Transient strong nonlinear characteristic signal time domain analysis method

Abstract

The invention discloses a transient strong nonlinear characteristic signal time domain analysis method, and relates to the field of nonlinear dynamic signal analysis. An existing analysis method based on a time domain or a frequency domain is difficult to describe remarkable nonlinear evolution characteristics in strong impact response, the sensitivity of a system to initial conditions and a mechanism that a dynamic state changes along with time cannot be quantitatively revealed, and the similarity of a model test and a prototype test is difficult to effectively evaluate on a nonlinear level. According to the method, the strong impact response time domain signal is subjected to preprocessing and stage division, the maximum Lyapunov index and the maximum Hurst index are calculated in stages, the evolution law of the maximum Lyapunov index and the maximum Hurst index along with the change of stages and scale ratios is analyzed, a nonlinear dynamic state plane is constructed, and quantitative analysis of strong nonlinear response evolution paths and model similarity is achieved. The method is suitable for structural dynamics analysis, model test verification and engineering safety evaluation under strong impact working conditions such as explosion and collision.

Description

Technical Field

[0002] This research relates to the field of nonlinear dynamic signal analysis, specifically a time-domain analysis method for transient, strongly nonlinear, and nonstationary dynamic response signals generated by a structure under strong impact loads. Background Technology

[0003] In engineering fields such as shipbuilding, marine engineering, aerospace, and vehicle collision, when structures are subjected to strong impact loads such as explosions and collisions, their dynamic response signals typically exhibit significant transient, strong nonlinear, and non-stationary characteristics. This type of response is highly sensitive to initial conditions, boundary conditions, and system parameters; even small perturbations can trigger abrupt changes, bifurcation, or even chaotic phenomena in dynamic behavior, and its evolution process has significant uncertainties.

[0004] For the analysis of impact response signals, current research and engineering practice mainly employ time-domain and frequency-domain analysis methods within the framework of linear system theory. For example, this involves comparing the peak value, time history envelope, and attenuation characteristics of acceleration or strain signals, or comparing and analyzing results from different experiments or models based on Fourier transforms and self-power spectral density. These methods are applicable to certain stationary linear or weakly nonlinear response conditions and can reflect the overall energy distribution and dominant frequency characteristics of the structure; therefore, they are widely used in conventional vibration analysis and engineering verification.

[0005] In addition, some studies have attempted to introduce time-frequency analysis methods such as short-time Fourier transform and wavelet analysis to enhance the ability to describe the local characteristics of non-stationary signals. However, these methods are essentially still based on the assumptions of linear superposition and determinism, and their analysis results mainly reflect the energy change characteristics of the signal in the time or frequency dimension, making it difficult to reveal the inherent nonlinear dynamic mechanism of the system.

[0006] Under severe impact conditions, structural dynamic responses often exhibit significant initial value sensitivity and path dependence. The system may undergo a complex evolution from a strongly nonlinear transient response to a weakly nonlinear or quasi-periodic state. Current technologies generally lack methods to quantitatively characterize this evolution process, failing to effectively distinguish the dynamic states corresponding to different stages of the response signal, and further struggling to quantify the system's sensitivity to initial conditions and the persistence and memory characteristics of the response sequence. In model and prototype testing, existing methods often rely on waveform similarity or spectral consistency as criteria, making it difficult to determine whether the model truly reflects the prototype behavior at the nonlinear dynamic mechanism level. This is especially problematic under strongly nonlinear conditions, easily leading to misjudgments and posing risks to engineering predictions and safety assessments.

[0007] In summary, existing technologies lack a method for quantitatively analyzing transient, highly nonlinear, and nonstationary dynamic behavior in stages based directly on measured response signals under strong impact loads. This makes it difficult to reveal the system's sensitivity to initial conditions, nonlinear intensity, and evolution path, and also prevents effective evaluation of the similarity between prototype and model test results at the nonlinear mechanism level. Summary of the Invention

[0008] To address the existing technical problems of lacking a method for directly and quantitatively analyzing transient, highly nonlinear, and nonstationary dynamic behavior in stages based on measured response signals under strong impact loads, making it difficult to reveal the system's sensitivity to initial conditions, nonlinear intensity, and evolution path, and also failing to effectively assess the similarity between prototype and model test results at the nonlinear mechanism level, the technical solution provided by this invention is as follows: A time-domain analysis method for transient strongly nonlinear characteristic signals includes: The steps include acquiring the time-domain signal of the transient dynamic response of the structure under strong impact load, filtering and denoising the signal, and performing baseline correction to obtain a stable discrete-time series as the analysis input. Based on the dynamic characteristics of the impact response, the complete time-domain signal is divided into an early strong nonlinear stage, a non-stationary transition stage, and a late quasi-periodic stage according to the discrete-time series, and the corresponding sub-sequences of each stage are extracted as the objects of subsequent analysis. The steps of performing phase space reconstruction based on each stage subsequence and calculating the maximum Lyapunov exponent are used to quantify the system’s sensitivity to initial conditions and nonlinear intensity. The step of calculating the Hurst exponent for each stage subsequence while keeping the stage division results unchanged is to characterize the persistence and memory features of the response signal. The steps for analyzing the evolution of the structural dynamic response from a transient strongly nonlinear state to a quasi-periodic deterministic state under strong impact based on the maximum Lyapunov exponent and Hearst exponent obtained at each stage. The steps of constructing a nonlinear dynamic state plane using the inverse of the structural scaling ratio as the independent variable and the maximum Lyapunov exponent and Hearst exponent corresponding to each stage as state variables are used to realize the similarity assessment between the prototype and the scaling model at the nonlinear dynamic mechanism level.

[0009] Furthermore, in a preferred embodiment, the transient dynamic response time-domain signal is the acceleration signal or strain signal acquired by the structure under strong impact load, and the preprocessed discrete time series is used as unified input data for subsequent stage division and nonlinear feature analysis.

[0010] Furthermore, in a preferred embodiment, the stage division is based on the amplitude variation characteristics, frequency distribution characteristics, and fluctuation stability characteristics of the response signal to determine the start and end times of the early strong nonlinear stage, the non-stationary transition stage, and the later quasi-periodic stage.

[0011] Furthermore, in a preferred embodiment, the calculation of the maximum Lyapunov exponent includes reconstructing the phase space of each stage subsequence and constructing a multidimensional phase space trajectory that reflects the dynamic behavior of the system by determining the time delay and the embedding dimension.

[0012] Furthermore, in a preferred embodiment, the maximum Lyapunov exponent is obtained by tracking the divergence of adjacent orbits in phase space over time and fitting their average divergence rate, and is used to characterize the system's sensitivity to initial conditions and nonlinear intensity.

[0013] Furthermore, in a preferred embodiment, the Hearst exponent is obtained by performing multi-scale rescaled range analysis on the subsequences of each stage, and is used to characterize the persistence, memory and long-range correlation features of the response signal.

[0014] Based on the same inventive concept, the present invention also provides a time-domain analysis device for transient strongly nonlinear characteristic signals, comprising: A module that acquires the time-domain signal of the transient dynamic response of the structure under strong impact load, performs filtering and noise reduction and baseline correction to obtain a stable discrete time series as the analysis input; Based on discrete time series, the module divides the complete time domain signal into an early strong nonlinear stage, a non-stationary transition stage, and a late quasi-periodic stage according to the dynamic characteristics of the impact response, and extracts the corresponding sub-sequences of each stage as the objects of subsequent analysis. The module quantifies the system's sensitivity to initial conditions and nonlinear strength by performing phase space reconstruction based on each stage subsequence and calculating the maximum Lyapunov exponent. While keeping the stage division results unchanged, the Hurst exponent is calculated for each stage subsequence to characterize the persistence and memory characteristics of the response signal. This module analyzes the evolution of the structural dynamic response from a transient, strongly nonlinear state to a quasi-periodic, deterministic state under strong impact based on the maximum Lyapunov exponent and Hearst exponent obtained at each stage. A module is used to construct a nonlinear dynamic state plane with the inverse of the structural scaling ratio as the independent variable and the maximum Lyapunov exponent and Hearst exponent corresponding to each stage as state variables, thereby realizing the similarity assessment of the prototype and the scaling model at the nonlinear dynamic mechanism level.

[0015] Based on the same inventive concept, the present invention also provides a computer storage medium for storing a computer program, wherein when the computer program is read by a computer, the computer executes the method described thereon.

[0016] Based on the same inventive concept, the present invention also provides a computer, including a processor and a storage medium, wherein when the processor reads a computer program stored in the storage medium, the computer executes the method described thereon.

[0017] Based on the same inventive concept, the present invention also provides a computer program product, which, when executed, implements the method described.

[0018] Compared with the prior art, the advantages of the technical solution provided by the present invention are as follows: By introducing preprocessing and stage division steps for the impact response signal into the method—specifically dividing the complete time-domain signal into an early strongly nonlinear stage, a non-stationary transition stage, and a later quasi-periodic stage—the transient impact signal, which was originally treated as a whole, is deconstructed into multiple dynamic sub-processes with clear physical meaning. This avoids the problem in existing techniques of treating the entire time-domain signal equally, thus masking the details of nonlinear evolution. This stage division feature directly originates from the response stage division part of the method steps. Its effect is to provide a physically consistent analytical basis for the targeted calculation of subsequent nonlinear indices, ensuring that the dynamic states of different stages no longer interfere with each other. This is something that traditional peak comparison or overall spectrum analysis methods cannot achieve.

[0019] By introducing the calculation steps of the maximum Lyapunov exponent at each stage, a quantitative characterization of the initial value sensitivity and nonlinear intensity in the structural impact response is achieved. This effect stems from the phase space reconstruction and trajectory divergence rate calculation steps in the method, allowing the system's state of strong nonlinearity or chaos to be directly determined by the sign and magnitude of λmax. Compared to existing studies that rely solely on waveform "disorder" or spectral broadening for qualitative judgment, this feature can clearly reflect the system's ability to amplify small initial disturbances with explicit numerical values, thus effectively avoiding the misjudgment of strong nonlinear responses as linear or weakly nonlinear responses in engineering analysis.

[0020] By introducing a phased calculation step of the Hearst exponent within the same stage, a quantitative characterization of the time correlation, memory, and persistence characteristics of the impact response signal is achieved. This effect stems from the Hearst exponent calculation process based on rescaled range analysis, which clearly distinguishes whether the response sequence exhibits randomness, anti-persistence, or strong persistence. Compared with commonly used time-frequency analysis or statistical feature extraction methods in the prior art, this feature can reflect the inherent trend of the system's dynamic state evolution over time, rather than just local energy or amplitude changes.

[0021] By jointly analyzing the maximum Lyapunov exponent and the Hearst exponent and comparing their evolution patterns at different stages, a holistic reconstruction of the system's dynamic evolution path was achieved. This effect stems from the method's comprehensive analysis of the changes in λmax and H with different stages, clearly revealing the complete evolution process of the structural response from an early strongly nonlinear transient state, through a non-stationary transition, and finally approaching a quasi-periodic state. Compared to existing studies that use nonlinear indices in isolation, this joint feature avoids the problem of insufficient explanatory power of a single index and improves the reliability of discriminating the dynamic essence of complex impact responses.

[0022] By constructing a dynamic state plane with the reciprocal of the scaling ratio as the independent variable and nonlinear dynamic indices as the state variables, a visual comparison and quantitative evaluation of prototype experiments and scaled-down model experiments at the nonlinear mechanism level is achieved. This effect stems from the steps related to state plane construction and trajectory analysis in the method, allowing for intuitive identification of whether the model truly reproduces the nonlinear dynamic behavior of the prototype at different response stages. Compared to existing model experiments that rely on waveform, peak value, or spectral similarity for judgment, this feature can reveal the applicability of model similarity at the dynamic mechanism level, thereby significantly reducing the engineering risk of model experiment failure or misjudgment under strongly nonlinear conditions.

[0023] It is applicable to research and engineering applications of nonlinear mechanism analysis, model test similarity verification, and engineering safety assessment of structural dynamic response signals under strong impact loads such as explosions and collisions. Attached Figure Description

[0024] Figure 1 Steps for time-domain analysis of transient strongly nonlinear characteristic signals; Figure 2 Test flowchart; Figure 3 Acceleration signal at prototype measurement point A1; Figure 4 Reconstruction phase space delay time τ With Embedding Dimension m ; Figure 5 Maximum Lyapunov exponent in the early stage of the prototype test acceleration signal A1 λ max ; Figure 6 Hearst exponent in the early stage of the prototype test acceleration signal A1 H ; Figure 7 Evolution of the maximum Lyapunov exponent in scaled model experiments; Figure 8 Evolution of the Hearst exponent in scaled-down model experiments. Detailed Implementation

[0025] To make the advantages and benefits of the technical solution provided by the present invention clearer, the technical solution provided by the present invention will now be described in further detail with reference to the accompanying drawings, specifically: Implementation Method 1: This implementation method provides a time-domain analysis method for transient strongly nonlinear characteristic signals, including: The steps include acquiring the time-domain signal of the transient dynamic response of the structure under strong impact load, filtering and denoising the signal, and performing baseline correction to obtain a stable discrete-time series as the analysis input. Based on the dynamic characteristics of the impact response, the complete time-domain signal is divided into an early strong nonlinear stage, a non-stationary transition stage, and a late quasi-periodic stage according to the discrete-time series, and the corresponding sub-sequences of each stage are extracted as the objects of subsequent analysis. The steps of performing phase space reconstruction based on each stage subsequence and calculating the maximum Lyapunov exponent are used to quantify the system’s sensitivity to initial conditions and nonlinear intensity. The step of calculating the Hurst exponent for each stage subsequence while keeping the stage division results unchanged is to characterize the persistence and memory features of the response signal. The steps for analyzing the evolution of the structural dynamic response from a transient strongly nonlinear state to a quasi-periodic deterministic state under strong impact based on the maximum Lyapunov exponent and Hearst exponent obtained at each stage. The steps of constructing a nonlinear dynamic state plane using the inverse of the structural scaling ratio as the independent variable and the maximum Lyapunov exponent and Hearst exponent corresponding to each stage as state variables are used to realize the similarity assessment between the prototype and the scaling model at the nonlinear dynamic mechanism level.

[0026] The transient dynamic response time-domain signal is the acceleration or strain signal acquired by the structure under strong impact load, and the preprocessed discrete time series is used as the unified input data for subsequent stage division and nonlinear characteristic analysis.

[0027] The phase division is based on the amplitude variation characteristics, frequency distribution characteristics, and fluctuation stability characteristics of the response signal to determine the start and end times of the early strong nonlinear phase, the non-stationary transition phase, and the later quasi-periodic phase.

[0028] The calculation of the maximum Lyapunov exponent involves reconstructing the phase space of each stage subsequence and constructing a multidimensional phase space trajectory that reflects the dynamic behavior of the system by determining the time delay and the embedding dimension.

[0029] The maximum Lyapunov exponent is obtained by tracking the divergence of adjacent orbits in phase space over time and fitting their average divergence rate. It is used to characterize the system's sensitivity to initial conditions and nonlinear intensity.

[0030] The Hearst exponent is obtained by performing multi-scale rescaled range analysis on subsequences at each stage, and is used to characterize the persistence, memory, and long-range correlation of the response signal.

[0031] A time-domain analysis device for transient strongly nonlinear characteristic signals is also provided, comprising: A module that acquires the time-domain signal of the transient dynamic response of the structure under strong impact load, performs filtering and noise reduction and baseline correction to obtain a stable discrete time series as the analysis input; Based on discrete time series, the module divides the complete time domain signal into an early strong nonlinear stage, a non-stationary transition stage, and a late quasi-periodic stage according to the dynamic characteristics of the impact response, and extracts the corresponding sub-sequences of each stage as the objects of subsequent analysis. The module quantifies the system's sensitivity to initial conditions and nonlinear strength by performing phase space reconstruction based on each stage subsequence and calculating the maximum Lyapunov exponent. While keeping the stage division results unchanged, the Hurst exponent is calculated for each stage subsequence to characterize the persistence and memory characteristics of the response signal. This module analyzes the evolution of the structural dynamic response from a transient, strongly nonlinear state to a quasi-periodic, deterministic state under strong impact based on the maximum Lyapunov exponent and Hearst exponent obtained at each stage. A module is used to construct a nonlinear dynamic state plane with the inverse of the structural scaling ratio as the independent variable and the maximum Lyapunov exponent and Hearst exponent corresponding to each stage as state variables, thereby realizing the similarity assessment of the prototype and the scaling model at the nonlinear dynamic mechanism level.

[0032] A computer storage medium is also provided for storing a computer program, which, when read by the computer, executes the method.

[0033] A computer is also provided, including a processor and a storage medium, wherein the computer executes the method when the processor reads a computer program stored in the storage medium.

[0034] A computer program product is also provided, which, when executed, implements the method described.

[0035] Implementation Method Two: This implementation method is a further detailed description of the technical solution provided in Implementation Method One, specifically: The transient dynamic response signal generated by the structure under strong impact load is used as the analysis object. The response signal can be an acceleration signal or a strain signal, and its source can be data collected by sensors in the test system or time-domain response results obtained by numerical simulation. Since the original signal under strong impact conditions is usually superimposed with high-frequency noise and low-frequency drift components, the original time-domain signal is first preprocessed. The main frequency components related to the structural impact response are retained by bandpass filtering, while high-frequency noise introduced by the test environment and the acquisition system is suppressed. Then, the low-frequency trend term caused by sensor zero drift or loading environment is eliminated by baseline correction, so that the processed signal exhibits zero-mean characteristics in a statistical sense, thereby obtaining a continuous, stable discrete-time series that can be used for nonlinear analysis. This discrete-time series serves as the basic input for subsequent analysis processes.

[0036] Based on the preprocessed discrete-time series, and considering the physical evolution characteristics of the structural dynamic response under strong impact loads, the complete time-domain signal is divided into stages. Specifically, according to the amplitude variation, frequency distribution, and fluctuation characteristics of the response signal on the time axis, the signal is divided into three consecutive stages: an early stage, a transition stage, and a late stage. The early stage corresponds to the transient response generated by the structure at the initial application of the impact load. Signals in this stage typically exhibit abrupt amplitude changes, significant high-frequency components, and drastic data fluctuations with strong randomness. The late stage corresponds to the gradual decay and stabilization of the structural dynamic response, characterized by a deterministic response with low-frequency dominance, gradual amplitude changes, and quasi-periodic characteristics. The transition stage, located between these two stages, reflects the non-stationary process of the system evolving from a strongly nonlinear state to a weakly nonlinear or deterministic state; its signal characteristics do not possess a uniform statistical regularity over time. By determining the start and end times of each stage, corresponding stage sub-sequences are extracted from the complete time series, providing input data for the staged nonlinear dynamic characteristic analysis.

[0037] For the subsequences of the early, transition, and late stages, the maximum Lyapunov exponent is calculated to characterize the system's sensitivity to initial conditions and nonlinear intensity. Specifically, the phase space of the stage subsequences is first reconstructed. A suitable time delay is determined by analyzing the correlation characteristics of the time series itself, ensuring minimal redundancy between the reconstructed components. Simultaneously, the complexity variation of the phase space trajectory under different embedding dimensions is analyzed to determine the embedding dimension that can fully unfold the system's dynamic characteristics, thus constructing a multidimensional phase space trajectory that reflects the system's true dynamic behavior. After phase space reconstruction, the nearest neighbor state point satisfying the time interval constraint is searched for each state point in the phase space to avoid interference from time correlation. Subsequently, the distance changes between the state point pairs during system evolution are tracked, and the average divergence trend of each pair over time is statistically analyzed. An interval with a clear linear characteristic is selected from the divergence trend and time relationship for fitting, and the fitting slope is used as the maximum Lyapunov exponent corresponding to that stage subsequence. Through this process, it is possible to determine whether the system exhibits chaotic characteristics at different stages and quantify its nonlinear intensity.

[0038] Based on the calculation of the maximum Lyapunov exponent, the Hearst exponent is further calculated for the same stage subsequence to characterize the long-range correlation and persistence of the response signal over time. Specifically, the stage subsequence is divided into several continuous sub-intervals according to different time scales. The average value of the signal is calculated within each sub-interval, and a cumulative deviation sequence of the signal relative to the mean is constructed based on this average value, thereby obtaining the maximum cumulative deviation range and the dispersion of the signal within that sub-interval. By statistically analyzing the ratio of the cumulative deviation range to the dispersion, and repeating the above calculation at multiple time scales, the data relationship of the rescaled range changing with the time scale is obtained. This relationship is linearly fitted on a logarithmic scale, and the slope of the fit is the Hearst exponent of the signal in that stage, used to quantitatively reflect whether the signal exhibits randomness, anti-persistence, or obvious persistence and memory characteristics.

[0039] After obtaining the maximum Lyapunov exponent and Hearst exponent for each stage, the variation patterns of these two types of nonlinear dynamic indices with each stage are comprehensively analyzed. By comparing the value ranges and trends of the indices at different stages, the complete dynamic process of the system under strong impact load can be identified, from the early transient response state with high initial value sensitivity, strong nonlinearity, and weak memory, to the nonstationary evolution state in the transition stage, and then to the later quasi-periodic response state with low initial value sensitivity, weak nonlinearity, and strong persistence. This allows for a quantitative characterization of the nonlinear evolution mechanism of the structural impact response.

[0040] Based on the above analysis, to evaluate the consistency between prototype tests and model tests at different scales at the nonlinear dynamic mechanism level, a nonlinear dynamic state plane is constructed using the reciprocal of the structural scale ratio as the independent variable and the maximum Lyapunov exponent and Hearst exponent calculated at each stage as state variables. The nonlinear dynamic indices corresponding to the prototype structure and models at different scales at each response stage are plotted on the same state plane. By comparing whether their evolution trajectories with scaling ratio are consistent, the degree of reproduction of the prototype response by the model test at the strong nonlinear dynamic mechanism level is determined. This provides a reliable basis for validating the model test's effectiveness, determining its similarity, and assessing engineering safety under strong impact conditions.

[0041] Implementation Method 3, in conjunction with Appendix Figure 1-8 This embodiment describes the technical solution provided above in further detail through specific examples. Specifically: This implementation addresses the shortcomings of existing technologies in characterizing the nonlinear dynamics of structural responses under strong impact loads and verifying the effectiveness of model tests. It proposes a time-domain analysis method for transient strong nonlinear characteristic signals that is both mechanistically clear and quantitatively accurate. This method abandons the traditional simple comparison of time domain or power spectra. Based on nonlinear dynamic system theory, it analyzes two kinetic exponents with clear physical meaning (the maximum Lyapunov exponent and the Hearst exponent) to deeply characterize the intrinsic evolution of the nonlinear characteristics of the impact response signal. This provides a quantitative criterion that directly addresses the nonlinear mechanism for the verification and evaluation of model tests.

[0042] This embodiment adopts the following technical solution: (1) Acquisition and preprocessing of raw signals. Acquire the raw time-domain signals of transient impact response obtained from sensor measurements or numerical simulations. x ( t This signal is an acceleration or strain time series. x ( t Preprocessing is performed, including frequency domain filtering to suppress high-frequency noise, and polynomial fitting to remove baseline drift. The resulting discrete signal sequence is then... xi}, i =1,2,…, N ,in N This represents the total length of the signal.

[0043] (2) Nonlinear dynamic response stage division. Based on the physical characteristics of the impact response signal, the complete time domain signal is divided into three characteristic time periods. The structural acceleration and strain response time series of different scaling ratio models at different measurement points all show strong nonlinear and non-stationary characteristics.

[0044] The early stage represents the transient, strongly nonlinear response phase of the structure, characterized by high frequency, drastic amplitude variations, and significant randomness and anti-persistence. The later stage represents the approaching steady-state response of the structural dynamics system, exhibiting low frequency and relatively gradual quasi-periodic amplitude variations. The intermediate stage between the early and late stages represents a non-stationary transition from the transient, strongly nonlinear response state to a quasi-periodic or deterministic state; this transition process does not exhibit a uniform evolutionary pattern from the perspective of data fluctuations. The specific divisions are as follows: (1) in t 1 represents the end time of the early signal. t 2 represents the end time of the transition segment signal. t 3 represents the end time of the complete signal. For each stage, extract its corresponding subsequence { x i (s)},s∈{ e , t , l}, their lengths are respectively M e , M t , M l .

[0045] (3) Calculate the maximum Lyapunov index in stages λ max For each stage subsequence { x i (s) The following calculation process will be executed: 1. Phase space reconstruction: The phase space is reconstructed using the coordinate delay method. First, the autocorrelation function of the time series is calculated according to the formula. R ( τ To determine the delay time τ : (2) in x This is the sequence mean. Select... R ( τ First decay to 1 / e time τ The value represents the optimal delay time. The embedding dimension is determined using the saturation correlation dimension method. m : (3) in It is a Heaviside step function. Y iThese are the reconstructed phase space points. r Using distance as the scale. Calculate the correlation dimension. D ( m ),when D ( m ) m When it increases to the point of saturation, the corresponding minimum m This is the optimal embedding dimension. The reconstructed dimension... m The point set of the phase space is: (4) 2. Calculate the local divergence rate of the evolving trajectory: for each point in phase space Y j Find its nearest neighbor. Y It must meet the following requirements: (5) in The average period of the sequence can be estimated from the signal power spectrum. The distance of this point pair as the system evolves is then traced. k Each time step Δ t Then, the distance becomes: (6) 3. Calculate the maximum Lyapunov exponent: For all valid point pairs, calculate the average logarithmic divergence rate. (7) in M s ′ represents the total number of valid point pairs. S ( k Relative to k From the curve, select the region with good linearity for least-squares linear fitting: (8) The slope of the fitted line is the maximum Lyapunov exponent of the signal in that stage. λ max . λ max A value greater than 0 indicates that the system response at this stage exhibits chaotic characteristics, and its value quantifies the system's sensitivity to initial conditions and the intensity of nonlinearity.

[0046] (4) Calculate the Hearst exponent in stages H For the same stage subsequence { x i (s)}, perform the following based on rescaled range R / S analyze: 1. [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context M sThe sequence is divided into d A length of n continuous subintervals ( n (For variable scales). For the first m Sub-intervals ( m (e.g., 1, 2, ..., d), calculate its mean. E m : (9) Cumulative deviation X k,m : (10) Range R m : (11) Standard deviation S m : (12) 2. Calculate the average rescaled range at each scale: the rescaled range for each subinterval is ( R / S ) m = R m / S m For all lengths of n The arithmetic mean of the subintervals is taken to obtain the average rescaled range at this scale: (13) 3. Hearst exponent fitting: changing the scale n Repeat steps 1 and 2 to obtain a series of data points. n ,( R / S ) n Hearst's empirical law shows the existence of a power-law relationship: ( R / S ) n ∝ n H Take the logarithm of both sides: (14) The least squares method is used to analyze the data point set {ln( n ),ln(( R / S ) n The slope of the fitted line obtained by performing linear fitting on the signal is the Hurst exponent of that stage. HIt is used to quantify the long-range correlation of time series and reveal the persistent state of the system's response evolution behavior.

[0047] (5) Analysis of nonlinear dynamic evolution. The above steps were repeated for the response signals at different stages to calculate the maximum Lyapunov exponent. λ max (s) and Hearst index H (s) Based on this, the nonlinear dynamic evolution characteristics of the system can be analyzed. The response signal in the early stage corresponds to... λ max ∈(0.7,1) and H The value ∈(0.5, 0.6) indicates that the system at this stage is extremely sensitive to initial conditions, exhibits strong nonlinear dynamic behavior, has weak response sequence memory, and its state evolution exhibits strong uncertainty, consistent with the characteristics of the structural transient response under initial impact load; the transition stage corresponds to λ max ∈(0.3,0.7) and H ∈(0.6, 0.8), reflecting the system's non-stationary transition from chaos to weak nonlinearity, with decreased sensitivity to initial conditions and gradually enhanced persistence and memory of the sequence, characterizing the complex dynamic state of multimodal energy coupling within the system; the later stage corresponds to λ max ∈(0,0.3) and H ∈(0.8, 1.0), indicating that the system is no longer sensitive to initial conditions, the dynamic behavior approaches a deterministic response, the sequence exhibits strong persistence and significant long-range correlation, corresponding to the stage where the structure tends to quasi-periodic vibration or steady-state response, as shown in Equation (15).

[0048] (15) To intuitively represent the relationship between the above evolutionary patterns and model similarity, a maximum Lyapunov index is constructed with the reciprocal of the scaling ratio (1 / cl) as the horizontal axis. λ max With Hearst index H The two-dimensional dynamic state plane represents the vertical axis. The nonlinear dynamic indices of the early, transition, and late stages are plotted on this plane. Their continuous evolution trajectory clearly reveals the changes in the system's dynamic state with the structural scale, thus providing a visual and quantifiable analytical basis for evaluating the similarity of nonlinear mechanisms between models and prototypes at different scaling ratios.

[0049] In a specific embodiment: This embodiment uses a reinforced cylindrical shell structure with a length of 2.46 m, a diameter of 1.60 m, a rib spacing of 0.19 m, and a rib height of 62.8 mm as a prototype. Based on the equality of single-valued physical quantities in the experimental system, prototype tests of the reinforced cylindrical shell and similar scale model tests at ratios of 1:2.5, 1:4.5, 1:6.25, and 1:7.2 were designed and carried out. The center of the blast-facing surface of the pressure shell was selected as the measuring point A1, and the acceleration response was measured using an electrical measurement method. Both the prototype reinforced cylindrical shell structure and the scale model structure were placed 13 m underwater in an explosion water tank to conduct prototype tests of the reinforced cylindrical shell and tests of each scale model. The specific test flowchart is as follows. Figure 1 As shown.

[0050] Obtain the raw acceleration time-domain signals for each experiment. a ( t After that, the signal is fitted with a third-order polynomial using the least squares method. The fitted curve is used as the baseline and subtracted from the original signal to obtain a zero-mean signal. A Butterworth bandpass filter is used, with the passband set to [55Hz, ... f 截至 This preprocessing is performed to preserve the main impulse frequency components and suppress high-frequency noise. After preprocessing, a clean, analyzable discrete-time series {ai}, i=1,2,…, is obtained. N ,in N This is the signal length.

[0051] Taking the prototype test acceleration signal A1 as an example, based on the physical characteristics of the impact response signal, the complete time-domain signal is divided into three characteristic periods. The early dynamic response exhibits uncertain characteristics such as high frequency, drastic amplitude changes, and significant data randomness. The later dynamic response exhibits quasi-periodic characteristics with low frequency and relatively gentle amplitude changes. The transition process between the early and later periods does not show a relatively unified evolutionary pattern from the perspective of data fluctuation. The specific division of the acceleration signal is as follows: Figure 2 .

[0052] Calculate the maximum Lyapunov exponent of the early stage of the prototype test acceleration signal A1. λ max First, the phase space is reconstructed using the coordinate delay method. Then, the autocorrelation function of the time series is calculated according to formula (2). R ( τ To determine the delay time τ =220, select R ( τ When the value first decays to the minimum trough τ The value is the optimal delay time. The correlation dimension is calculated using formula (3). D ( m ),when D ( m )m When it increases to the point of saturation, the corresponding minimum m This is the optimal embedding dimension; determine the embedding dimension. m =2. The specific calculation is as follows: Figure 3 .

[0053] According to formula (5), for each point in phase space Y j Find its nearest neighbor. Y And according to formula (6), the distance of the point pair as the system evolves is tracked, and the average logarithmic divergence rate is calculated for all effective point pairs. In the curve of S(k) relative to k, a region with good linearity is selected for least squares linear fitting, and the slope of the fitted line is the maximum Lyapunov exponent λ of the signal in this stage. max =0.83, such as Figure 4 As shown.

[0054] Repeat the above steps. Obtain the maximum Lyapunov exponent λ of the prototype test acceleration signal A1 during the transition phase. max =0.43, the maximum Lyapunov exponent λ in the later stage max =0.16.

[0055] Calculate the Hearst exponent of the early stage of the prototype test acceleration signal A1. H According to formula (9), the length is... M s The sequence is divided into d A length of n continuous subintervals ( n (For variable scales). For the first m Sub-intervals ( m Calculate the mean of each of the given numbers (e.g., 1, 2, ..., d). E m Cumulative deviation X k,m Extreme R m Standard deviation S m The rescaled range of each subinterval is ( R / S ) m = R m / S m According to formula (13), for all lengths... n The arithmetic mean of the subintervals is taken to obtain the average rescaled range at that scale. Changing the scale... n A series of data points were obtained. n ,( R / S) n Hearst's empirical law shows the existence of a power-law relationship: ( R / S ) n ∝ n H Taking the logarithm of both sides according to formula (14), the least squares method is used to calculate the data point set {ln( n ),ln(( R / S ) n Linear fitting was performed to obtain the Hearst exponent of the early stage of the prototype test acceleration signal A1. H =0.58, such as Figure 5 As shown.

[0056] Repeat the above steps to obtain the Hearst exponent of the transition phase of the prototype test acceleration signal A1. H =0.69, the Hearst exponent in the later stage H =0.82.

[0057] Based on the analysis method of the above examples, nonlinear characteristic signal time-domain analysis was performed on the acceleration signals of the same measuring points on the blast face in the similar scale model tests of 1:2.5, 1:4.5, 1:6.25, and 1:7.2, respectively, to calculate the maximum Lyapunov exponent at different stages. λ max and Hearst index H The results are shown in Table 1 and Table 2.

[0058]

[0059]

[0060] The maximum Lyapunov index obtained based on the above calculations λ max and Hearst index H The nonlinear dynamic response law of the system under strong impact is analyzed. It satisfies formula (15), indicating that the dynamic response system has undergone a complete dynamic process from an early strong nonlinear transient response, through a non-stationary transition stage, and finally into a later state approaching a quasi-periodic deterministic state. In order to intuitively reveal the similarity between the model and the prototype at different scaling ratios in terms of nonlinear mechanism, a model is constructed with different scaling ratios reciprocals (1 / cl () is the horizontal axis. λ max , H The vertical axis represents the two-dimensional dynamic state plane. For example... Figure 6 , Figure 7As shown in the figure, the exponential evolution trajectory of each stage can clearly reflect the change law of system dynamic characteristics with structural scaling ratio, thus providing a visual basis for the similarity assessment of model tests.

[0061] This embodiment uses the acceleration response signal of a stiffened cylindrical shell structure under underwater explosion impact as the analysis object. The method of this embodiment is applied to systematically analyze the test data of the prototype and four scaled models (1:2.5, 1:4.5, 1:6.25, and 1:7.2). The maximum Lyapunov exponent is calculated. λ ma With Hearst index H It is clearly shown that the dynamic state of the system follows λ max The response gradually decreases, and H The evolutionary pattern of gradually increasing response indicates that the system behavior transitions from an early strongly nonlinear state to a later weakly nonlinear quasi-periodic state, reflecting the increasing memory and persistence of the system. This pattern quantitatively reveals the complete path of the impact response's evolution from an early strongly nonlinear transient state, through a non-stationary transition, to a later quasi-periodic deterministic state. Evolutionary trajectory analysis based on characteristic indices shows that the maximum Lyapunov exponent of the 1:2.5 and 1:6.25 scaled models varies across the early, transitional, and later stages. λ max With Hearst index H The trend of change is highly consistent with the prototype. Figure 6 , Figure 7 This demonstrates that both models accurately reflect the response behavior of the prototype in terms of nonlinear dynamics. In contrast, the 1:7.2 scale model shows a significant deviation from the prototype's dynamic evolution trajectory during the transition phase, indicating that under this scale condition, the model struggles to fully reproduce the nonlinear dynamic process of the prototype under complex fluid-structure interaction. This reveals the applicability of different scale similarities in strongly nonlinear response analysis. In summary, the method proposed in this embodiment utilizes the maximum Lyapunov exponent. λ max and Hearst index H These two quantitative indicators, with clear physical meaning, successfully extracted and characterized the evolution law of transient strong nonlinear signals, and constructed a quantifiable and comparable method for nonlinear dynamic similarity in model experiments. This invention overcomes the limitations of traditional methods in analyzing strong nonlinear signals in the time domain, providing a reliable analytical tool for shock-resistant structure design, model test research, and safety assessment.

[0062] The above description of several specific embodiments further details the technical solution provided by the present invention in order to highlight the advantages and benefits of the technical solution provided by the present invention. However, the above-described specific embodiments are not intended to limit the present invention. Any reasonable modifications and improvements to the present invention, combinations of embodiments, and equivalent substitutions based on the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A time-domain analysis method for transient strongly nonlinear characteristic signals, characterized in that, include: The steps include acquiring the time-domain signal of the transient dynamic response of the structure under strong impact load, filtering and denoising the signal, and performing baseline correction to obtain a stable discrete-time series as the analysis input. Based on the dynamic characteristics of the impact response, the complete time-domain signal is divided into an early strong nonlinear stage, a non-stationary transition stage, and a late quasi-periodic stage according to the discrete-time series, and the corresponding sub-sequences of each stage are extracted as the objects of subsequent analysis. The steps of performing phase space reconstruction based on each stage subsequence and calculating the maximum Lyapunov exponent are used to quantify the system’s sensitivity to initial conditions and nonlinear intensity. The step of calculating the Hurst exponent for each stage subsequence while keeping the stage division results unchanged is to characterize the persistence and memory features of the response signal. The steps for analyzing the evolution of the structural dynamic response from a transient strongly nonlinear state to a quasi-periodic deterministic state under strong impact based on the maximum Lyapunov exponent and Hearst exponent obtained at each stage. The steps of constructing a nonlinear dynamic state plane using the inverse of the structural scaling ratio as the independent variable and the maximum Lyapunov exponent and Hearst exponent corresponding to each stage as state variables are used to realize the similarity assessment between the prototype and the scaling model at the nonlinear dynamic mechanism level.

2. The time-domain analysis method for transient strongly nonlinear characteristic signals according to claim 1, characterized in that, The transient dynamic response time-domain signal is the acceleration or strain signal acquired by the structure under strong impact load, and the preprocessed discrete time series is used as the unified input data for subsequent stage division and nonlinear characteristic analysis.

3. The time-domain analysis method for transient strongly nonlinear characteristic signals according to claim 1, characterized in that, The phase division is based on the amplitude variation characteristics, frequency distribution characteristics, and fluctuation stability characteristics of the response signal to determine the start and end times of the early strong nonlinear phase, the non-stationary transition phase, and the later quasi-periodic phase.

4. The time-domain analysis method for transient strongly nonlinear characteristic signals according to claim 1, characterized in that, The calculation of the maximum Lyapunov exponent involves reconstructing the phase space of each stage subsequence and constructing a multidimensional phase space trajectory that reflects the dynamic behavior of the system by determining the time delay and the embedding dimension.

5. The time-domain analysis method for transient strongly nonlinear characteristic signals according to claim 1, characterized in that, The maximum Lyapunov exponent is obtained by tracking the divergence of adjacent orbits in phase space over time and fitting their average divergence rate. It is used to characterize the system's sensitivity to initial conditions and nonlinear intensity.

6. The time-domain analysis method for transient strongly nonlinear characteristic signals according to claim 1, characterized in that, The Hearst exponent is obtained by performing multi-scale rescaled range analysis on subsequences at each stage, and is used to characterize the persistence, memory, and long-range correlation of the response signal.

7. A time-domain analysis device for transient strongly nonlinear characteristic signals, characterized in that, include: A module that acquires the time-domain signal of the transient dynamic response of the structure under strong impact load, performs filtering and noise reduction and baseline correction to obtain a stable discrete time series as the analysis input; Based on discrete time series, the module divides the complete time domain signal into an early strong nonlinear stage, a non-stationary transition stage, and a late quasi-periodic stage according to the dynamic characteristics of the impact response, and extracts the corresponding sub-sequences of each stage as the objects of subsequent analysis. The module quantifies the system's sensitivity to initial conditions and nonlinear strength by performing phase space reconstruction based on each stage subsequence and calculating the maximum Lyapunov exponent. While keeping the stage division results unchanged, the Hurst exponent is calculated for each stage subsequence to characterize the persistence and memory characteristics of the response signal. This module analyzes the evolution of the structural dynamic response from a transient, strongly nonlinear state to a quasi-periodic, deterministic state under strong impact based on the maximum Lyapunov exponent and Hearst exponent obtained at each stage. A module is used to construct a nonlinear dynamic state plane with the inverse of the structural scaling ratio as the independent variable and the maximum Lyapunov exponent and Hearst exponent corresponding to each stage as state variables, thereby realizing the similarity assessment of the prototype and the scaling model at the nonlinear dynamic mechanism level.

8. A computer storage medium for storing computer programs, characterized in that, When the computer program is read by the computer, the computer executes the method of claim 1.

9. A computer, comprising a processor and a storage medium, characterized in that, When the processor reads the computer program stored in the storage medium, the computer executes the method of claim 1.

10. A computer program product, as a computer program, is characterized by: When the computer program is executed, it implements the method of claim 1.

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