A high-precision hysteresis characteristic prediction method for a metal-rubber vibration isolation system
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- GENERAL ENG RES INST CHINA ACAD OF ENG PHYSICS
- Filing Date
- 2024-05-27
- Publication Date
- 2026-07-21
Smart Images
Figure CN118553351B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of nonlinear structural dynamics, and in particular to a high-precision method for predicting the hysteresis characteristics of a metal-rubber vibration isolation system. Background Technology
[0002] Metal rubber is a homogeneous porous material composed of wound metal wires. Under vibration loads, it exhibits high damping performance and is therefore used to manufacture metal rubber vibration isolators. To improve reliability and environmental adaptability, metal rubber vibration isolators are widely used to mitigate the vibration response of precision electronic equipment in aerospace, nuclear energy, and military fields. However, due to the internal frictional contact behavior of the metal rubber and the large geometric deformation mechanism of the metal wires, the isolators exhibit strong nonlinear hysteresis characteristics. This is caused by the hysteresis relationship between the contact force and relative deformation of the metal rubber. This nonlinear hysteresis characteristic hinders further theoretical research on metal rubber in engineering fields.
[0003] The hysteresis characteristics of metal-rubber vibration isolators have been described by scholars using various nonlinear models and parameter identification methods.
[0004] However, the identification of model parameters mainly focuses on the hysteresis curve at a specific excitation level, while ignoring the influence of different excitation levels on dynamic characteristics. Furthermore, the complexity of parameter identification methods increases the difficulty of theoretical analysis.
[0005] Furthermore, the hysteresis characteristic makes it extremely difficult to solve the nonlinear dynamic response of complex metal-rubber vibration isolation systems. Most existing methods are based on equivalent linearization to solve their steady-state response. Currently, a complete and high-precision method for predicting the hysteresis characteristics of metal-rubber vibration isolation systems is still lacking. Summary of the Invention
[0006] To address the problems mentioned in the background art, this invention provides a high-precision hysteresis characteristic prediction method for metal rubber vibration isolation systems. This method solves the problems that existing identification methods ignore the influence of different excitation levels on dynamic characteristics, and that the complexity of parameter identification methods also increases the difficulty of theoretical analysis.
[0007] To achieve the above objectives, the present invention provides the following technical solution:
[0008] A high-precision method for predicting the hysteresis characteristics of a metal-rubber vibration isolation system includes the following steps:
[0009] S1: Applying multiple sinusoidal displacement excitations of different amplitudes to a single metal-rubber vibration isolator yields multiple sets of different hysteresis curves;
[0010] S2: Establish a modified Bouc-Wen model, use the parameter separation identification method and the equivalent linearization method to identify the parameters of all hysteresis curves, obtain different model parameters, and form a parameter set;
[0011] S3: Establish an explicit evolution law function through the parameter set, and add parameter constraints to the modified Bouc-Wen model. Using the explicit evolution law function and the constrained modified Bouc-Wen model, predict the nonlinear hysteresis characteristics of the metal rubber vibration isolator within the preset excitation amplitude range.
[0012] S4: Based on the predicted nonlinear hysteresis characteristics, the harmonic balance method combined with time-frequency conversion analysis is used to predict the hysteresis dynamic steady-state response of the metal rubber vibration isolation system.
[0013] Preferably, in S1, the hysteresis curve expresses the correspondence between F and δ, where F is the total restoring force of the metal rubber vibration isolator and δ is the relative displacement of the metal rubber vibration isolator.
[0014] Preferably, the specific steps of S2 include:
[0015] S2.1: Decompose the total restoring force F of the metal-rubber vibration isolator at time t into a superposition of nonlinear elastic force and pure hysteresis force, and establish the modified Bouc-Wen model of the metal-rubber vibration isolator:
[0016] F(t)=F2(t)[z(t)+F1(t)]+h (1);
[0017] Where h is the vertical control parameter;
[0018] F1(t) is the high-order nonlinear elastic force function of the vibration isolator;
[0019] F1(t)=k1δ(t)+k3δ 3 (t) (2);
[0020] k1 is the first-order elastic coefficient; k3 is the third-order elastic coefficient, δ 3 (t) represents the cubic value of the relative displacement;
[0021] F2(t) is a symmetric control function:
[0022] F2(t)=b δ (t) (3);
[0023] b is a symmetric control parameter;
[0024] z(t) is the pure hysteresis force function of the metal-rubber vibration isolator, i.e., the Bouc-Wen model, and its differential form is as follows:
[0025]
[0026] This is the differential form of the Bouc-Wen model. Let be the relative deformation velocity of the metal-rubber vibration isolator, be the first derivative of δ(t), and α, β, γ, and n be the parameters of the Bouc-Wen model.
[0027] S2.2: Identify parameters h and b based on the geometric characteristics of the hysteresis curve;
[0028]
[0029] Among them, F + (δ=0) and F - (δ=0) represents the total restoring force of the upper and lower half-branch hysteresis curves when the relative displacement is 0, respectively.
[0030] b = exp(ln(-F) max / F min ) / (δ max -δ min )) (6);
[0031] Where the subscripts max and min represent the maximum and minimum values of the variable, respectively;
[0032] S2.3: Decompose the nonlinear elastic force from the total restoring force F, and identify the parameters k1 and k3 using the least squares method;
[0033] [k1, k3] = (C K T C K ) -1 C K T F K (7);
[0034] Among them, C K =[δ,δ 3 ] represents the relative displacement matrix, F K This is the extracted nonlinear elastic force matrix, with the superscript T indicating the transpose of the matrix;
[0035] S2.4: Identify the Bouc-Wen model parameters α, β, γ, n representing pure hysteresis force using the equivalent linearization method;
[0036] By arbitrarily selecting a value of n within a preset range as the prior input, identifying the parameters α, β, and γ, and integrating equation (4), we obtain:
[0037]
[0038] Among them, t i and t eThese represent the start and end times of the integration, respectively;
[0039] Equation (8) can be rewritten as:
[0040]
[0041] in represent In t i and t e The integral value within;
[0042]
[0043] These represent the corresponding equivalent transformations;
[0044] Let θ = [α, β, γ] T θ is the set of parameters to be identified;
[0045] λ=[y 1 y 2 y 3 ] T y is the y corresponding to the selected end time of integration. 1 y 2 , and y 3 A set;
[0046] Identifying parameters using the least squares method:
[0047] θ=(A T A) -1 A T y set (10);
[0048] Where A = [λ1, λ2, ..., λ] N ] T , representing the set of λ values at the selected N time points;
[0049] y set = [y1, y2…y N ] T , represents the set of y values at N selected times;
[0050] Using the transformation relationship of differentials:
[0051]
[0052] Substituting into equation (4), we get:
[0053]
[0054] The complete hysteresis curve is obtained using the Runge-Kutta method based on the above formula.
[0055] S2.5: Within a preset range of n values, select the set of parameters [α, β, γ, n] that has the smallest root mean square error compared to the experimental true value as the final identification result; the root mean square error is defined as follows:
[0056]
[0057] Among them, F exp and F sim These represent the total resilience in the experiment and simulation, respectively; N represents the sample size.
[0058] The parameter set {P} of the metal-rubber vibration isolator is formed by using the model parameters identified from multiple sets of hysteresis curves. h P b ,P k1 P k3 P α P β P γ ,P n}
[0059] Preferably, the specific steps of S3 are as follows:
[0060] S3.1: Based on the identified parameter set {P h P b P k1 P k3 P α P β P γ P n The parameter sets are established using exponential and polynomial functions to measure the relative displacement amplitude δ of the vibration isolator. max The explicit evolutionary law function;
[0061] S3.2: Use an exponential function to establish the amplitude evolution law of the pure hysteresis force as an additional parameter constraint for the Bouc-Wen model:
[0062]
[0063] S3.3: Using the evolution law of the parameter set of the modified Bouc-Wen model and the additional constraints, predict the nonlinear hysteresis characteristics of metal rubber vibration isolators over a wide excitation amplitude range.
[0064] Preferably, S4 includes the following sub-steps:
[0065] S4.1: Establish the dynamic control equations for a vibration isolation system composed of multiple metal-rubber vibration isolators:
[0066]
[0067] In the formula, m is the load-bearing mass of the vibration isolator; ρ is the response acceleration of the load-bearing mass; p is the number of vibration isolators installed in the vibration isolation system; z is the Bouc-Wen model, containing four parameters: α, β, γ, and n; b, h, k1, k3, α, β, γ, and n are the predicted values of the relative displacement amplitude δ in S3. max The corresponding model parameters;
[0068] Equation (15) is transformed into an equivalent form:
[0069]
[0070] In the formula, The acceleration signal of the vibrating foundation; u b ω represents the displacement signal amplitude of the vibrating foundation; ω represents the signal frequency of the vibrating foundation.
[0071] S4.2: Using the harmonic balance method, the displacement response of the metal-rubber vibration isolator is expressed as a truncated harmonic order:
[0072]
[0073] In the formula: H is the truncation order; and These are the cosine and sine components of the k-th truncated series, respectively.
[0074] Perform the following equivalent transformation:
[0075]
[0076] Where F e For equivalent incentive; F r For equivalent contact force; F a This is the equivalent inertial force;
[0077] Construct the time-domain residual function R using the above formula lt (t, δ(t)):
[0078] R lt (t,δ(t))=(F e -F a )-F r (19);
[0079] S4.3: The preset frequency domain harmonic coefficients of the relative displacement are:
[0080]
[0081] in The frequency domain harmonic coefficients of the preset relative displacement;
[0082] The above equation is transformed to the time domain using the Fast Fourier Transform (FFT) for calculating the residual function. The residual function is then transformed back to the frequency domain using the Inverse Fast Fourier Transform (IFFT).
[0083]
[0084] In the formula: R lf It is a residual function in the frequency domain;
[0085] Analytical solutions for relative displacements are obtained using Newton's iteration method:
[0086]
[0087] In the formula, The inverse of the Jacobian matrix in the (i-1)th iteration:
[0088]
[0089] The above formula is calculated using the finite difference method:
[0090]
[0091] S4.4: Predict the steady-state hysteresis dynamic response of the metal rubber vibration isolation system by continuously solving the analytical solution of the displacement response over a wide frequency range.
[0092] Compared with the prior art, the beneficial effects of the present invention are:
[0093] 1. By establishing a modified Bouc-Wen model with a metal-rubber vibration isolator, and using the equivalent linearization method that limits the parameter range combined with the idea of separating parameters, the model parameters with practical analytical significance were quickly and accurately identified.
[0094] 2. The evolution of the model parameter set was quantified using exponential and polynomial models, and an additional parameter constraint was added to the modified Bouc-Wen model. With only a few sets of experimental results, the nonlinear hysteresis characteristics of the metal-rubber vibration isolator over a wide amplitude range were predicted with high accuracy.
[0095] 3. Using the harmonic balance method based on time-frequency conversion, the steady-state hysteresis dynamic response of the metal rubber vibration isolation system is solved by selecting model parameter set data, and the resonant frequency position and resonant response peak of the metal rubber vibration isolation system are accurately predicted. Attached Figure Description
[0096] Figure 1 Here is a flowchart illustrating the method implementation of this application;
[0097] Figure 2 This is a drawing of the quasi-static test fixture used in this application;
[0098] Figure 3 Hysteresis curves of metal-rubber vibration isolators with different excitation amplitudes obtained in this application;
[0099] Figure 4 This is a breakdown diagram of the total restoring force of the metal-rubber vibration isolator in this application;
[0100] Figure 5 This is a comparison diagram of the test results and identification results in this application;
[0101] Figure 6 The hysteresis curve of the metal-rubber vibration isolator predicted in this application over a wide excitation amplitude range;
[0102] Figure 7 This is a diagram of the metal-rubber vibration isolation system device in this application;
[0103] Figure 8 A comparison diagram of the hysteresis dynamic steady-state response of the metal-rubber vibration isolation system tested and predicted in this application;
[0104] The image is labeled as follows:
[0105] 1-Frame; 2-Upper pull rod; 3-Clamp; 4-Lower support; 5-Lower pull rod; 6-Metal rubber vibration isolator specimen; 7-Bearing mass component of the vibration isolator; 8-Vibration generating device; 9-Metal rubber vibration isolator specimen. Detailed Implementation
[0106] To facilitate understanding of the technical content of this invention by those skilled in the art, the invention will be further described in detail below with reference to the accompanying drawings and specific examples. It should be understood that the specific examples described herein are merely illustrative and not intended to limit the scope of the invention.
[0107] Example 1
[0108] like Figure 1 As shown, a high-precision hysteresis characteristic prediction method for a metal-rubber vibration isolation system includes the following steps:
[0109] S1: Apply sinusoidal displacement excitation to a single metal rubber vibration isolator to obtain multiple sets of restoring force-displacement curves, i.e. hysteresis curves, under different excitation amplitudes. The hysteresis curves express the correspondence between F and δ, where F is the total restoring force of the metal rubber vibration isolator and δ is the relative displacement of the metal rubber vibration isolator.
[0110] F-δ curves under different excitation amplitudes (50N, 100N, 200N, 300N, 400N) were obtained through quasi-static tensile-compression tests on typical metal-rubber vibration isolators. The test fixtures were as follows: Figure 2As shown, the test fixture, from top to bottom, consists of a frame 1, an upper pull rod 2 connected to the top and bottom of the frame 1, a clamp 3 connected to the bottom of the upper pull rod 2, a metal-rubber vibration isolator specimen 6 connected to the clamp 3, a lower support 4 connected to the lower end of the metal-rubber vibration isolator 6, a lower pull rod 5 connected to the lower support 4, and the bottom end of the lower pull rod 5 connected to the top of the bottom of the frame 1. The F-δ hysteresis curve is shown below. Figure 3 As shown.
[0111] S2: Establish a modified Bouc-Wen model, and use the idea of separating parameters and the equivalent linearization method to identify the parameters of the above hysteresis curve to obtain the model parameters under different excitation amplitudes and form a parameter set.
[0112] S2 includes the following sub-steps:
[0113] S2.1: The total restoring force of the metal-rubber vibration isolator is decomposed into a superposition of nonlinear elastic force and pure hysteresis force, such as... Figure 4 As shown, a modified Bouc-Wen model of the metal-rubber vibration isolator is established to form a theoretical analysis model:
[0114] F(t)=F2(t)[z(t)+F1(t)]+h (1);
[0115] In the formula: h is the vertical control parameter, which controls the hysteresis curve to be symmetrical about the origin; F2(t) is the symmetrical control function, which controls the hysteresis curve to be symmetrical about the origin; z(t) is the pure hysteresis force function of the vibration isolator; F1(t) is the high-order nonlinear elastic force function of the vibration isolator.
[0116] The specific representations of each function are as follows:
[0117] F1(t)=k1δ(t)+k3δ 3 (t) (2);
[0118] k1 is the first-order elastic coefficient; k3 is the third-order elastic coefficient, δ 3 (t) represents the cubic value of the relative displacement;
[0119] F2(t)=b δ (t) (3);
[0120] In the formula: b is the symmetric control parameter.
[0121] z(t) is the pure hysteresis force function of the metal-rubber vibration isolator, i.e., the Bouc-Wen model. Its differential form is as follows:
[0122]
[0123] This is the differential form of the Bouc-Wen model. Let be the relative deformation velocity of the metal-rubber vibration isolator, be the first derivative of δ(t), and α, β, γ, and n be the parameters of the Bouc-Wen model.
[0124] S2.2: Identify parameters h and b based on the geometric characteristics of the hysteresis curve;
[0125]
[0126] Among them, F + (δ=0) and F - (δ=0) represents the total restoring force of the upper and lower half-branch hysteresis curves when the relative displacement is 0, respectively.
[0127] b = exp(ln(-F) max / F min ) / (δ max δ min )) (6);
[0128] Where the subscripts max and min represent the maximum and minimum values of the variable, respectively;
[0129] S2.3: Separate the nonlinear elastic force from the total restoring force F;
[0130] F K =F + (δ≥0)-F + (δ=0) (7);
[0131] In the formula: F + (δ≥0) is the restoring force matrix of the upper half-branch hysteresis curve when the displacement is non-negative;
[0132] The parameters k1 and k3 are identified using the least squares method.
[0133]
[0134] Among them, C K =[δ,δ 3 ] represents the relative displacement matrix, F K This is the extracted nonlinear elastic force matrix, with the superscript T indicating the transpose of the matrix;
[0135] S2.4: Identify the Bouc-Wen model parameters α, β, γ, n representing pure hysteresis force using the equivalent linearization method;
[0136] The Bouc-Wen model diverges when the parameter n is large. To make the identified parameters more useful and analytical, the range of parameter n is pre-limited to (0,5] (or other intervals, ensuring that the right endpoint of the interval does not diverge; here, (0,5] is taken as an example). An arbitrary interval of n value is selected as the prior input, and parameters α, β, and γ are identified first. Integrating equation (4):
[0137]
[0138] Among them, t i and t e These represent the start and end times of the integration, respectively;
[0139] Equation (8) can be rewritten as:
[0140]
[0141] in represent In t i and t e The integral value within;
[0142]
[0143] These represent the corresponding equivalent transformations;
[0144] Let θ = [α, β, γ] T θ is the set of parameters to be identified;
[0145] λ=[y 1 y 2 y 3 ] T y is the y corresponding to the selected end time of integration. 1 y 2 , and y 3 A set;
[0146] Identifying parameters using the least squares method:
[0147] θ=(A T A) -1 A T y set (10);
[0148] Where A = [λ1, λ2, ..., λ] N ] T , representing the set of λ values at the selected N time points;
[0149] y set = [y1, y2…y N ] T, represents the set of y values at N selected times;
[0150] Using the transformation relationship of differentials:
[0151]
[0152] Substituting into equation (4), we get:
[0153]
[0154] The complete hysteresis curve is obtained using the Runge-Kutta method.
[0155] S2.5: Within a preset range of n values, select the set of parameters [α, β, γ, n] that has the smallest root mean square error compared to the experimental true value as the final identification result; the root mean square error is defined as follows:
[0156]
[0157] Among them, F exp and F sim These represent the total resilience in the experiment and simulation, respectively; N represents the sample size.
[0158] The parameter set {P} of the metal-rubber vibration isolator is formed by using the model parameters identified from multiple sets of hysteresis curves. h P b P k1 P k3 P α P β P γ P n}
[0159] The final identification results are compared with the experimental curve, as follows: Figure 5 As shown, the modified Bouc-Wen model can accurately characterize the nonlinear hysteresis characteristics of metal-rubber vibration isolators.
[0160] S3: Based on the identified model parameters, perform statistical analysis, establish explicit evolution law functions for each parameter set, and add an additional parameter constraint to the modified Bouc-Wen model to predict the nonlinear hysteresis characteristics of the metal rubber vibration isolator within a wide excitation amplitude range.
[0161] Step S3 includes the following sub-steps:
[0162] S3.1: Based on the identified parameter set {P h P b P k1 P k3 P α P β P γ P nThe parameter sets are established using exponential and polynomial functions to measure the relative displacement amplitude δ of the vibration isolator. max The explicit evolutionary law function;
[0163]
[0164] k1 = -304.9145 + 1.2849 × 10 3 ·exp(-17.2369·δ max )+7.6579×10 2 ·exp(-0.1734·δ ma x) (17);
[0165] k3 = 739.6584 + 1.5868 × 10 3 ·exp(-11.8562·δ max -3.5795×10 5 ·exp(-40.7031·δ max (18);
[0166]
[0167] γ = -0.0892 - 5.602 × 10 -1 ·exp(10.1794·δ max -1.8364×10 -4 ·exp(23.5127·δ max ) (twenty two);
[0168] S3.2: The amplitude evolution law of pure hysteresis force is controlled by the following formula:
[0169]
[0170] The magnitude evolution law of pure hysteresis force is established using an exponential function as an additional parameter constraint for the Bouc-Wen model:
[0171] z max =28.8264 + 9.5339 × 10 0 ·exp(1.6109·δ max -3.7518×10 1 ·exp(-12.0196·δ max ) (twenty four);
[0172] Using the model parameter set and additional constraints, input the displacement amplitude δ max The corresponding model parameters can then be obtained, thereby predicting the nonlinear hysteresis curve.
[0173] S3.3: Utilizing the evolution law of the modified Bouc-Wen model parameter set and additional constraints, accurately predict the nonlinear hysteresis characteristics of metal-rubber vibration isolators within a wide excitation amplitude range (60N-450N), such as... Figure 6 As shown.
[0174] S4: Based on the predicted nonlinear hysteresis characteristics of the metal rubber vibration isolator, the harmonic balance method combined with time-frequency conversion analysis is used to predict the hysteresis dynamic steady-state response of the metal rubber vibration isolation system.
[0175] Step S4 includes the following sub-steps:
[0176] S4.1: Establish the dynamic control equations for a vibration isolation system composed of multiple metal-rubber vibration isolators:
[0177]
[0178] In the formula, m is the load-bearing mass of the vibration isolator; ρ is the response acceleration of the load-bearing mass; p is the number of vibration isolators installed in the vibration isolation system; z is the Bouc-Wen model, containing four parameters: α, β, γ, and n; b, h, k1, k3, α, β, γ, and n are the predicted values of the relative displacement amplitude δ in S3. max The corresponding model parameters;
[0179] In this embodiment, four vibration isolators are used. A schematic diagram of the metal-rubber vibration isolation system is shown below. Figure 7 As shown, the metal rubber vibration isolation system includes a bearing mass component 7 of the vibration isolator, a vibration generating device 8, and a metal rubber vibration isolator specimen 9.
[0180] Perform an equivalent transformation on the dynamic control equations:
[0181]
[0182] In the formula, The acceleration signal of the vibrating foundation; u b ω represents the displacement signal amplitude of the vibrating foundation; ω represents the signal frequency of the vibrating foundation.
[0183] S4.2: Using the harmonic balance method, the displacement response of the metal-rubber vibration isolator is expressed as a truncated harmonic order:
[0184]
[0185] In the formula: H is the truncation order; and These are the cosine and sine components of the k-th truncated series, respectively.
[0186] Perform the following equivalent transformation:
[0187]
[0188] Where F e For equivalent incentive; F r For equivalent contact force; F a This is the equivalent inertial force;
[0189] Construct the time-domain residual function using the above formula:
[0190] R lt (t,δ(t))=(F e -F a )-F r (29);
[0191] S4.3: The initial frequency domain solution for the preset relative displacement is:
[0192]
[0193] in The frequency domain harmonic coefficients of the preset relative displacement;
[0194] The above equation is transformed to the time domain using the Fast Fourier Transform (FFT) for calculating the residual function. The residual function is then transformed back to the frequency domain using the Inverse Fast Fourier Transform (IFFT).
[0195]
[0196] In the formula: R lf It is a residual function in the frequency domain.
[0197] Analytical solutions for relative displacements are obtained using Newton's iteration method:
[0198]
[0199] In the formula, The inverse of the Jacobian matrix in the (i-1)th iteration:
[0200]
[0201] Furthermore, the above formula is calculated using the finite difference method:
[0202]
[0203] S4.4: Based on 100N of data in the model parameter set, the steady-state hysteresis dynamic response of the metal rubber vibration isolation system is predicted by continuously solving the analytical solution of the displacement response over a wide frequency range.
[0204] In this embodiment, the excitation magnitudes for the frequency sweep test were 0.5g and 1.0g, and the excitation frequency range was 50Hz-400Hz. The final identification results were compared with the experimental curves, as shown below. Figure 5As shown, the modified Bouc-Wen model and the proposed prediction method can accurately characterize the hysteresis dynamic response amplitude and resonant frequency of the metal-rubber vibration isolation system.
[0205] This application establishes a modified Bouc-Wen model for metal-rubber vibration isolators. Utilizing an equivalent linearization method with limited parameter ranges combined with the concept of parameter separation, it quickly and accurately identifies model parameters with practical analytical significance. Furthermore, this application uses exponential and polynomial models to quantify the evolution of the model parameter set and adds an additional parameter constraint to the modified Bouc-Wen model. With only a few sets of experimental results, it accurately predicts the nonlinear hysteresis characteristics of metal-rubber vibration isolators over a wide amplitude range. This application employs a harmonic balance method based on time-frequency conversion, selecting model parameter set data to solve the steady-state hysteresis dynamic response of the metal-rubber vibration isolation system, accurately predicting the resonant frequency location and peak resonant response of the system.
[0206] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any person skilled in the art may make technical modifications based on the above-disclosed technical content. However, any simple modifications, equivalent changes, and alterations 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 high-precision method for predicting the hysteresis characteristics of a metal-rubber vibration isolation system, characterized in that, Includes the following steps: S1: Apply multiple sinusoidal displacement excitations of different amplitudes to a single metal rubber vibration isolator to obtain multiple sets of different hysteresis curves. The hysteresis curves express the correspondence between F and δ, where F is the total restoring force of the metal rubber vibration isolator and δ is the relative displacement of the metal rubber vibration isolator. S2: Establish a modified Bouc-Wen model, use the parameter separation identification method and the equivalent linearization method to identify the parameters of all hysteresis curves, obtain different model parameters, and form a parameter set; S3: Establish an explicit evolution law function through the parameter set, and add parameter constraints to the modified Bouc-Wen model. Using the explicit evolution law function and the constrained modified Bouc-Wen model, predict the nonlinear hysteresis characteristics of the metal rubber vibration isolator within the preset excitation amplitude range. S4: Based on the predicted nonlinear hysteresis characteristics, the harmonic balance method combined with time-frequency conversion analysis is used to predict the hysteresis dynamic steady-state response of the metal rubber vibration isolation system. The specific steps of S2 include: S2.1: Decompose the total restoring force F of the metal-rubber vibration isolator at time t into a superposition of nonlinear elastic force and pure hysteresis force, and establish the modified Bouc-Wen model of the metal-rubber vibration isolator: （1）； in, For vertical control parameters; This represents the higher-order nonlinear elastic force function of the vibration isolator; （2）； It is the first-order elastic coefficient; The elastic coefficients are third-order. This represents the cube of the relative displacement. For symmetric control functions: （3）； b is a symmetric control parameter; z(t) is the pure hysteresis force function of the metal-rubber vibration isolator, i.e., the Bouc-Wen model, and its differential form is as follows: （4）； Differential form of the Bouc-Wen model Let be the relative deformation velocity of the metal-rubber vibration isolator, be the first derivative of δ(t), and α, β, γ, and n be the parameters of the Bouc-Wen model. S2.2: Identify parameters based on the geometric characteristics of the hysteresis curve. and b; （5）； in, and These are the total restoring forces of the upper and lower half-branch hysteresis curves when the relative displacement is 0, respectively. （6）； Where the subscripts max and min represent the maximum and minimum values of the variable, respectively; S2.3: Decompose the nonlinear elastic force from the total restoring force F, and identify the parameters using the least squares method. and ; （7）； in, , representing the relative displacement matrix, This is the extracted nonlinear elastic force matrix, with the superscript T indicating the transpose of the matrix; S2.4: Identify the Bouc-Wen model parameters α, β, γ, n representing pure hysteresis force using the equivalent linearization method; By arbitrarily selecting a value of n within a preset range as the prior input, identifying the parameters α, β, and γ, and integrating equation (4), we obtain: （8）； in, and These represent the start and end times of the integration, respectively; Equation (8) can be rewritten as: （9）； in ,represent exist and The integral value within; ， ， , respectively, represent the corresponding equivalent transformations; make: , The set of parameters to be identified; , is the time corresponding to the selected end of integration. , ,and A set; Identifying parameters using the least squares method: （10）； in , representing the set of λ values at the selected N time points; , represents the set of y values at N selected times; Using the transformation relationship of differentials: （11）； Substituting into equation (4), we get: （12）； The complete hysteresis curve is obtained using the Runge-Kutta method based on the above formula. S2.5: Select a set of parameters within a preset range of n values that minimizes the root mean square error compared to the experimental true value. As the final identification result, the root mean square error is defined as follows: （13）； in, and These represent the total resilience in the experiment and simulation, respectively; N represents the sample size. The parameter set of the metal-rubber vibration isolator is formed by using the model parameters identified from multiple sets of hysteresis curves. .
2. The high-precision hysteresis characteristic prediction method for a metal-rubber vibration isolation system according to claim 1, characterized in that, The specific steps for S3 are as follows: S3.1: Based on the parameter set obtained from identification The amplitude of each parameter set as a function of the relative displacement of the vibration isolator is established using exponential and polynomial functions. The explicit evolutionary law function; S3.2: Use an exponential function to establish the amplitude evolution law of the pure hysteresis force as an additional parameter constraint for the Bouc-Wen model: （14）； S3.3: Using the evolution law of the parameter set of the modified Bouc-Wen model and the additional constraints, predict the nonlinear hysteresis characteristics of metal rubber vibration isolators over a wide excitation amplitude range.
3. The high-precision hysteresis characteristic prediction method for a metal-rubber vibration isolation system according to claim 2, characterized in that, S4 includes the following sub-steps: S4.1: Establish the dynamic control equations for a vibration isolation system composed of multiple metal-rubber vibration isolators: （15）； In the formula, m is the load-bearing mass of the vibration isolator; The response acceleration for bearing the mass; The number of vibration isolators installed in the vibration isolation system; z is the Bouc-Wen model, containing four parameters: α, β, γ, and n; b, h, k1, k3, α, β, γ, and n are the predicted values of the relative displacement amplitude in S3. The corresponding model parameters; Equation (15) can be transformed into an equivalent form: （16）； In the formula, The acceleration signal of the vibrating foundation; ω represents the displacement signal amplitude of the vibrating foundation; ω represents the signal frequency of the vibrating foundation. S4.2: Using the harmonic balance method, the displacement response of the metal-rubber vibration isolator is expressed as a truncated harmonic order: （17）； In the formula: H is the truncation order; and These are the cosine and sine components of the k-th truncated series, respectively. Perform the following equivalent transformation: , , （18）； in For equivalent incentive force; Equivalent contact force; This is the equivalent inertial force; Construct the time-domain residual function using the above formula : （19）； S4.3: The preset frequency domain harmonic coefficients of the relative displacement are: （20）； in The frequency domain harmonic coefficients of the preset relative displacement; The above equation is transformed to the time domain using the Fast Fourier Transform (FFT) for calculating the residual function. The residual function is then transformed back to the frequency domain using the Inverse Fast Fourier Transform (IFFT). （21）； In the formula: It is a residual function in the frequency domain; Analytical solutions for relative displacements are obtained using Newton's iteration method: （22）； In the formula, The inverse of the Jacobian matrix in the (i-1)th iteration: （23）； The above formula is calculated using the finite difference method: （24）； S4.4: Predict the steady-state hysteresis dynamic response of the metal rubber vibration isolation system by continuously solving the analytical solution of the displacement response over a wide frequency range.