Optimal frequency selection method for online monitoring bolt torque based on ultrasonic guided waves

Abstract

The invention provides an optimal frequency selection method for online monitoring bolt torque based on ultrasonic guided waves, belongs to the technical field of intelligent sensing systems, and solves the problems that the frequency of the ultrasonic guided waves in bolt connection detection is mostly determined according to experience and cannot fully excite all dynamic characteristics of the system, so that the real performance of the system cannot be comprehensively reflected by a measurement result, and the reliability of the system is influenced. The method comprises the following steps: establishing a two-dimensional frequency domain finite element model based on physical field simulation software; perfect matching layers are added to the two sides of the established two-dimensional frequency domain finite element model; placing a piezoelectric wafer as an excitation source in a transmission area of the two-dimensional frequency domain finite element model added with the perfect matching layer, and generating an ultrasonic guided wave signal; setting a contact pair and a boundary pair in a contact area of the two-dimensional frequency domain finite element model; frequency scanning analysis is carried out on the basis of the content, and the excitation center frequency corresponding to the change curve peak value of the total transmission energy TE is selected as the optimal excitation center frequency.

Description

Technical Field

[0001] The present invention relates to an optimal frequency selection method for online monitoring of bolt torque based on ultrasonic guided waves, belonging to the technical field of intelligent sensing systems. Background Art

[0002] Bolt connections are basic components in aerospace, machinery, and civil engineering. They not only play a role in connection and fixation but also bear the weight and load of equipment and ensure the safe operation of equipment under extreme conditions. However, with the accumulation of service time, bolt connections face multiple challenges such as mechanical vibration, chemical erosion, and external stress, which are extremely likely to cause loosening phenomena, thereby weakening the overall performance of fasteners and posing a serious threat to the stability and safety of system operation. Therefore, realizing online monitoring and real-time management of bolt torque has become an indispensable important link to ensure the long-term stable operation of the system and guarantee public safety. In recent years, the ultrasonic guided wave method has been widely used in the online monitoring of fasteners due to its excellent penetration, long-distance propagation ability, high sensitivity, and wide applicability.

[0003] Among them, the energy-based ultrasonic guided wave active sensing method has become a research hotspot due to its intuitive principle, simple operation, and significant advantages in multi-bolt loosening detection. For example, Du Fei et al. proposed an ultrasonic guided wave monitoring method for bolt pre-tightening torque based on an improved time reversal method in patent CN107192492A. This method collects the response signal in the healthy state through a sensor, reverses the signal in the time domain, and uses it as the time reversal retransmission signal to collect the re-focused signals under different torques. Finally, by comprehensively considering the amplitude and phase of the time reversal reconstructed signal, the problem of insensitive detection of early bolt loosening is solved to a certain extent, and the monitoring range and accuracy of bolt pre-tightening torque are effectively improved. Duan Yuanfeng et al. proposed a bolt group damage identification device and method combining ultrasonic guided wave normalized energy transmittance and neural network in patent CN116340806A. This method constructs a multi-path guided wave monitoring network by arranging a small number of sensors on both sides of the bolt connection area, calculates the normalized energy transmittance of each path to construct characteristic parameters, and combines simulated damage conditions to generate a sample data set to train the neural network model. It can effectively realize bolt group damage location, local damage degree evaluation, and overall health state prediction.

[0004] However, in the above monitoring method, the guided wave excitation frequency is usually determined by experimental experience. Although in many measurement cases, it is reasonable to determine the guided wave excitation frequency based on experience, there is still a risk of adversely affecting the measurement accuracy of bolt torque. Moreover, an inappropriate excitation frequency cannot fully stimulate all the dynamic characteristics of the system, resulting in the measurement results being unable to comprehensively reflect the true performance of the system, thereby reducing the bolt torque detection accuracy. Therefore, the selection of the optimal excitation frequency of ultrasonic guided waves is of great significance for improving the bolt torque monitoring accuracy. Summary of the Invention

[0005] To solve the problem that in the detection of bolt connections, the frequency of ultrasonic guided waves is mostly determined based on experience, which cannot fully stimulate all the dynamic characteristics of the system, resulting in the measurement results being unable to comprehensively reflect the true performance of the system, thereby reducing the bolt torque detection accuracy, the present invention further provides an optimal frequency selection method for online monitoring of bolt torque based on ultrasonic guided waves.

[0006] The technical solution adopted by the present invention to solve the above problems is as follows: The present invention includes the following steps:

[0007] Step 1: Establish a two-dimensional frequency-domain finite element model based on physical field simulation software. Simplify the bolt connection into two identical aluminum plates in contact. Furthermore, the propagation process of ultrasonic guided waves divides the entire bolt connection into a transmission region, a contact region, and a receiving region.

[0008] Step 2: Add a perfectly matched layer on both sides of the established two-dimensional frequency-domain finite element model to make the ultrasonic guided waves gradually attenuate during the propagation process.

[0009] Step 3: Place a piezoelectric wafer as an excitation source in the transmission region of the two-dimensional frequency-domain finite element model after adding the perfectly matched layer to generate ultrasonic guided wave signals.

[0010] Step 4: Set contact pairs and boundary pairs in the contact region of the two-dimensional frequency-domain finite element model, and respectively analyze the propagation characteristics of guided wave signals in the upper and lower aluminum plates and the guided wave transmission analysis in the contact simulation of the upper and lower aluminum plates.

[0011] Step 5: Based on Step 4, perform frequency scanning analysis to obtain the change curve of the total transmission energy TE of ultrasonic guided waves in the receiving region, and select the excitation center frequency corresponding to the peak of the curve as the optimal excitation center frequency.

[0012] Further, Step 1 specifically includes:

[0013] Based on the two-dimensional frequency-domain finite element model, establish a wave equation applicable to the propagation of ultrasonic guided waves in the transmission region, contact region, and receiving region in the frequency domain.

[0014] The expression of the wave equation is:

[0015]

[0016] In formula (1), ρ is the density of the aluminum plate, ω is the angular frequency, u is the displacement field, and C is the elastic tensor.

[0017] Further, step 2 specifically includes:

[0018] Step 2.1: Calculate the dilation factors in the x-direction and y-direction of the ultrasonic guided wave propagation path, and convert the coordinates defining the perfectly matched layer region into complex coordinates based on the dilation factors in the x-direction and y-direction;

[0019] Step 2.2: Introduce complex coordinates, obtain the transformed gradient operator in the wave equation, and then obtain the modified perfectly matched layer wave equation;

[0020] Step 2.3: Directly relate the attenuation characteristics and propagation coefficients of the ultrasonic guided wave by introducing complex coordinates, and in the perfectly matched layer, adjust the wave number k to a complex number to obtain a complex wave number, and introduce the imaginary part of the complex wave number into the attenuation of the ultrasonic guided wave, so that the amplitude of the ultrasonic guided wave decreases exponentially with the increase of distance in the perfectly matched layer region;

[0021] The expression for the coordinate transformation of the perfectly matched layer region is:

[0022]

[0023] In formula (2), s x is the dilation factor in the x-direction of the guided wave propagation path, s y is the dilation factor in the y-direction of the guided wave propagation path, s x and s y are complex numbers, and their calculation formulas are:

[0024] s x (x) = 1 + iδ x (x), s y (y) = 1 + iδ y (y)(3);

[0025] In formula (3), δ x (x) and δ y (y) are both damping functions used to control the absorption characteristics of the perfectly matched layer;

[0026] For the two-dimensional wave equation propagating in the x-direction, the calculation formula for the damping coefficient of the perfectly matched layer is:

[0027]

[0028] In formula (4), δ0 is the maximum damping coefficient of the perfectly matched layer, L xis the length of the perfectly matched layer, and n is the exponent controlling the attenuation intensity, usually taking 1 or 2;

[0029] The calculation formula of the transformed gradient operator in the wave equation is:

[0030]

[0031] The modified wave equation of the perfectly matched layer is:

[0032]

[0033] The expression for adjusting the wave number k to a complex number is:

[0034]

[0035] The expression for introducing the attenuation of ultrasonic guided waves by the imaginary part of the complex wave number is:

[0036]

[0037] Furthermore, the expression for the excitation form of the excitation source in step 3 is:

[0038] u(t) = Acos(ωt) (9);

[0039] In formula (9), A is the excitation amplitude and ω is the excitation angular frequency.

[0040] Furthermore, step 4 specifically includes:

[0041] Step 4.1: Set up contact pairs at the structural contact boundaries in the contact area of the two-dimensional frequency-domain finite element model. Since there is a gap between the upper and lower aluminum plates, it ensures that the vibrations of the upper and lower aluminum plates are independent, thus eliminating any interaction between the two aluminum plates. Then, analyze the propagation characteristics of the guided wave signals in the upper and lower aluminum plates. Among them, the analysis results include: the stress t1 on the contact surface of the lower aluminum plate is the result of internal fluctuations, that is The stress t2 on the upper plate contact surface is the result of internal fluctuations, that is No constraints are applied at the boundaries of the upper and lower plates, and the stress is discontinuous, that is t1 ≠ t2;

[0042] Step 4.2: Establish consistent boundary pairs on both sides of the contact area of the two-dimensional frequency-domain finite element model to simulate the complete contact of the upper and lower aluminum plates, and then promote the deletion of ultrasonic guided waves to the receiving area. Among them, the displacements of the upper and lower aluminum plates at the contact surface are equal, that is u1 = u2, and the normal stresses are equal, that is σ n,1 = σ n,2 , and the tangential stresses are equal, that is τ t,1 = τ t,2 .

[0043] Furthermore, step 5 specifically includes:

[0044] Step 5.1: Set the excitation voltage of the transducer and the increment during the frequency sweep analysis, and perform frequency sweep analysis on the bolt connection within the preset excitation center frequency range;

[0045] Step 5.2: Calculate the total transmission energy TE of the ultrasonic guided wave at different excitation center frequencies and draw a variation curve, and select the excitation center frequency corresponding to the peak of the curve as the optimal excitation center frequency.

[0046] Furthermore, the calculation of the total transmission energy TE of the ultrasonic guided wave in Step 5.2 specifically includes:

[0047] Step 5.2.1: The transmission energy of the ultrasonic guided wave in the receiving area includes potential energy and kinetic energy. Without considering material damping, the potential energy and kinetic energy are equal in the time-average sense, and the strain energy density is calculated based on the stress tensor and the strain tensor;

[0048] Step 5.2.2: Decompose the strain energy calculation formula in the two-dimensional frequency domain model;

[0049] Step 5.2.3: On the basis of Step 5.2.2, integrate the strain energy density along the thickness direction of the receiving area to obtain the potential energy, and then calculate the total transmission energy TE of the ultrasonic wave;

[0050] The calculation formula of the strain energy density is:

[0051]

[0052] In formula (10), σ is the stress tensor, ε is the strain tensor, and W h is the strain energy density;

[0053] In the two-dimensional frequency domain model, the strain energy density calculation formula is decomposed into:

[0054]

[0055] The calculation formula of the total transmission energy TE of the ultrasonic wave is:

[0056] TE = 2∫ h W h dy (12).

[0057] The beneficial effects of the present invention are:

[0058] 1. Effectively realize the selection of the best excitation frequency of the guided wave: Based on the strongest transmission energy of the guided wave, the present invention realizes the optimal frequency selection analysis through simulation analysis, and the sorting result is accurate, effectively improving the monitoring ability of the bolt torque;

[0059] 2. The model structure is simple and the simulation efficiency is high: A two-dimensional frequency-domain finite element model is established, and the effectiveness of the model and the analysis of the transmission energy spectrum can be verified by setting boundaries, enhancing the reliability and applicability of the model. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 It is a schematic flowchart of an optimal frequency selection method for on-line monitoring of bolt torque based on ultrasonic guided waves provided by the present invention;

[0061] Figure 2 It is a schematic diagram of the propagation characteristics of guided waves in a free aluminum plate at two excitation frequencies of 150 kHz and 250 kHz provided by the present invention. Figure 2 In it, (a) is the stress distribution nephogram of the free aluminum plate at an excitation frequency of 150 kHz, (b) is the spatial strain distribution diagram of the free aluminum plate at an excitation frequency of 150 kHz, (c) is the wavenumber spectrum result at an excitation frequency of 150 kHz, (d) is the stress distribution nephogram of the free aluminum plate at an excitation frequency of 250 kHz, (e) is the spatial strain distribution diagram of the free aluminum plate at an excitation frequency of 250 kHz, and (f) is the wavenumber spectrum result at an excitation frequency of 250 kHz;

[0062] Figure 3 It is a schematic diagram of the steady-state vibration mode at an excitation frequency of 250 kHz provided by the present invention;

[0063] Figure 4 It is a schematic diagram of the variation relationship of the normalized transmission energy with frequency provided by the present invention;

[0064] Figure 5 It is a schematic diagram of simplifying the bolt connection into two contacting aluminum plates through a two-dimensional frequency-domain finite element model provided by the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0065] DETAILED DESCRIPTION OF THE INVENTION 1: In combination with Figure 1 and Figure 5 to illustrate this embodiment. As Figure 1 shown, the steps of an optimal frequency selection method for on-line monitoring of bolt torque based on ultrasonic guided waves described in this embodiment include:

[0066] S1: Establish a two-dimensional frequency-domain finite element model based on physical field simulation software, and simplify the bolt connection into two completely identical contacting aluminum plates;

[0067] As Figure 5As shown in the figure, in order to analyze the propagation characteristics of guided waves in bolted joints, a two-dimensional frequency-domain finite element model is established using the multi-physics simulation software Comsol Multiphysics. The bolted joint is simplified as two aluminum plates in contact, effectively reducing the computational cost. The propagation process of elastic waves divides the entire bolted joint into three different regions: the transmission region, the contact region, and the receiving region. Considering that the roughness and air gap size of the contact interface between the two aluminum plates are usually in the micron range, much smaller than the wavelength of the guided wave, the contact region is set as a smooth surface, ignoring rough contact. In the simulation model, a certain gap is maintained between the two aluminum plates, and then the two sides of the contact region are set as structural contact boundary pairs, and different boundary conditions are set to simulate the two cases of contact and separation.

[0068] In the case of two-dimensional plane elastic waves, the propagation of ultrasonic guided waves is controlled by the elastic wave equation. In the frequency domain, the displacement field u follows the following wave equation:

[0069]

[0070] In formula (1), ρ is the density of the aluminum plate, ω is the angular frequency, u is the displacement field, and C is the elastic tensor. This equation is applicable to the transmission region, the contact region, and the receiving region.

[0071] S2: Add a perfectly matched layer on both sides of the established two-dimensional frequency-domain finite element model to gradually attenuate the ultrasonic guided wave during propagation;

[0072] To eliminate the interference of boundary reflected waves, a perfectly matched layer is added on both sides of the model in this embodiment. The purpose of the perfectly matched layer is to introduce an imaginary component or a damping term through a mathematical coordinate transformation. This transformation to complex coordinates effectively simulates the equivalent damping effect in physical space, causing the wave to gradually attenuate during propagation. Define the coordinate transformation of the perfectly matched layer region as:

[0073]

[0074] In formula (2), s x is the dilation factor in the x direction of the guided wave propagation path, s y is the dilation factor in the y direction of the guided wave propagation path, s x and s y are complex numbers, and their calculation formula is:

[0075] s x (x) = 1 + iδ x (x), s y (y) = 1 + iδ y (y) (3);

[0076] In formula (3), δ x (x) and δy (y) are both damping functions used to control the absorption characteristics of the perfectly matched layer;

[0077] For the two-dimensional wave equation propagating in the x direction, the calculation formula for the damping coefficient of the perfectly matched layer is:

[0078]

[0079] In formula (4), δ0 is the maximum damping coefficient of the perfectly matched layer, L x is the length of the perfectly matched layer, and n is the exponent controlling the attenuation intensity, usually taking 1 or 2;

[0080] By introducing complex coordinates, the gradient operator in the wave equation becomes:

[0081]

[0082] The modified wave equation of the perfectly matched layer is:

[0083]

[0084] According to formula (6), the introduction of complex coordinates directly relates the attenuation characteristics of the wave to the propagation coefficient. In the perfectly matched layer, the wave number k is adjusted to a complex number:

[0085]

[0086] The imaginary part of the complex wave number introduces wave attenuation, so the amplitude of the wave exponentially decreases with distance in the perfectly matched layer region:

[0087]

[0088] S3: Place a piezoelectric wafer as an excitation source in the transmission region of the two-dimensional frequency-domain finite element model after adding the perfectly matched layer to generate a guided wave signal;

[0089] In this embodiment, a piezoelectric wafer (PZT-5A) with a width of 8 mm and a thickness of 1.5 mm is placed as an excitation source in the transmission region of the finite element model to generate a guided wave signal. The excitation form is:

[0090] u(t) = Acos(ωt) (9);

[0091] In formula (9), A is the excitation amplitude and ω is the excitation angular frequency.

[0092] S4: Set contact pairs and boundary pairs in the contact region of the two-dimensional frequency-domain finite element model to analyze the propagation characteristics of the guided wave signal in the upper and lower aluminum plates and the guided wave transmission in the contact simulation of the upper and lower aluminum plates respectively;

[0093] S401: In this embodiment, the structural contact boundary pairs in the model contact area are set as contact pairs. Since there is a gap between the two aluminum plates, the contact surfaces of the upper and lower plates only conform to their respective wave conditions and do not interact with each other, so as to analyze the propagation characteristics of guided waves in the free aluminum plates.

[0094] The stress t1 on the lower plate contact surface is the result of its internal wave:

[0095]

[0096] The stress t2 on the upper plate contact surface is the result of its internal wave:

[0097]

[0098] No constraint is applied to the lower plate boundary, and the stress is discontinuous, that is:

[0099] t1≠t2(12);

[0100] This structure ensures that the vibrations of the upper and lower aluminum plates remain independent, thus eliminating any interaction between the two aluminum plates. This setting is conducive to analyzing the wave propagation characteristics in the free aluminum plates.

[0101] S402: In this embodiment, consistent boundary pairs are established on both sides of the model contact area. This structure ensures that the upper and lower aluminum plates are considered to be in full contact, thus promoting the continuity of displacement and stress on the contact surface. Specifically, the displacements of the upper and lower aluminum plates at the contact surface are equal, and the normal stress and shear stress are also equal:

[0102] u1 = u2(13);

[0103] σ n,1 = σ n,2 , τ t,1 = τ t,2 (14);

[0104] Thus, the contact between the upper and lower aluminum plates is effectively simulated to promote the transmission of guided waves to the receiving area.

[0105] S5: Based on S4, frequency sweep analysis is carried out to obtain the change curve of the total transmission energy TE of the ultrasonic guided wave in the receiving area, and the excitation center frequency corresponding to the peak of the curve is selected as the optimal excitation center frequency;

[0106] In this embodiment, the excitation voltage of the transducer is set to Vp-p = 100V, and then a frequency scan analysis is performed on the bolt connection within the excitation center frequency range of (50 - 320) kHz, with an increment of 5kHz. The transmitted energy of the ultrasonic guided wave in the receiving area includes potential energy (strain energy) and kinetic energy. Without considering material damping, the potential energy and kinetic energy always change periodically according to the law of conservation of energy, so the potential energy and kinetic energy are always equal in the sense of time average. In an elastic material, the strain energy density is expressed as:

[0107]

[0108] In formula (10), σ is the stress tensor, ε is the strain tensor, and W h is the strain energy density;

[0109] In the two-dimensional frequency domain model, formula (10) can be decomposed into:

[0110]

[0111] Based on formula (11), by integrating the strain energy density along the thickness direction of the receiving area, the potential energy can be obtained. Therefore, the total transmitted energy TE of the ultrasonic guided wave can be calculated as:

[0112] TE = 2∫ h W h dy (12).

[0113] Specific Embodiment 2: In combination with Figures 2 - 4 to illustrate this embodiment. To verify the technical effects in the specific embodiment, the following simulation experiment is designed in this embodiment:

[0114] In this embodiment, the material parameters and size parameters of the aluminum plate used in the two-dimensional frequency domain finite element model are shown in Table 1:

[0115] Table 1

[0116]

[0117] Based on the above conditions, this embodiment calculates the guided wave propagation characteristics in a free aluminum plate at two excitation frequencies of 150kHz and 250kHz. As Figure 2 shown, the positive strain components along the propagation direction at (100 - 300) mm in the transmission area of the aluminum plate at the two frequencies are extracted respectively, and the stress distribution cloud diagram and spatial strain distribution are as Figure 2 (a), (b), (d), (e) shown. From Figure 2 (a) and (d), it can be seen that in the perfectly matched layer area, the guided wave signal is quickly attenuated and absorbed. Therefore, there is basically no interference from the reflected wave at both ends of the aluminum plate, and the guided wave in the plate is completely excited by the piezoelectric wafer. InFigure 2 (b) and (e), although the strain components show a certain periodic pattern with the change of spatial position, due to the aliasing of multimodal signals, it is impossible to directly reflect each modal information. Therefore, the Fourier transform is used to convert the spatial domain signal to the wavenumber domain for analysis. The wavenumber spectrum results at two frequencies are respectively as Figure 2 (c) and (f) shown. It can be seen from the wavenumber diagram that within the first-order cut-off frequency range, only two modes can be excited, namely the A0 mode and the S0 mode. The two modal signals can be effectively distinguished. The one with a larger wavenumber corresponds to the A0 mode, and the one with a smaller wavenumber corresponds to the S0 mode. Moreover, the wavenumber of the same mode increases with the increase of frequency, which is consistent with the theoretical expectation.

[0118] The steady-state vibration mode at the excitation frequency of 250 kHz is as Figure 3 shown. It can be seen from Figure 3 that under the effective excitation of the transducer, the guided wave propagates from the transmission area and successfully reaches the receiving area through the contact area.

[0119] The variation relationship of the normalized transmission energy with frequency is as Figure 4 shown. Figure 4 It presents a significant trend: with the increase of the excitation center frequency, the transmission energy of the guided wave first increases and then decreases, reaching the peak at the frequency of 230 kHz, indicating that the highest transmission energy can be provided at this frequency. Therefore, this frequency is determined as the optimal excitation center frequency. Thus, it provides strong support for the method of detecting bolt torque using ultrasonic guided wave energy.

[0120] The above is only a preferred embodiment of the present invention, and it does not impose any form of limitation on the present invention. Although the present invention has been disclosed above with the preferred embodiment, it is not intended to limit the present invention. Any person skilled in the art can make some changes or modifications to the above-disclosed technical content to be equivalent embodiments with equivalent changes within the scope of the technical solution of the present invention. However, as long as it does not depart from the content of the technical solution of the present invention and is based on the technical essence of the present invention, any simple modification, equivalent replacement, and improvement made to the above embodiments still fall within the protection scope of the technical solution of the present invention.

Claims

1. An optimal frequency selection method for on-line monitoring of bolt torque based on ultrasonic guided waves, characterized in that The steps of the optimal frequency selection method for online monitoring of bolt torque based on ultrasonic guided waves are as follows: Step 1: Establish a two-dimensional frequency-domain finite element model based on physical field simulation software. Simplify the bolt connection as two identical aluminum plates in contact. Then, the propagation process of ultrasonic guided waves divides the entire bolt connection into a transmission region, a contact region, and a receiving region; Step 2: Add perfectly matched layers on both sides of the established two-dimensional frequency-domain finite element model to make the ultrasonic guided waves gradually attenuate during propagation; Step 3: Place a piezoelectric wafer as an excitation source in the transmission region of the two-dimensional frequency-domain finite element model after adding the perfectly matched layers to generate ultrasonic guided wave signals; Step 4: Set contact pairs and boundary pairs in the contact region of the two-dimensional frequency-domain finite element model, and analyze the propagation characteristics of guided wave signals in the upper and lower aluminum plates and the guided wave transmission analysis in the contact simulation of the upper and lower aluminum plates respectively; Step 5: On the basis of Step 4, perform frequency sweep analysis, obtain the change curve of the total transmission energy TE of ultrasonic guided waves in the receiving region, and select the excitation center frequency corresponding to the peak of the curve as the optimal excitation center frequency.

2. The optimal frequency selection method for on-line monitoring of bolt torque based on ultrasonic guided waves according to claim 1, characterized in that Step 1 specifically includes: On the basis of the two-dimensional frequency-domain finite element model, establish a wave equation applicable to the propagation of ultrasonic guided waves in the transmission region, contact region, and receiving region in the frequency domain; The expression of the wave equation is: In formula (1), ρ is the density of the aluminum plate, ω is the angular frequency, u is the displacement field, and C is the elastic tensor.

3. An optimal frequency selection method for on-line monitoring of bolt torque based on ultrasonic guided waves according to claim 1, characterized in that, Step 2 specifically includes: Step 2.1: Calculate the expansion factors in the x-direction and y-reverse direction of the propagation path of ultrasonic guided waves, and convert the coordinates defining the perfectly matched layer region into complex coordinates based on the expansion factors in the x-direction and y-reverse direction; Step 2.2: Introduce complex coordinates, obtain the transformed gradient operator in the wave equation, and then obtain the modified wave equation of the perfectly matched layer; Step 2.3: Directly relate the attenuation characteristics and propagation coefficient of ultrasonic guided waves by introducing complex coordinates, and in the perfectly matched layer, adjust the wave number k to a complex number to obtain a complex wave number. Introduce the imaginary part of the complex wave number into the attenuation of ultrasonic guided waves, so that the amplitude of ultrasonic guided waves decreases exponentially with the increase of distance in the perfectly matched layer region; The expression of the coordinate transformation of the perfectly matched layer region is: In formula (2), s x is the expansion factor in the x - direction of the guided - wave propagation path, s y is the expansion factor in the y - direction of the guided - wave propagation path, s x and s y are complex numbers, and their calculation formulas are as follows: s x (x) = 1 + iδ x (x), s y (y) = 1 + iδ y (y)(3); In formula (3), δ x (x) and δ y (y) are both damping functions used to control the absorption characteristics of the perfectly matched layer; For the two-dimensional wave equation propagating in the x-direction, the calculation formula of the damping coefficient of the perfectly matched layer is: In formula (4), δ0 is the maximum damping coefficient of the perfectly matched layer, L x is the length of the perfectly matched layer, and n is the exponent for controlling the attenuation intensity, usually taking 1 or 2; The calculation formula of the transformed gradient operator in the wave equation is: The modified wave equation of the perfectly matched layer is: The expression of adjusting the wave number k to a complex number is: The expression of introducing the imaginary part of the complex wave number into the attenuation of ultrasonic guided waves is:

4. An optimal frequency selection method for online monitoring of bolt torque based on ultrasonic guided waves according to claim 1, characterized in that The expression of the excitation form of the excitation source in Step 3 is: u(t) = Acos(ωt) (9); In formula (9), A is the excitation amplitude, and ω is the excitation angular frequency.

5. An optimal frequency selection method for on-line monitoring of bolt torque based on ultrasonic guided waves according to claim 1, characterized in that Step 4 specifically includes: Step 4.1: Set up contact pairs at the structural contact boundaries in the contact area of the two-dimensional frequency-domain finite element model. Since there is a gap between the upper and lower aluminum plates, the vibrations of the upper and lower aluminum plates are ensured to be independent, thus eliminating any interaction between the two aluminum plates, and then analyzing the propagation characteristics of the guided wave signals in the upper and lower aluminum plates. Among them, the analysis results include: the stress t1 on the contact surface of the lower aluminum plate is the result of internal fluctuations, that is the stress t2 on the contact surface of the upper plate is the result of internal fluctuations, that is No constraints are applied at the boundaries of the upper and lower plates, and the stress is discontinuous, that is, t1 ≠ t2; Step 4.2: Establish consistent boundary pairs on both sides of the contact area of the two-dimensional frequency-domain finite element model to simulate the complete contact between the upper and lower aluminum plates, thereby facilitating the deletion of ultrasonic guided waves to the receiving area, where the displacements of the upper and lower aluminum plates at the contact surface are equal, i.e., u1 = u2, the normal stresses are equal, i.e., σ n,1 = σ n,2 , and the shear stresses are equal, i.e., τ t,1 = τ t,2 .

6. An optimal frequency selection method for online monitoring of bolt torque based on ultrasonic guided waves according to claim 1, characterized in that Step 5 specifically includes: Step 5.1: Set the excitation voltage of the transducer and the increment during the frequency sweep analysis, and perform frequency sweep analysis on the bolt connection within the preset excitation center frequency range; Step 5.2: Calculate the total transmission energy TE of ultrasonic guided waves at different excitation center frequencies and draw a change curve, and select the excitation center frequency corresponding to the peak of the curve as the optimal excitation center frequency.

7. An optimal frequency selection method for on-line monitoring of bolt torque based on ultrasonic guided waves according to claim 6, characterized in that The calculation of the total transmission energy TE of the ultrasonic guided wave in Step 5.2 specifically includes: Step 5.2.1: The transmission energy of the ultrasonic guided wave in the receiving area includes potential energy and kinetic energy. Without considering material damping, the potential energy and kinetic energy are equal in the time-average sense, and the strain energy density is calculated based on the stress tensor and the strain tensor; Step 5.2.2: Decompose the strain energy calculation formula in the two-dimensional frequency domain model; Step 5.2.3: On the basis of Step 5.2.2, integrate the strain energy density along the thickness direction of the receiving area to obtain the potential energy, and then calculate the total transmission energy TE of the ultrasonic wave; The calculation formula for the strain energy density is: In Equation (10), σ is the stress tensor, ε is the strain tensor, and W h is the strain energy density; In the two-dimensional frequency domain model, the strain energy density calculation formula is decomposed into: The calculation formula for the total transmission energy TE of the ultrasonic wave is: TE = 2∫ h W h dy(12).

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