Nonlinear lamb wave and bayesian inference based non-reference damage imaging method

Abstract

The application provides a non-reference damage imaging method based on nonlinear Lamb waves and Bayesian inference, relates to the field of damage imaging, and specifically comprises the following steps: collecting nonlinear Lamb wave signals of multiple propagation paths; obtaining enhanced second harmonic signals through a pulse inversion technique; extracting time-of-flight characteristics of envelope signals; obtaining frequency spectra of the nonlinear Lamb wave signals by using fast Fourier transform, and calculating nonlinear coefficient damage index characteristics; performing feature-level fusion based on Bayesian estimation; solving a posterior probability density function of a damage position parameter by using Hamilton Monte Carlo sampling, and performing damage probability imaging positioning according to the posterior probability distribution. The technical scheme of the application overcomes the problem in the prior art that a large number of sensors usually need to be deployed, and it is difficult to meet the dual requirements of high-precision damage detection and minimization of the number of sensors in engineering practice.

Description

Technical Field

[0001] This invention relates to the field of damage imaging, and more specifically to a reference-free damage imaging method based on nonlinear Lamb waves and Bayesian inference. Background Technology

[0002] Under harsh conditions such as long-term high-load operation, stress concentration, and chemical corrosion, engineering structures are highly susceptible to various forms of damage, including cracks, corrosion, delamination, and debonding, both on their surfaces and internally. Timely and accurate detection and identification of these potential defects are of significant engineering importance for preventing catastrophic accidents and ensuring structural safety. Ultrasonic Lamb waves, due to their long propagation distance and sensitivity to internal structural defects, have become an important technical means in the fields of structural health monitoring and non-destructive testing assessment.

[0003] Nonlinear Lamb wave technology, with its unique ability to capture nonlinear effects such as higher harmonics and frequency mixing modulation, has demonstrated significant technical advantages in the field of early structural damage detection. Furthermore, this technology overcomes the dependence on baseline data in traditional linear Lamb wave damage detection methods. However, current nonlinear Lamb wave damage imaging methods still have significant shortcomings in utilizing feature information: existing research is mostly limited to damage imaging based on single nonlinear features, such as trajectory imaging methods based on second harmonic time-of-flight differences and defect probability detection reconstruction algorithms based on nonlinear coefficient damage exponents, failing to effectively integrate multiple nonlinear feature information, resulting in limited damage imaging resolution. In addition, existing methods typically require the deployment of a large number of sensors, making it difficult to meet the dual requirements of high-precision damage detection and minimizing the number of sensors in engineering practice.

[0004] Therefore, there is a need for a reference-free damage imaging method based on nonlinear Lamb waves and Bayesian inference that can improve the accuracy and reliability of damage detection without requiring the deployment of a large number of sensors. Summary of the Invention

[0005] The main objective of this invention is to provide a reference-free damage imaging method based on nonlinear Lamb waves and Bayesian inference, in order to solve the problem that existing technologies usually require the deployment of a large number of sensors, which makes it difficult to meet the dual requirements of high-precision damage detection and minimization of the number of sensors in engineering practice.

[0006] To achieve the above objectives, this invention provides a reference-free damage imaging method based on nonlinear Lamb waves and Bayesian inference, specifically including the following steps:

[0007] S1. A sparse piezoelectric sensor network is deployed on the surface of the structure under test. All sparse piezoelectric sensors take turns to excite and receive signals, forming several damage detection paths and collecting nonlinear Lamb wave signals from multiple propagation paths.

[0008] S2 obtains an enhanced second harmonic signal through pulse inversion technology.

[0009] S3 filters noise signals in the second harmonic signal through continuous Shannon wavelet transform, and calculates the filtered envelope signal using Hilbert transform, thereby extracting the time-of-flight characteristics of the envelope signal.

[0010] S4. The spectrum of the nonlinear Lamb wave signal is obtained by using the Fast Fourier Transform, and the damage index characteristics of the nonlinear coefficient are calculated.

[0011] S5 utilizes the time-of-flight difference feature combined with the trajectory imaging positioning principle to construct a likelihood function, utilizes the nonlinear coefficient damage index feature combined with the defect probability detection reconstruction algorithm principle to construct a prior probability density function, and constructs a posterior probability density function of the damage location parameter based on Bayes' theorem for feature-level data fusion.

[0012] S6 uses Hamiltonian Monte Carlo sampling to solve the posterior probability density function of the damage location parameters, and performs damage probability imaging localization based on the posterior probability distribution.

[0013] Furthermore, step S2 specifically includes the following steps:

[0014] S2.1, due to the presence of nonlinear damage, the excitation and response of the system are simplified to:

[0015] (1);

[0016] in, Indicates the excitation signal. For system response, , and These are the amplitude coefficients for the fundamental wave, second harmonic, and third harmonic, respectively.

[0017] S2.2, based on pulse inversion technology, firstly, the sensor array is excited in positive phase and the response signal is acquired; then, the excitation signal is excited in reverse phase, and the two acquired time-domain signals are superimposed; assuming the positive phase excitation is... The opposite incentive is - The responses obtained from the two stimuli are as follows:

[0018] (2);

[0019] in, and These represent the positive-phase response signal and the negative-phase response signal, respectively. Superimposing the two response signals yields:

[0020] (3).

[0021] Furthermore, step S3 specifically includes the following steps:

[0022] S3.1, noise signals in the second harmonic signal are filtered out by continuous Shannon wavelet transform. For the second harmonic signal in real space... The continuous Shannon wavelet transform (WT) is defined as follows:

[0023] (4);

[0024] in, As a scale factor, The translation factor; Represents the space of real numbers, where "¯" represents conjugate. Shannon mother wavelet function:

[0025] (5);

[0026] in, Represents the imaginary unit. Indicates time, This represents the bandwidth of the mother wavelet. is the center frequency of the Shannon mother wavelet.

[0027] S3.2, the relationship between the center frequency of the second harmonic signal and the scale factor is as follows:

[0028] (6);

[0029] in, The center frequency of the second harmonic. The sampling frequency of the nonlinear Lamb wave signal. is the scale factor.

[0030] S3.3, Calculate the envelope of the second harmonic signal using the Hilbert transform:

[0031] (7);

[0032] in, It is a second harmonic signal. for Hilbert transform, This is the corresponding envelope signal.

[0033] S3.4 Extract the time-of-flight characteristics of the second harmonic based on the peak value of the envelope signal.

[0034] Furthermore, step S4 specifically includes:

[0035] The spectrum of the positive-phase excitation nonlinear Lamb wave response signal is obtained using Fast Fourier Transform, and the nonlinear coefficient damage index characteristic (NCDI) is calculated according to the following formula:

[0036] (8);

[0037] in, The fundamental amplitude in the spectrum. This represents the amplitude of the second harmonic in the frequency spectrum.

[0038] Furthermore, step S5 specifically includes the following steps:

[0039] S5.1, assuming the center coordinates of the damage are... Grouping two different receiving sensors corresponding to the same exciter into a set, there are a total of Group; According to the trajectory imaging method, the second harmonic is related to the first harmonic; Set up two sensors, sensor and The theoretical calculation value of the flight time difference is :

[0040] (9);

[0041] in, and Sensors and coordinates and These are the second harmonics in the sensor and Group velocity in the direction.

[0042] S5.2, the measured time difference of flight is expressed as :

[0043] (10);

[0044] in, The expression is uncertain and follows a mean of 0 and a variance of . The Gaussian distribution.

[0045] S5.3, based on the measured second harmonic flight time difference characteristic data Construct the likelihood function using formula (9) :

[0046] (11);

[0047] in, It is an exponential function.

[0048] Furthermore, step S5 also includes the following steps:

[0049] S5.4, Construct the prior distribution of damage location parameters based on the defect probability detection and reconstruction algorithm. :

[0050] (12);

[0051] in, For the first The nonlinear coefficient of the excitation receiving path is the damage exponent. To incentivize the total number of receiving paths; For elliptic distribution functions:

[0052] (13);

[0053] in, These are the parameters for controlling the shape of the ellipse. The expression is:

[0054] (14);

[0055] The coordinates of the excitation sensor and the receiving sensor are respectively... and .

[0056] S5.5, According to Bayes' theorem, the posterior distribution of the damage location parameters is... satisfy:

[0057] (15);

[0058] in, It represents a direct proportion to.

[0059] The posterior distribution of the damage location parameters was obtained. :

[0060] (16).

[0061] Furthermore, step S6 specifically includes:

[0062] The probability distribution of the damage location within the monitoring area can be written in the standard form of a two-dimensional normal distribution. :

[0063] (17);

[0064] in, Damage location parameters The mean vector; It is the covariance matrix; Covariance matrix The determinant of; The coordinates corresponding to the maximum value are the predicted damage location.

[0065] The present invention has the following beneficial effects:

[0066] (1) The present invention uses pulse inversion technology and continuous Shannon wavelet transform technology, which not only enhances the second harmonic component but also reduces the interference of noise on damage identification.

[0067] (2) The present invention takes into account the uncertainties in the characteristic measurement of nonlinear Lamb wave signals and the process of damage location identification, thereby enhancing the reliability and robustness of damage localization.

[0068] (3) This invention makes full use of the characteristic information of nonlinear Lamb waves, and has excellent damage localization accuracy compared with traditional methods when a small number of sensors are deployed.

[0069] (4) The present invention does not require prior measurement of the health reference signal of the structure as a reference, and can overcome the challenges brought about by the time-varying characteristics of the actual service environment of the engineering structure. Attached Figure Description

[0070] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. In the drawings:

[0071] Figure 1 A flowchart of a reference-free damage imaging method based on nonlinear Lamb waves and Bayesian inference according to the present invention is shown.

[0072] Figure 2 A schematic diagram of the trajectory imaging principle is shown.

[0073] Figure 3 A schematic diagram illustrating the principle of the defect probability detection and reconstruction algorithm is shown.

[0074] Figure 4 The sensor array and damage diagram of the structure under test are shown.

[0075] Figure 5 The phase velocity dispersion curve of the structure under test in the 0° direction is shown.

[0076] Figure 6 A schematic diagram of the sensor excitation signal is shown.

[0077] Figure 7 The time-domain response signal of 0° phase excitation is shown.

[0078] Figure 8 The spectrum of the 0° phase excitation response signal is shown.

[0079] Figure 9 The time-domain response signal of 180° phase excitation is shown.

[0080] Figure 10 The spectrum of the 180° phase excitation response signal is shown.

[0081] Figure 11 The time-domain signals at 0° and 180° superimposed are shown.

[0082] Figure 12 The superimposed spectra at 0° and 180° are shown.

[0083] Figure 13 A schematic diagram of time-of-flight feature extraction is shown.

[0084] Figure 14 The group velocity profile of the second harmonic is shown.

[0085] Figure 15 The posterior distribution of the damage location parameter along the x-axis is shown.

[0086] Figure 16 The posterior distribution of the damage location parameter along the y-axis is shown.

[0087] Figure 17 An error graph of the trajectory imaging method for damage imaging localization is shown.

[0088] Figure 18 An error graph is shown for the defect probability detection reconstruction method used for damage imaging localization.

[0089] Figure 19 An error graph of the method proposed in this invention for damage imaging localization is shown. Detailed Implementation

[0090] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0091] like Figure 1 The reference-free damage imaging method based on nonlinear Lamb waves and Bayesian inference, as shown, specifically includes the following steps:

[0092] S1. A sparse piezoelectric sensor network is deployed on the surface of the structure under test. All sparse piezoelectric sensors take turns to excite and receive signals, forming several damage detection paths and collecting nonlinear Lamb wave signals from multiple propagation paths.

[0093] S2 obtains an enhanced second harmonic signal through pulse inversion technology.

[0094] S3 filters noise signals in the second harmonic signal through continuous Shannon wavelet transform, and calculates the filtered envelope signal using Hilbert transform, thereby extracting the time-of-flight characteristics of the envelope signal.

[0095] S4. The spectrum of the nonlinear Lamb wave signal is obtained by using the Fast Fourier Transform, and the damage index characteristics of the nonlinear coefficient are calculated.

[0096] S5 utilizes the time-of-flight difference feature combined with trajectory imaging localization principle to construct a likelihood function, utilizes the nonlinear coefficient damage exponent feature combined with defect probability detection and reconstruction algorithm principle to construct a prior probability density function, and constructs the posterior probability density function of damage location parameters according to Bayes' theorem for feature-level data fusion. That is: Bayesian feature-level fusion.

[0097] S6 uses Hamiltonian Monte Carlo sampling to solve the posterior probability density function of the damage location parameters, and performs damage probability imaging localization based on the posterior probability distribution.

[0098] Specifically, step S2 includes the following steps:

[0099] S2.1, due to the presence of nonlinear damage, the excitation and response of the system are simplified to:

[0100] (1);

[0101] in, Indicates the excitation signal. For system response, , and These are the amplitude coefficients for the fundamental wave, second harmonic, and third harmonic, respectively.

[0102] S2.2, based on pulse inversion technology, firstly, the sensor array (i.e., a sparse piezoelectric sensor network deployed on the surface of the structure under test) is excited with positive phase (phase angle of 0°) and the response signal is acquired; then, the excitation signal is excited with inverse phase (phase angle of 180°), and the two acquired time-domain signals are superimposed; assuming the positive phase excitation is... The opposite incentive is - The responses obtained from the two stimuli are as follows:

[0103] (2);

[0104] in, and These represent the positive-phase response signal and the negative-phase response signal, respectively. Superimposing the two response signals yields:

[0105] (3).

[0106] Therefore, a significantly enhanced second harmonic signal can be obtained through pulse inversion technology. The amplitudes of fourth-order and higher harmonics are much lower than those of the second harmonic and can be ignored.

[0107] Specifically, step S3 includes the following steps:

[0108] S3.1, noise signals in the second harmonic signal are filtered out by continuous Shannon wavelet transform. For the second harmonic signal in real space... The continuous Shannon wavelet transform (WT) is defined as follows:

[0109] (4);

[0110] in, As a scale factor, The translation factor; Represents the space of real numbers, where "¯" represents conjugate. Shannon mother wavelet function:

[0111] (5);

[0112] in, Represents the imaginary unit. Indicates time, This represents the bandwidth of the mother wavelet. is the center frequency of the Shannon mother wavelet.

[0113] S3.2, a purer second harmonic signal is extracted using continuous Shannon wavelet transform. The relationship between the center frequency and the scaling factor of the second harmonic signal is as follows:

[0114] (6);

[0115] in, The center frequency of the second harmonic. The sampling frequency of the nonlinear Lamb wave signal. is the scale factor.

[0116] Therefore, by adjusting the scaling factor Shannon mother wavelet center frequency bandwidth of the mother wavelet This allows for the extraction of a purer second harmonic signal.

[0117] S3.3, Calculate the envelope of the second harmonic signal using the Hilbert transform:

[0118] (7);

[0119] in, It is a second harmonic signal. for Hilbert transform, This is the corresponding envelope signal.

[0120] S3.4 Extract the time-of-flight characteristics of the second harmonic based on the peak value of the envelope signal.

[0121] Specifically, step S4 is as follows:

[0122] The spectrum of the positive-phase excitation nonlinear Lamb wave response signal is obtained using Fast Fourier Transform, and the nonlinear coefficient damage index characteristic (NCDI) is calculated according to the following formula:

[0123] (8);

[0124] in, The fundamental amplitude in the spectrum. This represents the amplitude of the second harmonic in the frequency spectrum.

[0125] Specifically, step S5 includes the following steps:

[0126] S5.1, such as Figure 2 As shown, assuming the center coordinates of the damage are... Grouping two different receiving sensors corresponding to the same exciter into a set, there are a total of Group; According to the trajectory imaging method, the second harmonic is related to the first harmonic; Set up two sensors, sensor and The theoretical calculation value of the flight time difference is :

[0127] (9);

[0128] in, and Sensors and coordinates and These are the second harmonics in the sensor and Group velocity in the direction.

[0129] S5.2, Due to the presence of random and systematic errors, the measured flight time difference is uncertain. The measured flight time difference is expressed as... :

[0130] (10);

[0131] in, The expression is uncertain and follows a mean of 0 and a variance of . The Gaussian distribution.

[0132] S5.3, based on the measured second harmonic flight time difference characteristic data Construct the likelihood function using formula (9) :

[0133] (11);

[0134] in, It is an exponential function.

[0135] Specifically, step S5 also includes the following steps:

[0136] S5.4, such as Figure 3 The diagram shown illustrates the principle of the defect probability detection reconstruction algorithm. Assume the coordinate distributions of the excitation sensor and the receiving sensor are as follows: and .Depend on Figure 3 It can be seen that when the nonlinear coefficient of the damage exponent in the excitation receiving path is significant, the probability of the damage occurring on the direct propagation path is the highest. Meanwhile, as the distance between the path and the direct propagation path increases, the probability of damage occurring gradually decreases. A prior distribution of the damage location parameters is constructed based on the defect probability detection and reconstruction algorithm. :

[0137] (12);

[0138] in, For the first The nonlinear coefficient of the excitation receiving path is the damage exponent. To incentivize the total number of receiving paths; For elliptic distribution functions:

[0139] (13);

[0140] in, The parameter for controlling the shape of the ellipse is 1.015, which is empirically determined. The expression is:

[0141] (14);

[0142] The coordinates of the excitation sensor and the receiving sensor are respectively... and .

[0143] S5.5, According to Bayes' theorem, the posterior distribution of the damage location parameters is... satisfy:

[0144] (15);

[0145] in, It represents a direct proportion to.

[0146] The posterior distribution of the damage location parameters was obtained. :

[0147] (16);

[0148] Therefore, according to formula (16), feature-level data fusion of two nonlinear Lamb wave features, namely flight time difference and nonlinear coefficient damage index, was achieved.

[0149] Specifically, step S6 is as follows:

[0150] The probability distribution of the damage location within the monitoring area can be written in the standard form of a two-dimensional normal distribution. :

[0151] (17);

[0152] in, Damage location parameters The mean vector; It is the covariance matrix: ; Covariance matrix The determinant of; The coordinates corresponding to the maximum value are the predicted damage location.

[0153] To verify the effectiveness of the method of this invention, nonlinear Lamb wave detection data for delamination damage in composite material plate structures were generated using numerical simulation. The dimensions of the structure under test, sensor layout, and damage settings are as follows: Figure 4 As shown. Figure 4 T1, T2, T3, and T4 are piezoelectric sensors. The composite material plate has a geometric dimension of 400 mm × 400 mm × 2 mm. The composite material type is AS4M3502, and the layup is [0° / 90°]. 2s Each layer is 0.25 mm thick, and the center of the delamination damage is located at (200 mm, 150 mm), between the bottommost single-layer board and the second-to-last single-layer board. For example... Figure 5 The figure shows the phase velocity dispersion curve of the composite material plate. According to the dispersion curve, only two modes, A0 and S0, exist in the lower frequency range, and there is a nonlinear cumulative effect of the S0 mode. Therefore, considering all factors, as... Figure 6 As shown, the excitation signal is a 10-cycle sine wave signal modulated by a Hanning window with a center frequency of 150 kHz. Figures 7-12 As shown, the pulse inversion technique significantly enhances the second harmonic component. Furthermore, continuous Shannon wavelet transform is applied to filter the second harmonic signal, reducing noise interference. For example... Figure 13 As shown, the envelope signal is calculated using Hilbert transform and the flight time is extracted, thus obtaining the time difference characteristics. The spectrum of the nonlinear Lamb wave signal is obtained using fast Fourier transform, and the nonlinear coefficient damage index characteristics are calculated according to formula (8). Figure 14 As shown, this is the group velocity profile of the second harmonic. Considering the anisotropy of the composite material, the posterior probability density function of the damage location parameter is established according to formula (16), thus completing the nonlinear feature data fusion. Figure 15 and Figure 16 As shown, the posterior distribution of the damage location parameters obtained by solving formula (16) using Hamiltonian Monte Carlo sampling is presented. Finally, the probability distribution of the damage location obtained according to formula (17) is compared with that of traditional methods. Figures 17-19 As shown, the trajectory imaging positioning error is 14.6 mm, the defect probability detection reconstruction algorithm positioning error is 36.4 mm, while the error of the method of the present invention is only 3.0 mm, indicating that the damage positioning accuracy of the method of the present invention is higher.

[0154] Of course, the above description is not intended to limit the present invention, and the present invention is not limited to the examples given above. Any changes, modifications, additions or substitutions made by those skilled in the art within the scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A non-linear Lamb wave and Bayesian inference based reference-free damage imaging method, characterized in that, Specifically comprising the following steps: S1, deploying sparse piezoelectric sensor network on the surface of the structure to be tested, all sparse piezoelectric sensors are stimulated and received in turn to form several damage detection paths, and the nonlinear Lamb wave signals of multiple propagation paths are collected; S2, obtaining enhanced second harmonic signals through pulse inversion technology; S3, filtering noise signals in the second harmonic signals through continuous Shannon wavelet transform, and calculating the envelope signal after filtering by applying Hilbert transform, and then extracting the time-of-flight feature of the envelope signal; S4, obtaining the frequency spectrum of the nonlinear Lamb wave signal by using fast Fourier transform, and calculating the nonlinear coefficient damage index feature; S5, constructing a likelihood function by using the time-of-flight difference feature combined with the trajectory imaging positioning principle, constructing a prior probability density function by using the nonlinear coefficient damage index feature combined with the defect probability detection reconstruction algorithm principle, and constructing a posterior probability density function of the damage position parameter according to Bayes theorem for feature level data fusion; S6, solving the posterior probability density function of the damage position parameter by using Hamilton Monte Carlo sampling, and performing damage probability imaging positioning according to the posterior probability distribution.

2. The non-linear Lamb wave and Bayesian inference based reference-free damage imaging method according to claim 1, characterized in that, Step S2 specifically comprises the following steps: S2.1, due to the existence of nonlinear damage, the excitation and response of the system are simplified as: （1）； wherein, represents the excitation signal, is the system response, , and are the amplitude coefficients of the fundamental, second and third harmonic, respectively. S2.2, based on the pulse inversion technique, first excite the sensor array with positive phase and collect the response signal; then excite the sensor array with negative phase, and superimpose the two time-domain signals obtained. Assuming that the positive phase excitation is and the negative phase excitation is , then the responses obtained from the two excitations are respectively: （2）； wherein and respectively represent the positive phase response signal and the inverted phase response signal, and the two response signals are superimposed to obtain: （3）。 3. The non-linear Lamb wave and Bayesian inference based reference-free damage imaging method of claim 1, wherein, Step S3 specifically comprises the following steps: S3.1, filtering noise signals in the second harmonic signal by a continuous Shannon wavelet transform, for the continuous Shannon wavelet transform WT of the second harmonic signal in the real number space is defined as follows: （4）； where is a scale factor, is a translation factor; denotes the real space, " " denotes the conjugate, is the Shannon mother wavelet function: （5）； wherein, denotes the imaginary unit, denotes time, denotes the bandwidth of the mother wavelet, is the center frequency of the Shannon mother wavelet; S3.2, the relationship between the center frequency of the second harmonic signal and the scale factor is as follows: （6）； wherein, is the center frequency of the second harmonic, is the sampling frequency of the nonlinear Lamb wave signal, is a scaling factor; S3.3, the envelope of the second harmonic signal is calculated by applying Hilbert transform: （7）； wherein is a second harmonic signal, is is a Hilbert transform of is a corresponding envelope signal; S3.4, the time-of-flight feature of the second harmonic is extracted according to the peak value of the envelope signal.

4. The non-linear Lamb wave and Bayesian inference based reference-free damage imaging method of claim 1, wherein, Step S4 is specifically: The frequency spectrum of the positive phase excitation nonlinear Lamb wave response signal is obtained by using fast Fourier transform, and the nonlinear coefficient damage index feature NCDI is calculated according to the following formula: （8）； wherein, is the fundamental amplitude in the frequency spectrum, is the second harmonic amplitude in the frequency spectrum.

5. The non-linear Lamb wave and Bayesian inference based reference-free damage imaging method of claim 1, wherein, Step S5 specifically comprises the following steps: S5.1, assuming the center coordinates of the lesion are Two different receiving sensors corresponding to the same exciter are a group, and there are groups in total; according to the trajectory imaging method, the second harmonic for the two sensors of the group, the time difference of the flight of the sensors and is theoretically calculated as : （9）； wherein and are the coordinates of the sensor and respectively, and are the group velocities of the second harmonic in the direction of the sensor and respectively. S5.2, the measured time-of-flight difference is expressed as : （10）； wherein, is an uncertainty, subject to a Gaussian distribution with mean 0 and variance . S5.3, the measured second harmonic time-of-flight difference characteristic data and equation (9) to construct a likelihood function : （11）； wherein is an exponential function.

6. The non-linear Lamb wave and Bayesian inference based reference-free damage imaging method of claim 5, wherein, Step S5 further comprises the following steps: S5.

4. Constructing the prior distribution of damage location parameters from the defect probability detection reconstruction algorithm : （12）； wherein, is the number of the first nonlinear coefficient impairment index of the mth is the total number of the excitation receiving paths; is an elliptic distribution function: （13）； wherein is an elliptical shape control parameter, The expression is: （14）； wherein the coordinates of the excitation sensor and the receiving sensor are respectively and ; S5.5, According to Bayes' theorem, the posterior distribution of the damage location parameter satisfies: （15）； wherein proportional to; obtaining a posterior distribution of the damage location parameter : （16）。 7. The non-linear Lamb wave and Bayesian inference based reference-free damage imaging method of claim 1, wherein, Step S6 is specifically: The probability distribution of the location of the lesion within the monitoring region is written in the standard form of a two-dimensional normal distribution : （17）； wherein, is the mean vector of the damage location parameters ; is the covariance matrix: is the determinant of the covariance matrix ; The coordinates corresponding to the maximum value are the predicted damage location.

Images