Bolt axial force measuring method based on non-uniform temperature field

Abstract

The invention discloses a bolt axial force measurement method based on a non-uniform temperature field, and relates to the technical field of bolt axial force measurement. Taking the bolt as a cylinder, establishing a temperature field model in a radial-axial section, converting the temperature field model into a two-dimensional model under a Cartesian coordinate system, describing the distribution of a temperature field to obtain a temperature field equation, setting a bolt boundary to meet a Dirichlet boundary condition, and obtaining a temperature distribution analytical solution of the bolt according to a superposition principle and a variable separation method; establishing a propagation time functional from an incidence point to a receiving point based on the Fermat principle, solving an extreme value and obtaining a control equation, and solving a propagation path in combination with a preset boundary condition; the propagation path length is corrected, the equivalent temperature is solved, and the axial force is calculated by adopting a double-wave method in combination with the sound time-axial force-temperature relation and utilizing the equivalent temperature. By solving the bolt temperature field distribution, the ultrasonic propagation path and the equivalent temperature, accurate compensation of the temperature error is realized, and the bolt axial force measurement accuracy can be effectively improved.

Description

Technical Field

[0001] This invention relates to the field of bolt axial force measurement technology, specifically a bolt axial force measurement method based on a non-uniform temperature field. Background Technology

[0002] Bolt axial force is a core parameter for ensuring the reliability of equipment connections, and ensuring that the bolt axial force is within the specified range is fundamental to the safe operation of the equipment. Currently, ultrasonic bolt axial force measurement methods based on the acoustoelastic principle are widely used. This method indirectly reflects the magnitude of the bolt axial force by detecting the change in acoustic time of ultrasonic waves propagating within the bolt.

[0003] However, in practical applications, both temperature and axial force can cause changes in ultrasonic wave duration. Moreover, the change in sound velocity caused by temperature is comparable in magnitude to the change in duration caused by axial force. Therefore, it is necessary to compensate for the time error caused by temperature in order to achieve accurate measurement of bolt axial force.

[0004] During service, bolts often exhibit a non-uniform temperature distribution. Existing bolt axial force temperature compensation methods typically use local bolt temperature measurements as compensation values ​​or simplify by assuming uniform radial temperature. Neither of these methods accurately reflects the overall non-uniform temperature distribution of the bolt, leading to inaccurate temperature compensation and consequently, significant errors in bolt axial force measurements, making it difficult to meet high-precision measurement requirements. Summary of the Invention

[0005] To address the shortcomings of the prior art, this invention provides a bolt axial force measurement method based on a non-uniform temperature field. By solving for the bolt temperature field distribution, ultrasonic wave propagation path, and equivalent temperature, it achieves accurate compensation for temperature errors, effectively improving the measurement accuracy of bolt axial force.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: a method for measuring bolt axial force based on a non-uniform temperature field, comprising the following steps:

[0007] Step 1: Solving the temperature field distribution of the bolt: Treat the bolt as a cylinder and establish a temperature field model within its radial-axial cross-section, then convert it to a Cartesian coordinate system. The following two-dimensional model describes the temperature field using the Laplace equation. The temperature field equation is derived from the distribution of the bolts. Assuming the bolt boundary satisfies the Dirichlet boundary conditions, the temperature field equation is decomposed into four subproblems according to the superposition principle. Each subproblem is solved using the method of separation of variables. The solutions of the four subproblems are then superimposed to obtain the analytical solution for the temperature distribution of the bolts. ;

[0008] Step 2: Solving the ultrasonic wave propagation path: Based on Fermat's principle, establish the path of the ultrasonic wave from the incident point... to the receiving point The propagation time functional is obtained, and the Euler-Lagrange equations are applied to solve for the extrema to obtain the governing equations for the ultrasonic wave propagation path. Combined with pre-defined boundary conditions, the propagation path of the ultrasonic wave is then obtained. ;

[0009] Step 3: Ultrasonic solid temperature equivalence and compensation: based on the bolt thermal expansion coefficient The propagation path length is corrected, and the propagation time of ultrasound is the same under non-uniform temperature fields and equivalent temperatures. The equivalent temperature is then solved. Transverse waves were selected as the temperature equivalent of ultrasound waves. A dual-wave method was employed, combining the acoustic time-axial force-temperature relationship, and utilizing the equivalent temperature... Calculated axial force The value of .

[0010] Furthermore, in step one, the Laplace equation in the Cartesian coordinate system... The Chinese character is represented as:

[0011]

[0012] The Dirichlet boundary conditions are expressed as follows:

[0013]

[0014] In the formula, For the width of the two-dimensional model, For the height of the two-dimensional model, , , and The temperature fields to be determined are as follows: The upper boundary condition, left boundary condition, lower boundary condition, and right boundary condition.

[0015] Furthermore, in step one, the four sub-problems are as follows: Sub-problem 1 is denoted as... Represents: the temperature field distribution when boundary condition ① is not 0 and all other boundary conditions are 0; subproblem 2 is denoted as Represents: the temperature field distribution when boundary condition ② is not 0 and all other boundary conditions are 0; subproblem 3 is denoted as Represents: the temperature field distribution when boundary condition ③ is not 0 and all other boundary conditions are 0; subproblem 4 is denoted as This indicates the temperature field distribution when boundary condition ④ is not 0 and the other boundary conditions are 0.

[0016] Furthermore, in step two, the propagation time of the ultrasonic wave from the excitation point to the receiving point is expressed as:

[0017]

[0018] in, ;

[0019] make The propagation time can then be expressed as a functional as follows:

[0020]

[0021] In the formula, for The first derivative, This represents the functional relationship between sound speed and temperature distribution.

[0022] Furthermore, in step two, the functional... Solving for the extrema using the Euler-Lagrange equations, expressed as:

[0023]

[0024] In the formula, for Tiny perturbations;

[0025] Let the actual path be The disturbance path is ,in, It is a smooth function and satisfies ,but Represented as:

[0026]

[0027] because Simplified representation:

[0028]

[0029] Finally, the governing equations for the ultrasonic wave propagation path are obtained. .

[0030] Furthermore, in step two, the preset boundary conditions are the known incident point and incident angle.

[0031] Furthermore, in step three, during the process of correcting the propagation path length, the propagation path is subdivided into... The change in the total length of the propagation path is expressed as: .

[0032] Furthermore, in step three, the acoustic time-axial force-temperature relationship is expressed as follows:

[0033]

[0034] In the formula, and These represent the axial forces at the reference temperature. The propagation time of the two ultrasound waves is 0. and These represent the isothermal axial force calibration coefficients for the two types of ultrasound. and These represent the temperature calibration coefficients for two different ultrasonic axial forces when they are both zero. Indicates the amount of temperature change. , , .

[0035] Compared with the prior art, the beneficial effects of the present invention are: by solving the bolt temperature field distribution, ultrasonic propagation path and equivalent temperature, the present invention achieves accurate compensation for temperature error, can accurately measure the bolt axial force under non-uniform temperature field, solves the problem of inaccurate measurement caused by the consideration of non-uniform temperature distribution in the existing temperature compensation method, and effectively improves the measurement accuracy of bolt axial force. Attached Figure Description

[0036] Figure 1 This is a schematic diagram of the temperature field modeling of the bolt and its boundary in the method of the present invention;

[0037] Figure 2 This is a schematic diagram of the decomposition and superposition of the temperature field model in the method of this invention;

[0038] Figure 3 This is a flowchart illustrating the calculation of the ultrasonic wave propagation path in the method of this invention. Detailed Implementation

[0039] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the invention, not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0040] like Figures 1-3 As shown, a method for measuring bolt axial force based on a non-uniform temperature field includes the following steps:

[0041] Step 1: Solving for the temperature field distribution of the bolts

[0042] S1.1 Bolt Temperature Field Modeling: To solve for the temperature distribution of the bolt, the bolt is considered as a cylinder, combined with... Figure 1 As shown, a schematic diagram of the temperature field modeling of the bolt and its boundary is presented, through its radial-axial ( Solving the heat conduction equation within the cross-section yields the two-dimensional temperature distribution of the bolt. The cross-section is then modeled and converted to a Cartesian coordinate system. The following two-dimensional model, temperature field The distribution is described by the Laplace equation, then the temperature field equation in Cartesian coordinates... The Chinese character is represented as:

[0043]

[0044] Assuming the bolt boundary satisfies the Dirichlet boundary conditions, i.e., the temperature at the boundary is known and constant, then the boundary conditions in the two-dimensional model are expressed as follows:

[0045]

[0046] In the formula, For the width of the two-dimensional model, For the height of the two-dimensional model, , , and The temperature fields to be determined are as follows: The upper boundary condition, left boundary condition, lower boundary condition, and right boundary condition.

[0047] S1.2 Subproblem Decomposition: Based on the superposition principle, the temperature field equation is decomposed into four subproblems, combined with... Figure 2 As shown, subproblem 1 is denoted as Represents: the temperature field distribution when boundary condition ① is not 0 and all other boundary conditions are 0; subproblem 2 is denoted as Represents: the temperature field distribution when boundary condition ② is not 0 and all other boundary conditions are 0; subproblem 3 is denoted as Represents: the temperature field distribution when boundary condition ③ is not 0 and all other boundary conditions are 0; subproblem 4 is denoted as This indicates the temperature field distribution when boundary condition ④ is not 0 and the other boundary conditions are 0.

[0048] S1.3, Subproblem Solving: Each subproblem is solved using the method of separation of variables, as detailed below:

[0049] For subproblem 1, the general solution of the equation is expressed as:

[0050]

[0051] Using boundary conditions ②, ③, and ④ as 0, the specific solution to the equation is as follows:

[0052]

[0053] Using boundary condition ①, we obtain:

[0054]

[0055] Similarly, for subproblem 3, the general solution of the equation is expressed as:

[0056]

[0057] Using boundary conditions ①, ②, and ④ as 0, the specific solution to the equation is as follows:

[0058]

[0059] Using boundary condition ③, we obtain:

[0060]

[0061] because and symmetry, and Symmetry, and To substitute, and By substitution, subproblems 2 and 4 can be solved, yielding:

[0062]

[0063]

[0064]

[0065]

[0066] In the formula, These are characteristic roots. , , and For each of the undetermined coefficients, To solve for the coefficients.

[0067] S1.4 Temperature Distribution Synthesis: Finally, the analytical solution to the temperature distribution of the bolt, which is the superposition of the solutions to the four subproblems, is expressed as follows: .

[0068] Step 2: Solving the ultrasonic wave propagation path

[0069] S2.1 Basic Settings and Parameter Definitions: Let the ultrasonic incident point be... The receiving point is The transmission path is The speed of sound and temperature distribution satisfy a functional relationship. According to Fermat's principle, the propagation path of a sound wave is the path that minimizes the propagation time. Solving for the propagation path is transformed into solving the corresponding functional extremum problem.

[0070] S2.2, Establishment of the propagation time functional: The propagation time of ultrasound from the excitation point to the receiving point is expressed as:

[0071]

[0072] in, .

[0073] make The propagation time can then be expressed as a functional as follows:

[0074]

[0075] In the formula, for The first derivative of represents the slope of the path curve.

[0076] S2.3 Solving variational problems: For functionals Solving for the extrema using the Euler-Lagrange equations, expressed as:

[0077]

[0078] In the formula, for Tiny perturbations.

[0079] Let the actual path be The disturbance path is ,in, It is a smooth function and satisfies ,but It can be represented as:

[0080]

[0081] because It can be simplified as follows:

[0082]

[0083] For the above formula to hold true, the following conditions must be met to constrain the formula:

[0084]

[0085] In the formula, , ,

[0086] .

[0087] Finally, the governing equations for the ultrasonic wave propagation path are obtained. .

[0088] S2.4 Boundary Conditions: The boundary conditions for the governing equations of the ultrasonic wave propagation path are divided into two categories: 1. The locations of the incident point and the receiver point are known; 2. The incident point and the incident angle are known. In actual solutions, the incident point and the incident angle are often known, i.e., the second type of boundary condition is used.

[0089] S2.5 Path Determination: The propagation path of the ultrasonic wave can be solved by combining the governing equations of the ultrasonic wave propagation path and the second type of boundary conditions. Figure 3 As shown, it includes the following steps:

[0090] 1. Input parameter: Incident point and angle of incidence Temperature distribution and the relationship between sound speed and temperature ;

[0091] 2. Calculate the slope at the current point. Temperature partial derivative and With the derivative of the speed of sound ;

[0092] 3. Determine the slope If the coordinate system is infinite, rotate the coordinate system by 90° and update the coordinates of the point; otherwise, proceed directly to the next step.

[0093] 4. Based on the information at the current point, use the governing equations of the ultrasonic wave propagation path to calculate the coordinates of the next point, i.e., the [number]th [point]. The coordinates of the points;

[0094] 5. If coordinate system rotation was performed during the calculation, it is necessary to convert all calculated path point coordinates back to the original coordinate system according to the inverse relationship of rotation to ensure that the final result is under a unified original reference system;

[0095] 6. Determine if the coordinates of the point are within the domain. If yes, return to step 3 and repeat the iterative calculation. If no, output the propagation path. .

[0096] Step 3: Ultrasonic Solid Temperature Equivalence and Compensation

[0097] S3.1, Propagation path length correction: Known temperature distribution and transmission path Then, based on the fact that the propagation time of sound waves at the equivalent temperature is the same as that in a non-uniform temperature field, the equivalent temperature is calculated. Since the bolt undergoes thermal expansion due to temperature, causing a change in the propagation path length, the propagation path is subdivided into... The change in the total length of the propagation path is expressed as: , This is the coefficient of thermal expansion of the bolt.

[0098] S3.2 Calculation of Actual Propagation Time: Since temperature causes changes in sound speed and propagation path length, the actual propagation time of a sound wave can be expressed as: .

[0099] S3.3, Solving for the equivalent temperature: Under the equivalent temperature condition, the propagation time of the sound wave can be expressed as: ,according to By combining the conditional formulas in step S2.3 with the governing equations for the ultrasonic wave propagation path, the equivalent temperature can be obtained. .

[0100] S3.4 Axial Force Calculation: Existing research shows that the dual-wave method is less affected by temperature than the single-wave method, while the transverse wave is more sensitive to temperature. Therefore, the transverse wave is selected as the temperature-equivalent ultrasonic wave, and the dual-wave method is chosen for bolt axial force measurement. The acoustic time of the ultrasonic wave (transverse wave, longitudinal wave, or mode-converted wave) after being affected by temperature and axial force is expressed as follows:

[0101]

[0102]

[0103] The acoustic time-axial force-temperature relationship combined by the dual-wave method is as follows:

[0104]

[0105] In the formula, and These represent the axial forces at the reference temperature. The propagation time of the two ultrasound waves is 0. and These represent the isothermal axial force calibration coefficients for the two types of ultrasound. and These represent the temperature calibration coefficients for two different ultrasonic axial forces when they are both zero. Indicates the amount of temperature change. , , .

[0106] Using the above formula, we first perform a uniform temperature field calibration to obtain... and Then, the equivalent temperature was calculated based on the ultrasonic time. The axial force can then be calculated. The value of .

[0107] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered illustrative and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of the equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

[0108] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A method for measuring bolt axial force based on a non-uniform temperature field, characterized in that: Includes the following steps: Step 1: Solving the temperature field distribution of the bolt: Treat the bolt as a cylinder and establish a temperature field model within its radial-axial cross-section, then convert it to a Cartesian coordinate system. The following two-dimensional model describes the temperature field using the Laplace equation. The temperature field equation is derived from the distribution of the bolts. Assuming the bolt boundary satisfies the Dirichlet boundary conditions, the temperature field equation is decomposed into four subproblems according to the superposition principle. Each subproblem is solved using the method of separation of variables. The solutions of the four subproblems are then superimposed to obtain the analytical solution for the temperature distribution of the bolts. ; Step 2: Solving the ultrasonic wave propagation path: Based on Fermat's principle, establish the path of the ultrasonic wave from the incident point... to the receiving point The propagation time functional is obtained, and the Euler-Lagrange equations are applied to solve for the extrema to obtain the governing equations for the ultrasonic wave propagation path. Combined with pre-defined boundary conditions, the propagation path of the ultrasonic wave is then obtained. ; Step 3: Ultrasonic solid temperature equivalence and compensation: based on the bolt thermal expansion coefficient The propagation path length is corrected, and the propagation time of ultrasound is the same under non-uniform temperature fields and equivalent temperatures. The equivalent temperature is then solved. Transverse waves were selected as the temperature equivalent of ultrasound waves. A dual-wave method was employed, combining the acoustic time-axial force-temperature relationship, and utilizing the equivalent temperature... Calculated axial force The value of .

2. The bolt axial force measurement method based on a non-uniform temperature field according to claim 1, characterized in that: In step one, the Laplace equation in the Cartesian coordinate system The Chinese character is represented as: The Dirichlet boundary conditions are expressed as follows: In the formula, For the width of the two-dimensional model, For the height of the two-dimensional model, , , and The temperature fields to be determined are as follows: The upper boundary condition, left boundary condition, lower boundary condition, and right boundary condition.

3. The bolt axial force measurement method based on a non-uniform temperature field according to claim 2, characterized in that: In step one, the four sub-problems are as follows: Sub-problem 1 is denoted as... Represents: the temperature field distribution when boundary condition ① is not 0 and all other boundary conditions are 0; subproblem 2 is denoted as Represents: the temperature field distribution when boundary condition ② is not 0 and all other boundary conditions are 0; subproblem 3 is denoted as Represents: the temperature field distribution when boundary condition ③ is not 0 and all other boundary conditions are 0; subproblem 4 is denoted as This indicates the temperature field distribution when boundary condition ④ is not 0 and the other boundary conditions are 0.

4. The bolt axial force measurement method based on a non-uniform temperature field according to claim 1, characterized in that: In step two, the propagation time of the ultrasonic wave from the excitation point to the receiving point is expressed as: in, ; make The propagation time can then be expressed as a functional as follows: In the formula, for The first derivative, This represents the functional relationship between sound speed and temperature distribution.

5. The bolt axial force measurement method based on a non-uniform temperature field according to claim 4, characterized in that: In step two, the functional Solving for the extrema using the Euler-Lagrange equations, expressed as: In the formula, for Tiny perturbations; Let the actual path be The disturbance path is ,in, It is a smooth function and satisfies ,but Represented as: because Simplified representation: Finally, the governing equations for the ultrasonic wave propagation path are obtained. .

6. The bolt axial force measurement method based on a non-uniform temperature field according to claim 1, characterized in that: In step two, the preset boundary conditions are the known incident point and incident angle.

7. The bolt axial force measurement method based on a non-uniform temperature field according to claim 1, characterized in that: In step three, during the process of correcting the propagation path length, the propagation path is subdivided into... The change in the total length of the propagation path is expressed as: .

8. The bolt axial force measurement method based on a non-uniform temperature field according to claim 1, characterized in that: In step three, the acoustic time-axial force-temperature relationship is expressed as follows: In the formula, and These represent the axial forces at the reference temperature. The propagation time of the two ultrasound waves is 0. and These represent the isothermal axial force calibration coefficients for the two types of ultrasound. and These represent the temperature calibration coefficients for two different ultrasonic axial forces when they are both zero. Indicates the amount of temperature change. , , .

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