from the random spatial contact of the internal metal filaments, thus lacking universality. There is an urgent need to construct a new, highly accurate, and universal dynamic model based on the contact friction characteristics between the metal filaments within the metal rubber. Summary of the Invention

[0003] The purpose of this invention is to provide a hysteresis dynamics modeling method for metal-rubber materials based on high-order nonlinear friction. This method is beneficial for establishing a metal-rubber dynamics model with high accuracy and versatility.

[0004] To achieve the above objectives, the technical solution adopted by this invention is: a method for modeling the hysteresis dynamics of metal-rubber materials based on high-order nonlinear friction, comprising the following steps:

[0005] Step S1: Construct a high-order nonlinear friction force model based on the contact pattern and spatial angle distribution of the metal wires inside the metal rubber;

[0006] Step S2: Construct a nonlinear elastic restoring force model, a nonlinear damping force model, and a hysteresis damping force model based on the traditional dynamic model, and decompose the hysteresis damping force using Chebyshev polynomials;

[0007] Step S3: Substitute the high-order nonlinear friction model into the traditional dynamic model to construct a metal-rubber hysteresis dynamic model based on high-order nonlinear friction;

[0008] Step S4: Use the Lsqlin solver, which is based on the internal point algorithm of nonlinear boundary condition constraints, to identify parameters of the metal-rubber hysteresis dynamics model;

[0009] Step S5: Substitute the identified parameters back into the metal-rubber hysteresis dynamics model to construct the NARMAX error model to represent the error that affects the test results outside the dynamics model;

[0010] Step S6: Use the OLS algorithm to identify the parameters of the error model, and use the APRESS statistic to determine the optimal number of model terms to be included in the model. If the identified error model APRESS > 0.95, the determination of the error model is completed; otherwise, return to step S5 to correct the number of error model terms.

[0011] Step S7: Complete the optimal selection of error model parameters and calculate the model accuracy index R of the high-order nonlinear metal-rubber hysteresis dynamics model based on the error model. 2 When R 2 >0.9 Complete the high-order nonlinear metallic rubber hysteresis dynamic model; otherwise, return to step S2 and reconstruct the traditional dynamic model.

[0012] Furthermore, in step S1, the formula for the higher-order nonlinear friction force model is as follows:

[0013]

[0014] In the formula, f fH For high-order nonlinear friction force, ε is the deformation displacement of the metal-rubber, and F is the high-order nonlinear friction force. N Let μ be the equivalent elastic force acting on the metal wire, μ be the coefficient of friction of the metal wire, and k′ be the equivalent elastic force. m ′ represents the equivalent stiffness in the form of point contact between adjacent metal wires, k′ m " is the equivalent stiffness of the contact between adjacent metal wire surfaces, and d is the probability distribution multinomial coefficient.

[0015] Furthermore, in step S2, a nonlinear elastic restoring force model, a nonlinear damping force model, and a hysteresis damping force model are constructed based on the traditional dynamic model. The hysteresis damping force is decomposed using Chebyshev polynomials, and the formula is as follows:

[0016]

[0017] In the formula, f k (y(t)) represents the nonlinear elastic restoring force, y(t) represents the deformation displacement ε of the metal-rubber, and t represents time. This is a nonlinear damping force model, where z(t) is the hysteretic damping force, n1 is the number of polynomial terms in the nonlinear elastic restoring force, and k... 2i-1 n is the stiffness coefficient, n² is the number of terms in the polynomial of the nonlinear damping force, and c is the number of terms in the polynomial of the nonlinear damping force. 2i-1 Let a0 be the damping coefficient and a be the damping coefficient. n Let n be the coefficients of the Chebyshev polynomial, and n³ be the number of terms in the Chebyshev polynomial. Let Δy be the derivative of the displacement, i.e., the velocity, and let Δy be the maximum deformation displacement of the metal rubber.

[0018] Furthermore, in step S3, the formula for the metal-rubber hysteresis dynamics model based on high-order nonlinear friction is as follows:

[0019]

[0020] In the formula, F(t) is the resultant external force on the metal rubber.

[0021] Further, in step S4, parameter identification is performed on the metal-rubber hysteresis dynamics model of high-order nonlinear friction, and the identified parameters are denoted as:

[0022]

[0023] In the formula, Θ is the parameter to be identified, [θ1,θ2,...,θ n ] is a vector composed of parameters to be identified.

[0024] Furthermore, the formula for the error model NARMAX is as follows:

[0025]

[0026] In the formula, Δ is the error value, and ξ is the error value. i L represents the residual term of the model. ε N represents the number of parameter terms in the residual. ξ The number of terms in the error model. These are the coefficients of the error model.

[0027] Furthermore, in step S1, the contact form of the metal wire is as follows:

[0028]

[0029] In the formula, P k′ This represents the probability that adjacent metal wires in a metallic rubber assembly do not come into contact. Let P be the probability of adjacent metal wires making contact in a metal-rubber composite. k″′ This represents the probability of contact between adjacent metal wires in a metallic rubber.

[0030] The spatial angular distribution probability is:

[0031]

[0032] In the formula, Let be the spatial distribution angle, randsrc be the probability function for the random distribution of the spatial angle, and n be the number of micro-spring units in the metal-rubber compound. It represents the discrete spatial angular distribution.

[0033] Further, in step S6, the APRESS statistic is:

[0034] APRESS(n) = c(n)MSE(n) (8)

[0035] In the formula, MSE(q) = 1 - ERR j The output variance is the error, and c(q) is the complexity cost function. α is a number greater than 1, N is the number of terms in the maximum model, and q is the number of terms required to meet the requirements.

[0036] Further, in step S7, the model correlation coefficient R 2 for:

[0037]

[0038] In the formula, RSS is the sum of squares of the residuals; TSS refers to the total sum of squares; n is the number of sample observations; y i Observations; This indicates that the sample is a prediction from the model; This represents the average value of the sample observations.

[0039] Compared with the prior art, the present invention has the following beneficial effects: Compared with the existing nonlinear hysteresis dynamic modeling of metal rubber, this method proposes a high-order nonlinear friction force for the first time based on the spatial tilt angle and contact form of the metal wires inside the metal rubber. Combined with nonlinear elastic restoring force, nonlinear damping force and Coulomb hysteresis friction force, a more accurate metal rubber hysteresis dynamic model is constructed, which improves the problem of insufficient universality of traditional metal rubber dynamic models and optimizes the model accuracy of the equivalent modeling method. Attached Figure Description

[0040] Figure 1 This is a flowchart illustrating the method implementation of an embodiment of the present invention.

[0041] Figure 2 These are three types of annular metal-rubber samples with different densities in the embodiments of this invention.

[0042] Figure 3 This is a diagram of the test fixture in an embodiment of the present invention.

[0043] Figure 4 This is a comparison chart of the experimental results and model curves in the embodiments of the present invention.

[0044] Figure 5 This is a timing diagram of the error model in an embodiment of the present invention.

[0045] Figure 6 This is a comparison diagram of the time-domain plot of the theoretical model and the time-domain plot of the experimental results in an embodiment of the present invention.

[0046] Figure 7 This is a hysteresis loop diagram of the metal-rubber dynamics model in this embodiment of the invention.

[0047] Figure 3 In the middle: 1-force sensor; 2-lower pressure block; 3-metal rubber sample; 4-frame; 5-upper pressure block; 6-extension plate; 7-displacement sensor; 8-excitation force. Detailed Implementation

[0048] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0049] It should be noted that the following detailed descriptions are exemplary and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0050] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments according to this application. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0051] like Figure 1 As shown, this embodiment provides a method for modeling the hysteresis dynamics of metal-rubber materials based on high-order nonlinear friction, including the following steps:

[0052] Step S1: Construct a high-order nonlinear friction force model based on the contact pattern and spatial angle distribution of the metal wires inside the metal rubber. The formula is as follows:

[0053]

[0054] In the formula, f fH For high-order nonlinear friction force, ε is the deformation displacement of the metal-rubber, and F is the high-order nonlinear friction force. N Let be the equivalent elastic force acting on the metal wire, μ be the coefficient of friction of the metal wire (taken as 0.3 in this embodiment), and k″ be the equivalent elastic force acting on the metal wire. m k′″ represents the equivalent stiffness in the form of point contact between adjacent metal wires. m Let d be the equivalent stiffness of the contact between adjacent metal wire surfaces, and d be the coefficient of the probability distribution multinomial.

[0055] Step S2: Construct a nonlinear elastic restoring force model, a nonlinear damping force model, and a hysteresis damping force model based on the traditional dynamic model. Decompose the hysteresis damping force using Chebyshev polynomials, as shown in the following formula:

[0056]

[0057]

[0058] In the formula, f k (y(t)) represents the nonlinear elastic restoring force, y(t) represents the deformation displacement ε of the metal-rubber, and t represents time. This is a nonlinear damping force model, where z(t) is the hysteretic damping force, n1 is the number of polynomial terms in the nonlinear elastic restoring force, and k... 2i-1 n is the stiffness coefficient, n² is the number of terms in the polynomial of the nonlinear damping force, and c is the number of terms in the polynomial of the nonlinear damping force. 2i-1 Let a0 be the damping coefficient and a be the damping coefficient. n Let n be the coefficients of the Chebyshev polynomial, and n³ be the number of terms in the Chebyshev polynomial. Let Δy be the derivative of the displacement, i.e., the velocity, and let Δy be the maximum deformation displacement of the metal rubber.

[0059] Step S3: Substitute the higher-order nonlinear friction model into the traditional dynamic model to construct a metal-rubber hysteresis dynamic model based on higher-order nonlinear friction, the formula of which is as follows:

[0060]

[0061] In the formula, F(t) is the resultant external force on the metal rubber.

[0062] Step S4: Use the Lsqlin solver based on the internal point algorithm of nonlinear boundary condition constraints to identify parameters of the metal-rubber hysteresis dynamics model, and record the identified parameters as follows:

[0063]

[0064] In the formula, Θ is the parameter to be identified, [θ1,θ2,...,θ n ] is a vector composed of parameters to be identified.

[0065] Step S5: Substitute the identified parameters back into the metal-rubber hysteresis dynamics model to construct an error model NARMAX (Nonlinear Auto Regressive Moving Average with Xogenous input) to represent the error affecting the experimental results outside the dynamics model, denoted as:

[0066]

[0067] In the formula, Δ is the error value, and ξ is the error value. i L represents the residual term of the model. ε N represents the number of parameter terms in the residual. ξ The number of terms in the error model. These are the coefficients of the error model.

[0068] Step S6: Use the OLS algorithm to identify the parameters of the error model, and use the APRESS statistic to determine the optimal number of model terms to be included in the model. If the identified error model APRESS > 0.95, the determination of the error model is completed; otherwise, return to step S5 to correct the number of error model terms.

[0069] Step S7: Complete the optimal selection of error model parameters and calculate the model accuracy index R of the high-order nonlinear metal-rubber hysteresis dynamics model based on the error model. 2 When R 2 >0.9 Complete the high-order nonlinear metal-rubber hysteresis dynamics model; otherwise, return to step S2 and reconstruct the traditional metal-rubber dynamics model.

[0070] In step S1, the contact pattern of the metal wire is as follows:

[0071]

[0072] In the formula, P k′ This represents the probability that adjacent metal wires in a metallic rubber assembly do not come into contact. Let P be the probability of adjacent metal wires making contact in a metal-rubber composite. k″′ This represents the probability of contact between adjacent metal wires in a metallic rubber.

[0073] The spatial angular distribution probability is:

[0074]

[0075] In the formula, Let be the spatial distribution angle, randsrc be the probability function for the random distribution of the spatial angle, and n be the number of micro-spring units in the metal-rubber compound. It represents the discrete spatial angular distribution.

[0076] The APRESS statistic is:

[0077] APRESS(n) = c(n)MSE(n) (8)

[0078] In the formula, MSE(q) = 1 - ERR j The error output variance, c(q), is the complexity cost function, typically set as follows: α is a number greater than 1, N is the number of terms in the maximum model, and q is the number of terms required to meet the requirements.

[0079] The correlation coefficient R of the model 2 for:

[0080]

[0081] In the formula, RSS is the sum of squares of the residuals; TSS refers to the total sum of squares; n is the number of sample observations; y i Observations; This indicates that the sample is a prediction from the model; This represents the average value of the sample observations.

[0082] In this embodiment, the proposed kinetic model was verified through a metal-rubber kinetic experiment. A ring-shaped metal-rubber sample with an outer diameter of 20 mm, an inner diameter of 8.5 mm, and a height of 15 mm was prepared using 304 stainless steel as the raw material. A 0.8 g / cm³ metal-rubber sample was used. 3 1.0g / cm 3 and 1.2g / cm 3 Three densities were used to verify the model's generality, such as... Figure 2 As shown in Table 1, the preparation parameters for the ring-shaped metal rubber are as follows. The MR was tested on a dynamic testing system at room temperature, and the testing fixtures are as follows. Figure 3 As shown.

[0083] Table 1. Characteristics and preparation parameters of MR

[0084]

[0085] The variance analysis of the high-order nonlinear friction dynamic model is shown in Table 2. The closer the correlation index is to 1, the greater the model accuracy. The correlation index for all models under different densities is 0.999, infinitely close to 1, indicating that the high-order nonlinear friction dynamic model has excellent model versatility and accuracy. Comparing the experimental results with the model curves, as shown... Figure 4 As shown, the model curves for different densities closely match their experimental results and are highly consistent with the results of the analysis of variance.

[0086] 2. Analysis of Variance Table

[0087]

[0088] The dynamic model and error model of the metal-rubber material are analyzed. The error model with optimized error parameters can be expressed as:

[0089] ε i =β1ξ i-1 +β2ξ i-5 +β3ξ i-4 +β4ξ i-3

[0090] Where β1 = -1.3153, β2 = 0.9484, β3 = 0.2556, and β4 = -0.0744. A time series diagram of the error model is plotted based on the polynomial error model, as shown below. Figure 5As shown. The error force is within -0.6 ≤ ε i The value is ≤0.6, which is much smaller than the excitation force of the model, reflecting that the dynamic model based on high-order nonlinear friction has a high signal-to-noise ratio and high accuracy.

[0091] The parameters of the dynamic model are solved using a linear least squares solver with boundary or linear constraints, and the dynamic model is finally determined.

[0092]

[0093] Where y0 = 2, k1 = 16.5049 N / mm, k3 = 1.8261 N / mm -3 k5 = 651.4864 N / mm -5 ,c1=-0.0879s·N / mm,c3=-3.8022×10 -4 s·N / mm,c5=3.5855×10 -7 s·N / mm,k s = 30.1153 N / mm,z s =1.7322N, d1=-0.3723, d2=0.1500, d3=-2.1664, d4=-1.5123, d5=0.0204, d6=0.8118.

[0094] Based on the identified high-order nonlinear friction dynamic model, a time-series curve of the metal-rubber dynamics was constructed. The time-domain plots of the theoretical model and the experimental results were compared. The positions of the points in the experimental results accurately fell within the theoretical time-domain plot. Figure 6 As shown. By introducing displacement response, the time series of the metal-rubber dynamics model is transformed into a hysteresis loop, as shown... Figure 7 As shown. Combined with Figure 6 and Figure 7 The analysis results show that the model has high accuracy.

[0095] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A method for modeling the hysteresis dynamics of metal-rubber materials based on high-order nonlinear friction, characterized in that, Includes the following steps: Step S1: Construct a high-order nonlinear friction force model based on the contact pattern and spatial angle distribution of the metal wires inside the metal rubber; Step S2: Construct a nonlinear elastic restoring force model, a nonlinear damping force model, and a hysteresis damping force model based on the traditional dynamic model, and decompose the hysteresis damping force using Chebyshev polynomials; Step S3: Substitute the high-order nonlinear friction model into the traditional dynamic model to construct a metal-rubber hysteresis dynamic model based on high-order nonlinear friction; Step S4: Use the Lsqlin solver, which is based on the internal point algorithm of nonlinear boundary condition constraints, to identify parameters of the metal-rubber hysteresis dynamics model; Step S5: Substitute the identified parameters back into the metal-rubber hysteresis dynamics model to construct the NARMAX error model to represent the error that affects the test results outside the dynamics model; Step S6: Use the OLS algorithm to identify the parameters of the error model, and use the APRESS statistic to determine the optimal number of model terms to be included in the model. If the identified error model APRESS > 0.95, the determination of the error model is completed; otherwise, return to step S5 to correct the number of error model terms. Step S7: Complete the optimal selection of error model parameters and calculate the model accuracy index R of the high-order nonlinear metal-rubber hysteresis dynamics model based on the error model. 2 When R 2 >0.9 Complete the high-order nonlinear metallic rubber hysteresis dynamics model; otherwise, return to step S2 and reconstruct the traditional dynamics model. In step S1, the formula for the high-order nonlinear friction model is as follows: （1） In the formula, f fH For high-order nonlinear friction, ε is the deformation displacement of the metal-rubber, and F N Let μ be the equivalent elastic force acting on the metal wire, and μ be the coefficient of friction of the metal wire. The equivalent stiffness for point contact between adjacent metal wires denoted as the equivalent stiffness of the contact between adjacent metal wire surfaces, and d is the probability distribution multinomial coefficient; In step S2, a nonlinear elastic restoring force model, a nonlinear damping force model, and a hysteresis damping force model are constructed based on the traditional dynamic model. The hysteresis damping force is decomposed using Chebyshev polynomials, and the formula is as follows: （2） In the formula, f k (y(t)) represents the nonlinear elastic restoring force, y(t) represents the deformation displacement ε of the metal-rubber, and t represents time. This is a nonlinear damping force model, where z(t) is the hysteresis damping force, n1 is the number of polynomial terms in the nonlinear elastic restoring force, and k... 2i-1 n is the stiffness coefficient, n² is the number of terms in the polynomial of the nonlinear damping force, and c is the number of terms in the polynomial of the nonlinear damping force. 2i-1 Let a0 be the damping coefficient and a be the damping coefficient. n Here, n represents the coefficients of the Chebyshev polynomial, and n³ represents the number of terms in the Chebyshev polynomial. Let Δy be the derivative of the displacement, i.e., the velocity, and Δy be the maximum deformation displacement of the metal rubber. In step S3, the formula for the metal-rubber hysteresis dynamics model based on high-order nonlinear friction is as follows: （3） In the formula, F(t) is the resultant external force acting on the metal rubber; In step S1, the contact pattern of the metal wire is as follows: （6） In the formula, , , These represent the probabilities of no contact, point contact, and surface contact between adjacent metal wires in a metallic rubber, respectively.

2. The method for modeling the hysteresis dynamics of metal-rubber materials based on high-order nonlinear friction according to claim 1, characterized in that, In step S4, the parameters of the metal-rubber hysteresis dynamics model of high-order nonlinear friction are identified, and the identified parameters are denoted as: （4） In the formula, For the parameters to be identified, It is a vector consisting of parameters to be identified.

3. The hysteresis dynamics modeling method for metal-rubber materials based on high-order nonlinear friction according to claim 2, characterized in that, The formula for the error model NARMAX is as follows: （5） In the formula, Δ is the error value, and ξ is the error value. i L represents the residual term of the model. ε N represents the number of parameter terms in the residual. ξ The number of terms in the error model. These are the coefficients of the error model.

4. The method for modeling the hysteresis dynamics of metal-rubber materials based on high-order nonlinear friction according to claim 1, characterized in that, In step S1, the spatial angle distribution probability is: （7） In the formula, Let be the spatial distribution angle, randsrc be the probability function for the random distribution of the spatial angle, and m1 be the number of micro-spring units in the metal-rubber compound. It represents the discrete spatial angular distribution.

5. The method for modeling the hysteresis dynamics of metal-rubber materials based on high-order nonlinear friction according to claim 1, characterized in that, In step S6, the APRESS statistic is: （8） In the formula, MSE(q) = 1 - ERR j The output variance is the error, and c(q) is the complexity cost function. α is a number greater than 1, N is the number of terms in the maximum model, and q is the number of terms required to meet the requirements.

6. The method for modeling the hysteresis dynamics of metal-rubber materials based on high-order nonlinear friction according to claim 1, characterized in that, In step S7, the model accuracy index R 2 for: （9） In the formula, RSS is the sum of squares of the residuals; TSS refers to the sum of the total squares; m2 is the number of sample observations; y i Observations; This indicates that the sample is a prediction from the model; This represents the average value of the sample observations.

7. The method for modeling the hysteresis dynamics of metal-rubber materials based on high-order nonlinear friction according to claim 1, characterized in that, The constructed high-order nonlinear friction hysteresis dynamic model of metal-rubber material is applied to the vibration analysis of metal-rubber material. After the high-order nonlinear friction hysteresis dynamic model of metal-rubber material is completed, the vibration data of metal-rubber material is input into the high-order nonlinear friction hysteresis dynamic model of metal-rubber material to obtain the vibration reduction frequency response data of metal-rubber material. The vibration reduction effect of metal-rubber material is judged based on the vibration reduction frequency response data of metal-rubber material.

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