A nonlinear modal test method based on adaptive feedforward control technology

Abstract

A nonlinear modal testing method based on adaptive feedforward control technology is characterized by the following steps: Step 1, establishing system and calibration connections; Step 2, setting reference force; Step 3, implementing an adaptive feedforward control strategy; Step 4, conducting nonlinear modal testing based on the above feedforward control strategy; and Step 5, acquiring surface reconstruction and fixed-frequency branch data based on controllable force spectrum data. This invention can suppress high-order harmonic injection and phase drift caused by coupled dynamics at the source, improve excitation purity, and enhance comparability and repeatability between operating conditions. It can stably obtain consistent, repeatable, and comparable target input force spectrum-target response data under multi-frequency and multi-amplitude operating conditions, improving the availability and reliability of unstable branches and critical boundaries near resonance, and providing a reliable experimental basis for nonlinear modal parameter identification and subsequent frequency domain model updates / multi-parameter corrections.

Description

Technical Field

[0001] This invention relates to control technology, and more particularly to an exciter-structure coupling control technology, specifically a nonlinear modal testing method based on adaptive feedforward control technology. Background Technology

[0002] Currently, nonlinear modal testing generally faces the following shortcomings in strongly nonlinear structures:

[0003] Firstly, due to the coupling between the exciter and the structure, single-frequency sinusoidal voltage drive cannot guarantee that the input force at the structure end is in the form of a single harmonic. The actual input force of the structure will change with the frequency and amplitude, resulting in phase shift and high-order harmonic pollution, which causes the external excitation conditions to deviate from the preset, introduces systematic errors, and weakens the accuracy and comparability of response prediction and modal parameter identification.

[0004] Secondly, while fixed-frequency testing can obtain S-curves and reveal unstable branches, the coupling effect can cause deviations in the measured location, width, and stability judgment of critical boundaries and unstable branches, reducing the comparability and repeatability of the results.

[0005] Third, the step-sine constant amplitude test and response surface reconstruction are highly sensitive to the purity of the excitation force spectrum. If the force spectrum is contaminated by coupled harmonics, the forced form under the "constant response" condition has changed, the geometric meaning of the response surface is weakened, and the error of the response surface slice reconstruction increases. Therefore, it is necessary to improve it. Summary of the Invention

[0006] The purpose of this invention is to address the problems of exciter-structure coupling leading to distortion of input force at the structural end, incomparable data, and difficulty in stably obtaining unstable branches in nonlinear modal testing of strongly nonlinear structures. The invention proposes a nonlinear modal testing method based on adaptive feedforward control technology, which is a nonlinear modal parameter acquisition method that integrates excitation force purification based on adaptive feedforward elimination control with step sine / fixed frequency testing.

[0007] The technical solution of this invention is:

[0008] A nonlinear modal testing method based on adaptive feedforward control technology, characterized by the following steps:

[0009] Step 1: Establish system and calibration connections:

[0010] Step 2, Setting the Reference Force:

[0011] Step 3: Adaptive Feedforward Control Strategy

[0012] Step 4: Nonlinear modal testing based on the above feedforward control strategy;

[0013] Step 5: Surface reconstruction and fixed-frequency branch data acquisition based on controllable force spectrum data.

[0014] The beneficial effects of this invention are:

[0015] This invention expresses the input voltage as a truncated Fourier series synchronized with the excitation frequency and updates the adaptive coefficients, making the structural input force distorted by coupling effects approximate a preset reference. This suppresses the injection of high-order harmonics and phase drift caused by coupling dynamics at the source, improving excitation purity and enhancing comparability and repeatability between operating conditions. Furthermore, this control strategy is organically combined with step-sinusoidal steady-state acquisition logic and fixed-frequency amplitude sweep, stably obtaining consistent, repeatable, and comparable target input force spectrum-target response data under multi-frequency and multi-amplitude operating conditions. This improves the availability and reliability of unstable branches and critical boundaries near resonance, providing a reliable experimental basis for nonlinear modal parameter identification and subsequent frequency domain model updates / multi-parameter corrections. Attached Figure Description

[0016] Figure 1 This is a flowchart of the feedforward control strategy of the present invention.

[0017] Figure 2 A schematic diagram of the feedforward control compensation force drop phenomenon of the present invention.

[0018] Figure 3 This is a comparison chart of the system response with and without the feedforward control strategy of this invention.

[0019] Figure 4 This is a schematic diagram of the harmonic analysis comparison results of the exciter-structure coupling system of the present invention. In the figure: (a) open-loop uncontrolled system and (b) adaptive feedforward control system. Detailed Implementation

[0020] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0021] The structures, proportions, and sizes illustrated in the accompanying drawings are merely for illustrative purposes and to aid those skilled in the art in understanding and reading the invention. They are not intended to limit the scope of the invention and therefore have no substantial technical significance. Any modifications to the structure, changes in proportions, or adjustments to size, without affecting the effectiveness and purpose of the invention, should still fall within the scope of the technical content disclosed herein. Furthermore, terms such as "upper," "lower," "left," "right," and "middle" used in this specification are merely for clarity and not intended to limit the scope of implementation. Changes or adjustments to their relative relationships, without substantially altering the technical content, should also be considered within the scope of the invention's implementation.

[0022] like Figure 1-4 As shown.

[0023] A nonlinear modal testing method based on adaptive feedforward control technology, such as Figure 1 As shown, it includes the following steps:

[0024] Step 1: System Composition and Connection Calibration

[0025] The nonlinear modal testing and force control system of this invention consists of an excitation link, a measurement link, a control and acquisition link, and a connecting fixture, forming a coupled system of exciter-connector-tested structure. The excitation link includes an electric exciter and a power amplifier; the power amplifier outputs a control voltage to drive the exciter to apply excitation. The measurement link includes force measurement and response measurement. Force measurement preferably uses an impedance head / force sensor installed at the connection between the push rod and the structure to obtain the actual input force F(t) at the structural end. Response measurement can use laser displacement, acceleration, or strain sensors to obtain the response at the driving point / key measurement point. The control and acquisition link consists of a control and data acquisition unit, used to perform adaptive feedforward control coefficient updates, step sine waves, and fixed-frequency sweep process management.

[0026] Step 2, Setting the Reference Force:

[0027] This step is used to set the structural end reference input force at each test frequency, providing a target for subsequent feedforward control. At each test frequency f, the structural end reference input force F is defined. ref (t). The reference force can be targeted only at the fundamental frequency, or it can be extended to target multiple harmonics to achieve force spectrum purification and shaping. For the fundamental frequency only:

[0028]

[0029] Where A1 is the target fundamental wave force amplitude, and ϕ1 is the target phase (phase not controlled by default, phase controlled if necessary). For multi-harmonic propagation:

[0030]

[0031] Where h=1 represents the fundamental frequency, and h≥2 represents higher harmonics. In engineering, a "fundamental frequency as target + higher harmonics as upper limit" approach can be adopted, that is, given h1 as the target value, while simultaneously constraining... This is to reduce total harmonic distortion (THD) and ensure consistent external excitation conditions under different operating conditions.

[0032] Step 3: Adaptive Feedforward Control Strategy

[0033] Power amplifier input / drive voltage Use preset voltage and compensation voltage It is expressed as the sum of its components.

[0034]

[0035] The compensation voltage is expressed as a linear combination of sine and cosine basis functions synchronized with the reference frequency using a truncated Fourier series:

[0036]

[0037] Among them, 𝛼 ℎ , 𝛽 ℎ These are adaptive coefficients that are updated online.

[0038] The aforementioned driving voltage is applied at each frequency point within the set test frequency range to drive the exciter. Due to the coupling effect of the exciter-connector-structure, the actual input force at the structural end may deviate from the preset reference in amplitude, phase, and harmonic components. Therefore, it is necessary to synchronously acquire the force signal F(t) and compare it with the reference force spectrum to construct the error, which serves as the basis for subsequent adaptive coefficient updates. Among these, synchronous detection / FFT is used to calculate the harmonic index and tolerance band criterion; the adaptive coefficient update is preferably implemented by correlation calculation of time-domain error and sine and cosine basis functions.

[0039] Real-time acquisition of the actual input force at the structural end; selection of a steady-state window (window length 𝑇) containing an integer number of excitation cycles at the current frequency. 𝑤 =𝑁 𝑝 𝑇, where 𝑇=2𝜋 / 𝜔, 𝑁 𝑝 (Number of periods). Perform synchronous detection / FFT on the signal within the window to extract the complex amplitude value of the ℎth harmonic:

[0040]

[0041] The corresponding reference force spectrum complex amplitude is defined as:

[0042]

[0043] Based on this, harmonic errors (in complex form, for easy uniform handling of amplitude and phase) are constructed:

[0044]

[0045] To simultaneously consider both multi-harmonic objectives and suppression requirements, a weighted L2 objective function is preferably constructed:

[0046]

[0047] Among them, 𝑤 ℎAs weights, 𝑤1 is used to ensure fundamental frequency tracking accuracy, and weights ℎ≥2 are used to enhance harmonic suppression.

[0048] The entry into the tolerance band criterion can be defined as: within a continuous N... hold Within a steady-state window, simultaneously satisfying

[0049]

[0050] After satisfying the tolerance band, wait for N. delay Each excitation cycle enters the final steady-state acquisition window to avoid contamination of the frequency response function / harmonic parameters by transient residuals. A time-domain error signal is constructed in the k-th steady-state window:

[0051]

[0052] For each harmonic coefficient, a correlation-based / LMS-based adaptive update is used (the most common method for discrete implementation). Let the sampling points within the window be... Sampling period 𝑠 The corresponding adaptive coefficient is:

[0053]

[0054]

[0055] Among them, 𝜇 ℎ To update the step size (gain), used to balance convergence speed and stability; can reduce α when there is strong nonlinearity, near the jump boundary, or significant harmonic enhancement. ℎ In addition, coefficient limiting / voltage limiting is used to avoid coefficient oscillation or divergence. Normalized LMS or adaptive scaling of the step size based on the measured force amplitude is preferred to avoid convergence speed differences at different frequencies / amplitude levels. The above updates are performed iteratively in "window" units until the tolerance band criterion is met and the steady-state acquisition phase is entered.

[0056] Step 4: Nonlinear modal testing based on the above feedforward control strategy:

[0057] This section will describe the nonlinear modal testing process combining adaptive feedforward control strategy with stepped sinusoidal response amplitude control. This step superimposes the stepped sinusoidal test process on the "feedforward force control inner loop" to achieve consistent steady-state force input at each frequency point, thereby outputting comparable frequency response function (FRF) and harmonic characteristics. When using response amplitude controlled stepped sinusoidal (RCT), a "response outer loop" is further introduced to keep the force spectrum controllable while maintaining a constant target response amplitude, thereby improving the feasibility and efficiency of testing under strongly nonlinear conditions.

[0058] In the step sine test, the selected frequency sequence is 𝑓 𝑖 (Frequency step size can be set as needed). For each frequency point, execute the following logic: First, maintain the sinusoidal excitation of the current frequency point and run the adaptive feedforward update in step three to ensure that the force error enters and remains within the tolerance band; then, enter the steady-state acquisition stage, that is, after the error meets the tolerance band criterion, continue to wait for N. delay One incentive cycle, then in N consecutive cycles avg The frequency domain parameters of a given frequency point are acquired and calculated within each excitation cycle, and then the process is advanced to the next frequency point. For ease of implementation, this invention defines the "steady-state window acquisition" as an integer period window to avoid spectral leakage affecting FRF estimation and harmonic parameters.

[0059] Within the steady-state acquisition window, extract the fundamental wave and necessary order harmonic complex amplitude values ​​of the structural end forces and responses. , It outputs amplitude, phase, and THD indicators as unified data input for subsequent response surface construction, fixed-frequency S-curve determination, and nonlinear modal parameter identification. When using Response Amplitude Stepped Sine (RCT), an outer loop control is added to the above "frequency point advancement - steady-state acquisition" framework: a target response fundamental amplitude Xtar is set (multiple amplitude levels can be set), and at each frequency point, the reference force fundamental target 𝐴1 is adjusted to make the response fundamental amplitude enter the tolerance band. At the same time, the feedforward force control inner loop still runs according to step three to ensure that the actual input force spectrum at the structural end meets the preset. The error of the outer loop of the response can be defined as:

[0060]

[0061] To achieve a simple and controllable propulsion strategy, the target force amplitude can be updated using a discrete iterative approach (example format):

[0062]

[0063] Where σ is the outer loop gain, and σ represents the number of outer loop iterations. The result is σ. 𝑋 After the tolerance band is established, the process of "delay period + steady-state window acquisition" is the same as that of the step sine wave, thus obtaining the FRF points and stress spectrum data under constant response conditions. Through the structure of "outer loop control of response, inner loop purification of stress spectrum," the drift caused by external excitation conditions due to relying solely on response amplitude control can be avoided, thereby improving the comparability and repeatability of data at different response levels. Therefore, at each frequency point, its corresponding {ω, , The fundamental wave data points and indicators such as THD are used to form a unified dataset that can be used for surface reconstruction and nonlinear modal recognition.

[0064] Step 5: Surface reconstruction and fixed-frequency branch data acquisition based on controllable force spectrum data:

[0065] Data points {𝑓,} were collected at multiple target response levels. , The force-frequency-response mapping surface is obtained through interpolation / regression fitting, for example:

[0066]

[0067] During the fitting process, outlier removal and smoothing constraints on the frequency direction can be used to improve the stability of the slices and their reproducibility across operating conditions. After constructing the response surface, it can be sliced ​​according to different constraints to obtain two-dimensional curves: for example, fixing the excitation force level yields the corresponding "amplitude-frequency curve", and fixing the response amplitude yields the corresponding "force-frequency curve". Repeated slicing for different force levels or different response levels can form a set of comparable feature curves for subsequent identification and correction.

[0068] This invention constructs a response surface based on stepped sinusoidal amplitude data and obtains a set of frequency-varying characteristic curves through different slices to reflect the overall amplitude-frequency behavior of the system on stable branches. However, for strongly nonlinear structures, frequency-advancing tests alone are often insufficient to reliably cover the jump region and multivalued solution region near resonance, especially unstable branches and their critical boundaries, which require gradual changes in excitation level near a fixed frequency for more reliable capture. Therefore, this invention introduces a fixed-frequency amplitude sweep test as a supplement while maintaining controllable feedforward force spectrum: by gradually changing the excitation level at a selected frequency, the S-curve, jump threshold, and unstable branch-related data are directly obtained, thus complementing the constructed surface slice results and jointly supporting subsequent parameter identification.

[0069] The target fundamental wave force level is gradually changed at a fixed frequency, and the steady-state fundamental wave amplitude of the actual force and actual response is collected after each convergence, and plotted. The relationship is used to obtain the fixed-frequency S-curve; at the same time, the critical moment when the jump occurs is recorded by up-scan / down-scan. and Correspondingly, information related to the jump boundary and unstable branch is obtained. This dataset can be directly used for skeleton line extraction, nonlinear modal parameter identification, and subsequent correction of the dry friction damping multi-parameter model. Since coupling effects may cause deviations between the settings and actual conditions, the measured actual values ​​are used instead. Using the horizontal axis as the axis ensures the comparability and reproducibility of the S-curve with the critical threshold. This dataset can be directly used for skeleton line extraction, nonlinear modal parameter identification, and subsequent correction of multi-parameter dry friction damping models.

[0070] Details are as follows:

[0071] The following study focuses on a blade-shaped cantilever beam-exciter coupled structure system. The adaptive feedforward elimination control proposed in this invention is applied to purify the input force at the structural end and obtain comparable nonlinear modal test data. The system consists of a power amplifier, an electric exciter, a push rod connector, and a blade-shaped cantilever beam. A force sensor is placed at the connection between the push rod and the structure to measure the actual input force F(t) at the structural end, and an acceleration sensor is placed at the drive point to measure the response x(t). The frequency scanning range is set to 40–65 Hz, and the reference input force at the structural end is set as the fundamental target F. ref (t). The power amplifier drive voltage is expressed as a truncated Fourier form synchronized with the excitation frequency (harmonic order is taken as 5). For each frequency point, the force spectrum of each order is extracted by synchronous detection / FFT within an integer period steady-state window. The feedforward coefficients (α) are iteratively updated according to the error index. h ,β h Once the error enters the tolerance band and remains there, steady-state acquisition begins, outputting the fundamental force, fundamental response, and FRF data for that frequency point. Comparing the "no control" and "feedforward control" operating conditions, the results are as follows: Figure 2 , Figure 3 As shown. Without feedforward control, the actual input force at the structural end near resonance shows a significant drop, indicating that the exciter-structure coupling causes the structural end force to fail to maintain the target. With feedforward control, the input force at the structural end stably tracks the reference force level across the entire frequency band, and the force drop near resonance is significantly suppressed, achieving force spectrum purification. Figure 2 For FRF comparison, compared with the input force distortion in the uncontrolled state and the broadening and distortion phenomenon in the resonance region, the input force conditions at each frequency point of the structure are consistent under the feedforward control strategy, and the FRF curve is more stable, clearer and more repeatable, which can be used for subsequent nonlinear modal parameter identification.

[0072] Figure 4 The results show a comparison of voltage, excitation force, and displacement responses under uncontrolled (open-loop) and adaptive feedforward control conditions, which can clearly reflect the changes in excitation quality before and after coupling compensation.

[0073] like Figure 4 As shown in (a), under open-loop uncontrolled conditions, the input voltage is essentially sinusoidal. However, due to the significant coupling between the exciter and the structure, the output excitation force no longer maintains the ideal single-frequency characteristics. The time-domain curve shows significant distortion, and the frequency-domain results contain third and fifth harmonic components in addition to the fundamental frequency. This indicates that the structural reaction is fed back to the excitation end through electromechanical coupling, causing the excitation force to be "contaminated." This contamination is further transmitted to the structural response, resulting in a certain noise component remaining in the displacement spectrum besides the fundamental frequency. This causes the experimental input to deviate from the expected pure harmonic excitation, making it difficult to accurately identify the system's intrinsic nonlinearity.

[0074] The harmonic analysis results after introducing the adaptive feedforward control strategy are as follows: Figure 4 As shown in (b), the voltage signal has been adjusted from a single sine wave to a non-sine wave containing a compensation term. The controller does not simply suppress the output, but actively constructs a compensation input that cancels out the coupling disturbance by iteratively updating the control coefficients in the voltage signal. Although the voltage waveform is more complex, the excitation force time-domain curve is significantly more regular, the fundamental frequency component dominates in the frequency domain, and higher harmonics are effectively suppressed, indicating that the force output by the exciter to the structure after control is closer to the ideal pure excitation.

[0075] From the perspective of displacement response, the position of the system's principal response frequency did not change fundamentally before and after control, indicating that feedforward control did not disrupt the original structural dynamics or artificially alter the system's inherent nonlinear mechanism. On the contrary, due to the reduced coupling contamination, the structural response reflects more of the specimen's true nonlinear characteristics than the additional distortion introduced by the exciter-structure coupling. Therefore, with adaptive feedforward control, the experimental force input is "cleaner," and the response results are closer to the actual forced vibration state of the structure.

[0076] As can be seen from the above examples, the method of the present invention can be implemented by relying only on conventional experimental hardware and real-time acquisition and calculation. It can effectively suppress force distortion near resonance and improve the comparability of FRF in the blade cantilever beam-exciter coupling system, thus proving that the patent has clear engineering feasibility.

[0077] The above embodiments are merely preferred embodiments of the present invention. It should be noted that those skilled in the art can make several improvements and equivalent substitutions without departing from the principle of the present invention. All such improvements and equivalent substitutions to the claims of the present invention fall within the protection scope of the present invention.

[0078] All parts not covered in this invention are the same as or can be implemented using existing technologies.

Claims

1. A nonlinear modal testing method based on adaptive feedforward control technology, characterized by: Includes the following steps: Step 1: Establish system and calibration connections: The system consists of an excitation link, a measurement link, a control and acquisition link, and connecting fixtures, forming a coupled system of exciter-connector-structure under test. The excitation link includes an electric exciter and a power amplifier, with the power amplifier outputting a control voltage to drive the exciter to apply excitation. The measurement link includes force measurement and response measurement. Force measurement preferably uses an impedance head / force sensor installed at the connection between the push rod and the structure to obtain the actual input force F(t) at the structure end. Response measurement uses laser displacement, acceleration, or strain sensors to obtain the response at the drive point / key measurement point. The control and acquisition link consists of a control and data acquisition unit, used to perform adaptive feedforward control coefficient updates, step sine and fixed-frequency sweep process management. Step 2, Setting the Reference Force: This is used to set the structural end reference input force at each test frequency point, providing a target for subsequent feedforward control; at each test frequency point f, the structural end reference input force F is defined. ref (t); The reference force is set only for the fundamental frequency, or extended to multiple harmonic targets to achieve force spectrum purification and shaping; when the fundamental frequency is set only: Where A1 is the target fundamental wave power amplitude, and ϕ1 is the target phase; For targets with multiple harmonics: Where h=1 represents the fundamental frequency, and h≥2 represents higher harmonics; in engineering, the approach of "setting the target value for the fundamental frequency + setting the upper limit for higher harmonics" is adopted, that is, giving H1 as the target value, while simultaneously constraining... To reduce total harmonic distortion (THD) and ensure consistent external excitation conditions under different operating conditions; Step 3: Adaptive Feedforward Control Strategy Power amplifier input / drive voltage Use preset voltage and compensation voltage Represented by the sum of; The compensation voltage is expressed as a linear combination of sine and cosine basis functions synchronized with the reference frequency using a truncated Fourier series: Among them, 𝛼 ℎ , 𝛽 ℎ These are adaptive coefficients that are updated online. The aforementioned driving voltage is applied at each frequency point within the set test frequency range to drive the exciter to load. Due to the coupling effect of the exciter-connector-structure, the actual input force at the structural end deviates from the preset reference in terms of amplitude, phase, and harmonic components. Therefore, it is necessary to synchronously acquire the force signal F(t) and compare it with the reference force spectrum to construct the error, which serves as the basis for subsequent adaptive coefficient updates. Among these, synchronous detection / FFT is used to calculate the harmonic index and tolerance band criterion. The adaptive coefficient update is achieved by using the correlation calculation between the time domain error and the sine and cosine basis functions. The actual input force at the structural end is acquired in real time, and a steady-state window containing an integer number of excitation cycles is selected at the current frequency, with a window length of 𝑇. 𝑤 =𝑁 𝑝 𝑇, where 𝑇=2𝜋 / 𝜔, 𝑁 𝑝 (where the number of periods is the threshold); perform synchronous detection / FFT on the signal within the window to extract the complex amplitude value of the ℎth harmonic: The corresponding reference force spectrum complex amplitude is defined as: Based on this, harmonic errors are constructed: To simultaneously consider both multi-harmonic objectives and suppression requirements, a weighted L2 objective function is constructed: Among them, 𝑤 ℎ As weights, 𝑤1 is used to ensure fundamental frequency tracking accuracy, and weights ℎ≥2 are used to enhance harmonic suppression; The entry into the tolerance band criterion can be defined as: within a continuous N... hold Simultaneously satisfying the following within a steady-state window: After satisfying the tolerance band, wait for N. delay The excitation cycle enters the final steady-state acquisition window to avoid the contamination of the frequency response function / harmonic parameters by transient residuals; the time-domain error signal is constructed in the k-th steady-state window: For each harmonic coefficient, a correlation / LMS adaptive update is used; the sampling points within the window are set. Sampling period 𝑠 The corresponding adaptive coefficient is: Among them, 𝜇 ℎ To update the step size, balancing convergence speed and stability; can be reduced when there is strong nonlinearity, near the jump boundary, or significant harmonic enhancement. ℎ In addition, coefficient limiting / voltage limiting is used to avoid coefficient oscillation or divergence; the update is performed iteratively in "window" units until the tolerance band criterion is met and the steady-state acquisition stage is entered; Step 4: Nonlinear modal testing based on the above feedforward control strategy; Step 5: Surface reconstruction and fixed-frequency branch data acquisition based on controllable force spectrum data: Data points {𝑓,} were collected at multiple target response levels. , The force-frequency-response mapping surface is obtained through interpolation / regression fitting: During the fitting process, outlier removal and smoothing constraints on the frequency direction can be used to improve the stability of the slices and the reproducibility across operating conditions. After constructing the response surface, it is sliced ​​according to different constraints to obtain two-dimensional curves. A set of characteristic curves that "change with frequency" is obtained through different slices to reflect the overall amplitude-frequency law of the system on the stable branch. However, for strongly nonlinear structures, it is often difficult to stably cover the jump region and multi-valued solution region near resonance by relying solely on frequency-progressive testing. Unstable branches and their critical boundaries need to be captured more reliably by gradually changing the excitation level near a fixed frequency. Based on this, while keeping the feedforward force spectrum controllable, a fixed-frequency amplitude sweep test is introduced as a supplement: the excitation level is gradually changed at the selected frequency to directly obtain the S-curve, jump threshold, and unstable branch related data, thus complementing the constructed surface slicing results and jointly supporting subsequent parameter identification. The target fundamental wave force level is gradually changed at a fixed frequency, and the steady-state fundamental wave amplitude of the actual force and actual response is collected after each convergence, and plotted. The relationship is used to obtain the fixed-frequency S-curve; at the same time, the critical moment when the jump occurs is recorded by up-scan / down-scan. and Correspondingly, information related to the jump boundary and unstable branch is obtained; this dataset is directly used for skeleton line extraction, nonlinear modal parameter identification, and subsequent dry friction damping multi-parameter model correction; due to the coupling effect causing deviation between the setting and the actual situation, the measured actual situation is used. Using the horizontal axis, the comparability and reproducibility of the S-curve and the critical threshold are ensured; this dataset is directly used for skeleton line extraction, nonlinear modal parameter identification, and subsequent dry friction damping multi-parameter model correction.

2. The method according to claim 1, characterized in that: To avoid coefficient oscillations or divergence, normalized LMS or adaptive scaling of the step size based on the measured force amplitude can be used to avoid convergence speed differences at different frequencies / amplitude levels.

3. The method according to claim 1, characterized in that: The nonlinear modal testing process, which combines adaptive feedforward control strategy with step sine response amplitude control, superimposes the step sine test process on the basis of "feedforward force control inner loop" to achieve consistent steady-state force input at each frequency point, thereby outputting comparable frequency response function (FRF) and harmonic characteristics. When using Response Controlled Amplitude Stepped Sine Wave (RCT), an "outer response loop" is introduced to maintain a constant target response amplitude while keeping the force spectrum controllable, thereby improving the feasibility and efficiency of testing under strongly nonlinear conditions. In the stepped sine wave test, a frequency sequence ω is selected. 𝑖 For each frequency point, the following logic is followed: First, maintain the sinusoidal excitation at the current frequency point and run the adaptive feedforward update in step three to ensure that the force error enters and remains within the tolerance band; then, enter the steady-state acquisition stage, that is, after the error meets the tolerance band criterion, continue to wait for N. delay One incentive cycle, then in N consecutive cycles avg The frequency domain parameters of each frequency point are collected and calculated within each excitation cycle, and then the process is advanced to the next frequency point. The "steady-state window acquisition" is defined as an integer period window to avoid spectral leakage affecting FRF estimation and harmonic parameters. Within the steady-state acquisition window, the fundamental wave and necessary order harmonic complex amplitude values ​​of the structural end forces and responses are extracted. , It outputs amplitude, phase, and THD indicators as unified data inputs for subsequent response surface construction, fixed-frequency S-curve determination, and nonlinear modal parameter identification. When using Response Amplitude Stepped Sine (RCT), an outer loop response control is added to the above "frequency point advancement - steady-state acquisition" framework: a target response fundamental amplitude Xtar is set, and at each frequency point, the reference force fundamental target φ1 is adjusted to make the response fundamental amplitude enter the tolerance band. At the same time, the inner loop of the feedforward force control still runs according to step three to ensure that the actual input force spectrum at the structural end meets the preset. The error of the outer loop response can be defined as: To achieve a simple and controllable propulsion strategy, a discrete iterative approach is used to update the target force amplitude: Where 𝜅 is the outer loop gain, and 𝑚 represents the number of outer loop iterations; reaching 𝑒 𝑋 After the tolerance band, it enters the same "delay period + steady-state window acquisition" process as the stepping sine wave, thereby obtaining the FRF points and stress spectrum data under constant response conditions; through the structure of "outer loop control response, inner loop purification stress spectrum", the drift of external excitation conditions caused by relying solely on response amplitude control is avoided, thereby improving the comparability and repeatability of data at different response levels; Therefore, at each frequency point, the corresponding {𝑓, , The fundamental wave data points and indicators such as THD are used to form a unified dataset that can be used for surface reconstruction and nonlinear modal recognition.

4. The method according to claim 1, characterized in that: The two-dimensional curves are obtained by using a fixed excitation force level to obtain the corresponding "amplitude-frequency curve" and a fixed response amplitude to obtain the corresponding "force-frequency curve". By repeatedly slicing for different force levels or different response levels, a set of comparable feature curves are formed for subsequent identification and correction.

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