CN116956644B - Rail longitudinal stress detection method based on ultrasonic guided wave characteristics - Google Patents
Rail longitudinal stress detection method based on ultrasonic guided wave characteristics Download PDFInfo
- Publication number
- CN116956644B CN116956644B CN202311212618.XA CN202311212618A CN116956644B CN 116956644 B CN116956644 B CN 116956644B CN 202311212618 A CN202311212618 A CN 202311212618A CN 116956644 B CN116956644 B CN 116956644B
- Authority
- CN
- China
- Prior art keywords
- steel rail
- rail
- guided wave
- variation
- receiving
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
- 238000001514 detection method Methods 0.000 title claims abstract description 56
- 229910000831 Steel Inorganic materials 0.000 claims abstract description 196
- 239000010959 steel Substances 0.000 claims abstract description 196
- 238000000034 method Methods 0.000 claims abstract description 116
- 239000006185 dispersion Substances 0.000 claims abstract description 75
- 230000005284 excitation Effects 0.000 claims abstract description 36
- 230000008569 process Effects 0.000 claims description 31
- 239000011159 matrix material Substances 0.000 claims description 27
- 230000009471 action Effects 0.000 claims description 22
- 239000013598 vector Substances 0.000 claims description 20
- 238000004364 calculation method Methods 0.000 claims description 18
- 230000008859 change Effects 0.000 claims description 17
- 238000006073 displacement reaction Methods 0.000 claims description 10
- 238000001228 spectrum Methods 0.000 claims description 8
- 230000005624 perturbation theories Effects 0.000 claims description 7
- 230000003595 spectral effect Effects 0.000 claims description 7
- 230000001902 propagating effect Effects 0.000 claims description 4
- 230000010354 integration Effects 0.000 claims description 3
- 230000017105 transposition Effects 0.000 claims description 3
- 238000004458 analytical method Methods 0.000 abstract description 4
- 238000012544 monitoring process Methods 0.000 abstract description 4
- 230000009286 beneficial effect Effects 0.000 abstract description 2
- 238000010586 diagram Methods 0.000 description 4
- 238000005516 engineering process Methods 0.000 description 4
- 238000009659 non-destructive testing Methods 0.000 description 4
- 238000012360 testing method Methods 0.000 description 3
- 238000012423 maintenance Methods 0.000 description 2
- 230000004048 modification Effects 0.000 description 2
- 238000012986 modification Methods 0.000 description 2
- 230000000644 propagated effect Effects 0.000 description 2
- 230000005540 biological transmission Effects 0.000 description 1
- 238000011088 calibration curve Methods 0.000 description 1
- 230000003247 decreasing effect Effects 0.000 description 1
- 230000007547 defect Effects 0.000 description 1
- 238000011549 displacement method Methods 0.000 description 1
- 238000005315 distribution function Methods 0.000 description 1
- 230000000694 effects Effects 0.000 description 1
- 230000006872 improvement Effects 0.000 description 1
- 238000005259 measurement Methods 0.000 description 1
- 238000012545 processing Methods 0.000 description 1
- 230000035945 sensitivity Effects 0.000 description 1
Classifications
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
- Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation
Abstract
The application discloses a method for detecting longitudinal stress of a steel rail based on ultrasonic guided wave characteristics, which comprises the following steps: establishing a rail fluctuation characteristic equation to obtain a rail guided wave dispersion characteristic equation; establishing a steel rail two-dimensional Gao Jiepu element method model to obtain a steel rail guided wave dispersion curve; according to the analysis of the rail guided wave frequency dispersion curve, the rail detection frequency and the specific mode are obtained, and the guided wave excitation signal and the receiving signal in the free state of the rail and the locked state of the rail are obtained; acquiring a difference value of the guided wave dispersion characteristic between the free state of the steel rail and the locked state of the steel rail through the dispersion characteristic parameter of the guided wave, wherein the difference value is the variation of the guided wave dispersion characteristic; and constructing a perturbation equation of the longitudinal stress variation of the steel rail and the guided wave dispersion characteristic variation to calculate the longitudinal stress of the steel rail to be measured. The application has the beneficial effects that: the number of times of steel rail detection is reduced, the stress condition of the seamless line steel rail can be simply and accurately measured, and the method is suitable for long-distance detection and real-time monitoring of the longitudinal stress of the seamless line steel rail.
Description
Technical Field
The application relates to the technical field of nondestructive testing and monitoring, in particular to a steel rail longitudinal stress detection method based on ultrasonic guided wave characteristics.
Background
The seamless rail is an important component of the high-speed rail, namely the rail without gaps between standard length rails, so that the high-speed rail can run more stably, the safety is higher, and the operation and maintenance cost can be reduced. However, the steel rail is seamless and has no joint, so that when the temperature is increased, the steel rail cannot be stretched, internal stress is generated, and the steel rail is easy to bend, twist, crack and the like after exceeding the strength of the steel rail. When the temperature is reduced, the steel rail can loosen, and stability and safety of driving are affected. Under the action of huge temperature stress, the seamless steel rail can even induce accidents such as rail expansion, rail breakage and the like, and potential safety hazards are brought to driving. Therefore, a nondestructive testing method capable of accurately determining the longitudinal stress of the steel rail in real time is urgently needed, and safe operation of a high-speed railway is guaranteed.
The traditional steel rail longitudinal stress nondestructive testing technology such as a steel rail displacement method and the like is greatly influenced by human factors, has poor real-time performance, is difficult to meet the modern operation, maintenance and management requirements of a high-speed railway line, and is difficult to widely popularize due to the problems of poor anti-interference capability, small detection range and the like of the novel steel rail longitudinal stress nondestructive testing technology such as an X-ray method, a magneto-elastic method, a magneto-noise method and the like. The ultrasonic guided wave detection technology has the advantages of sensitivity to stress, large detection range, long transmission distance, high detection speed and the like, and has great advantages for detecting the longitudinal stress of the seamless rail.
The existing ultrasonic guided wave steel rail longitudinal stress detection technology needs to measure and determine a stress calibration curve of the ultrasonic guided wave propagation speed according to the actual detection process for multiple times, the process is complicated, accurate stress calibration cannot be given to any guided wave mode, certain errors exist, and popularization and application are not facilitated.
According to the method, the steel rail guided wave characteristic analysis is carried out by a semi-analytic spectral element method, the correlation between the longitudinal stress of the steel rail and the guided wave characteristic change is proposed by utilizing a nonlinear perturbation theory, and a method for calculating a related guided wave characteristic perturbation theoretical formula of the longitudinal stress of the steel rail is obtained; the application can accurately measure the stress condition of the seamless rail line, is suitable for long-distance detection and real-time monitoring of the longitudinal stress of the seamless rail, provides a new thought for related departments to solve the potential safety hazard of the seamless rail due to stress change, and can early warn in time.
Disclosure of Invention
In order to overcome the defects in the prior art, the application aims to provide the rail longitudinal stress detection method based on the ultrasonic guided wave characteristics, so that the real-time rail longitudinal stress of the seamless line can be effectively detected, the stress condition of the seamless rail line can be accurately measured, and the method is suitable for long-distance detection of the longitudinal stress of the seamless line.
In order to achieve the above purpose, the present application adopts the following technical scheme: a steel rail longitudinal stress detection method based on ultrasonic guided wave characteristics comprises the following steps:
step S1, an ultrasonic guided wave semi-analytic finite element method is used for establishing a rail fluctuation characteristic equation, and a rail guided wave dispersion characteristic equation is obtained;
step S2, a two-dimensional Gao Jiepu element method model of the steel rail is established, steel rail parameters and a steel rail guided wave dispersion characteristic equation in the step S1 are input, and a wave number dispersion curve, a phase velocity dispersion curve and a group velocity dispersion curve of the steel rail are obtained;
s3, analyzing and obtaining steel rail detection frequency and steel rail specific modes according to the wave number dispersion curve, the phase velocity dispersion curve and the group velocity dispersion curve of the steel rail in the step S2;
s4, inputting the steel rail detection frequency in the step S3 into an ultrasonic guided wave detection system for detection, and respectively acquiring guided wave excitation signals and receiving signals in a steel rail free state and a steel rail locking state under a specific steel rail mode;
s5, identifying the time interval of ultrasonic guided wave receiving signals of two receiving transducers of the ultrasonic guided wave detection system and the interval between the two receiving transducers, and obtaining the dispersion characteristic parameters of guided waves in the steel rail;
step S6, obtaining the difference value of the guided wave dispersion characteristics of the steel rail in the free state and the locked state of the steel rail as the variation of the guided wave dispersion characteristics of the steel rail through the relation between the guided wave excitation signal and the receiving signal in the step S4;
s7, constructing a perturbation equation of the longitudinal stress variation of the steel rail and the variation of the guided wave dispersion characteristic of the steel rail by utilizing a guided wave nonlinear perturbation theory;
and S8, calculating the longitudinal stress of the steel rail to be measured by utilizing a perturbation equation of the longitudinal stress variation of the steel rail and the guided wave dispersion characteristic variation of the steel rail constructed in the step S7.
Further, in step S1, a rail fluctuation characteristic equation is established, and is shown in a formula (1);
（1）；
in the method, in the process of the application,；
in the method, in the process of the application,；
in the method, in the process of the application,and->Three different total stiffness matrices of the rail, respectively +.>For the total mass matrix of the rail->For wave number, < >>Is angular frequency; />To eliminate imaginary number->Is>For the left-hand multiplication auxiliary matrix->All displacement vectors of all nodes in the steel rail of the guided wave after the guiding>For guiding the total displacement vector of each node in the rail, < >>As an auxiliary matrix +.>For the transposition of the auxiliary matrix of the rail +.>Is an imaginary number;
introducing auxiliary matrixThen, the formula (1) is re-expressed as a first-order eigenvalue equation, which is shown in the formula (2);
（2）；
wherein:two symmetric matrixes which are first-order eigenvalue equations; wherein:
。
further, in step S2, a two-dimensional Gao Jiepu element method model of the steel rail is established, which specifically comprises:
according to the existing semi-analytic finite element method, a spectrum unit is introduced, a total stiffness matrix and a total mass matrix are calculated, and the total stiffness matrix and the total mass matrix are expressed by a shape function through a formula (3):
(3)；
wherein:is a unit-shaped function, wherein->Parameter variables for two directions of basis function in physical space,/->Two basis functions in two directions of physical space;
the W-order basis function based on Gaussian-lobar-Legendre integral configuration points in a one-dimensional reference coordinate system is obtained by applying Legendre polynomial interpolation, and is as follows:
(4)；
wherein:for the point of integration based on Gauss-lobar-Legend +.>W is the node order of the spectral element method, ++>For node->Values of the Legend's orthonormal polynomial of order N,/>Is an N-order Legend orthogonal polynomialFirst derivative of>Is zero point of formula (5), namely node in spectral element method, ++>；
(5)；
According to the wave characteristic equation of the steel rail after the introduction of the spectrum unit, the wave number dispersion curve, the phase velocity dispersion curve and the group velocity dispersion curve of the steel rail are calculated, and the wave number dispersion curve, the phase velocity dispersion curve and the group velocity dispersion curve are specifically:
the phase velocity and group velocity of guided wave propagating in the steel rail are shown in formula (6) and formula (7);
（6）；
（7）；
in the method, in the process of the application,for the phase velocity of the rail%>For group velocity of rail->The left characteristic vector of the steel rail and the right characteristic vector of the steel rail are respectively;
solving the first-order eigenvalue equation of formula (2) to obtain all eigenvalue solutions, namely wave numbersSolving the wave number obtained by solving, a calculation formula (6) of the phase velocity of the steel rail and a calculation formula (7) of the group velocity of the steel rail to obtain the wave numbers under different frequenciesWave number dispersion curve, phase velocity dispersion curve and group velocity dispersion curve of steel rail.
Further, in step S3, the wave number dispersion curve, the phase velocity dispersion curve and the group velocity dispersion curve of the steel rail are analyzed to obtain an optimal detection frequency of 20-40 kHz of the steel rail, and a specific mode corresponding to the steel rail under different frequencies is obtained.
In the step S4, the rail detection frequency in the step S3 is input into an ultrasonic guided wave detection system for detection, and guided wave excitation signals and receiving signals in a free state and a locked state of the rail are respectively obtained in a specific mode of the rail; the method comprises the following steps:
the ultrasonic guided wave detection system comprises an excitation transducer and two receiving transducers, wherein the two receiving transducers are a 1 st receiving transducer and a 2 nd receiving transducer respectively, the excitation transducer and the two receiving transducers which are arranged through the ultrasonic guided wave detection system are respectively applied to different positions in a steel rail free state and a steel rail locking state, an excitation signal of guided wave propagation in the steel rail free state, a receiving signal of the 1 st receiving transducer in the steel rail free state and a receiving signal of the 2 nd receiving transducer in the steel rail free state are obtained, and an excitation signal of guided wave propagation in the steel rail locking state, a receiving signal of the 1 st receiving transducer in the steel rail locking state and a receiving signal of the 2 nd receiving transducer in the steel rail locking state are obtained.
Further, in step S5, identifying a time interval between ultrasonic guided wave receiving signals of two receiving transducers of the ultrasonic guided wave detecting system and a space between the two receiving transducers, and obtaining a dispersion characteristic parameter of guided waves in the steel rail; the method comprises the following steps:
three group velocities of the rail in the rail locked state:
（8）；
in the method, in the process of the application,receiving the change of the guide wave of the steel rail at the 1 stGroup velocity of propagation between the energy source and the 2 nd receiving transducer,>group velocity propagation between excitation transducer and 2 nd receiving transducer for rail guided wave, +.>For the propagation group velocity of the guided wave of the steel rail between the exciting transducer and the 1 st receiving transducer, d1 is the distance from the 1 st receiving transducer to the 2 nd receiving transducer, d2 is the distance from the exciting transducer to the 2 nd receiving transducer, d is the distance from the exciting transducer to the 1 st receiving transducer>For the propagation time of the 1 st receiving transducer to the 2 nd receiving transducer,/for the first receiving transducer>For the propagation time of the excitation transducer to the 2 nd receiving transducer, < >>To excite the transducer to the 1 st receiving transducer, there are:
（9）；
in the method, in the process of the application,for the rail guided wave acquired, 1 st moment of receiving the signal by the transducer, < >>Time of receiving signal of 2 nd receiving transducer for acquired rail guided wave, +.>The moment of the guided wave excitation signal for the acquired steel rail;
by solving for the mean value of the three group velocities of the rail, i.e. in the locked stateGroup velocity of steel railThe method comprises the following steps:
（10）。
further, in step S6, the difference value of the guided wave dispersion characteristics in the free state of the rail and the locked state of the rail is obtained through the relation between the guided wave excitation signal and the received signal in step S4, and is the variation of the guided wave dispersion characteristics of the rail; the method comprises the following steps:
the group velocity of the steel rail in the free state obtained by adopting the ultrasonic guided wave detection system is recorded asThe group velocity of the rail in the locked state of the rail is recorded as +.>Longitudinal stress in rail locked state>Longitudinal stress in the free state of the rail>The difference of the two is that the longitudinal stress of the steel rail to be measured is marked as +.>Analyzing signals obtained by an ultrasonic guided wave detection system to obtain the steel rail group velocity variation +.>。
Further, in step S7, a perturbation equation of the longitudinal stress variation of the steel rail and the dispersion characteristic variation of the steel guided wave rail is constructed, specifically:
back calculation solution is carried out by utilizing wave number variation caused by steel rail under the action of longitudinal stress variationStress is recorded asSee formula (11);
（11）；
in the method, in the process of the application,for the density of the rail->Defined as a coefficient related to modal guided wave characteristics:，/>wavenumber of mth order mode of rail, < +.>For the longitudinal stress variation of the rail>Wave number variation of mth order mode is caused by action, < >>For the total mass matrix of the rail->Transpose of left eigenvector of mth order mode of rail,/for example>Right eigenvector of the m-th order mode of the steel rail;
the stress obtained by back calculation and solving of the phase velocity variation caused by the action of the longitudinal stress variation of the steel rail is recorded asSee formula (12);
（12）；
in the method, in the process of the application,for the longitudinal stress variation of the rail>The phase velocity variation is caused under the action;
the stress obtained by back calculation and solving of group velocity variation caused by the action of longitudinal stress variation of the steel rail is recorded asSee formula (13);
（13）；
in the method, in the process of the application,for the longitudinal stress variation of the rail>Under the action of which the group velocity change is caused, < >>For the longitudinal stress variation of the rail>Wave number variation caused by action, +.>Transpose of left eigenvector of rail, +.>Is the variation of the right characteristic vector of the steel rail.
The application has the beneficial effects that: (1) The method is simple and convenient to operate, the optimal excitation point, detection frequency and specific mode are obtained through theory, and only one excitation point and two receiving points are required to be arranged in actual test measurement; and theoretically obtaining the change amount of the guided wave frequency dispersion characteristic of the steel rail to be detected, and obtaining the longitudinal stress of the steel rail. Moreover, the longitudinal stress of the steel rail after the temperature change can be accurately obtained, the longitudinal stress of the section of steel rail at any moment can be more easily measured, the measuring range is large, the measuring result is accurate, and the method is suitable for long-distance continuous detection of the longitudinal stress of the steel rail of the ballastless track and the ballastless track; (2) The application reduces the detection times, can simply and accurately measure the stress condition of the seamless rail line, is suitable for long-distance detection and real-time monitoring of the longitudinal stress of the seamless line, provides a new thought for solving the potential safety hazard of the seamless line caused by stress change, and gives early warning in time.
Drawings
FIG. 1 is a schematic flow chart of the method of the present application.
Fig. 2 is a two-dimensional Gao Jiepu-element method model diagram of a steel rail in an embodiment of the application.
FIG. 3 is a graph showing the wave number dispersion of the steel rail in the embodiment of the application.
Fig. 4 is a schematic view of a dispersion curve of the phase velocity of the rail in the embodiment of the present application.
Fig. 5 is a schematic diagram of a group velocity dispersion curve of a rail in an embodiment of the present application.
FIG. 6 is a graph of calculated stress versus assumed stress for a modal 24 group velocity variation at 35KHz in an embodiment of the present application.
FIG. 7 is a schematic diagram of an ultrasonic guided wave detection system in an embodiment of the application.
Fig. 8 is a schematic diagram of an excitation signal obtained by detecting a steel rail test piece, a receiving signal of a 1 st receiving transducer, and a receiving signal of a 2 nd receiving transducer in the embodiment of the present application.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present application more apparent, the technical solutions of the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present application, and it is apparent that the described embodiments are some embodiments of the present application, but not all embodiments of the present application. All other embodiments, which can be made by those skilled in the art based on the embodiments of the application without making any inventive effort, are intended to be within the scope of the application.
As shown in fig. 1, which is a schematic flow chart of the method of the present application, in the method for detecting longitudinal stress of a rail based on ultrasonic guided wave dispersion characteristics in this embodiment, the existing semi-analytical finite element method is improved by aiming at a geometric structure with a complex section such as a rail, and a spectrum unit is introduced to form a two-dimensional semi-analytical spectrum element method suitable for a long tie bar waveguide medium; the rail guide wave dispersion characteristic is combined with the nonlinear perturbation theory, and the longitudinal stress of the rail is estimated and calculated by utilizing the wave number variation, the phase velocity variation and the group velocity variation.
Further, in step S1, the semi-analytic finite element method is a semi-analytic method for solving the dispersion problem in the complex cross-section waveguide medium. When the half-resolution finite element method is adopted to solve the guided wave dispersion characteristic solution, only finite element dispersion is needed to be carried out on the cross section of the waveguide medium, and simple harmonic vibration modes are carried out on the propagation direction of the waveguide medium to carry out resolution processing. Taking an elastically isotropic CHN60 rail as an example, the cross section is defined as the y-z plane and the direction of propagation of the guided wave along the rail longitudinal direction is defined as the x direction. Available spatial distribution functionTo represent the displacement value of any point in the waveguide medium, expressed as:
(14)；
wherein:for the displacement of the joints of the rail sections at any point in the Cartesian coordinate system, N is the form function of the unit, < ->The unit node on the section of the steel rail is +.>Coordinate value of axial direction>The unit nodes on the section of the steel rail are respectively +.>Coordinate value of direction, ++>For time (I)>For the unit node displacement vector, +.>For wave number, < >>For angular frequency +.>Representing an imaginary number.
Further, in step S1, a rail wave characteristic equation is established, as follows;
（1）；
in the method, in the process of the application,；
in the method, in the process of the application,；
in the method, in the process of the application,and->Three different total stiffness matrices of the rail, respectively +.>For the total mass matrix of the rail->For wave number, < >>Is angular frequency; />To eliminate imaginary number->Is>For the left-hand multiplication auxiliary matrix->All displacement vectors of all nodes in the steel rail of the guided wave after the guiding>For guiding the total displacement vector of each node in the rail, < >>In order to assist in the matrix,for the transposition of the auxiliary matrix of the rail +.>Is imaginary.
Further, in step S2, a two-dimensional Gao Jiepu element method model of the steel rail is established, a spectrum unit is introduced according to the existing semi-analytic finite element method, and a total stiffness matrix and a total mass matrix are calculated and expressed by adopting a shape function by the following formula:
(3)；
wherein: n is a form function of the unit, whereinIs a parameter variable of two directions of a basis function in physical space,two basis functions in two directions of physical space;
the W-order basis function based on Gaussian-lobar-Legendre integral configuration points in a one-dimensional reference coordinate system is obtained by applying Legendre polynomial interpolation, and is as follows:
(4)；
wherein:for the point of integration based on Gauss-lobar-Legend +.>W is the node order of the spectral element method, ++>For node->Values of the Legend's orthonormal polynomial of order N,/>Is an N-order Legend orthogonal polynomialFirst derivative of>Is zero point of formula (5), namely node in spectral element method, ++>；
(5)。
Further, in step S2, the wave number and the eigenvector at a specific frequency can be solved by the formula (1), and the relationship between the phase velocity and the group velocity propagating in the rail along with the frequency and the wave number can be obtained by a general eigenvalue problem solving method, where the relationship is:
（6）；
（7）；
in the method, in the process of the application,for the phase velocity of the rail%>For group velocity of rail->The left characteristic vector of the steel rail and the right characteristic vector of the steel rail are respectively.
Further, in step S7, when the rail is subjected to longitudinal stressIn operation, the quadratic term of the strain displacement relationship must be considered. The polynomial eigenvalue wave equation containing the longitudinal stress of the waveguide can be obtained by simplifying the Hamiltonian principle, and the first-order eigenvalue equation is rewritten as follows:
(15)；
in the method, in the process of the application,two symmetric matrices for first order eigenvalue equations, < ->Feature vector moment, which is a first order feature equation, wherein:
(16)；
(17)；
in the method, in the process of the application,longitudinal stress to the rail +.>For an additional stiffness matrix induced by longitudinal stress, +.>For the total stiffness matrix->A stiffness matrix after the additional stress due to the longitudinal stress;
obtaining the phase velocity of guided wave propagating in steel railAs in equation (6), group velocity +.>The method comprises the following steps:
(18)。
further, step S7, when the structural parameters (rigidity and mass) of the rail system are at a given angular frequencyAfter the perturbation is generated, the quality of the steel rail is kept unchanged, and structural parameters such as rigidity and the like of the steel rail system are changed, as follows:
(19)；
(20)；
(21)；
wherein:for the longitudinal stress variation of the rail>The amount of stiffness change induced->The change amounts of the three total rigidities after rail perturbation are respectively +.>Three total stiffness matrices after perturbation of the rail system, < > are given>For the longitudinal stress variation of the rail>Cause->Stiffness variation of>For longitudinal stress variation->Under the action cause->Wave number variation of order mode, +.>For longitudinal stress variation->Under the action cause->Eigenvalue vector matrix variation of order mode, +.>For longitudinal stress variation->Under the action cause->Characteristic value vector variation of order mode, +.>Is +.>Wavenumber of order mode, < >>Is +.>Eigenvector moment of first order eigenvector equation of order modality,/>Is +.>Feature vectors of the order modality.
Further, in step S7, based on the nonlinear perturbation theory, there are:
(22)；
(23)；
wherein:corresponding matrix after perturbation of rail system>Is a variable amount of (a).
Further, in step S7, the frequency is obtained from the relationship between the shot amount of the simultaneous structure parameter and the shot amount of the guided wave characteristicThe amount of the shot of the following wave numbers is:
(24)。
further, in step S7, at a given angular frequencyThe phase velocity and group velocity of the rail system after perturbation are as follows:
(25)；
in the method, in the process of the application,for the phase velocity of the rail system after perturbation, < >>For the post-perturbation group velocity of the rail system, < >>Is a characteristic vector of the rail system after perturbation, < +.>Transpose of eigenvectors after perturbation of rail system, < >>Is wave number after perturbation of the rail system.
Further, in step S7, considering the stiffness change caused by the longitudinal stress, the wavenumber and the group velocity after perturbation are respectively:
(26)；
in the method, in the process of the application,post-perturbation of the rail system>Wavenumbers of the order modes.
Further, in step S7, when the longitudinal stress changes, the initial stress is relatively highThere is a stress variation->The wave number variation can be obtained>Phase velocity variation->And group velocity variation->The following formulas are respectively given:
(27)；
(28)；
(29)；
in the method, in the process of the application,post-perturbation of the rail system>Right eigenvector of order modality,>transpose left eigenvector after perturbation of rail system,/->Is the right eigenvector of the rail system after perturbation.
Further, in step S7, higher order terms are eliminated, and the wave number variation can be obtainedPhase velocity variationAnd group velocity variation->Inverse calculation to obtain longitudinal stress variation>Additionally set an initial stress->=0, then there is:
stress obtained by back calculation and solving of wave number variation caused by the action of longitudinal stress variation of steel rail is recorded asSee formula (11);
（11）；
the stress obtained by back calculation and solving of the phase velocity variation caused by the action of the longitudinal stress variation of the steel rail is recorded asSee formula (12);
（12）；
the stress obtained by back calculation and solving of group velocity variation caused by the action of longitudinal stress variation of the steel rail is recorded asSee formula (13);
（13）；
in the embodiment, a CHN60 steel rail is taken as an example, and the steel rail is utilized in a certain stress variationThe lower result is a group velocity change +.>And performing back calculation to solve the stress. Elastic modulus e=210 GPa, poisson's ratio +.>=0.3, density->=7800kg/m 3 The cross section of the steel rail is discretized by adopting a fourth-order spectrum unit as shown in fig. 2, and parameters of the steel rail are brought into (6), (7) and (15) to obtain wave number, phase velocity and group velocity dispersion curves of the steel rail, as shown in fig. 3-5. From the dispersion curves, each dispersion curve represents the same guided wave propagation mode. For the same guided wave mode, the propagation speed tends to be stable along with the increase of the frequency, namely the dispersion phenomenon is weakened. And the number of the guided wave modes which can be transmitted in the steel rail under the same frequency gradually increases along with the increase of the excitation frequency. When the guided wave is used for detecting the longitudinal stress of the steel rail, the excitation frequency is optimally within the range of 20-40 kHz. Taking the example of a high frequency of 35kHz, fig. 3-5 show a total of 26 modes at this frequency. Table 1 shows the change in group velocity of 26 modes under a tensile stress of 200 MPa. According to table 1, at 35kHz frequency, the group velocity of each mode varies differently under the influence of longitudinal stress, possibly increasing or decreasing, with the group velocity variation of mode 24 being the greatest and most sensitive to rail longitudinal stress.
TABLE 1 group velocity variation for each mode with a frequency of 35kHz and a tensile stress of 200MPa
And selecting a detection frequency of 35KHz and a sensitive mode 24, analyzing the relation of the change quantity of the group velocity to the change quantity of the longitudinal stress of the steel rail, substituting the obtained change quantity of the group velocity into a wave guiding characteristic perturbation theoretical formula (13), and calculating the longitudinal stress in the steel rail. FIG. 6 is a graph of inversion calculation results of different group velocity variation of the mode 24 at a detection frequency of 35kHz for different rail longitudinal stresses versus assumed stresses. Fig. 6 shows the relationship between the calculation result of the mode 24 at 35kHz and the assumed stress of different steel rail longitudinal stresses through the wave guiding characteristic perturbation theoretical formula (13). The calculation result shows that the seamless steel rail longitudinal stress back calculated by the group velocity variation by utilizing the wave guiding characteristic perturbation theory has good effect, and the correctness and the accuracy of the wave guiding characteristic perturbation theory are fully explained.
In the embodiment, as shown in fig. 7, in order to obtain the longitudinal stress of the rail formed by the actually operated seamless rail under the action of temperature, it is necessary to detect the group velocities at different temperatures, i.e. different longitudinal stresses, and the group velocities which are not subjected to the longitudinal stress of the rail in the free state of the rail by using an ultrasonic guided wave detection system, and solve the difference value of the group velocities to obtain the variation of the group velocities. Fig. 8 shows an excitation signal obtained by detecting a steel rail test piece, a receiving signal of a 1 st receiving transducer and a receiving signal of a 2 nd receiving transducer, in an ultrasonic guided wave detection system shown in fig. 7, the acoustic guided wave detection system includes one excitation transducer and two receiving transducers, the two receiving transducers are the 1 st receiving transducer and the 2 nd receiving transducer, and the one excitation transducer and the two receiving transducers arranged by the ultrasonic guided wave detection system are respectively applied to different positions in a steel rail free state and a steel rail locking state to obtain an excitation signal propagated by a guided wave in the steel rail free state, a receiving signal of the 1 st receiving transducer in the steel rail free state and a receiving signal of the 2 nd receiving transducer in the steel rail free state, and obtain an excitation signal propagated by a guided wave in the steel rail locking state, a receiving signal of the 1 st receiving transducer in the steel rail locking state and a receiving signal of the 2 nd receiving transducer in the steel rail locking state.
Further, the method comprises the following steps. Identifying the time interval of ultrasonic guided wave received signals of two receiving transducers of the ultrasonic guided wave detection system and the interval between the two receiving transducers, and obtaining the group velocity of guided wave propagation in the steel rail; the method comprises the following steps:
three group velocities of the rail in the rail locked state:
（8）；
in the method, in the process of the application,group velocity between the 1 st receiving transducer and the 2 nd receiving transducer for guiding the rail>Group velocity propagation between excitation transducer and 2 nd receiving transducer for rail guided wave, +.>For the propagation group velocity of the guided wave of the steel rail between the exciting transducer and the 1 st receiving transducer, d1 is the distance from the 1 st receiving transducer to the 2 nd receiving transducer, d2 is the distance from the exciting transducer to the 2 nd receiving transducer, d is the distance from the exciting transducer to the 1 st receiving transducer>For the propagation time of the 1 st receiving transducer to the 2 nd receiving transducer,/for the first receiving transducer>For the propagation time of the excitation transducer to the 2 nd receiving transducer, < >>To excite the transducer to the 1 st receiving transducer, there are:
（9）；
in the method, in the process of the application,for the rail guided wave acquired, 1 st moment of receiving the signal by the transducer, < >>Time of receiving signal of 2 nd receiving transducer for acquired rail guided wave, +.>The moment of the guided wave excitation signal for the acquired steel rail;
by solving for the average of the three group velocities of the rail, i.e. the group velocity of the rail in the locked stateThe method comprises the following steps:
（10）；
the group velocity of the steel rail in the free state (in the state of being free from the longitudinal stress of the steel rail) is recorded asThere is longitudinal stress of the rail in the locked state +.>And the longitudinal stress in the free state (at this time +.>) The difference value of (2) is the longitudinal stress of the steel rail to be measured>The change of the group velocity of the steel rail in two states can be obtained through experimental analysisThe longitudinal stress of the rail is then back calculated by equation (13).
The foregoing is merely exemplary of the present application and is not intended to limit the present application. Various modifications and variations of the present application will be apparent to those skilled in the art. Any modification, equivalent replacement, improvement, etc. which come within the spirit and principles of the application are to be included in the scope of the claims of the present application.
Claims (5)
1. A steel rail longitudinal stress detection method based on ultrasonic guided wave characteristics is characterized by comprising the following steps of: the method comprises the following steps:
step S1, an ultrasonic guided wave semi-analytic finite element method is used for establishing a rail fluctuation characteristic equation, and a rail guided wave dispersion characteristic equation is obtained;
step S2, a two-dimensional Gao Jiepu element method model of the steel rail is established, steel rail parameters and a steel rail guided wave dispersion characteristic equation in the step S1 are input, and a wave number dispersion curve, a phase velocity dispersion curve and a group velocity dispersion curve of the steel rail are obtained;
s3, analyzing and obtaining steel rail detection frequency and steel rail specific modes according to the wave number dispersion curve, the phase velocity dispersion curve and the group velocity dispersion curve of the steel rail in the step S2;
s4, inputting the steel rail detection frequency in the step S3 into an ultrasonic guided wave detection system for detection, and respectively acquiring guided wave excitation signals and receiving signals in a steel rail free state and a steel rail locking state under a specific steel rail mode;
s5, identifying the time interval of ultrasonic guided wave receiving signals of two receiving transducers of the ultrasonic guided wave detection system and the interval between the two receiving transducers, and obtaining the dispersion characteristic parameters of guided waves in the steel rail;
step S6, obtaining the difference value of the guided wave dispersion characteristics of the steel rail in the free state and the locked state of the steel rail as the variation of the guided wave dispersion characteristics of the steel rail through the relation between the guided wave excitation signal and the receiving signal in the step S4;
s7, constructing a perturbation equation of the longitudinal stress variation of the steel rail and the variation of the guided wave dispersion characteristic of the steel rail by utilizing a guided wave nonlinear perturbation theory;
s8, calculating the longitudinal stress of the steel rail to be measured by utilizing a perturbation equation of the longitudinal stress variation of the steel rail and the guided wave dispersion characteristic variation of the steel rail constructed in the step S7;
in the step S1, a steel rail fluctuation characteristic equation is established, and is shown in a formula (1);
（1）；
in the method, in the process of the application,；
in the method, in the process of the application,；
in the method, in the process of the application,three different total stiffness matrices of the rail, respectively +.>For the total mass matrix of the rail->For wave number, < >>Is angular frequency; />To eliminate imaginary number->Is>For the left-hand multiplication auxiliary matrix->All displacement vectors of all nodes in the steel rail of the guided wave after the guiding>For guiding the total displacement vector of each node in the rail, < >>As an auxiliary matrix +.>For the transposition of the auxiliary matrix of the rail +.>Is an imaginary number;
introducing auxiliary matrixThen, the formula (1) is re-expressed as a first-order eigenvalue equation, which is shown in the formula (2);
（2）；
wherein:two symmetric matrixes which are first-order eigenvalue equations; wherein:
；
in the step S2, a two-dimensional Gao Jiepu element method model of the steel rail is established, and the method specifically comprises the following steps:
according to the semi-analytic finite element method, a spectrum unit is introduced, a total stiffness matrix and a total mass matrix are calculated, and the total stiffness matrix and the total mass matrix are expressed by a shape function through a formula (3):
(3)；
wherein:is a unit-shaped function, wherein->Is a parameter variable of two directions of a basis function in physical space,two basis functions in two directions of physical space;
interpolation using Legendre polynomials is obtained at oneGaussian-lobar-Legendre integral point-based configuration point in a dimensional reference frameWThe order basis functions are as follows:
(4)；
wherein:for the point of integration based on Gauss-lobar-Legend +.>A kind of electronic deviceWThe order basis function is used to determine,Wfor the node order of the spectral element method, +.>For node->A kind of electronic deviceNValues of the order Legendre orthopolynomial, < ->Is thatNLegend orthogonal polynomialFirst derivative of>Is zero point of formula (5), namely node in spectral element method, ++>；
(5)；
According to the wave characteristic equation of the steel rail after the introduction of the spectrum unit, the wave number dispersion curve, the phase velocity dispersion curve and the group velocity dispersion curve of the steel rail are calculated, and the wave number dispersion curve, the phase velocity dispersion curve and the group velocity dispersion curve are specifically:
the phase velocity and group velocity of guided wave propagating in the steel rail are shown in formula (6) and formula (7);
（6）；
（7）；
in the method, in the process of the application,for the phase velocity of the rail%>For group velocity of rail->The left characteristic vector of the steel rail and the right characteristic vector of the steel rail are respectively;
solving the first-order eigenvalue equation of formula (2) to obtain all eigenvalue solutions, namely wave numbersSolving the wave number, the phase velocity and the group velocity of the steel rail according to a calculation formula (6) for solving the wave number and the phase velocity of the steel rail and a calculation formula (7) for solving the wave number, the phase velocity and the group velocity of the steel rail under different frequencies;
in step S7, constructing a perturbation equation of the longitudinal stress variation of the steel rail and the guided wave dispersion characteristic variation of the steel rail, wherein the perturbation equation specifically comprises the following steps:
stress obtained by back calculation and solving of wave number variation caused by the action of longitudinal stress variation of steel rail is recorded asSee formula (11);
（11）；
in the method, in the process of the application,for the density of the rail->Defined as a coefficient related to modal guided wave characteristics:，/>wavenumber of mth order mode of rail, < +.>For the longitudinal stress variation of the rail>Wave number variation of mth order mode is caused by action, < >>For the total mass matrix of the rail->Transpose of left eigenvector of mth order mode of rail,/for example>Right eigenvector of the m-th order mode of the steel rail;
the stress obtained by back calculation and solving of the phase velocity variation caused by the action of the longitudinal stress variation of the steel rail is recorded asSee formula (12);
（12）；
in the method, in the process of the application,for the longitudinal stress variation of the rail>The phase velocity variation is caused under the action;
the stress obtained by back calculation and solving of group velocity variation caused by the action of longitudinal stress variation of the steel rail is recorded asSee formula (13);
（13）；
in the method, in the process of the application,for the longitudinal stress variation of the rail>Under the action of which the group velocity change is caused, < >>For the longitudinal stress variation of the rail>Wave number variation caused by action, +.>Transpose of left eigenvector of rail, +.>Is the variation of the right characteristic vector of the steel rail.
2. The method for detecting the longitudinal stress of the steel rail based on the ultrasonic guided wave characteristics, according to claim 1, is characterized in that: in the step S3, analyzing a wave number dispersion curve, a phase velocity dispersion curve and a group velocity dispersion curve of the steel rail to obtain an optimal detection frequency of the steel rail within a range of 20-40 kHz, and obtaining specific modes corresponding to the steel rail under different frequencies.
3. The method for detecting the longitudinal stress of the steel rail based on the ultrasonic guided wave characteristics, which is characterized in that in the step S4, the steel rail detection frequency in the step S3 is input into an ultrasonic guided wave detection system for detection, and guided wave excitation signals and receiving signals in a steel rail free state and a steel rail locking state are respectively obtained in a steel rail specific mode; the method comprises the following steps:
the ultrasonic guided wave detection system comprises an excitation transducer and two receiving transducers, wherein the two receiving transducers are a 1 st receiving transducer and a 2 nd receiving transducer respectively, the excitation transducer and the two receiving transducers which are arranged through the ultrasonic guided wave detection system are respectively applied to different positions in a steel rail free state and a steel rail locking state, an excitation signal of guided wave propagation in the steel rail free state, a receiving signal of the 1 st receiving transducer in the steel rail free state and a receiving signal of the 2 nd receiving transducer in the steel rail free state are obtained, and an excitation signal of guided wave propagation in the steel rail locking state, a receiving signal of the 1 st receiving transducer in the steel rail locking state and a receiving signal of the 2 nd receiving transducer in the steel rail locking state are obtained.
4. The method for detecting the longitudinal stress of the steel rail based on the ultrasonic guided wave characteristics, according to claim 1, is characterized in that in the step S5, the time interval of ultrasonic guided wave receiving signals of two receiving transducers of an ultrasonic guided wave detection system and the interval between the two receiving transducers are identified, and the dispersion characteristic parameters of guided waves in the steel rail are obtained; the method comprises the following steps:
three group velocities of the rail in the rail locked state:
（8）；
in the method, in the process of the application,group velocity between the 1 st receiving transducer and the 2 nd receiving transducer for guiding the rail>Group velocity propagation between excitation transducer and 2 nd receiving transducer for rail guided wave, +.>For the propagation group velocity of the guided wave of the steel rail between the exciting transducer and the 1 st receiving transducer, d1 is the distance from the 1 st receiving transducer to the 2 nd receiving transducer, d2 is the distance from the exciting transducer to the 2 nd receiving transducer, d is the distance from the exciting transducer to the 1 st receiving transducer>For the propagation time of the 1 st receiving transducer to the 2 nd receiving transducer,/for the first receiving transducer>For the propagation time of the excitation transducer to the 2 nd receiving transducer, < >>To excite the transducer to the 1 st receiving transducer, there are:
（9）；
in the method, in the process of the application,for the rail guided wave acquired, 1 st moment of receiving the signal by the transducer, < >>Time of receiving signal of 2 nd receiving transducer for acquired rail guided wave, +.>The moment of the guided wave excitation signal for the acquired steel rail;
by solving for the average of the three group velocities of the rail, i.e. the group velocity of the rail in the locked stateThe method comprises the following steps:
（10）。
5. the method for detecting the longitudinal stress of the steel rail based on the ultrasonic guided wave characteristics, which is characterized in that in the step S6, the difference value of the guided wave dispersion characteristics of the steel rail in the free state and the locked state of the steel rail is obtained as the variation of the guided wave dispersion characteristics of the steel rail through the relation between the guided wave excitation signal and the receiving signal in the step S4; the method comprises the following steps:
the group velocity of the steel rail in the free state obtained by adopting the ultrasonic guided wave detection system is recorded asThe group velocity of the rail in the locked state of the rail is recorded as +.>Longitudinal stress in rail locked state>Longitudinal stress in the free state of the rail>The difference value of (1) is the required to be measuredThe longitudinal stress of the rail is marked as->Analyzing signals obtained by an ultrasonic guided wave detection system to obtain the group velocity variation of the steel rail in the locked state and the free state of the steel rail。
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202311212618.XA CN116956644B (en) | 2023-09-20 | 2023-09-20 | Rail longitudinal stress detection method based on ultrasonic guided wave characteristics |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202311212618.XA CN116956644B (en) | 2023-09-20 | 2023-09-20 | Rail longitudinal stress detection method based on ultrasonic guided wave characteristics |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN116956644A CN116956644A (en) | 2023-10-27 |
| CN116956644B true CN116956644B (en) | 2023-12-08 |
Family
ID=88442842
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202311212618.XA Active CN116956644B (en) | 2023-09-20 | 2023-09-20 | Rail longitudinal stress detection method based on ultrasonic guided wave characteristics |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN116956644B (en) |
Families Citing this family (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN117214303B (en) * | 2023-11-07 | 2024-01-30 | 华东交通大学 | Guided wave nondestructive testing method for elastic modulus of solid material |
Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105571751A (en) * | 2015-12-14 | 2016-05-11 | 中国铁路总公司 | Seamless steel rail stress detection device and method based on ultrasonic guided-wave linear array |
| CN108168745A (en) * | 2017-12-14 | 2018-06-15 | 大连理工大学 | The axial stress monitoring method of symmetrical section stringer based on high-order acoustic elasticity supersonic guide-wave |
| CN112464524A (en) * | 2020-11-07 | 2021-03-09 | 西南交通大学 | Method for determining guided wave propagation characteristics of turnout variable cross-section steel rail |
| CN116165270A (en) * | 2022-12-05 | 2023-05-26 | 东莞理工学院 | Method for calculating ultrasonic guided wave dispersion characteristics of pipeline structure |
Family Cites Families (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US9410853B2 (en) * | 2012-06-21 | 2016-08-09 | Siemens Energy, Inc. | Guided wave thermography methods and systems for inspecting a structure |
| CN114487109B (en) * | 2022-01-19 | 2022-08-23 | 哈尔滨工业大学 | Non-baseline data stress online monitoring method, system, equipment and medium based on single-mode multi-frequency signal fusion |
-
2023
- 2023-09-20 CN CN202311212618.XA patent/CN116956644B/en active Active
Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105571751A (en) * | 2015-12-14 | 2016-05-11 | 中国铁路总公司 | Seamless steel rail stress detection device and method based on ultrasonic guided-wave linear array |
| CN108168745A (en) * | 2017-12-14 | 2018-06-15 | 大连理工大学 | The axial stress monitoring method of symmetrical section stringer based on high-order acoustic elasticity supersonic guide-wave |
| CN112464524A (en) * | 2020-11-07 | 2021-03-09 | 西南交通大学 | Method for determining guided wave propagation characteristics of turnout variable cross-section steel rail |
| CN116165270A (en) * | 2022-12-05 | 2023-05-26 | 东莞理工学院 | Method for calculating ultrasonic guided wave dispersion characteristics of pipeline structure |
Non-Patent Citations (4)
| Title |
|---|
| Numerical calculation of guided wave disperse curve in helix wire based on semi-analytical finite element method;Zhou Fangjun;IEEE 2011 10th International Conference on Electronic Measurement & Instruments;全文 * |
| 基于超声导波的钢轨完整性检测方法研究;朱力强;许西宁;余祖俊;史红梅;段翔宇;;仪器仪表学报(第07期);全文 * |
| 基于钢轨导波的无缝线路温度应力监测方法研究;郝浩业;中国优秀硕士学位论文全文数据库;全文 * |
| 高速铁路钢轨超声导波特性及应用;周杭;中国优秀硕士学位论文全文数据库;全文 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN116956644A (en) | 2023-10-27 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Inoue et al. | Review of inverse analysis for indirect measurement of impact force | |
| Ben et al. | Damage identification in composite materials using ultrasonic based Lamb wave method | |
| CN116956644B (en) | Rail longitudinal stress detection method based on ultrasonic guided wave characteristics | |
| CN106949861B (en) | A kind of method of non-linear ultrasonic on-line monitoring metal material strain variation | |
| Liu et al. | Measuring bulk material nonlinearity using harmonic generation | |
| CN106556647A (en) | A kind of impact echo data processing method | |
| CN114487109B (en) | Non-baseline data stress online monitoring method, system, equipment and medium based on single-mode multi-frequency signal fusion | |
| Piana et al. | Experimental modal analysis of straight and curved slender beams by piezoelectric transducers | |
| JP3313028B2 (en) | Measurement method of bending stiffness and tension of cable under tension | |
| Chen et al. | Modeling and simulation of static component generation of Lamb wave propagation in a layered plate | |
| Erofeev et al. | Ultrasonic sensing method for evaluating the limit state of metal structures associated with the onset of plastic deformation | |
| Li et al. | Mode pair selection of circumferential guided waves for cumulative second-harmonic generation in a circular tube | |
| Grishchenko et al. | Experimental investigation of the acoustic anisotropy field in the sample with a stress concentrator | |
| CN106680375A (en) | Air-coupled ultrasonic testing method for determining elastic modulus of material | |
| Periyannan et al. | Temperature dependent E and G measurement of materials using ultrasonic guided waves | |
| Jacquelin et al. | Force recovered from three recorded strains | |
| Brizard | An impact test to determine the wave speed in SHPB: Measurement and uncertainty | |
| Gokhale | Determination of applied stresses in rails using the acoustoelastic effect of ultrasonic waves | |
| Hull | An inverse method to measure the axial modulus of composite materials under tension | |
| CN113447163A (en) | Method for measuring locked rail temperature based on overall change of steel rail mode | |
| Zhang et al. | A new nondestructive technique for measuring pressure in vessels by surface waves | |
| JPH0493653A (en) | Dynamic response characteristic measuring method for ae sensor | |
| Fong et al. | Curvature effect on the properties of guided waves in plates | |
| JP2001337077A (en) | Device for non-destructively inspecting exfoliation in concrete structure | |
| CN117310006A (en) | Rail damage identification and positioning method |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |