A metal rubber dynamic test error compensation and adaptive unified modeling method based on frequency multiplication harmonic characteristics

Abstract

This invention relates to a method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics. The method includes: calibrating system time error and phase deviation using standard components; extracting the dominant frequency and harmonic characteristics of the displacement spectrum through pre-tests, constructing an excitation error compensation model and generating a compensation waveform to suppress resonance distortion; preprocessing the response data after the formal test; adaptively determining the Chebyshev series order based on the amplitude tolerance threshold of the force spectrum; and finally using the Chebyshev series to construct a unified mathematical model including elastic components, viscous damping components, and inviscous damping components. This invention can effectively eliminate test system errors and excitation rebound errors, achieving automated and standardized modeling of the dynamic characteristics of metal rubber, and significantly improving the model's versatility and engineering applicability.

Description

Technical Field

[0001] This invention relates to the field of dynamic mechanical property testing and modeling technology for materials, specifically to a method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics. Background Technology

[0002] Metal-rubber material is a highly elastic, high-damping structural material composed of woven metal wires, possessing both high load-bearing capacity and excellent corrosion resistance. As a key damping element in vibration isolation devices, it has been widely used in military, shipbuilding, and aerospace fields. However, due to the significant non-uniformity and randomness of the spatial distribution of its internal metal wires, metal-rubber material exhibits strong nonlinearity and significant dispersion in its macroscopic mechanical response, preventing the development of a unified constitutive model and state description framework. Therefore, the dynamic modeling of metal-rubber material is highly dependent on experimental data, typically requiring extensive experiments for different structural designs and process parameters, and the establishment of corresponding dedicated models, resulting in high modeling costs and limited versatility.

[0003] In existing dynamic testing methods, due to the strong nonlinear stiffness of metallic rubber, under high excitation force conditions, when the test frequency approaches the natural frequency of the overall system formed by the specimen and fixture, a resonance amplification effect easily occurs, causing the displacement response waveform to deviate from the ideal sinusoidal excitation form and introducing significant test errors. Furthermore, the test system itself has time synchronization errors and phase deviations, further affecting modeling accuracy. Therefore, there is an urgent need for a method that can effectively compensate for test errors, automatically extract characteristic parameters, and achieve universal modeling across structures. Summary of the Invention

[0004] The purpose of this invention is to provide a method for dynamic testing error compensation and adaptive unified modeling of metal rubber based on harmonic characteristics, so as to solve the technical problems of large dynamic testing error of metal rubber, modeling relying on empirical parameters, and poor model universality in the prior art.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: a method for dynamic testing error compensation and adaptive unified modeling of metal rubber based on harmonic characteristics, comprising the following steps:

[0006] Step S1: Calibrate the dynamic test system using standard linear components to obtain the system's time synchronization error and phase deviation parameters;

[0007] Step S2: Set the test parameters, conduct a preliminary test on the metal-rubber sample, and obtain the initial time-displacement response data and time-force response data;

[0008] Step S3: Perform Fourier transform on the time-displacement response data obtained from the pre-test to obtain the displacement spectrum, and extract the frequency characteristics of the main frequency component and each harmonic component.

[0009] Step S4: Based on the main frequency component and harmonic components, construct a displacement excitation error compensation model, correct the error of the displacement excitation signal, generate the compensated displacement driving waveform, and use it as the input excitation signal for the formal test.

[0010] Step S5: Under the compensated displacement excitation conditions, conduct a formal dynamic test on the metal-rubber sample to obtain high-precision time-displacement response data and time-force response data;

[0011] Step S6: Preprocess the data obtained from the formal dynamic experiment to form a standardized analytical data sequence;

[0012] Step S7: Perform Fourier transform on the standardized time-force response data to obtain the force spectrum, and simultaneously calculate the amplitude and phase information of the time-force response data and time-displacement response data at each harmonic to form a set of frequency domain characteristic parameters;

[0013] Step S8: By setting an amplitude tolerance threshold, the harmonic components of each order in the force spectrum are screened. Based on the number of effective harmonics with amplitude components higher than the tolerance threshold, the Chebyshev series order required for the fitting model is adaptively determined.

[0014] Step S9: Based on the determined series order and the phase and amplitude parameters of the corresponding harmonics, a dynamic fitting mathematical model of the metal rubber material is constructed using Chebyshev series as the function basis, so as to realize the unified modeling expression of its dynamic characteristics.

[0015] Furthermore, in step S1, the standard linear element is a standard linear helical spring; the time synchronization error and phase deviation parameters of the system are calibrated by analyzing the phase difference between the displacement response of the standard linear helical spring and the excitation signal.

[0016] Furthermore, the phase deviation parameter is calculated using the following formula: phase error φ calibrated Quantification:

[0017]

[0018] in, This represents the cross-power spectrum of displacement and force at the main excitation angular frequency ω. This indicates taking the imaginary part of a complex number, used to obtain the orthogonal components of the cross power spectrum. This indicates taking the real part of a complex number, used to obtain the in-phase component of the cross power spectrum;

[0019] The time synchronization error parameter is calculated using the following formula: time error Δt calibrated Quantification:

[0020]

[0021] Using time error Δt calibrated Interpolation corrections are applied to the force response data from subsequent tests.

[0022] Furthermore, in step S4, the displacement excitation error compensation model compensates for the component with a frequency twice the excitation angular frequency 2ω, compensating for the amplitude A2 and phase angle of the waveform. Calculated using the following formula:

[0023]

[0024]

[0025] Where N represents the number of discrete sampling points of the signal. This represents the frequency domain value corresponding to the second harmonic in the pre-test displacement spectrum. This represents the frequency index in the discrete Fourier transform spectrum corresponding to twice the excitation angular frequency 2ω.

[0026] Furthermore, in step S6, the preprocessing includes denoising, detrending, phase alignment, truncation normalization, and period consistency processing.

[0027] Further, in step S8, the adaptive determination of the Chebyshev series order required for the fitting model specifically involves:

[0028] Starting with the first harmonic corresponding to the fundamental frequency, the amplitude A of the j-th harmonic is compared sequentially. j Compared with the preset amplitude tolerance threshold ϵ r When A first appears j <ϵ r When the maximum effective harmonic order is determined to be j−1, this j−1 is determined as the order of the Chebyshev series.

[0029] Furthermore, in step S9, the dynamic fitting mathematical model of the metal-rubber material will incorporate the dynamic force f. MR Decomposed into elastic components Viscous damping component Non-viscous damping component The superposition of these elements is represented as:

[0030] .

[0031] Furthermore, the elastic component force Represented as:

[0032]

[0033] in, This represents the amplitude of the (2n-1)th harmonic. This represents the phase of the (2n-1)th harmonic. Let x represent a Chebyshev polynomial of the first kind, order 2n-1, where x represents displacement and x0 represents displacement amplitude.

[0034] Furthermore, the viscous damping component force Represented as:

[0035]

[0036] in, Indicates speed.

[0037] Furthermore, the nonviscous damping component force Represented as:

[0038]

[0039] in, This represents the amplitude of the 2nth harmonic. This represents the phase of the 2nth harmonic. This represents a Chebyshev polynomial of the second kind, order 2n-1. This represents a Chebyshev polynomial of the first kind, order 2n.

[0040] Compared with the prior art, the present invention has the following beneficial effects:

[0041] 1. This method can be integrated into existing testing software and control systems. It only requires two rounds of standardized testing procedures, namely "pre-test + formal test", to automatically obtain the dynamic response characteristic parameters of metal rubber under different frequency operating conditions. There is no need for repeated manual parameter adjustment or complex control strategy design, which significantly reduces the difficulty of test system integration and operation complexity.

[0042] 2. This method identifies waveform offset characteristics in displacement excitation through a pre-test process and constructs an error compensation waveform for reverse superposition correction, thereby achieving active compensation for the excitation input and effectively reducing the impact of rebound error and resonance distortion on the test results.

[0043] 3. This method is applicable to metal-rubber materials with various structural forms and process parameters. It uses Chebyshev series to construct mathematical models with clear physical meaning, which can handle complex dynamic behaviors with strong nonlinearity, strong hysteresis characteristics and high-order harmonic response. It has good adaptability and consistent modeling effect for metal-rubber components with different material systems, different geometries and different molding directions. Attached Figure Description

[0044] Figure 1 This is a flowchart illustrating the implementation of the method for dynamic testing error compensation and adaptive unified modeling of metal rubber based on harmonic characteristics provided in this embodiment of the invention. Detailed Implementation

[0045] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0046] 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.

[0047] 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.

[0048] like Figure 1 As shown in the figure, this embodiment provides a method for dynamic testing error compensation and adaptive unified modeling of metal rubber based on harmonic characteristics, and its specific implementation steps are as follows.

[0049] Step S1: Use standard linear components to calibrate the dynamic test system and obtain the system's time synchronization error and phase deviation parameters.

[0050] The standard linear element is a standard linear stiffness helical spring. The time synchronization error and phase deviation parameters of the system are calibrated by analyzing the phase difference between the displacement response of the standard linear helical spring and the excitation signal.

[0051] The phase deviation parameter is calculated using the following formula: phase error φ calibrated Quantification:

[0052]

[0053] in, This represents the cross-power spectrum of displacement and force at the main excitation angular frequency ω. This indicates taking the imaginary part of a complex number, used to obtain the orthogonal components of the cross power spectrum. This indicates taking the real part of a complex number, used to obtain the in-phase component of the cross power spectrum.

[0054] The time synchronization error parameter is calculated using the following formula: time error Δt calibrated Quantification:

[0055]

[0056] Using time error Δt calibrated Interpolation corrections are applied to the force response data from subsequent tests.

[0057] Step S2: Set the test parameters, conduct a preliminary test on the metal-rubber sample, and obtain the initial time-displacement response data and time-force response data.

[0058] Step S3: Perform Fourier transform on the time-displacement response data obtained from the pre-test to obtain the displacement spectrum, and extract the frequency characteristics of the main frequency component and each harmonic component.

[0059] Step S4: Based on the main frequency component and harmonic components, construct a displacement excitation error compensation model, correct the error of the displacement excitation signal, generate the compensated displacement driving waveform, and use it as the input excitation signal for the formal test.

[0060] The displacement excitation error compensation model compensates for the component at twice the excitation angular frequency 2ω, and the amplitude A2 and phase angle of the compensation waveform are calculated. Calculated using the following formula:

[0061]

[0062]

[0063] Where N represents the number of discrete sampling points of the signal. This represents the frequency domain value corresponding to the second harmonic in the pre-test displacement spectrum. This represents the frequency index in the discrete Fourier transform spectrum corresponding to twice the excitation angular frequency 2ω.

[0064] Step S5: Under the compensated displacement excitation conditions, conduct a formal dynamic test on the metal-rubber sample to obtain high-precision time-displacement response data and time-force response data.

[0065] Step S6: Preprocess the data obtained from the formal dynamic experiment to form a standardized analytical data sequence. The preprocessing includes denoising, detrending, phase alignment, truncation normalization, and period consistency processing.

[0066] Step S7: Perform Fourier transform on the standardized time-force response data to obtain the force spectrum, and simultaneously calculate the amplitude and phase information of the time-force response data and time-displacement response data at each harmonic to form a frequency domain characteristic parameter set.

[0067] Step S8: By setting an amplitude tolerance threshold, the harmonic components of each order in the force spectrum are screened. Based on the number of effective harmonics with amplitude components higher than the tolerance threshold, the Chebyshev series order required for the fitting model is adaptively determined.

[0068] Specifically, the adaptive determination of the Chebyshev series order required for the fitted model is as follows:

[0069] Starting with the first harmonic corresponding to the fundamental frequency, the amplitude A of the j-th harmonic is compared sequentially. j Compared with the preset amplitude tolerance threshold ϵ r When A first appears j <ϵ r When the maximum effective harmonic order is determined to be j−1, this j−1 is determined as the order of the Chebyshev series.

[0070] Step S9: Based on the determined series order and the phase and amplitude parameters of the corresponding harmonics, a dynamic fitting mathematical model of the metal rubber material is constructed using Chebyshev series as the function basis, so as to realize the unified modeling expression of its dynamic characteristics.

[0071] Specifically, the dynamic fitting mathematical model of the metal-rubber material will use the dynamic force f MR Decomposed into elastic components Viscous damping component Non-viscous damping component The superposition of these elements is represented as:

[0072] .

[0073] The elastic component force Represented as:

[0074]

[0075] in, This represents the amplitude of the (2n-1)th harmonic. This represents the phase of the (2n-1)th harmonic. Let x represent a Chebyshev polynomial of the first kind, order 2n-1, where x represents displacement and x0 represents displacement amplitude.

[0076] The viscous damping component Represented as:

[0077]

[0078] in, Indicates speed.

[0079] The non-viscous damping component Represented as:

[0080]

[0081] in, This represents the amplitude of the 2nth harmonic. This represents the phase of the 2nth harmonic. This represents a Chebyshev polynomial of the second kind, order 2n-1. This represents a Chebyshev polynomial of the first kind, order 2n.

[0082] This method, through the technical route of "system error calibration - excitation compensation - frequency domain feature extraction - adaptive series determination - Chebyshev series transformation", realizes the automation, standardization and model structure unification of the dynamic modeling process of metal rubber. It significantly reduces the dependence of modeling on a large number of empirical parameters and repeated experiments, and improves modeling efficiency, model stability and versatility across structural forms.

[0083] The implementation process of metal rubber modeling is described in further detail below.

[0084] (1) Perform a benchmark test on the helical spring sample and calibrate the time error of the test system.

[0085] During dynamic testing, especially under high-frequency conditions, the duration of a single cycle is on the order of milliseconds. This inevitably introduces time delays and phase shift errors in the data sampling and signal synchronization of the testing system, affecting test accuracy. To eliminate these systematic time errors, a benchmark test is first performed on a standard helical spring. Since the phase difference between the displacement response and the excitation signal of a rigid elastic element is theoretically zero under ideal conditions, phase analysis of the helical spring test results can be used to reverse-calibrate the time deviation and phase error introduced by the testing system. This yields the system time error parameters, providing a benchmark for subsequent data correction in dynamic testing of metal rubber.

[0086] To calculate the time error, we first calculate the Fourier transforms of displacement and force in the calibration test. The phase error φ between force and displacement is then calculated. calibrated The cross-power spectrum S of displacement and force data at the principal excitation angular frequency ω xf (ω) is calculated to yield:

[0087]

[0088] The time error Δt is obtained by calculating the phase error. calibrated :

[0089]

[0090] Time error Δt calibratedIt was used to correct the force data f(t) obtained from subsequent tests using interpolation methods. This represents the selected interpolation function, and the subscript "calibrated" indicates the corrected data.

[0091]

[0092] (2) Preliminary test

[0093] Because the metal rubber is made of woven metal wires, it has significant strong nonlinear stiffness characteristics. When working under high excitation force conditions, if the test frequency is close to the system's natural frequency, a resonance amplification effect is likely to occur, causing the displacement waveform excitation to deviate from the ideal sinusoidal excitation form, thus introducing significant test errors. Therefore, it is necessary to obtain the excitation error at various frequencies through a preliminary test. The frequency range and input displacement / force are set as test parameters in a conventional manner, and a preliminary test is conducted.

[0094] Taking time error Δt into account calibrated Subsequently, Fourier spectra were constructed for the time-displacement signals from the preliminary experiment. To calculate the excitation error, the DC flux was first removed to obtain the displacement after removing the DC flux. with force In the frequency domain, the discrete Fourier transform X of the displacement in the preliminary experiment is calculated. pre The discrete Fourier transform F of the force in the pre-test pre .

[0095] In the preliminary test, the angular frequency at which the amplitude reaches its maximum value in the displacement spectrum should be the excitation angular frequency ω, corresponding to a frequency of k. main :

[0096]

[0097] (3) By calculating and analyzing the vibration main frequency and its harmonic components, a displacement excitation error compensation model is constructed to correct the systematic deviation in the displacement excitation signal.

[0098] A corrected waveform is generated based on the compensation model and input to the drive mechanism of the test device as a control signal. The generated compensated waveform serves as the displacement excitation input signal in the formal test phase, so as to effectively suppress the test excitation error and improve the dynamic response accuracy.

[0099] As a preferred method, when calculating the displacement waveform error, it is only necessary to calculate the waveform when the angular frequency is twice the test frequency (the corresponding frequency is denoted as k2), and then generate compensation for the displacement error using the PWM method. The parameter calculation formula is as follows:

[0100]

[0101] In the formula, N represents the number of discrete sampling points of the signal. A2 represents the amplitude of the compensation waveform.

[0102] The phase angle at frequency k2 is calculated as follows:

[0103]

[0104] Based on the phase angle parameters, a composite sinusoidal displacement function model containing the main excitation angular frequency ω and a component of twice the excitation angular frequency 2ω is constructed. The amplitude of this displacement function is then normalized to obtain the target displacement function. This target displacement function serves as the standardized displacement input signal for the drive mechanism, used to generate the drive control waveform and implement formal experimental excitation.

[0105]

[0106] As a preferred method for establishing the excitation signal for the input drive mechanism, the PWM method is used, with a triangular wave as the carrier wave:

[0107]

[0108] During the formal testing phase, the input displacement function uses a PWM generation function processed by a low-pass filter (LPF) as the displacement excitation signal.

[0109]

[0110] (4) Formal test

[0111] The displacement excitation signal was input into the drive mechanism for formal testing.

[0112] (5) Experimental data processing

[0113] The collected displacement and force data are processed in a unified manner to eliminate the time error and phase deviation introduced by the test system. The displacement signal is also phase calibrated to adjust its initial phase to zero, thereby normalizing the signal starting point and aligning it with the time reference, providing a standardized data foundation for subsequent frequency domain analysis and model construction.

[0114] Including time error Δt calibrated Subsequently, the DC flux was removed from the displacement signal x[n] acquired in the formal experiment, and the signal spectrum was analyzed to establish a frequency domain transformation X. The frequency component with the largest amplitude in the amplitude spectrum was determined as the system's dominant frequency ω. The phase angle parameters when the angular frequency is the system's dominant frequency ω were then calculated.

[0115] Based on the time offset Δt and the time difference T between two adjacent sampling points s The number of cutoff points N is calculated. s :

[0116]

[0117] With the number of cutoff points N s As a time alignment reference, the acquired displacement and force data are truncated, retaining only the index located at N. s The subsequent valid data segments thus achieve signal start-up alignment and time base unification.

[0118] Since performing a new Discrete Fourier Transform after truncation still cannot guarantee that the phase is close enough to zero, a segment N of the data is selected to balance computational accuracy and efficiency. seg Perform a local Fourier transform again to calculate the local frequencies. , and local phase If the local phase is greater than the tolerance, the truncation continues, and this process is repeated until it is less than the tolerance.

[0119] The original data is denoted as:

[0120]

[0121] When the correction is applied to the force data, the resulting corrected force signal data and displacement signal data are denoted as:

[0122]

[0123] (6) Establish the Fourier spectrum of the time-force signal; calculate the phase and amplitude of each harmonic of the time-force signal and the time-displacement signal.

[0124] As a preferred method, for the truncated displacement data x align [τ], truncated and corrected force data f acali [τ] Applying the window function w[n] yields x zp [n] and f zp [n]. Then, to improve frequency resolution, zero-padding was applied to the data, expanding the data size to N. m indivual.

[0125] The displacement data is filled in as follows:

[0126]

[0127] For force data filling, the following is obtained:

[0128]

[0129] The frequency domain transform X is obtained by using the discrete Fourier transform on the filled displacement and force data. zp With F zp。

[0130] Calculate the gain error G introduced by the window function. ω The amplitude in the spectrum is corrected to obtain the corrected displacement amplitude A. X [k] and force amplitude A F [k]:

[0131]

[0132] Calculate the corresponding phase Φ in the spectrum X With Φ F Calculate the frequency increment Δf in the spectrum.

[0133] Based on the comparison results of the amplitude components of each harmonic in the spectrum with the preset tolerance threshold, the effective harmonic order j for modeling is determined, thereby adaptively determining the order of the series required for the fitting model. Assume that the amplitude at an excitation frequency ω that is j times the angular frequency is A. j The tolerance threshold is ε r When the following conditions are met:

[0134]

[0135] If it is determined to be the maximum order, frequency components less than j times ω are identified as effective harmonic components and included in the model construction.

[0136] (7) Generate a dynamic mechanical model for the metal rubber based on the series order number n, harmonic phase φ, harmonic amplitude A, displacement amplitude x0, and excitation angular frequency ω:

[0137]

[0138] Simplified trigonometric functions for calculating the formula of an nth-order Chebyshev series:

[0139]

[0140] According to relevant research on metallic rubber, its elastic force is related to an odd power function of displacement, and its viscous damping force is related to an odd power function of velocity. Therefore, the dynamic force f of metallic rubber... MR The terms of can be decomposed into the following components:

[0141]

[0142] Elastic component , representing the force related to elasticity in the dynamic vibration of metal rubber, the physical meaning of the coefficient is the strength of rigidity:

[0143]

[0144] Viscous damping component This represents the force positively correlated with the vibration velocity in the dynamic vibration of metallic rubber, and the physical meaning of the coefficient is the strength of viscosity:

[0145]

[0146] Non-viscous damping component This represents the force in the dynamic vibration of metallic rubber that is coupled with velocity and deformation, and its physical meaning is the degree of nonlinearity of the dynamic force.

[0147]

[0148] This model can quickly and accurately model metal rubber with different raw materials, structural forms, process parameters and molding directions, improving the efficiency of engineering design and data analysis.

[0149] This embodiment also provides a dynamic testing error compensation and adaptive unified modeling system for metal rubber based on harmonic characteristics, used to perform the above-described method.

[0150] This embodiment also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described method.

[0151] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0152] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0153] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0154] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0155] 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 error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics, characterized in that, Includes the following steps: Step S1: Calibrate the dynamic test system using standard linear components to obtain the system's time synchronization error and phase deviation parameters; Step S2: Set the test parameters, conduct a preliminary test on the metal-rubber sample, and obtain the initial time-displacement response data and time-force response data; Step S3: Perform Fourier transform on the time-displacement response data obtained from the pre-test to obtain the displacement spectrum, and extract the frequency characteristics of the main frequency component and each harmonic component. Step S4: Based on the main frequency component and harmonic components, construct a displacement excitation error compensation model, correct the error of the displacement excitation signal, generate the compensated displacement driving waveform, and use it as the input excitation signal for the formal test. Step S5: Under the compensated displacement excitation conditions, conduct a formal dynamic test on the metal-rubber sample to obtain high-precision time-displacement response data and time-force response data; Step S6: Preprocess the data obtained from the formal dynamic experiment to form a standardized analytical data sequence; Step S7: Perform Fourier transform on the standardized time-force response data to obtain the force spectrum, and simultaneously calculate the amplitude and phase information of the time-force response data and time-displacement response data at each harmonic to form a set of frequency domain characteristic parameters; Step S8: By setting an amplitude tolerance threshold, the harmonic components of each order in the force spectrum are screened. Based on the number of effective harmonics with amplitude components higher than the tolerance threshold, the Chebyshev series order required for the fitting model is adaptively determined. Step S9: Based on the determined series order and the phase and amplitude parameters of the corresponding harmonics, a dynamic fitting mathematical model of the metal rubber material is constructed using Chebyshev series as the function basis, so as to realize the unified modeling expression of its dynamic characteristics.

2. The method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics according to claim 1, characterized in that, In step S1, the standard linear element is a standard linear helical spring; the time synchronization error and phase deviation parameters of the system are calibrated by analyzing the phase difference between the displacement response of the standard linear helical spring and the excitation signal.

3. The method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics according to claim 2, characterized in that, The phase deviation parameter is calculated using the following formula: phase error φ calibrated Quantification: in, This represents the cross-power spectrum of displacement and force at the main excitation angular frequency ω. This indicates taking the imaginary part of a complex number, used to obtain the orthogonal components of the cross power spectrum. This indicates taking the real part of a complex number, used to obtain the in-phase component of the cross power spectrum; The time synchronization error parameter is calculated using the following formula: time error Δt calibrated Quantification: Using time error Δt calibrated Interpolation corrections are applied to the force response data from subsequent tests.

4. The method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics according to claim 1, characterized in that, In step S4, the displacement excitation error compensation model compensates for the component with a frequency twice the excitation angular frequency 2ω, compensating for the amplitude A2 and phase angle of the waveform. Calculated using the following formula: Where N represents the number of discrete sampling points of the signal. This represents the frequency domain value corresponding to the second harmonic in the pre-test displacement spectrum. This represents the frequency index in the discrete Fourier transform spectrum corresponding to twice the excitation angular frequency 2ω.

5. The method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics according to claim 1, characterized in that, In step S6, the preprocessing includes denoising, detrending, phase alignment, truncation normalization, and period consistency processing.

6. The method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics according to claim 1, characterized in that, In step S8, the adaptive determination of the Chebyshev series order required for the fitting model specifically involves: Starting with the first harmonic corresponding to the fundamental frequency, the amplitude A of the j-th harmonic is compared sequentially. j Compared with the preset amplitude tolerance threshold ϵ r When A first appears j <ϵ r When the maximum effective harmonic order is determined to be j−1, this j−1 is determined as the order of the Chebyshev series.

7. The method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics according to claim 1, characterized in that, In step S9, the dynamic fitting mathematical model of the metal-rubber material will use the dynamic force f MR Decomposed into elastic components Viscous damping component Non-viscous damping component The superposition of these elements is represented as: 。 8. The method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics according to claim 7, characterized in that, The elastic component force Represented as: in, This represents the amplitude of the (2n-1)th harmonic. This represents the phase of the (2n-1)th harmonic. Let x represent a Chebyshev polynomial of the first kind, order 2n-1, where x represents displacement and x0 represents displacement amplitude.

9. The method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics according to claim 7, characterized in that, The viscous damping component Represented as: in, Indicates speed.

10. The method for error compensation and adaptive unified modeling of dynamic testing of metal rubber based on harmonic characteristics according to claim 7, characterized in that, The non-viscous damping component Represented as: in, This represents the amplitude of the 2nth harmonic. This represents the phase of the 2nth harmonic. This represents a Chebyshev polynomial of the second kind, order 2n-1. This represents a Chebyshev polynomial of the first kind, order 2n.

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