A method for solving dispersion curves of ultrasonic guided waves in multi-layer spiral structures

Abstract

The present invention discloses a method for solving the dispersion curve of ultrasonic guided waves of a multilayer helical structure, comprising establishing a 3D model of a multilayer helical structure in which each helical line has a single pitch length; meshing the cross section; calculating the mesh size in the mesh sweep direction; mesh sweeping each helical line; solving the unit stiffness matrix and unit mass matrix of each node, using a rectangular coordinate system for straight lines and a spiral coordinate system for spiral lines; assembling the cross-sectional overall stiffness matrix and cross-sectional overall mass matrix of each line, and assembling the pitch overall stiffness matrix and pitch overall mass matrix; periodically replicating and expanding the pitch overall stiffness matrix and pitch overall mass matrix of each line to assemble the overall stiffness matrix and overall mass matrix of the multilayer helical structure; solving the wave number, group velocity, and phase velocity; and matching the guided wave eigenvectors based on orthogonality. The present invention can achieve a fast and accurate solution to the dispersion curve of a multilayer helical structure composed of different pitches, handedness, sizes, or materials.

Description

Technical Field

[0001] The present invention relates to the technical field of nondestructive testing, and in particular to a method for solving a dispersion curve of an ultrasonic guided wave with a multi-layer spiral structure. Background Art

[0002] Cables and cable strands are typical multi-layer spiral structures. Ultrasonic guided wave nondestructive testing of these structures is an important means of preventing accidents. Ultrasonic guided wave dispersion curves, which reflect the relationship between the number of ultrasonic guided wave modes, group velocity, and phase velocity and frequency, are a key theoretical foundation for ultrasonic guided wave damage detection. They provide a reference for selecting appropriate guided wave modes and excitation frequency ranges, and are a crucial basis for damage detection and location.

[0003] The existing dispersion curve solution method (CN202310431186.5) obtains a first rough solution of the dispersion matrix; obtains a preset number of first exact solutions based on the first rough solution; performs root tracing calculations based on the first exact solution to obtain a second rough solution; obtains a second exact solution based on the second rough solution; and plots a dispersion curve based on the first and second exact solutions. Obtaining the first rough solution of the dispersion matrix includes: setting the wave number to a preset fixed value; calculating a first reference value corresponding to the dispersion matrix based on an initial frequency value; determining a second frequency value based on the initial frequency value and a preset step size; calculating a second reference value corresponding to the dispersion matrix based on the second frequency value; and determining the first rough solution based on the first and second reference values. Obtaining a preset number of first precise solutions based on the first rough solution includes setting an initial interval with a preset frequency step size in the frequency increasing direction and the frequency decreasing direction of the first rough solution; obtaining the boundary values ​​corresponding to the dispersion matrix at the endpoints of the initial interval; selecting the endpoint with the smaller absolute value of the boundary value and the origin to perform an interval reduction operation to form a screening interval; repeating the interval reduction operation so that the screening interval is less than or equal to a preset error, thereby obtaining the first precise solution. The dispersion curve solution method proposed in the prior art does not require repeatedly obtaining the dispersion matrix for different frequency intervals. Instead, it obtains the dispersion curve by obtaining the rough value corresponding to the dispersion matrix a finite number of times and then performing an accurate solution and root tracing calculation. This prior art does not solve the curve of the spiral structure. Since the spiral structure is characterized by being composed of single wires with different pitches, different hands or different sizes, the solution should fully consider the contact between the single wires in each layer.

[0004] When solving the ultrasonic guided wave dispersion curve for a regular waveguide structure, the traditional semi-analytical finite element method uses an analytical solution in the wave propagation direction and a finite element method for the cross-section of the waveguide structure, reducing one dimension. However, due to the differences in pitch, handedness, and pitch-to-diameter ratio between the individual helical wires in a multilayered helical structure, it is difficult to unify the solutions for each helical wire along the wave propagation direction within a single coordinate system, resulting in inaccurate dispersion curve solutions. Summary of the Invention

[0005] In order to solve the above technical problems, the purpose of the present invention is to provide a method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure, so as to realize the solution of the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure composed of single wires with different pitches, different rotation directions, different sizes or different materials.

[0006] The present invention is achieved through at least one of the following technical solutions.

[0007] A method for solving a dispersion curve of an ultrasonic guided wave of a multi-layer spiral structure comprises the following steps:

[0008] A. Establish a 3D model of a multi-layer spiral structure in which each spiral line has a single pitch length;

[0009] B. Divide the mesh of each single line at the cross section and refine the mesh in the contact area;

[0010] C. Calculate the pitch of the shortest helical layer divided by the number of single lines in the adjacent helical layer. Set the minimum value of the calculated result to h. If h is greater than half the wavelength of the maximum frequency signal in the dispersion curve solution range, reduce h by integer multiples to obtain h', until h' is no greater than half the wavelength of the maximum frequency signal in the dispersion curve range;

[0011] D. Sweep the mesh of each spiral line along its centerline. The mesh size of each layer of spiral line in the sweeping direction is d. L =αh', where α is the twist coefficient of each layer, and no mesh sweeping is performed on straight single lines;

[0012] E. Solve the unit stiffness matrix k1, k2, k3 and unit mass matrix m of each node of each linear single-line cross-section mesh based on the rectangular coordinate system; solve the unit stiffness matrix and unit mass matrix of each node of each spiral single-line swept volume mesh based on the spiral coordinate system;

[0013] F. Assemble the overall stiffness matrix K of each single line in each section parallel to the z = 0 plane S1 , K S2 , K S3 and the cross-section overall mass matrix M S The overall stiffness matrix and overall mass matrix of each section of the spiral single line are assembled into the pitch overall stiffness matrix K V1 , KV2 , K V3 and the pitch overall mass matrix M V ;

[0014] G. Periodically replicate and extend the pitch overall stiffness matrix and pitch overall mass matrix of each single line to the same grid length and the length of the measured object that satisfies the dispersion curve solution, and assemble them into the multi-layer spiral structure overall stiffness matrix K1, K2, K3 and the multi-layer spiral structure overall mass matrix M;

[0015] H. According to the general homogeneous wave equation of ultrasonic guided waves, given the value of the series frequency ω with an interval of Δω, solve the wave number k and group velocity c g , phase velocity c p ;

[0016] I. Orthogonality-based guided wave eigenvectors of different modes: wave number k, group velocity c g , phase velocity c p to match.

[0017] Furthermore, in the 3D model of the multi-layer spiral structure in step A, the length of each layer of spiral single wire is the pitch of the single wire of this layer. For a straight single wire, only the cross-sectional dimensions of this layer are drawn.

[0018] Furthermore, in step B, each single line is meshed at the cross section, and the mesh size is no larger than half the wavelength of the maximum value frequency signal in the dispersion curve range.

[0019] Furthermore, in step E, the discrete grid units Ω of each straight line single-line cross-section grid are solved based on the rectangular coordinate system. e The solution formulas for the node element stiffness matrix k1, k2, k3 and the element mass matrix m are as follows:

[0020]

[0021] Where B1 and B2 are strain matrices, C is the elastic constant matrix of the material, N(x,y) is the shape function matrix of the node coordinates (x,y), and ρ is the material density.

[0022] Furthermore, the strain matrix B2=L z N(x,y),L x , L y , L z is the corresponding differential operator:

[0023]

[0024] Furthermore, in step E, the discrete grid Ω of each spiral single-line swept volume grid is solved based on the spiral coordinate system. eThe solution formulas for the node element stiffness matrix k1, k2, k3 and the element mass matrix m are as follows:

[0025]

[0026] Where C is the elastic constant matrix of the material, s is the arc length along the center line of the helix, L xy 、L s is the differential operator, and N(s) is the shape function matrix:

[0027]

[0028]

[0029]

[0030] Where l is the length of the curve of one spiral period, (e X ,e Y ,e Z ) are the orthogonal basis vectors in the Cartesian coordinate system, θ is the initial phase angle of the center line of the spiral line on the z = 0 plane, κ and τ are the curvature and torsion of the spiral line, respectively.

[0031] Furthermore, in step F:

[0032]

[0033] Where n is the number of nodes in the cross-section mesh. For a spiral single line, let the number of mesh nodes along the sweep direction of a spiral single line within a single pitch be P, and we have:

[0034]

[0035] Furthermore, the wave number k is solved in step H, and the general homogeneous wave equation of ultrasonic guided waves is obtained according to the Hamilton principle formula, {K1+ikK2+k 2 K3-ω 2 M}U=0, U is the particle displacement at the node position, i is the imaginary unit, the frequency range is solved by solving the ultrasonic guided wave dispersion curve, and the wave number k is solved by given the value of the series frequency ω with an interval of Δω.

[0036] Furthermore, in step H, the group velocity c is solved g , phase velocity c p , the solution formula is as follows:

[0037] c g =Δω / Δk,c p =ω / k

[0038] Where Δk is the difference in wavenumber k between two adjacent ω values.

[0039] Furthermore, the waveguide eigenvectors of different modes solved based on orthogonality in step I: wave number k, group velocity c g , phase velocity c p For matching, the orthogonality solution formula is as follows:

[0040]

[0041] Among them, ψ m and ψ n Represents the guided wave eigenvectors of different modes,

[0042] Compared with the existing technology, the beneficial effects of the present invention are:

[0043] The present invention takes into account the characteristics of multi-layer spiral structures that are composed of single wires with different pitches, different rotation directions or different sizes. The contact conditions between the single wires of each layer are fully considered, and an accurate and complete stiffness matrix and mass matrix are constructed through the single-pitch grid cycle of each single wire, thereby achieving a faster and more accurate solution of the dispersion curve of the multi-layer spiral structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Various other advantages and benefits will become apparent to those skilled in the art upon reading the detailed description of the preferred embodiment below. The accompanying drawings are for illustration purposes only and are not to be considered as limiting the present description. The same reference symbols are used throughout the drawings to represent the same components. In the drawings:

[0045] Figure 1 This is a flow chart of a method for solving a dispersion curve of an ultrasonic guided wave with a multi-layer spiral structure according to an embodiment of the present invention;

[0046] Figure 2a This is a schematic diagram of a 3D model of a steel-core aluminum stranded wire with 1 core and 4 layers in forward and reverse regular stranding according to an embodiment;

[0047] Figure 2b This is a schematic diagram of a 3D model of a 1-core 3-layer stranded wire with regular stranding in the same direction according to an embodiment;

[0048] Figure 3 yes Figure 2a Schematic diagram of cross-section mesh division of 3D model;

[0049] Figure 4a yes Figure 2a Schematic diagram of the overall mesh division of the 3D model;

[0050] Figure 4b yes Figure 4a A schematic diagram of a mesh showing only a single line in each layer of the mesh. DETAILED DESCRIPTION

[0051] To help those skilled in the art better understand the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It is apparent that the embodiments described are only a portion of the present invention, not all of the embodiments. All other embodiments derived by those skilled in the art based on the embodiments of the present invention without inventive effort are also within the scope of protection of the present invention.

[0052] like Figure 1 The method for solving the dispersion curve of ultrasonic guided waves in a multilayer helical structure shown in the figure includes the following steps: first, establishing a 3D model of a multilayer helical structure in which each helical line has a single pitch length; then, meshing the cross section; calculating the mesh size in the mesh sweep direction; meshing each helical line; solving the element stiffness matrix and element mass matrix of each node, using a rectangular coordinate system for straight lines and a helical coordinate system for spiral lines; assembling the cross-sectional global stiffness matrix and cross-sectional global mass matrix of each line, and assembling the pitch global stiffness matrix and pitch global mass matrix; periodically replicating and expanding the pitch global stiffness matrix and pitch global mass matrix of each line to assemble the multilayer helical structure global stiffness matrix and global mass matrix; solving the wave number, group velocity, and phase velocity; and matching the guided wave eigenvectors based on orthogonality. The present invention can achieve a fast and accurate solution to the dispersion curve of a multilayer helical structure composed of different pitches, handedness, sizes, or materials.

[0053] As a specific embodiment, Figure 2a and Figure 2b 3D model of a multi-layer spiral structure with a single spiral line of single pitch length. Figure 2a It is a 1-core 4-layer steel-cored aluminum stranded wire made of single wires of different materials and diameters, with regular twisting in both directions. The center layer is a straight steel single wire, the second layer is a steel spiral wire, the third and fourth layers are aluminum spiral wires, and the rotation directions of adjacent spiral layers are opposite. Figure 2b The method for solving the ultrasonic guided wave dispersion curve of the multi-layer helical structure specifically includes the following steps:

[0054] Step 1: Establish a 3D model of a multi-layer spiral structure for solving the dispersion curve of ultrasonic guided waves. The length of each layer of spiral single line is the pitch of the single line of this layer. For a straight single line, only the cross-sectional dimensions of this layer are drawn, such as Figure 2a and Figure 2b shown.

[0055] Step 2: Mesh each single line at the cross section. The mesh size is no larger than half the wavelength of the maximum frequency signal in the dispersion curve range. The mesh in the contact area is further refined. Figure 3 for Figure 2a Schematic diagram of the cross-sectional meshing of the model. Assume that the cross-sectional domain Ω can be discretized into multiple finite elements, and the discretized elements are denoted as Ω e .

[0056] Step 3: Calculate the pitch of the shortest helical layer and divide it by the number of single lines in the adjacent helical layer. Set the minimum value of the calculated result to h. If h is greater than half the wavelength of the maximum frequency signal within the dispersion curve range, reduce h by integer multiples to obtain h', until h' is no greater than half the wavelength of the maximum frequency signal within the dispersion curve range.

[0057] by Figure 2a Taking the steel-core aluminum stranded wire with regular forward and reverse stranding as shown in the figure as an example, the spiral layer with the shortest pitch is the second layer, its inner adjacent layer (i.e., the first layer) is a non-spiral layer, and the outer adjacent layer (i.e., the third layer) is a spiral layer. The spiral pitch of the second layer is calculated and divided by the number of single wires in the third layer, and the result is h; if the two adjacent layers of the spiral layer with the shortest pitch are both spiral layers, the pitch of the spiral layer with the shortest pitch is calculated separately and divided by the number of single wires in the two adjacent spiral layers, and the minimum value of the settlement result is set to h.

[0058] Step 4: Sweep the mesh of each spiral line along the center line of the line. The size of the mesh of each layer of spiral line in the sweeping direction is d L =αh', where α is the twist coefficient of each layer, and no grid sweeping is performed on straight single lines.

[0059] Figure 4a for Figure 2a The steel core aluminum stranded wire with regular twisting in the forward and reverse directions shown is Figure 3 The cross-sectional mesh shown is swept across the entire 3D model to generate the volume mesh. Figure 4b for Figure 4a In the grid, each layer only shows a schematic diagram of a single line grid.

[0060] Step 5 is solved based on the rectangular coordinate system to solve the unit stiffness matrix k1, k2, k3 and unit mass matrix m of each node of each straight line single-line section mesh. The solution formula is as follows:

[0061]

[0062] Where B1 and B2 are strain matrices, B2=L z N(x,y),L x , L y , L z is the corresponding differential operator, N(x,y) is the shape function matrix of the node coordinates (x,y), C is the elastic constant matrix of the material, ρ is the material density, Ω e is a discrete grid unit;

[0063]

[0064] Based on the spiral coordinate system, the element stiffness matrix and element mass matrix of each node of each spiral single-line swept volume mesh are solved. The solution formula is as follows:

[0065]

[0066] Where N(s) is the shape function matrix, L xy , L s is the differential operator, s is the arc length along the spiral centerline, l is the curve length of one spiral cycle, (e X ,e Y ,e Z ) are the orthogonal basis vectors in the Cartesian coordinate system, θ is the initial phase angle corresponding to the center line of the spiral line on the z = 0 plane;

[0067]

[0068] Where κ and τ are the curvature and torsion of the spiral line respectively.

[0069] As a specific embodiment, Figure 4b and Figure 4a Taking the mesh shown as an example, the center layer is a straight steel single wire. Substitute the relevant material and size parameters of the center layer straight steel single wire and solve the unit stiffness matrix k1, k2, k3 and unit mass matrix m of each node of the center layer cross-section mesh based on the rectangular coordinate system;

[0070] The other layers are spiral lines. Substitute the relevant material and dimensional parameters of each spiral line, and solve the unit stiffness matrix k1, k2, k3 and unit mass matrix m of all nodes of the spiral line parallel to the z=0 section one by one.

[0071] Step 6: Assemble the overall stiffness matrix K of each single line in each section parallel to the z=0 plane according to the mesh of each section in the mesh sweep direction. S1 , K S2 , K S3 and the cross-section overall mass matrix M S The overall stiffness matrix and overall mass matrix of each section of the spiral single line are assembled into the pitch overall stiffness matrix K V1 , K V2 , K V3 and the pitch overall mass matrix M V ;

[0072]

[0073] Where n is the number of nodes in the cross-section mesh.

[0074] For a spiral single line, let the number of mesh nodes along the sweep direction of a spiral single line in a single pitch be P, then:

[0075]

[0076] Step 7: Periodically replicate and extend the pitch global stiffness matrix and pitch global mass matrix of each single line to the same grid length, and the length of the measured object solved by the dispersion curve should be satisfied, and further assemble them into the multi-layer spiral structure global stiffness matrix K1, K2, K3 and the multi-layer spiral structure global mass matrix M;

[0077] Step 8: According to Hamilton's principle, the general homogeneous wave equation of ultrasonic guided waves is obtained, {K1+ikK2+k 2 K3-ω 2 M}U=0, U is the particle displacement at the node position, i is the imaginary unit, the frequency range is solved by solving the ultrasonic guided wave dispersion curve, and the wave number k is solved by given the value of the series frequency ω with an interval of Δω;

[0078] According to the wave number k at each frequency obtained, the group velocity c is solved respectively. g , phase velocity c p , the solution formula is as follows:

[0079] c g =Δω / Δk,c p =ω / k;

[0080] Where Δk is the difference in wavenumber k between two adjacent ω values.

[0081] Step 9: Based on the orthogonality, solve the waveguide eigenvectors of different modes: wave number k, group velocity c g , phase velocity c p For matching, the orthogonality solution formula is as follows:

[0082]

[0083] Among them, ψ m and ψ n Represents the guided wave eigenvectors of different modes,

[0084] Although the embodiments disclosed herein are as described above, the contents described herein are merely embodiments for facilitating understanding of the present invention and are not intended to limit the present invention. Any person skilled in the art may make any modifications and variations in the form and details of the embodiments without departing from the spirit and scope of the present invention. However, the scope of protection of the present invention shall remain subject to the scope defined by the appended claims.

Claims

1. A method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure, characterized in that: The following steps are involved: A. Establish a 3D model of a multi-layer spiral structure in which each spiral line has a single pitch length; B. Divide the mesh of each single line at the cross section and refine the mesh in the contact area; C. Calculate the pitch of the shortest helical layer divided by the number of single lines in the adjacent helical layer. Set the minimum value of the calculated result to h. If h is greater than half the wavelength of the maximum frequency signal in the dispersion curve solution range, reduce h by integer multiples to obtain h', until h' is no greater than half the wavelength of the maximum frequency signal in the dispersion curve range; D. Sweep the mesh of each spiral line along its centerline. The mesh size of each layer of spiral line in the sweeping direction is d. L =αh', where α is the twist coefficient of each layer, and no mesh sweeping is performed on straight single lines; E. Solve the unit stiffness matrix k1, k2, k3 and unit mass matrix m of each node of each linear single-line cross-section mesh based on the rectangular coordinate system; solve the unit stiffness matrix and unit mass matrix of each node of each spiral single-line swept volume mesh based on the spiral coordinate system; F. Assemble the overall stiffness matrix K of each single line in each section parallel to the z = 0 plane S1 , K S2 , K S3 and the cross-section overall mass matrix M S The overall stiffness matrix and overall mass matrix of each section of the spiral single line are assembled into the pitch overall stiffness matrix K V1 , K V2 , K V3 and the pitch overall mass matrix M V ; G. Periodically replicate and extend the pitch overall stiffness matrix and pitch overall mass matrix of each single line to the same grid length and the length of the measured object that satisfies the dispersion curve solution, and assemble them into the multi-layer spiral structure overall stiffness matrix K1, K2, K3 and the multi-layer spiral structure overall mass matrix M; H. According to the general homogeneous wave equation of ultrasonic guided waves, given the value of the series frequency ω with an interval of Δω, solve the wave number k and group velocity c g , phase velocity c p ; I. Orthogonality-based guided wave eigenvectors of different modes: wave number k, group velocity c g , phase velocity c p to match.

2. The method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure according to claim 1, characterized in that: In the 3D model of the multi-layer spiral structure in step A, the length of each layer of spiral single wire is the pitch of the single wire of that layer. For a straight single wire, only the cross-sectional dimensions of that layer are drawn.

3. The method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure according to claim 1, characterized in that: In step B, each single line is meshed at the cross section, and the mesh size is no larger than half the wavelength of the maximum frequency signal within the range of the dispersion curve to be solved.

4. The method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure according to claim 1, characterized in that: In step E, the discrete grid unit Ω of each straight line single line cross section grid is solved based on the rectangular coordinate system. e The solution formulas for the node element stiffness matrix k1, k2, k3 and the element mass matrix m are as follows: Where B1 and B2 are strain matrices, C is the elastic constant matrix of the material, N(x,y) is the shape function matrix of the node coordinates (x,y), and ρ is the material density.

5. The method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure according to claim 4, characterized in that: Strain Matrix B2=L z N(x,y),L x 、L y 、L z is the corresponding differential operator:

6. The method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure according to claim 1, characterized in that: In step E, the discrete grid Ω of each spiral single-line swept volume grid is solved based on the spiral coordinate system. e The solution formulas for the node element stiffness matrix k1, k2, k3 and the element mass matrix m are as follows: Where C is the elastic constant matrix of the material, s is the arc length along the center line of the helix, L xy 、L s is the differential operator, and N(s) is the shape function matrix: Where l is the length of the curve of one spiral period, (e X ,e Y ,e Z ) are the orthogonal basis vectors in the Cartesian coordinate system, θ is the initial phase angle of the center line of the spiral line on the z = 0 plane, κ and τ are the curvature and torsion of the spiral line, respectively.

7. The method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure according to claim 1, characterized in that: In step F: Where n is the number of nodes in the cross-section mesh. For a spiral single line, let the number of mesh nodes along the sweep direction of a spiral single line within a single pitch be P, and we have:

8. The method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure according to claim 1, characterized in that: Solve the wave number k in step H, and obtain the general homogeneous wave equation of ultrasonic guided waves according to the Hamilton principle formula, {K1+ikK2+k 2 K3-ω 2 M}U=0, U is the particle displacement at the node position, i is the imaginary unit, the frequency range is solved by solving the ultrasonic guided wave dispersion curve, and the wave number k is solved by given the value of the series frequency ω with an interval of Δω.

9. The method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure according to claim 1, characterized in that: Solve for the group velocity c in step H g , phase velocity c p , the solution formula is as follows: c g =Dω / Δk,c p =ω / k Where Δk is the difference in wavenumber k between two adjacent ω values.

10. The method for solving the dispersion curve of ultrasonic guided waves of a multi-layer spiral structure according to claim 1, characterized in that: The guided wave eigenvectors of different modes solved based on orthogonality in step I: wave number k, group velocity c g , phase velocity c p For matching, the orthogonality solution formula is as follows: Among them, ψ m and ψ n Represents the guided wave eigenvectors of different modes,

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