Impact echo method resonant frequency calculation method based on lamb wave standing wave effect

Abstract

The invention belongs to the technical field of engineering structure nondestructive testing, and provides an impact echo method resonant frequency calculation method for generating a standing wave effect at a first-order symmetric mode zero group velocity based on Lamb waves. A Rayleigh-Lamb wave frequency dispersion equation is analyzed, and an expression of Lamb wave first-order symmetric mode cut-off frequency calculated by longitudinal wave velocity or transverse wave velocity is given by taking Poisson's ratio 1 / 3 as a boundary; designing an example and utilizing Lamb wave frequency dispersion curve calculation software to solve the Lamb wave first-order symmetric mode cut-off frequency and the frequency (resonance frequency) when the velocity of the mode group is 0, defining beta as the ratio of the resonance frequency to the cut-off frequency, and solving the value of beta changing along with the Poisson's ratio in the plate structure; and a single-layer plate impact echo resonance frequency analytical calculation formula is given in combination with beta and a cut-off frequency expression. The impact echo resonance frequency calculation method is researched on the basis of the Lamb wave fluctuation theory, and the method has important engineering significance for improving the precision of concrete structure defect detection and component thickness measurement through the impact echo method.

Description

Technical Field

[0001] This invention belongs to the field of non-destructive testing technology for engineering structures, and proposes a method for calculating the resonance frequency of an impact echo based on the standing wave effect generated at the zero group velocity of the first-order symmetric mode of Lamb wave. Background Technology

[0002] The impact-echo method utilizes the transient resonance response induced by Lamb wave reflection. For detecting the thickness of plates and the depth of defects, relevant information can be obtained through the resonance frequency and its variation. The defect depth or component thickness can then be calculated using the resonance frequency calculation formula. Therefore, studying the impact-echo resonance response based on Lamb wave theory and exploring accurate and effective methods for calculating the resonance frequency are crucial for optimizing the calculation results of the detection data, improving the reliability and effectiveness of the impact-echo method's detection results, and promoting its development towards refinement and precision. This has significant engineering implications for improving the accuracy of concrete structure defect detection and component thickness measurement using the impact-echo method.

[0003] The modes of resonance response induced by impact echo are divided into thickness resonance mode and bending vibration mode. Among them, bending vibration mode is only excited in the medium between the plate surface and the top surface of the defect when the impact echo method is applied to detect hidden defects near the surface of the plate with a large width-to-thickness ratio. In most cases where the impact echo method is applied in engineering practice, the thickness resonance mode of the impact echo resonance response is studied.

[0004] In the study of thickness resonance frequency correlation using the impact echo method, the local transient resonance phenomenon generated by the impact echo is a standing wave effect formed when the velocity of the first-order symmetric mode group of the Lamb wave is 0, and the impact echo resonance frequency is the frequency when the velocity of the first-order symmetric mode group of the Lamb wave is 0. The shape factor correction parameter β in the traditional empirical formula for calculating the impact echo resonance frequency of a single-layer plate is empirically taken as 0.96. Although β of 0.96 is widely accepted in practice and included in the ASTM standard, it lacks rigorous theoretical basis, the empirical formula for calculating the resonance frequency is not rigorous enough, and the calculation results contain certain errors. Therefore, theoretically explaining the shape factor correction parameter β and giving a specific value, thereby obtaining a theoretically rigorous analytical formula for calculating the resonance frequency of the impact echo method, becomes the problem that this invention aims to solve. Summary of the Invention

[0005] The purpose of this application is to provide a method for calculating the resonance frequency of the impulse echo based on the Lamb wave standing wave effect, which aims to solve the problems in the prior art.

[0006] This application provides a method for calculating the resonance frequency based on the Lamb wave standing wave effect using the impulse echo method, as detailed below:

[0007] (1) Starting from the Rayleigh-Lamm wave dispersion equation, we introduce the constraint condition that the plate structure has uniform thickness resonance at the cutoff frequency of the symmetric modes other than the fundamental mode of the Lamb wave. Under this condition, the zero wave number obtained by the phase velocity of the Lamb wave tending to infinity is substituted into the dispersion equation to obtain the expression for the cutoff frequency.

[0008] (2) An expression for the cutoff frequency of the first-order symmetric mode, calculated from the longitudinal wave velocity or the transverse wave velocity, is given with the material Poisson's ratio of 1 / 3 as the boundary.

[0009] (3) Design a calculation example and use Lamb wave dispersion curve calculation software to find the cutoff frequency of the first-order symmetric mode of Lamb wave and the frequency (i.e., resonance frequency) when the group velocity of the mode is 0. Define β as the ratio of resonance frequency to cutoff frequency and find the value of β as a function of Poisson's ratio in the plate structure.

[0010] (4) Combining the analytical calculation formula of β and the cutoff frequency, the analytical calculation formula of the resonance frequency of the impact echo of a single-layer plate is given.

[0011] Furthermore, the wave characteristic equation of the Lamb wave is as follows:

[0012]

[0013]

[0014]

[0015] In the formula: h is the plate thickness, ω is the angular frequency (also known as the circular frequency), and V P V is the longitudinal wave velocity. S Let k be the transverse wave velocity and k be the wave number along the horizontal direction of the plate.

[0016]

[0017] In the formula: λ is the wavelength, V Ph Let f be the phase velocity, T be the wave period, and f be the wave frequency.

[0018] At the cutoff frequency, the phase velocity tends to infinity, therefore the wave number k = 0. Substituting k = 0 into the symmetric mode Rayleigh-Lamb wave equation, we get:

[0019]

[0020] Analysis yields the expression for the cutoff frequency of the Lamb wave symmetric mode related to the transverse wave velocity:

[0021]

[0022] In the formula: n1∈N + N + It is the set of positive integers;

[0023] Analysis yields the expression for the cutoff frequency of the Lamb wave symmetric mode related to the P-wave velocity:

[0024]

[0025] In the formula: n2∈N, N is the set of natural numbers.

[0026] Furthermore, considering the first-order mode, where n1 = 1 and n2 = 0, the cutoff frequency f of the first-order symmetric mode of the Lamb wave is... C f should be taken CS and f CP The smaller of the two values ​​is used to obtain the cutoff frequency f of the first-order symmetric mode of the Lamb wave. C The calculation expression:

[0027]

[0028] Furthermore, the value of the plate structure shape factor correction parameter β as a function of Poisson's ratio is calculated, and an image of the plate structure shape factor correction parameter β as a function of Poisson's ratio is obtained.

[0029] Furthermore, combining the analytical calculation formulas for β and the cutoff frequency, an analytical calculation formula for the resonance frequency of the impact echo of a single-layer plate is given:

[0030]

[0031] The beneficial effects of this invention are as follows: This invention studies the resonance response of impact echo based on Lamb wave theory, and the resonance frequency calculation method of impact echo method based on the standing wave effect generated at the zero group velocity of the first-order symmetric mode of Lamb wave promotes the optimization of the calculation results of the detection data, improves the reliability and effectiveness of the impact echo method detection results, and has important engineering significance for improving the accuracy of concrete structure defect detection and component thickness measurement using the impact echo method. Attached Figure Description

[0032] Figure 1 This is a graph showing the relationship between the shape factor correction parameter β and Poisson's ratio in the formula for calculating the resonance frequency of a single-layer plate impact echo in this invention.

[0033] Figure 2 This is a simplified finite element model diagram of an impact echo experiment.

[0034] Figure 3 This is a frequency domain diagram obtained by FFT transformation of the time domain signal simulating an impact echo under one condition.

[0035] Figure 4 This is the frequency domain diagram obtained by FFT transformation of the time domain signal simulating the impact echo in another case. Detailed Implementation

[0036] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. 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.

[0037] This application provides a method for calculating the resonance frequency based on the Lamb wave standing wave effect using the impulse echo method, as detailed below:

[0038] (1) Starting from the Rayleigh-Lamm wave dispersion equation, we introduce the constraint condition that the plate structure has uniform thickness resonance at the cutoff frequency of the symmetric modes other than the fundamental mode of the Lamb wave. Under this condition, the zero wave number obtained by the phase velocity of the Lamb wave tending to infinity is substituted into the dispersion equation to obtain the expression for the cutoff frequency.

[0039] (2) An expression for the cutoff frequency of the first-order symmetric mode, calculated from the longitudinal wave velocity or the transverse wave velocity, is given with the material Poisson's ratio of 1 / 3 as the boundary.

[0040] (3) Design a calculation example and use Lamb wave dispersion curve calculation software to find the cutoff frequency of the first-order symmetric mode of Lamb wave and the frequency (i.e., resonance frequency) when the group velocity of the mode is 0. Define β as the ratio of resonance frequency to cutoff frequency and find the value of β as a function of Poisson's ratio in the plate structure.

[0041] (4) Combining the analytical calculation formula of β and the cutoff frequency, the analytical calculation formula of the resonance frequency of the impact echo of a single-layer plate is given.

[0042] The specific implementation method is as follows:

[0043] The wave characteristic equation of the Lamb wave is as follows:

[0044]

[0045]

[0046]

[0047] In the formula: h is the plate thickness, ω is the angular frequency (also known as the circular frequency), and V P V is the longitudinal wave velocity. S Let k be the transverse wave velocity and k be the wave number along the horizontal direction of the plate.

[0048]

[0049] In the formula: λ is the wavelength, V Ph Let f be the phase velocity, T be the wave period, and f be the wave frequency.

[0050] At the cutoff frequency, the phase velocity tends to infinity, therefore the wave number k = 0. Substituting k = 0 into the symmetric mode Rayleigh-Lamb wave equation, we get:

[0051]

[0052] Analysis yields the expression for the cutoff frequency of the Lamb wave symmetric mode related to the transverse wave velocity:

[0053]

[0054] In the formula: n1∈N + N + It is the set of positive integers.

[0055] Analysis yields the expression for the cutoff frequency of the Lamb wave symmetric mode related to the P-wave velocity:

[0056]

[0057] In the formula: n2∈N, N is the set of natural numbers.

[0058] Consider the first-order mode, where n1 = 1 and n2 = 0. The cutoff frequency f of the first-order symmetric mode of the Lamb wave is... C f should be taken CS and f CP The smaller of the two values ​​is used to obtain the cutoff frequency f of the first-order symmetric mode of the Lamb wave. C The calculation expression:

[0059]

[0060] The value of the shape factor correction parameter β of the plate structure as a function of Poisson's ratio is calculated, and the graph of the shape factor correction parameter β as a function of Poisson's ratio is obtained.

[0061] Combining the analytical calculation formulas for β and cutoff frequency, the analytical calculation formula for the resonance frequency of a single-layer plate impact echo is given:

[0062]

[0063] A numerical simulation of an impact echo experiment was conducted using ABAQUS finite element software. The plate was set to have an infinitely large transverse dimension and no support. An impact load was applied to the center of the plate. The analysis focused on the right half of the plate, with the center as the axis. The plate thickness was 0.2 meters, and the longitudinal dimension of the selected portion of the plate was 2 meters. The impact echo study region was defined as the area within 1.4 meters of the impact point. A simplified finite element model of the impact echo experiment is shown below. Figure 2 Two sets of finite element models of single-layer slab structures were established: the first set of models was made of cement concrete with a density of 2400 kg / m³. 3With a Poisson's ratio of 0.2 and a transverse wave velocity of 2485 m / s in the plate, the frequency domain of the simulated impact echo signal obtained by FFT transformation is shown below. Figure 3 The precise resonant frequency is 9670Hz; the other set of models is made of asphalt with a density of 2300 kg / m³. 3 With a Poisson's ratio of 0.35 and a transverse wave velocity of 1200 m / s in the plate, the frequency domain of the simulated impact echo signal obtained by FFT transformation is shown below. Figure 4 The precise resonant frequency is 5552Hz.

[0064] The impact echo resonant frequency of the model is estimated using the traditional empirical formula for calculating the impact echo resonant frequency of a single-layer plate. The empirical formula for calculating the impact echo resonant frequency of a single-layer plate is as follows:

[0065]

[0066] In the empirical formula for calculating the impact echo resonance frequency of a single-layer slab, β = 0.96, estimating the impact echo resonance frequency of a cement concrete slab to be 9739 Hz, with an error of 0.71% compared to the accurate resonance frequency; the empirical formula for calculating the impact echo resonance frequency of a single-layer slab estimates the impact echo resonance frequency of an asphalt slab to be 5995 Hz, with an error of 7.98% compared to the accurate resonance frequency.

[0067] The impact echo resonance frequency of a single-layer plate is calculated using an analytical formula derived from the impact echo method based on the standing wave effect generated at the zero group velocity of the first-order symmetric mode of the Lamb wave. Figure 1 When Poisson's ratio is 0.2, β = 0.9530; when Poisson's ratio is 0.35, β = 0.9276. The analytical calculation formula for the impact echo resonance frequency of a single-layer slab yields an impact echo resonance frequency of 9668 Hz for cement concrete slabs, with an error of 0.02% compared to the accurate resonance frequency; the calculated impact echo resonance frequency for asphalt slabs is 5566 Hz, with an error of 0.25% compared to the accurate resonance frequency.

[0068] Comparing the calculation results of the analytical formula for calculating the resonance frequency of a single-layer plate impact echo in the method of this invention with the calculation results of the empirical formula for calculating the resonance frequency of a single-layer plate impact echo, the results show that the calculation accuracy of the method of this invention is tens of times higher than that of the traditional method for calculating the resonance frequency of impact echo.

[0069] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

Claims

1. A method for calculating the resonant frequency based on the Lamb wave standing wave effect using the impulse echo method, characterized in that, Specifically as follows: (1) Starting from the Rayleigh-Lamm wave dispersion equation, we introduce the constraint condition that the plate structure has uniform thickness resonance at the cutoff frequency of the symmetric modes other than the fundamental mode of the Lamb wave. Under this condition, the zero wave number obtained by the phase velocity of the Lamb wave tending to infinity is substituted into the dispersion equation to obtain the expression for the cutoff frequency. (2) An expression for the cutoff frequency of the first-order symmetric mode, calculated from the longitudinal wave velocity or the transverse wave velocity, is given with the material Poisson's ratio of 1 / 3 as the boundary. (3) Design a calculation example and use Lamb wave dispersion curve calculation software to find the cutoff frequency of the first-order symmetric mode of Lamb wave and the frequency (i.e., resonance frequency) when the group velocity of the mode is 0. Define β as the ratio of resonance frequency to cutoff frequency and find the value of β as a function of Poisson's ratio in the plate structure. (4) Combining the analytical calculation formula of β and the cutoff frequency, the analytical calculation formula of the resonance frequency of the impact echo of a single-layer plate is given.

2. The method for calculating the resonance frequency based on the Lamb wave standing wave effect according to claim 1, characterized in that, The wave characteristic equation of the Lamb wave is as follows: In the formula: h is the plate thickness, ω is the angular frequency, and V P V is the longitudinal wave velocity. S Let k be the transverse wave velocity and k be the wave number along the horizontal direction of the plate. In the formula: λ is the wavelength, V Ph Let f be the phase velocity, T be the wave period, and f be the wave frequency. At the cutoff frequency, the phase velocity tends to infinity, therefore the wave number k = 0. Substituting k = 0 into the symmetric mode Rayleigh-Lamb wave equation, we get: Analysis yields the expression for the cutoff frequency of the Lamb wave symmetric mode related to the transverse wave velocity: In the formula: n1∈N + N + It is the set of positive integers; Analysis yields the expression for the cutoff frequency of the Lamb wave symmetric mode related to the P-wave velocity: In the formula: n2∈N, N is the set of natural numbers.

3. The method for calculating the resonance frequency based on the Lamb wave standing wave effect according to claim 2, characterized in that, Consider the first-order mode, where n1 = 1 and n2 = 0. The cutoff frequency f of the first-order symmetric mode of the Lamb wave is... C f should be taken CS and f CP The smaller of the two values ​​is used to obtain the cutoff frequency f of the first-order symmetric mode of the Lamb wave. C The calculation expression:

4. The method for calculating the resonance frequency based on the Lamb wave standing wave effect according to claim 1, characterized in that, The value of the shape factor correction parameter β of the plate structure as a function of Poisson's ratio is calculated, and the graph of the shape factor correction parameter β as a function of Poisson's ratio is obtained.

5. The method for calculating the resonance frequency based on the Lamb wave standing wave effect according to claim 4, characterized in that, Combining the analytical calculation formulas for β and cutoff frequency, the analytical calculation formula for the resonance frequency of a single-layer plate impact echo is given:

Images