A Detection Method for Continuity and Icing of Steel Strand Based on Ultrasonic Guided Waves

Abstract

The present invention proposes a method for detecting the continuity and icing of steel strands based on ultrasonic guided waves, and the steps are as follows: First, convert the wave control equation established according to the boundary conditions and material properties of the steel strands into a characteristic equation, and introduce the Floquet periodic boundary condition for simplification; Second, solve the simplified characteristic equation based on the characteristic frequency solver of the finite element simulation software COMSOL to obtain the wave number-frequency relationship in the Cartesian coordinate system; Then, mathematically abstract the geometric structure of the steel strands to establish a torsional coordinate system; Convert the wave number-frequency relationship in the Cartesian coordinate system into the wave number-frequency relationship in the torsional coordinate system, and draw the dispersion curve of the ultrasonic guided wave propagating in the steel strands; Finally, select the guided wave mode and detection frequency according to the obtained dispersion curve to realize the detection of the continuity and icing of the steel strands. The dispersion characteristic analysis method of the present invention is simple, and the model establishment is easy; the detection mode selection is accurate and the detection accuracy is high.

Description

Technical Field

[0001] The present invention relates to the technical field of non-destructive testing, and particularly to a method for detecting the continuity and icing of steel strands based on ultrasonic guided waves. Background Art

[0002] With the development of industrial construction, steel strands, as load-bearing structures, are widely used in fields such as cable-stayed bridges, oil exploration, civil engineering, and wind turbine lightning protection cables. Steel strands are in a high-stress state for a long time during operation and are also affected by rain, sunlight, and environmental pollution. Therefore, in-service steel strands may suffer various forms of damage, such as corrosion, broken wires, and wear. As a load-bearing structure, the health status of steel strands directly affects the stability and safety of the entire structure. Therefore, it is of great significance to detect the damage of in-service steel strands.

[0003] Currently, the commonly used methods for detecting steel strand damage include the damage method, semi-damage method, and non-destructive method. The surface of in-service steel strands is usually galvanized to slow down the corrosion rate. To avoid damaging the anti-corrosion coating during the detection process, non-destructive testing methods should be given priority. Common non-destructive testing methods include eddy current testing, acoustic emission testing, ultrasonic guided wave testing, radiographic testing, visual inspection, and magnetic flux leakage testing, etc. Due to the skin effect, eddy current testing can only detect surface defects of steel strands and is not suitable for non-destructive testing of steel strands with a large diameter; acoustic emission signals are usually very weak, which brings great difficulties to signal acquisition and subsequent signal processing; the cost of radiographic testing is high, and it is harmful to the health of experimenters and prone to radiation pollution, so it cannot be widely promoted; visual inspection is only applicable to the detection of surface defects of steel strands and is greatly affected by surface coverings and lighting environments; magnetic flux leakage testing can only detect point by point, with low detection efficiency, and can only detect ferromagnetic materials and cannot detect the aluminum layer outside aluminum-clad steel strands.

[0004] Compared with traditional non-destructive testing methods, ultrasonic guided waves are very sensitive to fractures, corrosion, and mechanical fatigue of metal materials and are ideal for detecting long-distance and large-size metal structures. A method for solving the propagation characteristics of guided waves based on the semi-analytical finite element method can be used to analyze the dispersion characteristics of steel strands. A method for solving the propagation characteristics of guided waves based on the wave finite element method can be used to analyze the dispersion characteristics of steel strands. An ultrasonic guided wave detection method combining dispersion curves, continuous wavelet transform, and wave velocity measurement can quantitatively evaluate the corrosion damage of prestressed steel strands. A time-frequency energy analysis method based on the propagation of ultrasonic guided waves in steel strands is used to detect stress changes in steel strands. An ultrasonic guided wave detection method based on magnetostrictive sensors is used to detect defects in steel strands. Summary of the Invention

[0005] In view of the deficiencies in the above-mentioned background art, the present invention proposes a method for detecting the continuity and icing of steel strands based on ultrasonic guided waves, which solves the technical problems that the existing methods can only roughly analyze the propagation characteristics of ultrasonic guided waves in steel strands, and the process of analyzing the propagation characteristics is complex and the detection results are poor.

[0006] The technical solution of the present invention is realized as follows:

[0007] A method for detecting the continuity and icing of steel strands based on ultrasonic guided waves, the steps are as follows:

[0008] Step 1: Establish the Navier wave control equation for wave propagation according to the boundary conditions and material properties of the steel strand;

[0009] Step 2: Based on the finite element theory, transform the Navier wave control equation for wave propagation into a characteristic equation, and introduce the Floquet periodic boundary condition to simplify the characteristic equation;

[0010] Step 3: Solve the simplified characteristic equation based on the characteristic frequency solver of the finite element simulation software COMSOL to obtain the wave number-frequency relationship in the Cartesian coordinate system;

[0011] Step 4: Mathematically abstract the geometric structure of the steel strand to establish a torsional coordinate system;

[0012] Step 5: Convert the wave number-frequency relationship in the Cartesian coordinate system into the wave number-frequency relationship in the torsional coordinate system, and obtain the phase velocity and group velocity according to the wave number-frequency relationship in the torsional coordinate system. Draw the dispersion curve of the ultrasonic guided wave propagating in the steel strand from the phase velocity and group velocity;

[0013] Step 6: Select the guided wave mode and detection frequency according to the obtained dispersion curve to realize the detection of the continuity and icing of the steel strand.

[0014] Preferably, the method for establishing the Navier wave control equation for wave propagation is as follows:

[0015] The steel strand is an elastically isotropic steel strand, and the boundary condition of the steel strand is that the elastically isotropic steel strand satisfies the zero stress boundary condition on the boundary surface. The Navier wave control equation for wave propagation is expressed as:

[0016]

[0017] Where t is time; is the displacement field, which is a function of the position constant and time in the Cartesian coordinate system; ρ is the material density; μ and λ are both Lame constants; represents the Hamiltonian operator.

[0018] Preferably, in the second step, the specific conversion method is as follows:

[0019] Based on the finite element theory, without considering the external load, the Navier wave control equation for the propagation of guided waves can be rewritten as:

[0020]

[0021] where M is the mass matrix, C is the damping matrix, U represents the displacement matrix, and K represents the stiffness matrix; and C = αM + βK, where both α and β are weighting coefficients. In the ultrasonic frequency domain, the weighting coefficient α is ignored, and Equation (2) is rewritten as:

[0022]

[0023] For a non-damping problem with an angular frequency of ω, Equation (2) can be simplified to:

[0024] (K - ω 2 M)U = 0 (4);

[0025] There is an inherent connection between elastic waves and the vibration of elastic bodies. Based on the principle of mutual transformation between vibration solutions and wave solutions, the wave characteristics of a structure can be obtained by analyzing the vibration modes of the structure, and then the phase velocity and group velocity dispersion curves of a waveguide with any cross-section can be calculated; in the analysis of vibration solutions, the calculation of the dispersion curve is generally transformed into the solution of the zeros of the characteristic equation:

[0026] F(ω, k) = 0 (6);

[0027] where k is the wave number; ω is the angular frequency;

[0028] For a periodic waveguide, the Floquet periodic boundary condition is introduced to solve the characteristic equation (6). The Floquet periodic boundary condition is often used to solve ordinary differential equations in the form of Equation (7):

[0029]

[0030] where A(x) is a given continuous periodic function matrix with a period of L; F(x): R → C n is the unknown function; the Floquet periodic boundary condition states that the solutions of Equation (7) can all be expressed in the form of u(x)e kx where u(x) is a function with a period of L, and k is a complex scalar; for a periodic waveguide, the solutions of Equation (6) have the following form:

[0031]

[0032] where k F is the vector of the Floquet periodic boundary condition; uk (r) is a periodic function, i is the imaginary unit, r represents coordinates, and u(r,t) represents displacement.

[0033] Preferably, in step three, the specific method is as follows:

[0034] In the definition of COMSOL, the Floquet BC is applied to the source boundary and the target boundary:

[0035]

[0036] where u src is the displacement of the source boundary, and r src is the coordinate of the source boundary; u dst is the displacement of the target boundary, and r dst is the coordinate of the target boundary; k F has the following relationship with the wave number value k in the direction of guided wave propagation:

[0037]

[0038] where d is the distance between the source boundary and the target boundary to which the periodic boundary condition is applied, and n is the number of periods.

[0039] Preferably, the method for constructing the torsional coordinate system is as follows:

[0040] S41. Mathematically abstract the geometric structure of the steel strand;

[0041] A steel strand is a steel wire rope formed by helically winding multiple steel wires. The outer steel wire rope is tightly wound around the middle steel wire rope. A common steel strand is a seven-strand steel strand, and the geometric structure of its outer helical steel wire rope can be described by a cylindrical helix:

[0042]

[0043] where is the helical variable in the Cartesian coordinate system; L is the pitch of the helix; t ∈ [0, l], is the arc length corresponding to the helix pitch L; R is the helix radius of the helix center line; is the starting phase angle of different outer helices, n = 1,..., 6; the helix laying angle α' is For the helix, the curvature is The torsion is

[0044] S42. Establish a torsional coordinate system applicable to the steel strand;

[0045] Based on the Frenet-Serret criterion, the helix center line The unit vectors of the tangent T(t), normal N(t), and binormal B(t) are as follows:

[0046]

[0047] where e X , e Y , e Z all represent the standard orthogonal basis in the Cartesian coordinate system;

[0048] A helical coordinate system with an orthogonal basis of (N, B, T) is established. Based on the helical coordinate system, a torsional coordinate system that is applicable to both the outer helical steel wire rope and the central straight steel wire rope is established. The following changes are made compared to the helical coordinate system:

[0049]

[0050] The standard orthogonal basis of the torsional coordinate system is:

[0051]

[0052] where is the helical variable in the torsional coordinate system, and e x , e y , e z all represent the standard orthogonal basis in the torsional coordinate system.

[0053] Preferably, the wave number - frequency relationship in the torsional coordinate system is:

[0054]

[0055] where k t is the wave number in the torsional coordinate system.

[0056] Preferably, the phase velocity and group velocity are respectively expressed as:

[0057]

[0058] where Cp is the phase velocity, Cg is the group velocity, Δω represents the angular frequency change, and Δk t represents the wave number change in the torsional coordinate system.

[0059] Preferably, the on - off and icing detection of the steel strand is realized by characterizing the absorption change of the guided wave signal by the waveguide.

[0060] Compared with the prior art, the beneficial effects of the present invention are as follows: The present invention analyzes the dispersion characteristics of elastic waves propagating in a steel strand by using the mutual conversion principle of wave solutions and vibration solutions, and then uses the Floquet periodic boundary condition to reduce the model size and simplify the calculation process; the torsional coordinate system of the present invention relies on the geometric structure of the steel strand, has strong versatility, and the conversion relationship with the Cartesian coordinate system is simple; it is applicable to the analysis of the dispersion characteristics of irregular complex structures; the experimental results of the present invention can clearly judge the on-off and icing conditions of the steel strand, and the detection speed is fast and the detection distance is long. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0062] Figure 1 It is a flowchart in the present invention.

[0063] Figure 2 It is a result diagram of the steel strand model construction in the present invention.

[0064] Figure 3 It is a result diagram of the torsional coordinate system construction in the present invention.

[0065] Figure 4 It is a result diagram of the phase velocity and group velocity in the present invention.

[0066] Figure 5 It is a signal for exciting the L(0,1) mode in the present invention.

[0067] Figure 6 It is a result diagram of the icing detection in the present invention.

[0068] Figure 7 It is a diagram of the steel strand broken wire model.

[0069] Figure 8 It is a comparison diagram of the guided wave signals of the broken wire and the complete steel strand. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0070] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0071] As Figure 1As shown in the figure, an embodiment of the present invention provides a method for detecting the continuity and icing of steel strands based on ultrasonic guided waves. The characteristic equation of free vibration is obtained according to the mutual conversion principle of wave solutions and vibration solutions; the Floquet periodic boundary condition is used to simplify the calculation process and reduce the model size while; a torsional coordinate system suitable for the geometric structure of the steel strand is established; the wave number conversion relationship between the Cartesian coordinate system and the torsional coordinate system is constructed; the dispersion curve of the steel strand is drawn according to the obtained wave number-frequency relationship; finally, a suitable guided wave mode and detection frequency are selected for defect detection. The specific steps are as follows:

[0072] Step 1: Establish the Navier wave control equation for the propagation of guided waves according to the boundary conditions and material properties of the steel strand; establish a wave control equation based on the boundary conditions, material properties and geometric characteristics of the steel strand to describe the propagation characteristics of guided waves. When the elastic isotropic steel strand satisfies the zero stress (free) boundary condition on the boundary surface, the Navier wave control equation for the propagation of guided waves can be expressed as:

[0073]

[0074] where t is time; is the displacement field, which is a function of the position constant and time in the Cartesian coordinate system; ρ is the material density; μ and λ are both Lame constants; represents the Hamiltonian operator.

[0075] Step 2: Based on the finite element theory, transform the Navier wave control equation for the propagation of guided waves into a characteristic equation, and introduce the Floquet periodic boundary condition to simplify the characteristic equation; use the finite element method to divide the complex region into several finite element regions, and the situation in each region can be simplified into a simple load and boundary condition problem. Based on the finite element theory, without considering the external load, without considering the external load, the Navier wave control equation for the propagation of guided waves can be rewritten as:

[0076]

[0077] Equation (2) can be understood as a generalized Newton's second law. Where M is the mass matrix, C is the damping matrix, U represents the displacement matrix, and K represents the stiffness matrix; for Rayleigh damping, the damping coefficient C matrix is defined as a linear combination of the mass matrix and the stiffness matrix: C = αM + βK, where α and β are both weighting coefficients and can be obtained from experiments. In the ultrasonic frequency domain, α is approximately 0 and can be ignored. Equation (2) is rewritten as:

[0078]

[0079] For an undamped problem with an angular frequency of ω, Equation (2) can be simplified to:

[0080] (K-ω 2 M)U = 0 (4);

[0081] According to Equation (4), the propagation problem of ultrasonic guided waves can be solved in the frequency domain.

[0082] There is an inherent connection between elastic waves and the vibration of elastic bodies. Elastic waves can be regarded as the process of vibration propagation and a form of energy transfer. In the vibration solution, the wave function is expressed in the separated variable form of φ(x)q(t), and the vibration solution can be expressed in the form of an infinite series:

[0083] ∑ n φ n (x)q n (t) (5)

[0084] Each term in the equation represents a standing wave that has a fixed pattern in space and vibrates at a specific frequency. Therefore, based on the principle of mutual transformation between vibration solutions and wave solutions, the wave characteristics of the structure can be obtained by analyzing the vibration modes of the structure, and then the phase velocity and group velocity dispersion curves of the waveguide at any cross-section can be calculated; in the analysis of vibration solutions, the essence of the dispersion characteristics lies in establishing the wave number-frequency relationship. Therefore, the calculation of the dispersion curve is generally transformed into the solution of the zeros of the characteristic equation:

[0085] F(ω,k) = 0 (6);

[0086] where k is the wave number; ω is the angular frequency.

[0087] For a periodic waveguide, the Floquet periodic boundary condition is introduced to solve the characteristic equation (6). The Floquet periodic boundary condition is often used to solve ordinary differential equations in the form of Equation (7):

[0088]

[0089] where A(x) is a given continuous periodic function matrix with a period of L; F(x):R→C n is an unknown function; the Floquet periodic boundary condition states that the solutions of Equation (7) can all be expressed in the form of u(x)e kx , where u(x) is a function with a period of L and k is a complex scalar; this theory provides a method for solving ordinary differential equations from the perspective of eigenvalues. For a periodic waveguide, the solutions of Equation (6) have the following form:

[0090]

[0091] where, k F is the vector of the Floquet periodic boundary condition; u k(r) is a periodic function, i is the imaginary unit, r represents coordinates, and u(r,t) represents displacement.

[0092] Step Three: Solve the simplified characteristic equation based on the eigenfrequency solver of the finite element simulation software COMSOL to obtain the wave number-frequency relationship in the Cartesian coordinate system. The specific method is as follows:

[0093] In the definition of COMSOL, the Floquet BC is applied to the source boundary and the target boundary:

[0094]

[0095] where u src is the displacement of the source boundary, and r src is the coordinate of the source boundary; u dst is the displacement of the target boundary, and r dst is the coordinate of the target boundary; k F has the following relationship with the wave number value k in the direction of guided wave propagation:

[0096]

[0097] where d is the distance between the source boundary and the target boundary to which the periodic boundary condition is applied, and n is the number of periods.

[0098] Step Four: Mathematically abstract the geometric structure of the steel strand and establish a torsional coordinate system;

[0099] S41. Mathematically abstract the geometric structure of the steel strand;

[0100] The steel strand is a steel wire rope formed by helically winding multiple steel wires. The outer steel wire rope is tightly wound around the middle steel wire rope. The common steel strand is a seven-strand steel strand, and the geometric structure of its outer helical steel wire rope can be described by a cylindrical helix:

[0101]

[0102] where is the helical variable in the Cartesian coordinate system; L is the pitch of the helix; t ∈ [0, l], is the arc length corresponding to the helix pitch L; R is the helix radius of the helix center line; is the starting phase angle of different outer helices, n = 1,..., 6; the helix laying angle α' is For the helix, the curvature is The torsion is Figure 2 The actual structure of the steel strand and the COMSOL model are shown.

[0103] S42. Establish a torsional coordinate system applicable to the steel strand;

[0104] Based on the Frenet - Serret criterion, for the unit vectors of the tangent T(t), normal N(t), and binormal B(t) of the spiral centerline are as follows:

[0105]

[0106] where, e X and e Y and e Z all represent the standard orthogonal basis in the Cartesian coordinate system.

[0107] A spiral coordinate system with an orthogonal basis of (N, B, T) is established. It is similar to the Cartesian coordinate system and still satisfies the right - hand screw rule, but the plane varies helically along the spiral centerline; based on the spiral coordinate system, a torsion coordinate system applicable to both the outer spiral steel wire rope and the central straight steel wire is established. Figure 3 (a) is a schematic diagram of the torsion coordinate system. The following changes are made compared with the spiral coordinate system:

[0108]

[0109] The standard orthogonal basis of the torsion coordinate system is:

[0110]

[0111] where, is the spiral variable in the torsion coordinate system, and e x , e y , e z represent the standard orthogonal basis in the torsion coordinate system.

[0112] The radius of the outer spiral of the strand model is 2.63 mm, the helix angle is 7.9°, the pitch of the helix is 240 mm, the radius of the central straight steel wire is 2.7 mm, and the length is 240 mm. The mechanical properties of the strand model are E = 2.17e11 Pa, υ = 0.28, ρ = 7932 kg / m 3 .

[0113] A three - dimensional model and the minimum calculation unit of the strand are established in COMSOL. Figure 3 (b) is a structural diagram of the minimum calculation unit of the strand. The outer spiral of the complete model conforms to the cylindrical helix equation in step two, and the minimum calculation unit is a part of the strand with upper and lower cross - sections perpendicular to the (x, y) plane.

[0114] Step 5: Convert the wave number-frequency relationship in the Cartesian coordinate system into that in the torsional coordinate system, and obtain the phase velocity and group velocity according to the wave number-frequency relationship in the torsional coordinate system. Draw the dispersion curve of the ultrasonic guided wave propagating in the steel strand based on the phase velocity and group velocity;

[0115] Convert the wave number obtained in the Cartesian coordinate system into the wave number in the torsional coordinate system:

[0116]

[0117] where k t is the wave number in the torsional coordinate system.

[0118] Obtain the phase velocity and group velocity according to the wave number-frequency relationship in the torsional coordinate system:

[0119]

[0120] where Cp is the phase velocity, Cg is the group velocity, Δω represents the angular frequency change, and Δk t represents the wave number change in the torsional coordinate system.

[0121] Step 6: Select appropriate guided wave modes and detection frequencies according to the obtained dispersion curve to realize the on-off and icing detection of the steel strand.

[0122] Figure 4 The following is the dispersion characteristic result diagram, where Figure 4 (a) is the phase velocity diagram, Figure 4 (b) is the group velocity diagram, Figure 4 (c)-(d) are the vibration mode diagrams corresponding to points A, B, and C in the phase velocity diagram. According to the dispersion curve, the L(0,1) mode can be selected as the detection mode, and the detection frequency should avoid being in the missing frequency band shown in the red frame. The detection frequency of this model is 60KHz.

[0123] Figure 5 The following is the excitation signal determined according to the dispersion characteristics. Figure 5 (a) is the time-domain representation of the excitation signal, Figure 5 is the frequency-domain representation of the excitation signal. According to the vibration mode diagram of the L(0,1) mode, the particle vibration in this mode exists in the radial and axial directions. Therefore, axial boundary loads are applied to all nodes on the end face of the steel strand, and the excitation frequency is 60KHz. Since the frequency of the excited guided wave will affect the specific generated guided wave mode, a narrowband pulse signal should be used for excitation. The present invention selects the load signal as a sine wave signal modulated by a Hann window:

[0124]

[0125] where n is the number of cycles, take n = 10, f cis the center frequency, and f c = 60 KHz. t is the signal duration, and t = 80 μs.

[0126] Figure 6 The experimental results of ice detection on steel strands are shown as follows. Figure 6 (a) shows the ice detection results with a relatively small volume of 14 cm in length and 1.7 cm in thickness set at the end of the steel strand; Figure 6 (b) shows the ice detection results with a relatively large volume of 15 cm in length and 6 cm in thickness set at the end of the steel strand; Figure 6 (c) shows the comparison results of ice detection for the two volumes. It can be seen that ice causes obvious changes in the echo signals of guided waves. Comparing Figure 6 (b) and Figure 6 (c), it can be seen that the intensity of the echo signal of the ice layer with a larger thickness is less than that of the echo signal of the ice layer with a smaller thickness at 650 μs, indicating that the ice layer with a larger thickness absorbs more signals than the ice layer with a smaller thickness, which can be used to analyze the thickness of the ice layer.

[0127] Figure 7 (a) shows the schematic diagram of broken wires of the steel strand, Figure 7 (b) shows the vibration conditions of each point on the steel strand at a certain moment during the propagation process of the ultrasonic guided wave.

[0128] Figure 8 The experimental results of the continuity detection of the steel strand are shown as follows. It can be seen that the steel strand after wire breakage weakens the absorption of signals, and there are obvious differences between the guided wave signals in different wire breakage situations and the intact steel strand. Therefore, the dispersion characteristic analysis method and defect detection method proposed according to the present invention can effectively detect the wire breakage defects of the steel strand. The continuity of the steel strand can be detected through the signal changes during the propagation of the guided wave.

[0129] The above are only the preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for detecting the continuity and icing of steel strand based on ultrasonic guided wave, characterized in that, The steps are as follows: Step 1: Establish the Navier wave control equation for guided wave propagation based on the boundary conditions and material properties of the steel strand. Step 2: Based on the finite element theory, transform the Navier wave control equation for guided wave propagation into a characteristic equation, and introduce the Floquet periodic boundary condition to simplify the characteristic equation. Step 3: Solve the simplified characteristic equation using the characteristic frequency solver of the finite element simulation software COMSOL to obtain the wave number-frequency relationship in the Cartesian coordinate system. Step 4: Mathematically abstract the geometric structure of the steel strand and establish a torsional coordinate system. The construction method of the torsional coordinate system is as follows: S41: Mathematically abstract the geometric structure of the steel strand. The steel strand is a steel wire rope formed by helically winding multiple steel wires. The outer steel wire rope is tightly wound around the middle steel wire rope. The common steel strand is a seven-strand steel strand, and the geometric structure of its outer helical steel wire rope can be described by a cylindrical helix: Among them, is a spiral variable in the Cartesian coordinate system; L is the pitch of the spiral; t ∈ [0, l], is the arc length corresponding to the spiral pitch L; R is the spiral radius of the spiral center line; is the starting phase angle of different outer spiral lines, n = 1,..., 6; the spiral laying angle α' is For a spiral line, the curvature is The torsion is S42: Establish a torsional coordinate system applicable to the steel strand. Based on the Frenet-Serret criterion, the helix centerline The unit vectors of the tangent T(t), normal N(t), and binormal B(t) are as follows: wherein, e X , e Y , e Z all represent the standard orthogonal basis in the Cartesian coordinate system; Establish a helical coordinate system with an orthogonal basis of (N, B, T). On the basis of the helical coordinate system, establish a torsional coordinate system that is applicable to both the outer helical steel wire rope and the central straight steel wire rope. The following changes are made compared with the helical coordinate system: The orthonormal basis of the torsional coordinate system is: Among them, is the screw variable in the torsional coordinate system, e x , e y , e z all represent the orthonormal basis in the torsional coordinate system; Step 5: Convert the wave number-frequency relationship in the Cartesian coordinate system into the wave number-frequency relationship in the torsional coordinate system, and obtain the phase velocity and group velocity according to the wave number-frequency relationship in the torsional coordinate system. Draw the dispersion curve of the ultrasonic guided wave propagating in the steel strand from the phase velocity and group velocity. Step 6: Select the guided wave mode and detection frequency according to the obtained dispersion curve to realize the on-off and icing detection of the steel strand.

2. The method for detecting the continuity and icing of a steel strand based on ultrasonic guided waves according to claim 1, characterized in that The establishment method of the Navier wave control equation for guided wave propagation is as follows: The steel strand is an elastically isotropic steel strand. The boundary condition of the steel strand is that the elastically isotropic steel strand satisfies the zero stress boundary condition on the boundary surface. The Navier wave control equation for guided wave propagation is expressed as: where t is time; is the displacement field, which is a function of position constants and time in the Cartesian coordinate system; ρ is the material density; both μ and λ are Lamé constants; denotes the Hamiltonian operator.

3. The method for detecting the continuity and icing of steel strands based on ultrasonic guided waves according to claim 2, characterized in that In Step 2, the specific conversion method is as follows: Based on the finite element theory, without considering the external load, the Navier wave control equation for guided wave propagation can be rewritten as: where M is the mass matrix, C is the damping matrix, U represents the displacement matrix, and K represents the stiffness matrix; and C = αM + βK, where both α and β are weighting coefficients. In the ultrasonic frequency domain, the weighting coefficient α is ignored, and Equation (2) is rewritten as: For a non-damping problem with an angular frequency of ω, Equation (2) can be simplified to: (K-ω 2 M)U = 0 (4); There is an inherent connection between elastic waves and the vibration of elastic bodies. Based on the principle of mutual transformation between vibration solutions and wave solutions, the wave characteristics of the structure can be obtained by analyzing the vibration mode of the structure, and then the phase velocity and group velocity dispersion curves of any cross-sectional waveguide can be calculated; in the analysis of vibration solutions, the calculation of the dispersion curve is generally transformed into the solution of the zeros of the characteristic equation: F(ω,k) = 0 (6); where k is the wave number; ω is the angular frequency; For a periodic waveguide, the Floquet periodic boundary condition is introduced to solve the characteristic equation (6). The Floquet periodic boundary condition is often used to solve ordinary differential equations in the form of Equation (7): where \(A(x)\) is a given continuous periodic function matrix with period \(L\); \(F(x):\mathbb{R}\to\mathbb{C}\) n is an unknown function; the Floquet periodic boundary condition states that the solutions of equation (7) can all be expressed in the form of \(u(x)e\) kx where \(u(x)\) is a function with period \(L\) and \(k\) is a complex scalar; for a periodic waveguide, the solutions of equation (6) have the following form: where k F is a vector for the Floquet periodic boundary condition; u k (r) is a periodic function, i is the imaginary unit, r represents the coordinate, and u(r,t) represents the displacement.

4. The method for detecting the continuity and icing of a steel strand based on ultrasonic guided waves according to claim 3, characterized in that, In step three, the specific method is as follows: In the definition of COMSOL, the Floquet BC is applied to the source boundary and the target boundary: Among them, u src is the displacement of the source boundary, and r src is the coordinate of the source boundary; u dst is the displacement of the target boundary, and r dst is the coordinate of the target boundary; k F has the following relationship with the wave number k in the direction of guided wave propagation: where d is the distance between the source boundary and the target boundary to which the periodic boundary condition is applied, and n is the number of periods.

5. The method for detecting the continuity and icing of steel strands based on ultrasonic guided waves according to claim 4, wherein The wave number-frequency relationship in the twisted coordinate system is as follows: Among them, k t is the wave number in the torsional coordinate system.

6. The method for detecting the continuity and icing of steel strands based on ultrasonic guided waves according to claim 5, characterized in that, The phase velocity and the group velocity are respectively expressed as: where Cp is the phase velocity, Cg is the group velocity, △ω represents the angular frequency change, and △k t represents the wave number change in the torsional coordinate system.

7. The method for detecting the continuity and icing of a steel strand based on ultrasonic guided waves according to claim 1, wherein The on-off and ice coating detection of the steel strand are realized by characterizing the absorption change of the guided wave signal by the waveguide.

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