# Parameter Recognition of Mode-Converted Wave in Single-Source Ultrasound Using Gabor Transform for Bolt Axial Stress Evaluation

Authors: Ping Chen; Xingliang He; Wei Song
Journal: Journal of Sensors
 (2020)

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## Abstract

In this paper, a novel parameter recognition method of mode-converted wave in single-source ultrasound is put forward and applied to the estimation of axial stresses in bolts. To overcome the distortion and aliasing of the mode-converted wave in single-source ultrasonic signal, a time-frequency parameter recognition method based on the Gabor transform is introduced to recognize the accurate time of flight (TOF) of the mode-converted wave. Based on the mode conversion of the single-source longitudinal ultrasound, a new nonlinear evaluation model based on acoustoelastic equation is derived to determine the axial stress of bolt. The performance of the proposed method is evaluated by comparing it with the commonly used TOF ratio method of longitudinal wave and shear wave (L-S). The experiment result shows that the proposed method is more effective in detecting the connection status of bolted joints than the traditional L-S method.


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## Body

## 1. Introduction

Loosening is one kind of failure that can markedly impact the stiffness and reliability of bolted joints. Insufficient preload of bolt will significantly increase the risk of overall mechanical structural disintegration. One way to address the critical issues in maintaining the strength of the structure and ensuring the safety of engineers is to apply appropriate pretightening force to the bolted joints and conduct long-term online monitoring of their axial stress. The most frequently used nondestructive estimation methods of stress include the photoelasticity method [1], resistance strain gauge method [2], piezoresistance method [3], active sensing method [4, 5], and ultrasonic testing method [6–9]. Among all these methods, the ultrasonic testing method is considered as a potential alternative technique due to its convenience to measure bolt tension with better accuracy.Among the ultrasonic testing technologies, the acoustoelastic method by using TOF [6–12] is a typical stress measurement method. In the method, the TOF difference before and after bolt loading is calculated and multiplied with an acoustic-stress factor to achieve the axial stress. Whereas one shortcoming of this method is that it cannot be utilized to measure the stress of a fastened bolt. Yasui et al. [13] proposed the L-S method based on multisource ultrasound to solve this problem, in which the stress is measured according to the TOF ratio of longitudinal wave (L-wave) and shear wave (S-wave) by two separate transducers mounted on the two ends of the bolt with great care and precision. And the TOF ratio was proved to be independent of the original length of the bolt or the initial TOF. However, the coupling conditions (e.g., installation position, pressure, and coupling agent concentration) of the L-wave and S-wave transducers must be repeatable to ensure accuracy. Hence, this method is difficult to achieve reliable measurement results under practical conditions. Kim and Hong [14] proposed a stress measurement method using mode conversion of longitudinal waves to reduce the experimental error caused by different contact condition. However, compared with the multi-source method, owing to aliasing and distortion in the mode-converted wave signal used in Kim’s method, it’s difficult to obtain accurate TOF directly. They used a special designed probe with an acoustic lens to generate the mode-converted wave with high SNR (signal-noise ratio). In addition, it is necessary to provide enough space to install the specially designed probe, and the length of the workpiece cannot exceed the focal length of the lens. Therefore, a new parameter recognition method of the mode-converted wave in single-source ultrasound must be studied to promote the measurement efficiency of the bolt.In this paper, a novel axial stress evaluation method using the mode-converted wave generated by single-source ultrasound based on the Gabor transform is proposed. Firstly, the distortion and aliasing of mode-converted process in single-source ultrasonic longitudinal signal are analyzed. Then, the parameter recognition method based on the Gabor transform and Gaussian empirical model to obtain the accurate TOF of the mode-converted signal is introduced and its effectiveness is proved by simulation. Next, a new nonlinear model for evaluating the axial stress of bolt is derived based on the acoustoelastic equation. Finally, the performance of the proposed method is evaluated by comparing it with the commonly used TOF ratio method of longitudinal wave and shear wave. And the experiment result shows that the proposed method is more effective in detecting the connection status of bolted joints than the traditional L-S method.


## 2. TOF Recognition Method in Single-Source Ultrasound

### 2.1. Mode-Converted Wave in Ultrasonic Testing

The velocities and polarization directions of the S-wave and the L-wave are related to the elastic properties, anisotropy, and current stress state of materials. Both the L-wave and the S-wave can be transmitted in solid medium, while only the L-wave can be transmitted in fluid medium. When a longitudinal wave beam enters the solid medium from the fluid medium, it will not only be reflected at the interface but will also derive an orthogonal bipolarized transverse wave and some other waveforms near the interface, such as the Rayleigh wave and the Love wave. Similarly, when there exist ultrasound incidents from the solid medium to the fluid medium, reflection wave and orthogonal bipolarized transverse wave will occur. This phenomenon is called mode conversion. Figure1 shows a typical mode conversion process in a monostatic pulse-echo experiment, in which the red beam represents a longitudinal wave and the green beam represents a shear wave.Figure 1
Mode conversion in a pulse-echo experiment.In Figure1, Linc represents the incident single-source longitudinal wave generated by the transducer. At the liquid-solid interface, Linc derives a reflection wave Lrefl, a transmission L-wave (LL), and a transmission shear wave (LT). Suppose that the reflection angle of the longitudinal wave is α and the angle of shear wave is β, the relationship between them can be described by Snell’s law:
(1)sinαCL=sinβCT.In equation (1), CL and CT refer to the acoustical velocities of L-wave and S-wave in the solid medium, respectively. When LL and LT reach the other end of the mounted solid, the mode conversion occurs again. The LL generates the LLL wave and the LLT wave, and the LT generates the LTL wave and the LTT wave. At last, all the waves undergo mode conversion again after arriving at the left solid interface, and the part transmitted into the coupling layer will all be converted to longitudinal waves and captured by the transducer. It should be noted that the propagation distance of the LLT and the LTL is exactly the same theoretically.The transit time of acoustical wave can be rephrased as TOF, andTOFL and TOFT denote the transit time of L-wave and S-wave, respectively, in this paper. The relationship between the TOF and the sound velocity can be expressed as
(2)CL≈2DTOFL,CT≈2DTOFT,where D is the length of the solid medium. And the TOF of the converted waves (TLLL, TLLT, TLTL, and TLTT) in solid can be expressed as
(3)TLLL=2D/cosα2D/TOFL,TLLT=D/cosα2D/TOFL+D/cosβ2D/TOFT,TLTL=D/cosα2D/TOFL+D/cosβ2D/TOFT,TLTT=2D/cosβ2D/TOFT.When the incident direction is vertical to the interface, the anglesα and β are approximately equal to zero. So, their relationship can be described as
(4)TLLL<TLLT=TLTL<TLTT.Equations (3) and (4) can determine the positions of the converted waves in a time-domain signal approximately. In general, the LTT wave has serious attenuation in amplitude and falls in the vicinity of the second echo of LLL (because the travel speed of the L-wave is nearly two times than that of the S-wave). Therefore, the LTT wave is not suitable for stress evaluation. Owing to the relatively low attenuation, it is possible for the remaining LLT wave and the LTL wave to be adopted in stress evaluation. However, their waveforms are completely overlapped in the time domain because the LLT and LTL have exactly the same TOF in theory (here we use LC to represent the overlapped mode-converted waveforms). Moreover, LC often exhibits varying degrees of nonlinear distortion and aliasing caused by coupling conditions and multiple mode conversion. As a result, the direct acquisition of the LC TOF through the original signal is almost infeasible.


### 2.2. Parameter Recognition Method Using the Gabor Transform

The cross-correlation method is the most commonly used parameter recognition method to obtain the TOF of an ultrasonic signal, which can achieve high estimation accuracy under the condition of high signal-to-noise ratio (SNR). However, due to the nonlinear distortion and aliasing in LC, applying the cross-correlation method to this situation may cause extra error [15]. This paper introduces an ultrasonic signal processing method based on an empirical model [15, 16] and the Gabor transform [17] to recognize the parameter of the LC. The Gabor transform is a time-frequency transform method and widely used in signal parameter estimation for the ability of revealing the local frequency distribution of a signal or an image. From the perspective of time-frequency analysis, this paper applies the Gabor transform to estimate the parameter of the empirical model of LC. The authors use XLCt to express the signal of LC in time domain, and XLCt can be rephrased as a two-dimensional time-frequency expansion by applying the Gabor transform:
(5)XLCt=∑τ=−∞+∞∑ω=−∞+∞Cτ,ωhτ,ω=∑τ=−∞+∞∑ω=−∞+∞Cτ,ωht−τae−j2πbωt.In equation (5), τ and ω represent the time and frequency coordinates of the two-dimensional plane, a and b are the width of the time and frequency grid of the plane, respectively, Cτ,ω is the Gabor expansion coefficient, and hτ,ω is the Gabor generating function. What is more, Cτ,ω is also defined as the Gabor transform (GT) of XLCt:
(6)Cτ,ω=GTXLC=∫−∞∞XLCδh∗δ−τae−j2πωbδdδ.According to the physical characteristics of the ultrasonic transducer commonly used in bolt stress detection technology, the Gauss attenuation signal [18] is selected as the empirical model to describe LC:
(7)SLCt=A0e−dt−τ2sin2πωt−τ.In equation (7), A0 is the amplitude and d is the shape factor. Since the Gaussian function is still a Gaussian kernel function after Fourier transformation, its energy distribution is concentrated in both the frequency domain and the time domain, which can clearly reveal the local details of the signal. Considering that SLCt is a Gaussian model, an analysis window function with a Gaussian kernel is selected as the Gabor generating function:
(8)hτ,ω=ηe−r0t−aτ2+jω0t−aτ/bω.To ensure the energy uniformity of the window function, let(9)η=2r0π1/4.Then, the Euclidean distance of Gabor generating function and the empirical model is adopted to characterize the similarity betweenSLCt and hτ,ωt, which can be expressed as
(10)SLCt−hτ,ωt2=SLC2t+hτ,ω2t−2GTSLC.In fact, the purpose of finding the maximum similarity between the Gabor generating function and the empirical model is equivalent to obtaining the minimum value of equation (10). Since SLC2t and h2τ,ωt are both nonnegative and unrelated to τ and ω, the minimum value of equation (10) is transformed into the maximum value of ∣GTSLC∣. And ∣GTSLC∣ can be further written as
(11)GTSLC=fτ,ω=ηA0πd+r0e−4dr0τ−TOF2+ωc−ω0/ω2/4d+r0.In conclusion, the parameter recognition of the empirical model of LC can be described as(12)TOF,ωC=argmaxτ,ωGTSLC.


### 2.3. Parameter Recognition Using Particle Filter

Absolutely, the solution of equation (12) is a typical two-dimensional optimization problem. Taking both estimation accuracy and convergence speed into account, the particle filter method is selected as the solver of equation (12). The basic idea of the particle filter algorithm is the Monte Carlo method, which approximates and replaces the probability density function of the system through a large number of random samples and selects a subset of particles for sequential importance sampling according to the posterior probability in order to shrink the scope of estimation [19–22]. Assume the target vector VLC:
(13)VLC=TOF,ωC.The parameter recognition process of the empirical model of LC based on the particle filter method is as follows:(1)
Suppose the lengths of the time and the frequency axials of the Gabor expansion plane isM and N, respectively. Establish the initial particle set according to the number of particles K:(14)PV0LC=V0LCi,W0ii=0K,∑i=0NW0i=1.In equation (14), 0VLCi refers to the target vector of each particle in the initial set and W0 is the probability of each particle in the initial set. They can be expressed as
(15)0VLCi=TOF0i,ωC0i,TOF0i∈0,M,ωC0i∈0,N.To ensure generality, the uniform distribution is selected as the initial importance distribution:(16)W0i=1M,1N.(2)
Substituting the value ofTOF0i and ωC0i into equation (11) for each particle in the set, then the expression of ∣GTiSLC∣ can be derived. Next, we can calculate the state transition probability according to the likelihood between ∣GTSLC∣ and ∣GTiXLC∣. The normalized weight corresponding to the state transition probability of each particle is given by(17)wi=GTiSLC−GTXLC∑i=1KGTiSLC−GTXLC.(3)
Perform sequential importance resample according to the elimination threshold which equals to the average ofwi. The particles with low weight are eliminated and those with high weight are retained and copied. After resample, the new particle set can be derived as(18)PVjLC=VjLCi,Wjii=0K,∑i=0NWji=1.(4)
Repeat procedures (1) to (3) until the values of every particle are exactly the same as each other, and the target vector of the last set is expressed as(19)VLCest=TOFest,ωCest.TOFest and ωCest are corresponding to the TOF and the central frequency of mode-converted wave LC.


### 2.4. Simulation

To evaluate the performance of the proposed method, a group of double Gaussian echo signals with different TOF and central frequency are introduced to simulate the mode-converted wave LC. The general expression of the signals is(20)St=A1e−d1t−τ12sinfc1t−τ1+A2e−d2t−τ22sinfc2t−τ2,where A1 and A2 represent the amplitude of the echo envelope, τ1 and τ2 are the TOF, fc1 and fc2 are the central frequencies, and d1 and d2 are the shape factors. In each signal, A1=1V, A2=0.8V, d1=1.5×10−7s, and d2=1.4×10−7s. The values of the TOF and the central frequency of these signals are shown in Table 1. The initial set is made up of 10000 random chosen particles within the range of ±2×10−7sec and ±0.5 MHz around the preset values. After -10 dB~10 dB Gaussian white noise is applied to the signals to simulate different SNR conditions, the TOFs and the central frequencies of No. 1~No. 8 signals are recognized and each signal is calculated over 100 times. The No. 1 signal with 5 dB noise is shown in Figure 2(a), and its Gabor expansion is shown in Figure 2(b). It can be observed in Figure 2(a) that the time-domain signal is severely distorted and the echoes are submerged in clutter, while the echoes can be clearly recognized in the diagram of Gabor expansion (Figure 2(b)) and the recognized results are shown in Table 2.Table 1
Parameters of the simulation signals.
Numberfc1 (MHz)fc2 (MHz)τ1 (s)τ2 (s)14.0003.0001.900×10−62.100×10−624.5003.5002.000×10−62.200×10−635.0024.0002.100×10−62.300×10−645.5004.5002.200×10−62.400×10−656.0005.0002.300×10−62.500×10−666.5005.5002.400×10−62.600×10−677.0006.0002.500×10−62.700×10−687.5006.5002.600×10−62.800×10−6Figure 2
Simulation results.
(a)
Simulation signal(b)
Gabor expansion resultTable 2
Recognized results of the simulation signals.
Numberfc1 (MHz)fc2 (MHz)Average frequency errorτ1 (s)τ2 (s)Average TOF error14.0822.9392.04%1.901×10−62.106×10−60.17%24.4953.5070.15%1.987×10−62.198×10−60.76%35.0123.9780.37%2.102×10−62.308×10−60.22%45.4984.4870.24%2.204×10−62.389×10−60.32%55.9915.0120.19%2.297×10−62.486×10−60.35%66.5045.6120.89%2.406×10−62.587×10−60.38%76.9936.1241.00%2.507×10−62.712×10−60.36%87.5226.5140.25%2.602×10−62.791×10−60.20%The TOF difference caused by the acoustoelastic effect in an actual measurement is generally about 1%, while for the proposed method the measurement error of TOF is about 0.5% in average under the condition of low SNR, as shown in Table2. Therefore, the parameter recognition using the Gabor transform is efficient and can be applied to the TOF recognition of mode-converted wave in single-source ultrasound.


## 2.1. Mode-Converted Wave in Ultrasonic Testing

The velocities and polarization directions of the S-wave and the L-wave are related to the elastic properties, anisotropy, and current stress state of materials. Both the L-wave and the S-wave can be transmitted in solid medium, while only the L-wave can be transmitted in fluid medium. When a longitudinal wave beam enters the solid medium from the fluid medium, it will not only be reflected at the interface but will also derive an orthogonal bipolarized transverse wave and some other waveforms near the interface, such as the Rayleigh wave and the Love wave. Similarly, when there exist ultrasound incidents from the solid medium to the fluid medium, reflection wave and orthogonal bipolarized transverse wave will occur. This phenomenon is called mode conversion. Figure1 shows a typical mode conversion process in a monostatic pulse-echo experiment, in which the red beam represents a longitudinal wave and the green beam represents a shear wave.Figure 1
Mode conversion in a pulse-echo experiment.In Figure1, Linc represents the incident single-source longitudinal wave generated by the transducer. At the liquid-solid interface, Linc derives a reflection wave Lrefl, a transmission L-wave (LL), and a transmission shear wave (LT). Suppose that the reflection angle of the longitudinal wave is α and the angle of shear wave is β, the relationship between them can be described by Snell’s law:
(1)sinαCL=sinβCT.In equation (1), CL and CT refer to the acoustical velocities of L-wave and S-wave in the solid medium, respectively. When LL and LT reach the other end of the mounted solid, the mode conversion occurs again. The LL generates the LLL wave and the LLT wave, and the LT generates the LTL wave and the LTT wave. At last, all the waves undergo mode conversion again after arriving at the left solid interface, and the part transmitted into the coupling layer will all be converted to longitudinal waves and captured by the transducer. It should be noted that the propagation distance of the LLT and the LTL is exactly the same theoretically.The transit time of acoustical wave can be rephrased as TOF, andTOFL and TOFT denote the transit time of L-wave and S-wave, respectively, in this paper. The relationship between the TOF and the sound velocity can be expressed as
(2)CL≈2DTOFL,CT≈2DTOFT,where D is the length of the solid medium. And the TOF of the converted waves (TLLL, TLLT, TLTL, and TLTT) in solid can be expressed as
(3)TLLL=2D/cosα2D/TOFL,TLLT=D/cosα2D/TOFL+D/cosβ2D/TOFT,TLTL=D/cosα2D/TOFL+D/cosβ2D/TOFT,TLTT=2D/cosβ2D/TOFT.When the incident direction is vertical to the interface, the anglesα and β are approximately equal to zero. So, their relationship can be described as
(4)TLLL<TLLT=TLTL<TLTT.Equations (3) and (4) can determine the positions of the converted waves in a time-domain signal approximately. In general, the LTT wave has serious attenuation in amplitude and falls in the vicinity of the second echo of LLL (because the travel speed of the L-wave is nearly two times than that of the S-wave). Therefore, the LTT wave is not suitable for stress evaluation. Owing to the relatively low attenuation, it is possible for the remaining LLT wave and the LTL wave to be adopted in stress evaluation. However, their waveforms are completely overlapped in the time domain because the LLT and LTL have exactly the same TOF in theory (here we use LC to represent the overlapped mode-converted waveforms). Moreover, LC often exhibits varying degrees of nonlinear distortion and aliasing caused by coupling conditions and multiple mode conversion. As a result, the direct acquisition of the LC TOF through the original signal is almost infeasible.


## 2.2. Parameter Recognition Method Using the Gabor Transform

The cross-correlation method is the most commonly used parameter recognition method to obtain the TOF of an ultrasonic signal, which can achieve high estimation accuracy under the condition of high signal-to-noise ratio (SNR). However, due to the nonlinear distortion and aliasing in LC, applying the cross-correlation method to this situation may cause extra error [15]. This paper introduces an ultrasonic signal processing method based on an empirical model [15, 16] and the Gabor transform [17] to recognize the parameter of the LC. The Gabor transform is a time-frequency transform method and widely used in signal parameter estimation for the ability of revealing the local frequency distribution of a signal or an image. From the perspective of time-frequency analysis, this paper applies the Gabor transform to estimate the parameter of the empirical model of LC. The authors use XLCt to express the signal of LC in time domain, and XLCt can be rephrased as a two-dimensional time-frequency expansion by applying the Gabor transform:
(5)XLCt=∑τ=−∞+∞∑ω=−∞+∞Cτ,ωhτ,ω=∑τ=−∞+∞∑ω=−∞+∞Cτ,ωht−τae−j2πbωt.In equation (5), τ and ω represent the time and frequency coordinates of the two-dimensional plane, a and b are the width of the time and frequency grid of the plane, respectively, Cτ,ω is the Gabor expansion coefficient, and hτ,ω is the Gabor generating function. What is more, Cτ,ω is also defined as the Gabor transform (GT) of XLCt:
(6)Cτ,ω=GTXLC=∫−∞∞XLCδh∗δ−τae−j2πωbδdδ.According to the physical characteristics of the ultrasonic transducer commonly used in bolt stress detection technology, the Gauss attenuation signal [18] is selected as the empirical model to describe LC:
(7)SLCt=A0e−dt−τ2sin2πωt−τ.In equation (7), A0 is the amplitude and d is the shape factor. Since the Gaussian function is still a Gaussian kernel function after Fourier transformation, its energy distribution is concentrated in both the frequency domain and the time domain, which can clearly reveal the local details of the signal. Considering that SLCt is a Gaussian model, an analysis window function with a Gaussian kernel is selected as the Gabor generating function:
(8)hτ,ω=ηe−r0t−aτ2+jω0t−aτ/bω.To ensure the energy uniformity of the window function, let(9)η=2r0π1/4.Then, the Euclidean distance of Gabor generating function and the empirical model is adopted to characterize the similarity betweenSLCt and hτ,ωt, which can be expressed as
(10)SLCt−hτ,ωt2=SLC2t+hτ,ω2t−2GTSLC.In fact, the purpose of finding the maximum similarity between the Gabor generating function and the empirical model is equivalent to obtaining the minimum value of equation (10). Since SLC2t and h2τ,ωt are both nonnegative and unrelated to τ and ω, the minimum value of equation (10) is transformed into the maximum value of ∣GTSLC∣. And ∣GTSLC∣ can be further written as
(11)GTSLC=fτ,ω=ηA0πd+r0e−4dr0τ−TOF2+ωc−ω0/ω2/4d+r0.In conclusion, the parameter recognition of the empirical model of LC can be described as(12)TOF,ωC=argmaxτ,ωGTSLC.


## 2.3. Parameter Recognition Using Particle Filter

Absolutely, the solution of equation (12) is a typical two-dimensional optimization problem. Taking both estimation accuracy and convergence speed into account, the particle filter method is selected as the solver of equation (12). The basic idea of the particle filter algorithm is the Monte Carlo method, which approximates and replaces the probability density function of the system through a large number of random samples and selects a subset of particles for sequential importance sampling according to the posterior probability in order to shrink the scope of estimation [19–22]. Assume the target vector VLC:
(13)VLC=TOF,ωC.The parameter recognition process of the empirical model of LC based on the particle filter method is as follows:(1)
Suppose the lengths of the time and the frequency axials of the Gabor expansion plane isM and N, respectively. Establish the initial particle set according to the number of particles K:(14)PV0LC=V0LCi,W0ii=0K,∑i=0NW0i=1.In equation (14), 0VLCi refers to the target vector of each particle in the initial set and W0 is the probability of each particle in the initial set. They can be expressed as
(15)0VLCi=TOF0i,ωC0i,TOF0i∈0,M,ωC0i∈0,N.To ensure generality, the uniform distribution is selected as the initial importance distribution:(16)W0i=1M,1N.(2)
Substituting the value ofTOF0i and ωC0i into equation (11) for each particle in the set, then the expression of ∣GTiSLC∣ can be derived. Next, we can calculate the state transition probability according to the likelihood between ∣GTSLC∣ and ∣GTiXLC∣. The normalized weight corresponding to the state transition probability of each particle is given by(17)wi=GTiSLC−GTXLC∑i=1KGTiSLC−GTXLC.(3)
Perform sequential importance resample according to the elimination threshold which equals to the average ofwi. The particles with low weight are eliminated and those with high weight are retained and copied. After resample, the new particle set can be derived as(18)PVjLC=VjLCi,Wjii=0K,∑i=0NWji=1.(4)
Repeat procedures (1) to (3) until the values of every particle are exactly the same as each other, and the target vector of the last set is expressed as(19)VLCest=TOFest,ωCest.TOFest and ωCest are corresponding to the TOF and the central frequency of mode-converted wave LC.


## 2.4. Simulation

To evaluate the performance of the proposed method, a group of double Gaussian echo signals with different TOF and central frequency are introduced to simulate the mode-converted wave LC. The general expression of the signals is(20)St=A1e−d1t−τ12sinfc1t−τ1+A2e−d2t−τ22sinfc2t−τ2,where A1 and A2 represent the amplitude of the echo envelope, τ1 and τ2 are the TOF, fc1 and fc2 are the central frequencies, and d1 and d2 are the shape factors. In each signal, A1=1V, A2=0.8V, d1=1.5×10−7s, and d2=1.4×10−7s. The values of the TOF and the central frequency of these signals are shown in Table 1. The initial set is made up of 10000 random chosen particles within the range of ±2×10−7sec and ±0.5 MHz around the preset values. After -10 dB~10 dB Gaussian white noise is applied to the signals to simulate different SNR conditions, the TOFs and the central frequencies of No. 1~No. 8 signals are recognized and each signal is calculated over 100 times. The No. 1 signal with 5 dB noise is shown in Figure 2(a), and its Gabor expansion is shown in Figure 2(b). It can be observed in Figure 2(a) that the time-domain signal is severely distorted and the echoes are submerged in clutter, while the echoes can be clearly recognized in the diagram of Gabor expansion (Figure 2(b)) and the recognized results are shown in Table 2.Table 1
Parameters of the simulation signals.
Numberfc1 (MHz)fc2 (MHz)τ1 (s)τ2 (s)14.0003.0001.900×10−62.100×10−624.5003.5002.000×10−62.200×10−635.0024.0002.100×10−62.300×10−645.5004.5002.200×10−62.400×10−656.0005.0002.300×10−62.500×10−666.5005.5002.400×10−62.600×10−677.0006.0002.500×10−62.700×10−687.5006.5002.600×10−62.800×10−6Figure 2
Simulation results.
(a)
Simulation signal(b)
Gabor expansion resultTable 2
Recognized results of the simulation signals.
Numberfc1 (MHz)fc2 (MHz)Average frequency errorτ1 (s)τ2 (s)Average TOF error14.0822.9392.04%1.901×10−62.106×10−60.17%24.4953.5070.15%1.987×10−62.198×10−60.76%35.0123.9780.37%2.102×10−62.308×10−60.22%45.4984.4870.24%2.204×10−62.389×10−60.32%55.9915.0120.19%2.297×10−62.486×10−60.35%66.5045.6120.89%2.406×10−62.587×10−60.38%76.9936.1241.00%2.507×10−62.712×10−60.36%87.5226.5140.25%2.602×10−62.791×10−60.20%The TOF difference caused by the acoustoelastic effect in an actual measurement is generally about 1%, while for the proposed method the measurement error of TOF is about 0.5% in average under the condition of low SNR, as shown in Table2. Therefore, the parameter recognition using the Gabor transform is efficient and can be applied to the TOF recognition of mode-converted wave in single-source ultrasound.


## 3. Nonlinear Evaluation Model of Bolt Axial Stress

### 3.1. Nonlinear Evaluation Model Based on Mode-Converted Ultrasound

The existing evaluation models [6–14] of bolt axial stress are simplified and linear. And it is inevitable for them to induce extra fitting error. In this part, a new nonlinear evaluation model of bolt axial stress is proposed, its nonlinear feature is much better to meet the characteristic of acoustic elastic effect in theory. According to the theory of elasticity and nonlinear acoustic [23],
(21)CL2=CL02−σλ+2μ4λ+10μ+4m/μ+λ+2jρ03λ+2μ,CT2=CT02−σ2j−2m+λ+2μ/μρ03λ+2μ,where CL and CT are the velocities of longitudinal wave and shear wave in a solid medium under stress, respectively; CL0 and CT0 are their velocities in the stress-free state; λ and μ are the Lame coefficients; j and m are the third-order elastic constants and ρ0 is the density of the solid. Equation (21) can be rewritten as
(22)CL=CL01−kLσ1/2,CT=CT01−kTσ1/2,where
(23)kL=λ+2μ4λ+10μ+4m/μ+λ+2jρ03λ+2μ1/2/CL0,kT=2j−2m+λ+2μ/μρ03λ+2μ1/2/CT0.The propagation speed of elastic waves will change along with the stress status of the medium on the basis of equation (22), while in practical measurement, because the change of sound velocity is too weak to detect, TOF is often frequently used instead of velocity. Figure 3 shows a typical bolt connection structure, in which L and L0 are the equivalent stress length and overall length of the bolt, respectively. Since the stress distribution in a loaded bolt is inhomogeneous, the bolt can be simplified to an axisymmetric cylinder with uniform stress distribution, in which L represents the length of the cylinder. Then, the TOF can be expressed as
(24)TL=21+σE−1L+L0−LCL,TT=21+σE−1L+L0−LCS,where TL and TT are the TOF of L-wave and S-wave, respectively. Squaring the quotient of TL by TT, we can obtain
(25)RL‐S2=CL02KLCT02KTCT02KT/CL02KL−CT02/CL02CT02KT/CL02KLCT02/CL02−CT02/CL02KTσ.Figure 3
Bolt connection structure.Let(26)a=CT02KTCL02KL,b=CT02CL02,c=KT.Then, equation (25) can be simplified as
(27)RL‐S2=1a+a−bab−bcσ.Since the TOF of LC can be expressed as(28)TLC=TT+TL2=TL/x+T2,the TOF ratioRLC‐L of LC wave to LLL wave (LC-L method) is derived as
(29)RLC‐L=12RL‐S+12.Finally, the axial stressσ can be calculated by the following nonlinear model:
(30)σ=b−aabcRL‐S2−bc+1c=b−aabc1/2RLC‐L−12−bc+1c.Based on the derivations above, as long as the coefficients related to material properties and the TOF ratioRLC‐L are known, the axial stress of the bolt can be determined.


### 3.2. Flowchart of Stress Evaluation

The process of the measurement and calibration methods of bolt axial stress using single-source mode-converted ultrasound is shown in Figure4.
(1)
Mill the head of the tested bolt flat and mount a longitudinal transducer on it properly to ensure stable coupling status(2)
Obtain the original signal by conducting a monostatic pulse-echo experiment, and then resample the signal for at least 100 times to get the statistical data of the signal(3)
Extract waveform of LLL (SLLLt) and LC (SLCt) of each signal by a rectangular window Rτ(4)
Utilize the Gabor transform to get the Gabor expansion ofSLLLt and SLCt(5)
Calculate the TOF of LC and LLL by the particle filter optimization method(6)
Obtain the value ofRLC‐L of each signal by dividing the TOF of LC to the TOF of LLL(7)
Average the statisticalRLC‐L and get the best estimate value to reduce the random errors(8)
Change the load of the tested bolt and repeat operations 2–6 for at least 10 times to complete the calibration process of the tested bolt(9)
Correlate the stress andRLC‐L to get the calibration data. Then, nonlinear fitting is conducted to estimate the value of a, b, and c in equation (30) to get the evaluation model of axial stress of the bolt(10)
Evaluate the axial stress by the nonlinear evaluation modelFigure 4
Flowchart of the stress evaluation.


## 3.1. Nonlinear Evaluation Model Based on Mode-Converted Ultrasound

The existing evaluation models [6–14] of bolt axial stress are simplified and linear. And it is inevitable for them to induce extra fitting error. In this part, a new nonlinear evaluation model of bolt axial stress is proposed, its nonlinear feature is much better to meet the characteristic of acoustic elastic effect in theory. According to the theory of elasticity and nonlinear acoustic [23],
(21)CL2=CL02−σλ+2μ4λ+10μ+4m/μ+λ+2jρ03λ+2μ,CT2=CT02−σ2j−2m+λ+2μ/μρ03λ+2μ,where CL and CT are the velocities of longitudinal wave and shear wave in a solid medium under stress, respectively; CL0 and CT0 are their velocities in the stress-free state; λ and μ are the Lame coefficients; j and m are the third-order elastic constants and ρ0 is the density of the solid. Equation (21) can be rewritten as
(22)CL=CL01−kLσ1/2,CT=CT01−kTσ1/2,where
(23)kL=λ+2μ4λ+10μ+4m/μ+λ+2jρ03λ+2μ1/2/CL0,kT=2j−2m+λ+2μ/μρ03λ+2μ1/2/CT0.The propagation speed of elastic waves will change along with the stress status of the medium on the basis of equation (22), while in practical measurement, because the change of sound velocity is too weak to detect, TOF is often frequently used instead of velocity. Figure 3 shows a typical bolt connection structure, in which L and L0 are the equivalent stress length and overall length of the bolt, respectively. Since the stress distribution in a loaded bolt is inhomogeneous, the bolt can be simplified to an axisymmetric cylinder with uniform stress distribution, in which L represents the length of the cylinder. Then, the TOF can be expressed as
(24)TL=21+σE−1L+L0−LCL,TT=21+σE−1L+L0−LCS,where TL and TT are the TOF of L-wave and S-wave, respectively. Squaring the quotient of TL by TT, we can obtain
(25)RL‐S2=CL02KLCT02KTCT02KT/CL02KL−CT02/CL02CT02KT/CL02KLCT02/CL02−CT02/CL02KTσ.Figure 3
Bolt connection structure.Let(26)a=CT02KTCL02KL,b=CT02CL02,c=KT.Then, equation (25) can be simplified as
(27)RL‐S2=1a+a−bab−bcσ.Since the TOF of LC can be expressed as(28)TLC=TT+TL2=TL/x+T2,the TOF ratioRLC‐L of LC wave to LLL wave (LC-L method) is derived as
(29)RLC‐L=12RL‐S+12.Finally, the axial stressσ can be calculated by the following nonlinear model:
(30)σ=b−aabcRL‐S2−bc+1c=b−aabc1/2RLC‐L−12−bc+1c.Based on the derivations above, as long as the coefficients related to material properties and the TOF ratioRLC‐L are known, the axial stress of the bolt can be determined.


## 3.2. Flowchart of Stress Evaluation

The process of the measurement and calibration methods of bolt axial stress using single-source mode-converted ultrasound is shown in Figure4.
(1)
Mill the head of the tested bolt flat and mount a longitudinal transducer on it properly to ensure stable coupling status(2)
Obtain the original signal by conducting a monostatic pulse-echo experiment, and then resample the signal for at least 100 times to get the statistical data of the signal(3)
Extract waveform of LLL (SLLLt) and LC (SLCt) of each signal by a rectangular window Rτ(4)
Utilize the Gabor transform to get the Gabor expansion ofSLLLt and SLCt(5)
Calculate the TOF of LC and LLL by the particle filter optimization method(6)
Obtain the value ofRLC‐L of each signal by dividing the TOF of LC to the TOF of LLL(7)
Average the statisticalRLC‐L and get the best estimate value to reduce the random errors(8)
Change the load of the tested bolt and repeat operations 2–6 for at least 10 times to complete the calibration process of the tested bolt(9)
Correlate the stress andRLC‐L to get the calibration data. Then, nonlinear fitting is conducted to estimate the value of a, b, and c in equation (30) to get the evaluation model of axial stress of the bolt(10)
Evaluate the axial stress by the nonlinear evaluation modelFigure 4
Flowchart of the stress evaluation.


## 4. Experiment

### 4.1. Setup

The JSR PRC50 ultrasonic transceiving card with gain range from -14 dB to 60 dB is selected as the actuator for the ultrasonic transducers. And the AD-link PCIe-9852 oscilloscope with sampling rates upper to 200 MHz is selected as the data acquisition hardware. They are integrated into a portable industrial computer, as shown in Figure5. Then, the ultrasonic calibration/measurement software to implement excitation, acquisition, and processing of ultrasonic signals is programmed based on LABVIEW. The Dakota PT7 magnetic longitudinal ultrasonic transducer is chosen as the L-wave source and an Olympus V-156RM transverse ultrasonic transducer is chosen as the S-wave source. The central frequency of both probes is 5 MHz. Two types of bolts with different specifications and materials are selected as the test samples. The specification of the specimens is listed in Table 3. The CTM2200S tensile testing machine is utilized to simulate the loading condition with the maximum load 2000 kg and load accuracy ±0.1%.Figure 5
Experiment device.Table 3
Details of the samples.
TypeSpecificationGuaranteed loadSection areaMaterialHexagon head half thread boltM8×1042160 kg36.6mm2C45M12×2003200 kg84.3mm2C35


### 4.2. TOF Estimation

Figure6 shows the ultrasonic signal collected in sample M8×104 excited by the PT7 L-wave transducer. According to equations (3) and (4), the waveforms of LLL and LC can be recognized obviously, as shown in Figure 6. And the TOF of LLL can almost be obtained directly because its waveform is clearly a single sharp pulse. While the waveform of LC is consisted of several peaks with similar amplitudes and different phases, its TOF cannot be directly determined. Figure 7 shows the waveform of LC intercepted from the original signal and its Gabor expansion. From Figure 7(b), it can be observed that there is a sharp peak in the diagram of Gabor expansion, and the TOF of the intercepted waveform can be obtained by the proposed particle filter method.Figure 6
Ultrasound in aM8×104 bolt.Figure 7
LC and its Gabor expansion.
(a)
LC waveform(b)
Gabor expansion of LCIn order to eliminate the influence of temperature on the sound velocity, the whole experiment is carried out under constant temperature (20°C). The loading range is 0-190 MPa and the loading step is 10 MPa. The elongation of TOF caused by the acoustoelastic effect is around 1 nanosecond per MPa, but the minimum sampling period of PCIe-9852 is 5 ns. There is a certain probability that the change of the signal cannot be collected intact, so the one-dimensional fast Fourier interpolation method is introduced to enhance the sample rate. After being processed by ten-time upsampling, the time resolution of the system is upgraded to 0.5 ns. The estimated TOFs under different stress status using the proposed method are shown in Tables4 and 5, in which RL‐S is the time ratio of longitudinal wave (L) to shear wave (S) and RLC‐L is the time ratio of converted wave (LC) to longitudinal wave.Table 4
Estimated TOF of sampleM8×104.
Axial stress (MPa)TOF of L (μS)TOF of LC(μS)TOF of S(μS)RLC‐LRL‐S037.76352.27165.4171.384190.577271037.77052.27965.4271.384140.577282037.77852.28965.4351.384110.577343037.78552.29865.4491.384090.577324037.79352.30865.4601.384070.577345037.79952.31665.4711.384060.577346037.80852.32765.4801.384020.577407037.81552.33665.4911.384000.577418037.82252.34465.5021.383960.577429037.82952.35365.5131.383940.5774310037.83652.36265.5241.383920.5774411037.84252.37065.5341.383910.5774412037.84952.37965.5451.383890.5774513037.85652.38765.5561.383850.5774614037.86652.39965.5661.383800.5775215037.87452.40865.5771.383750.5775516037.88152.41665.5871.383700.5775717037.88752.42465.5981.383690.5775618037.89352.43165.6031.383660.5776119037.90152.43965.6211.383580.57757Table 5
Estimated TOF of sampleM12×200.
Axial stress (MPa)TOF of L (μS)TOF of LC(μS)TOF of S(μS)RLC‐LRL‐S076.194105.170133.3461.380290.571401076.207105.186133.3651.380270.571422076.222105.204133.3861.380230.571443076.236105.221133.4151.380200.571424076.250105.238133.4281.380170.571475076.265105.256133.4501.380140.571496076.279105.274133.4751.380120.571497076.292105.289133.4891.380080.571528076.305105.305133.5101.380050.571539076.315105.317133.5301.380030.5715210076.328105.333133.5451.380000.5715511076.343105.351133.5731.379970.5715512076.358105.368133.5911.379920.5715813076.373105.386133.6211.379890.5715614076.387105.403133.6391.379860.5715915076.402105.421133.6581.379820.5716216076.416105.438133.6851.379790.5716117076.430105.456133.7011.379770.5716518076.443105.471133.7211.379730.5716619076.452105.481133.7381.379700.57166


### 4.3. Calibration/Measurement

The axial stress calibration tests of bolts are carried out using the L-S method and the LC-L method, respectively. Figure8 shows the calibration curves obtained from nonlinear fitting by Levenberg-Marquardt method. It can be clearly observed that the results of the LC-L method are significantly better than the results of the L-S method from Figure 8. The correlation coefficients between the fitting model and original data of the LC-L method are 0.9957 for sample M8×104 and 0.9994 for sample M12×200. The correlation coefficients of the L-S method are 0.9487 for sample M8×104 and 0.94664 for sample M12×200.Figure 8
Calibration results.
(a)
SampleM8×104 result of the L-S method(b)
SampleM8×104 result of the LC-L method(c)
SampleM12×200 result of the L-S method(d)
SampleM12×200 result of the LC-L methodAnother two bolts exactly the same as the samples are chosen to test the validity of the proposed method. After loading by the tensile testing machine at the step of 20 MPa, the TOFs under different stress status using the proposed method are recognized. And then, the Levenberg-Marquardt models of calibration results are utilized to evaluate the stress of bolts. The evaluation results are shown in Tables6 and 7 and Figure 9. In Figure 9, the yellow bar represents the absolute error using the proposed method and the blue bar refers to the absolute error of the L-S method. According to the results, the average evaluation errors of the LC-L method and the L-S method for sample M8×104 are 1.789% and 4.671%, respectively. And the average errors of sample M12×200 are 2.582% and 4.634%. As a result, the proposed method has a higher accuracy as it utilizes a single longitudinal ultrasound. However, the accuracy of the L-S method is relatively low for the difference of the coupling conditions of L-wave and S-wave transducers.Table 6
Measurement result of sampleM8×104.
Preset stress (MPa)RLC‐LRL‐SResult of LC-L (MPa)Error of LC-LResult of L-S (MPa)Error of L-S201.384100.5773121.0125.06%23.53117.65%401.384070.5773338.975-2.56%41.8914.72%601.384030.5774161.2112.02%62.3013.83%801.383970.5774282.0072.50%82.9903.73%1001.383940.57744101.2501.25%95.699-4.30%1201.383880.57748118.271-1.43%122.2561.88%1401.383790.57751142.5710.11%134.368-3.88%1601.383720.57757163.2542.03%163.0211.88%1801.383630.57760180.2010.11%181.3360.74%2001.383570.57763198.367-0.82%208.2144.10%Table 7
Measurement result of sampleM12×200.
Preset stress (MPa)RLC‐LRL‐SResult of LC-L (MPa)Error of LC-LResult of L-S (MPa)Error of L-S201.380220.5714522.10310.51%17.655-11.72%401.380170.5714738.127-4.68%40.2390.59%601.380130.5714858.699-2.16%55.627-7.28%801.380040.5715481.4751.84%88.04410.05%1001.380010.5715796.583-3.42%105.2155.21%1201.379920.57159119.338-0.55%123.1442.62%1401.379850.57161140.152-0.10%144.8783.48%1601.379790.57162163.2542.03%157.716-1.42%1801.379740.57165179.563-0.24%182.7711.53%2001.379690.57164199.361-0.32%204.8852.44%Figure 9
Measurement error.
(a)
Error of the sampleM8×104(b)
Error of the sampleM12×200


## 4.1. Setup

The JSR PRC50 ultrasonic transceiving card with gain range from -14 dB to 60 dB is selected as the actuator for the ultrasonic transducers. And the AD-link PCIe-9852 oscilloscope with sampling rates upper to 200 MHz is selected as the data acquisition hardware. They are integrated into a portable industrial computer, as shown in Figure5. Then, the ultrasonic calibration/measurement software to implement excitation, acquisition, and processing of ultrasonic signals is programmed based on LABVIEW. The Dakota PT7 magnetic longitudinal ultrasonic transducer is chosen as the L-wave source and an Olympus V-156RM transverse ultrasonic transducer is chosen as the S-wave source. The central frequency of both probes is 5 MHz. Two types of bolts with different specifications and materials are selected as the test samples. The specification of the specimens is listed in Table 3. The CTM2200S tensile testing machine is utilized to simulate the loading condition with the maximum load 2000 kg and load accuracy ±0.1%.Figure 5
Experiment device.Table 3
Details of the samples.
TypeSpecificationGuaranteed loadSection areaMaterialHexagon head half thread boltM8×1042160 kg36.6mm2C45M12×2003200 kg84.3mm2C35


## 4.2. TOF Estimation

Figure6 shows the ultrasonic signal collected in sample M8×104 excited by the PT7 L-wave transducer. According to equations (3) and (4), the waveforms of LLL and LC can be recognized obviously, as shown in Figure 6. And the TOF of LLL can almost be obtained directly because its waveform is clearly a single sharp pulse. While the waveform of LC is consisted of several peaks with similar amplitudes and different phases, its TOF cannot be directly determined. Figure 7 shows the waveform of LC intercepted from the original signal and its Gabor expansion. From Figure 7(b), it can be observed that there is a sharp peak in the diagram of Gabor expansion, and the TOF of the intercepted waveform can be obtained by the proposed particle filter method.Figure 6
Ultrasound in aM8×104 bolt.Figure 7
LC and its Gabor expansion.
(a)
LC waveform(b)
Gabor expansion of LCIn order to eliminate the influence of temperature on the sound velocity, the whole experiment is carried out under constant temperature (20°C). The loading range is 0-190 MPa and the loading step is 10 MPa. The elongation of TOF caused by the acoustoelastic effect is around 1 nanosecond per MPa, but the minimum sampling period of PCIe-9852 is 5 ns. There is a certain probability that the change of the signal cannot be collected intact, so the one-dimensional fast Fourier interpolation method is introduced to enhance the sample rate. After being processed by ten-time upsampling, the time resolution of the system is upgraded to 0.5 ns. The estimated TOFs under different stress status using the proposed method are shown in Tables4 and 5, in which RL‐S is the time ratio of longitudinal wave (L) to shear wave (S) and RLC‐L is the time ratio of converted wave (LC) to longitudinal wave.Table 4
Estimated TOF of sampleM8×104.
Axial stress (MPa)TOF of L (μS)TOF of LC(μS)TOF of S(μS)RLC‐LRL‐S037.76352.27165.4171.384190.577271037.77052.27965.4271.384140.577282037.77852.28965.4351.384110.577343037.78552.29865.4491.384090.577324037.79352.30865.4601.384070.577345037.79952.31665.4711.384060.577346037.80852.32765.4801.384020.577407037.81552.33665.4911.384000.577418037.82252.34465.5021.383960.577429037.82952.35365.5131.383940.5774310037.83652.36265.5241.383920.5774411037.84252.37065.5341.383910.5774412037.84952.37965.5451.383890.5774513037.85652.38765.5561.383850.5774614037.86652.39965.5661.383800.5775215037.87452.40865.5771.383750.5775516037.88152.41665.5871.383700.5775717037.88752.42465.5981.383690.5775618037.89352.43165.6031.383660.5776119037.90152.43965.6211.383580.57757Table 5
Estimated TOF of sampleM12×200.
Axial stress (MPa)TOF of L (μS)TOF of LC(μS)TOF of S(μS)RLC‐LRL‐S076.194105.170133.3461.380290.571401076.207105.186133.3651.380270.571422076.222105.204133.3861.380230.571443076.236105.221133.4151.380200.571424076.250105.238133.4281.380170.571475076.265105.256133.4501.380140.571496076.279105.274133.4751.380120.571497076.292105.289133.4891.380080.571528076.305105.305133.5101.380050.571539076.315105.317133.5301.380030.5715210076.328105.333133.5451.380000.5715511076.343105.351133.5731.379970.5715512076.358105.368133.5911.379920.5715813076.373105.386133.6211.379890.5715614076.387105.403133.6391.379860.5715915076.402105.421133.6581.379820.5716216076.416105.438133.6851.379790.5716117076.430105.456133.7011.379770.5716518076.443105.471133.7211.379730.5716619076.452105.481133.7381.379700.57166


## 4.3. Calibration/Measurement

The axial stress calibration tests of bolts are carried out using the L-S method and the LC-L method, respectively. Figure8 shows the calibration curves obtained from nonlinear fitting by Levenberg-Marquardt method. It can be clearly observed that the results of the LC-L method are significantly better than the results of the L-S method from Figure 8. The correlation coefficients between the fitting model and original data of the LC-L method are 0.9957 for sample M8×104 and 0.9994 for sample M12×200. The correlation coefficients of the L-S method are 0.9487 for sample M8×104 and 0.94664 for sample M12×200.Figure 8
Calibration results.
(a)
SampleM8×104 result of the L-S method(b)
SampleM8×104 result of the LC-L method(c)
SampleM12×200 result of the L-S method(d)
SampleM12×200 result of the LC-L methodAnother two bolts exactly the same as the samples are chosen to test the validity of the proposed method. After loading by the tensile testing machine at the step of 20 MPa, the TOFs under different stress status using the proposed method are recognized. And then, the Levenberg-Marquardt models of calibration results are utilized to evaluate the stress of bolts. The evaluation results are shown in Tables6 and 7 and Figure 9. In Figure 9, the yellow bar represents the absolute error using the proposed method and the blue bar refers to the absolute error of the L-S method. According to the results, the average evaluation errors of the LC-L method and the L-S method for sample M8×104 are 1.789% and 4.671%, respectively. And the average errors of sample M12×200 are 2.582% and 4.634%. As a result, the proposed method has a higher accuracy as it utilizes a single longitudinal ultrasound. However, the accuracy of the L-S method is relatively low for the difference of the coupling conditions of L-wave and S-wave transducers.Table 6
Measurement result of sampleM8×104.
Preset stress (MPa)RLC‐LRL‐SResult of LC-L (MPa)Error of LC-LResult of L-S (MPa)Error of L-S201.384100.5773121.0125.06%23.53117.65%401.384070.5773338.975-2.56%41.8914.72%601.384030.5774161.2112.02%62.3013.83%801.383970.5774282.0072.50%82.9903.73%1001.383940.57744101.2501.25%95.699-4.30%1201.383880.57748118.271-1.43%122.2561.88%1401.383790.57751142.5710.11%134.368-3.88%1601.383720.57757163.2542.03%163.0211.88%1801.383630.57760180.2010.11%181.3360.74%2001.383570.57763198.367-0.82%208.2144.10%Table 7
Measurement result of sampleM12×200.
Preset stress (MPa)RLC‐LRL‐SResult of LC-L (MPa)Error of LC-LResult of L-S (MPa)Error of L-S201.380220.5714522.10310.51%17.655-11.72%401.380170.5714738.127-4.68%40.2390.59%601.380130.5714858.699-2.16%55.627-7.28%801.380040.5715481.4751.84%88.04410.05%1001.380010.5715796.583-3.42%105.2155.21%1201.379920.57159119.338-0.55%123.1442.62%1401.379850.57161140.152-0.10%144.8783.48%1601.379790.57162163.2542.03%157.716-1.42%1801.379740.57165179.563-0.24%182.7711.53%2001.379690.57164199.361-0.32%204.8852.44%Figure 9
Measurement error.
(a)
Error of the sampleM8×104(b)
Error of the sampleM12×200


## 5. Conclusions

In order to determine the parameters of mode-converted wave in single-source ultrasound for bolt stress evaluation, a time-frequency parameter recognition method based on the Gabor transform and a new bolt axial stress evaluation model are proposed in this paper. After analyzing the propagation and the mode conversion process of ultrasonic longitudinal wave in mounted solid, a parameter recognition method based on the Gabor transform to achieve the accurate TOF of the mode-converted wave is introduced to overcome the distortion and aliasing in single-source ultrasonic signal and its effectiveness is proved by simulation. Based on the mode conversion of single-source ultrasound and the acoustoelastic effect, a nonlinear evaluation model of the axial stress is put forward. The proposed method is compared with the commonly used L-S method by a series of calibrations and experiments. And the experiment results show that the proposed method can effectively detect the bolt axial stress ranging from 20 MPa to 200 MPa with a higher accuracy. As a result, the proposed method is more efficient and applicable in detecting the connection status of bolted joints than the traditional L-S method.



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*Source: 8883845-2020-08-20.xml*