Theoretical method and device for estimating fastening degree of swivel bolt based on vibration

Abstract

The invention belongs to the technical field of bolt physical modeling and fastening degree estimation, and particularly relates to a theoretical method and device for estimating the fastening degree of a rotating bolt based on vibration. According to the theoretical method, the Euler-Bernoulli beam theory is applied to perform physical modeling on the bolt, and after the natural vibration frequency of the bolt is measured, the natural vibration frequency can be substituted into a vibration equation to directly solve the estimated value of the bolt pre-tightening force. According to the method, only the physical quantity of the natural vibration frequency of the bolt needs to be measured, analysis on the mechanical process in the bolt is avoided, and the step of estimating the fastening degree of the bolt is simple. Compared with other bolt pre-tightening force estimation methods such as an ultrasonic method, the method has the advantages of being small in temperature influence, simple in device and the like.

Description

Technical Field

[0001] This invention belongs to the field of bolt physical modeling and tightness estimation technology, specifically involving a theoretical method and device for estimating the tightness of rotating bolts based on vibration. Background Technology

[0002] In the power systems of critical equipment such as aero-engines, high-speed railways, and wind power, bolts play crucial roles in power transmission, structural fixation, and vibration suppression. Their performance directly affects the operational stability and safety of the system. Bolt preload, a physical quantity, is an important indicator of bolt tightness and a vital reference for the safety and stability of the power system. Therefore, timely and accurate estimation of bolt preload is crucial for determining the safety of power system connections. However, direct measurement of bolt preload is physically difficult to achieve, and online estimation methods such as ultrasonic methods suffer from drawbacks such as temperature sensitivity and complex equipment, making them unsuitable for application in the power systems of critical equipment like aero-engines. Summary of the Invention

[0003] This invention provides a theoretical method and apparatus for estimating the tightness of a rotating bolt based on vibration. This theoretical method can accurately estimate the preload of the bolt through the bolt's natural vibration frequency.

[0004] To achieve the above objectives, the present invention adopts the following specific technical solution:

[0005] This invention provides a theoretical method for estimating the tightness of rotating bolts based on vibration, which includes the following steps:

[0006] Step 1: Establish a physical model of the bolt;

[0007] Step two: Measure the natural vibration frequency of the bolt;

[0008] Step 3: Parameter identification, to obtain the equation between the natural vibration frequency of the bolt and the preload, which contains specific stiffness parameter values;

[0009] Step four: Solve the equation to obtain a numerical solution, and estimate the bolt preload based on the measured natural vibration frequency.

[0010] Furthermore, in step one, the bolt is physically modeled using Euler-Bernoulli beam theory. The vibration equation of the Euler-Bernoulli beam is:

[0011]

[0012] In the above formula, E is the elastic modulus, I is the moment of inertia of the cross section, U is the deflection of the beam, ρ is the density of the beam, A is the cross-sectional area, x is the abscissa, and N is the axial tension of the beam.

[0013] After considering rotational and axial vibration excitation, the vibration equation becomes:

[0014]

[0015] In the above formula, R is the distance between the bolt shaft and the rotation shaft, and Ω is the rotational angular velocity;

[0016] For ease of solution, the vibration equation needs to be dimensionless:

[0017]

[0018] Where U0 is the centrifugal static displacement, T is the characteristic time, and Ω cr The critical speed is given by the following expressions:

[0019]

[0020]

[0021] The dimensionless vibration equation becomes:

[0022]

[0023] To solve for the vibration response using the method of separation of variables, let:

[0024] η(x,t)=φ(x)·Υ(t)

[0025] After separating the variables and rearranging, we obtain the intrinsic equations:

[0026]

[0027] The eigenvalue equation is an ordinary differential equation, which yields four characteristic roots, including two real roots and two imaginary roots. The four characteristic roots are as follows:

[0028]

[0029] Based on the four characteristic roots mentioned above, the general form of the general solution to the differential equation can be written:

[0030] η(ξ)=C1cosh(λ1ξ)+C2sinh(λ1ξ)+C3cos(λ2ξ)+C4sin(λ2ξ)

[0031] In the above formula, C1-C4 are four coefficients to be determined;

[0032] Substitute the boundary conditions again:

[0033] -η″′(0,τ)+pη′(0,τ)=K u η(0,τ)

[0034] η″(0,τ)=K θ η(0,τ)

[0035] -η″′(1,τ)pη′(1,τ)=-K u η(1,τ)

[0036] η″(1,τ)=-K θ n(1,τ)

[0037] Among them, K u For boundary tensile stiffness, K θ Boundary torsional stiffness, boundary tensile stiffness, and boundary torsional stiffness are two system parameters that need to be identified.

[0038] Solving the system of equations simultaneously, we obtain the matrix equation:

[0039] AC = 0

[0040]

[0041] Among them, s1=sinh(λ1), s2=cosh(λ1), s3=sinh(λ2), s4=cosh(λ2), c1=sin(λ1), c2=cos(λ1), c3=sin(λ2), c4=cos(λ2);

[0042] For the matrix equation to have non-zero solutions, the determinant of A must be equal to 0, resulting in the following equation:

[0043] |A|=0

[0044] In the formula, λ1 and λ2 are both functions of the dimensionless preload p and the natural vibration frequency w, thus obtaining an equation for the dimensionless preload p and the natural vibration frequency w.

[0045] Furthermore, step two specifically includes:

[0046] An accelerometer was used to measure the vibration response of the bolt.

[0047] The measured vibration response is subjected to FFT (fast Fourier transform) to obtain the natural vibration frequency of the bolt.

[0048] Furthermore, step three specifically includes:

[0049] Assume the system model is as follows:

[0050] f(ω,p,θ)=0

[0051] In the above formula, θ is a vector composed of parameters that need to be identified, p is the dimensionless preload, and w is the natural vibration frequency.

[0052] The sum of squared prediction errors is used as the loss function:

[0053]

[0054] Calculate the gradient of the loss function using the chain rule. Finally, update each parameter to be identified:

[0055]

[0056] During the iterative update process, an adaptive learning rate is used to accelerate model convergence.

[0057] The number of iterations or stopping conditions can be determined by setting a maximum number of iterations or setting a threshold.

[0058] Furthermore, step four employs Newton's iteration method to numerically solve the equation, gradually approximating the root of the equation through linear approximation. Specifically, this includes:

[0059] Initial conditions are set by selecting an initial value x0 close to the true root; a convergence threshold ∈ is set; and the maximum number of iterations N is set. max ;

[0060] Iteration,

[0061]

[0062] In the above formula, x n This is the approximate root of the equation obtained in the nth iteration;

[0063] Termination condition: The game will stop if any of the following conditions are met:

[0064] The function value is small enough: |f(x) n )∣<∈;

[0065] The iteration step size is small enough: |x n+1 -x n |<∈;

[0066] The measured natural vibration frequency is substituted into the preload estimation program to obtain an accurate estimate of the bolt preload, thereby measuring the tightness of the bolt connection.

[0067] In addition, the present invention also provides a theoretical device for estimating the tightness of a rotating bolt. The theoretical device adopts the above-mentioned theoretical method and includes a measurement module, a wireless transmission module, a parameter identification module, and a preload estimation module.

[0068] The measurement module is used to measure the natural vibration frequency of the bolt;

[0069] The parameter identification module is used to identify the parameters of the two boundary stiffnesses of the bolt model and obtain the equation between the natural vibration frequency and the preload of the bolt containing specific stiffness parameter values.

[0070] The wireless transmission module is used to transmit the natural vibration frequency measured by the measurement module to the parameter identification module and the preload estimation module;

[0071] The preload estimation module uses the natural vibration frequency as input and estimates the bolt preload by solving the equation between the natural vibration frequency and the preload.

[0072] Furthermore, the measurement module includes an accelerometer and an MCU with signal connections;

[0073] An accelerometer is used to measure the vibration response of a bolt;

[0074] The MCU is used to preprocess and save the vibration data measured by the accelerometer.

[0075] Furthermore, the wireless transmission module is a Bluetooth module.

[0076] Furthermore, the parameter identification module uses the gradient descent method to identify the parameters of the boundary stiffness.

[0077] Furthermore, the preload estimation module uses Newton's iterative method to solve the equations for exact numerical solutions.

[0078] Compared with the prior art, the technical solution of the present invention has the following beneficial effects:

[0079] 1. In the theoretical method of this invention, it is found through theoretical derivation that changes in bolt preload affect its natural vibration frequency. Therefore, the bolt preload is estimated using the natural vibration frequency of the bolt. This method avoids studying the complex mechanical processes inside the bolt, and its principle is simple and easy to implement. Even technicians unfamiliar with bolt modeling can quickly estimate the bolt preload using this method.

[0080] 2. In the theoretical method of this invention, the influence of rotational centrifugal force on bolt vibration is considered, thereby extending the vibration-based preload estimation method to rotational scenarios.

[0081] 3. In the theoretical method of this invention, the gradient descent method is used to effectively identify the two system parameters of torsional stiffness and tensile stiffness, and the Newton iteration method is used to accurately solve for the high-precision estimate of bolt preload.

[0082] 4. In the theoretical method of this invention, only two parameters need to be identified, and after the identification is completed, these two parameters generally do not change with the change of environment. After one parameter identification, the result can be used continuously, which has the advantages of being less affected by environmental changes and having good robustness.

[0083] 5. The theoretical device of the present invention can accurately estimate the preload by measuring the natural vibration frequency of the bolt and then solving the vibration equation using the Newton-Raphson iteration method. Attached Figure Description

[0084] Figure 1 A schematic diagram of a typical application scenario for bolts - a rotating bolt flange;

[0085] Figure 2 A schematic diagram of the Euler-Bernoulli beam model used for physical modeling;

[0086] Figure 3 A flowchart illustrating the theoretical method for estimating the tightness of fasteners;

[0087] Figure 4 A schematic diagram of the theoretical device for estimating the tightness. Detailed Implementation

[0088] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0089] Vibration is a common phenomenon in the use of fastening devices, and the vibration response is easy to measure. The natural vibration frequency can be obtained by performing FFT on the measured vibration response. Therefore, by physically modeling the bolt using Euler-Bernoulli beam theory, the equations for the bolt's natural vibration frequency and its preload can be found. Then, numerical analysis methods such as Newton's iteration method can be used to solve the equations to obtain numerical solutions. By measuring the natural vibration frequency, the bolt preload can be accurately estimated.

[0090] In some power systems, there are many such Figure 1 The rotating bolted flange structure shown presents a challenge in estimating bolt preload in such scenarios. To address this, the present invention first utilizes... Figure 2 The Euler-Bernoulli beam model shown is used to theoretically analyze bolt vibration. Based on this, a theoretical method and device for directly estimating the preload using the natural vibration frequency of the bolt are developed, which will be described in detail below.

[0091] Example 1

[0092] This embodiment provides a theoretical method for estimating the tightness of rotating bolts based on vibration. This method includes four steps: establishing a physical model of the bolt, measuring the bolt's natural vibration frequencies, parameter identification, and equation solving. Figure 3 As shown, it specifically includes the following:

[0093] Step 1: Establish the physical model of the bolt:

[0094] A physical model of bolt vibration behavior is constructed based on Euler-Bernoulli beam theory, yielding a partial differential equation for bolt vibration. By making this partial differential equation dimensionless, separating variables, and then substituting boundary conditions, a transcendental equation containing only preload and natural vibration frequency is obtained, specifically including:

[0095] Using Euler-Bernoulli beam theory to physically model the bolt, the vibration equation of the Euler-Bernoulli beam is:

[0096]

[0097] In the above formula, E is the elastic modulus, I is the moment of inertia of the cross section, U is the deflection of the beam, ρ is the density of the beam, A is the cross-sectional area, x is the abscissa, and N is the axial tension of the beam.

[0098] After considering rotational and axial vibration excitations, the above vibration equation becomes:

[0099]

[0100] In the formula, R is the distance between the bolt shaft and the rotating shaft, and Ω is the rotational angular velocity.

[0101] For ease of solution, the differential equation needs to be dimensionless:

[0102]

[0103] Where U0 is the centrifugal static displacement, T is the characteristic time, and Ω cr The critical speed is given by the following expressions:

[0104]

[0105]

[0106] The dimensionless vibration equation becomes:

[0107]

[0108] The vibration response is solved using the method of separation of variables, i.e., let: η(x,t)=φ(x)·Υ(t);

[0109] After separating the variables and rearranging, we obtain the intrinsic equations:

[0110]

[0111] The eigenvalue equation is an ordinary differential equation, which yields four characteristic roots, including two real roots and two imaginary roots. The four characteristic roots are as follows:

[0112]

[0113] Based on these four characteristic roots, the general form of the general solution to the differential equation can be written:

[0114] η(ξ)=C1cosh(λ1ξ)+C2sinh(λ1ξ)+C3cos(λ2ξ)+C4sin(λ2ξ)

[0115] In the above formula, C1-C4 are four undetermined coefficients.

[0116] Substitute the boundary conditions again:

[0117] -η″′(0,τ)+pη′(0,τ)=K u η(0,τ)

[0118] η″(0,τ)=K θ η(0,τ)

[0119] -η″′(1,τ)+pη′(1,τ)=-K u η(1,τ)

[0120] η″(1,τ)=-K θ n(1,τ)

[0121] Among them, K u For boundary tensile stiffness, K θ These two boundary stiffnesses are the two system parameters that need to be identified in this embodiment, namely the boundary torsional stiffness.

[0122] Solving the system of equations simultaneously, we obtain the matrix equation:

[0123] AC = 0

[0124] In the formula, A is the system matrix, and C is a column vector consisting of four undetermined coefficients.

[0125]

[0126] Among them, s1=sinh(λ1), s2=cosh(λ1), s3=sinh(λ2), s4=cosh(λ2), c1=sin(λ1), c2=cos(λ1), c3=sin(λ2), c4=cos(λ2).

[0127] For the matrix equation to have non-zero solutions, the determinant of A must be equal to 0, resulting in the following equation:

[0128] |A|=0

[0129] In the formula, λ1 and λ2 are both functions of the dimensionless preload p and the natural frequency w, thus obtaining an equation for p and w. However, this equation contains two unknown parameters, and the specific values ​​of these two parameters must be obtained through parameter identification before the preload can be estimated.

[0130] Step 2: Measure the natural vibration frequency of the bolt.

[0131] The vibration response of the bolt was measured using an accelerometer; the natural vibration frequency of the bolt was obtained by performing an FFT (fast Fourier transform) on the measured vibration response.

[0132] Step 3, parameter identification, to obtain the equation between the natural vibration frequency of the bolt and the preload, containing specific stiffness parameter values:

[0133] Assume the system model is as follows:

[0134] f(ω,p,θ)=0

[0135] In the above formula, θ is a vector composed of parameters that need to be identified, p is the dimensionless preload, and w is the natural vibration frequency.

[0136] The sum of squared prediction errors is used as the loss function:

[0137]

[0138] Calculate the gradient of the loss function using the chain rule. Finally, update each parameter to be identified:

[0139]

[0140] During iterative updates, an adaptive learning rate can be used to accelerate model convergence.

[0141] The number of iterations or stopping conditions can be determined by setting a maximum number of iterations or setting a threshold.

[0142] Step four: Solving the equation. The equation established above regarding the dimensionless preload p and the natural vibration frequency w is a transcendental equation, so numerical analysis methods are needed to find the numerical solution of the equation.

[0143] In this embodiment, Newton's iteration method is used to numerically solve the above equation, gradually approximating the root of the equation through linear approximation, specifically including:

[0144] Initial conditions are set by selecting an initial value x0 close to the true root; a convergence threshold ∈ is set; and the maximum number of iterations N is set. max ;

[0145] Iteration,

[0146]

[0147] In the above formula, x n This is the approximate root of the equation obtained in the nth iteration;

[0148] Termination condition: The game will stop if any of the following conditions are met:

[0149] The function value is small enough: |f(x) n )∣<∈;

[0150] The iteration step size is small enough: |x n+1 -x n |<∈;

[0151] The measured natural vibration frequency is substituted into the preload estimation program to obtain an accurate estimate of the bolt preload, thereby measuring the tightness of the bolt connection.

[0152] Example 2

[0153] Based on the above estimation method, this embodiment designs a theoretical device for estimating bolt preload. The principle of the theoretical device is as follows: Figure 4 As shown, the natural vibration frequency is determined by first measuring the vibration response of the bolt using an accelerometer, and then performing an FFT to directly obtain the natural vibration frequency of the bolt.

[0154] The theoretical device includes a measurement module, a wireless transmission module, a parameter identification module, and a preload estimation module, wherein:

[0155] The measurement module is used to measure the natural vibration frequency of the bolt. The module includes an accelerometer and an MCU with signal connections. The accelerometer measures the bolt's vibration response. The MCU preprocesses and stores the vibration data measured by the accelerometer and performs analog-to-digital conversion of the signal. The measurement module uses an accelerometer for frequency response measurement; this accelerometer can be a PCB 356A01 miniature piezoelectric triaxial accelerometer.

[0156] The parameter identification module is used to identify the parameters of the two boundary stiffnesses of the bolt model using the gradient descent method, and to obtain the equation between the natural vibration frequency and the preload of the bolt containing specific stiffness parameter values.

[0157] The wireless transmission module transmits the natural vibration frequency measured by the measurement module to the parameter identification module and the preload estimation module. The wireless transmission module can use a Bluetooth module or similar for ease of use. Specifically, the wireless transmission module can use the HC-05 Bluetooth module for data transmission. The Bluetooth module plugs into a specific pin of the STM32 series microcontroller, using this pin to transmit data sent from the microcontroller to Bluetooth. Bluetooth then sends the data to the PC (personal computer), and a serial port assistant on the PC integrates the data sent by Bluetooth.

[0158] The preload estimation module uses the natural vibration frequency as input and solves the equation between the natural vibration frequency and the preload using the Newton-Raphson iteration method to estimate the bolt preload.

[0159] The parameter identification module and the preload estimation module can be implemented by software programs installed on a PC. The parameter identification program executed by the parameter identification module can be written in Python, and the specific steps are as follows:

[0160] Assume the system model is as follows:

[0161] f(ω,p,θ)=0

[0162] In the above formula, θ is a vector consisting of the parameters to be identified. The sum of squared prediction errors is used as the loss function:

[0163]

[0164] Calculate the gradient of the loss function using the chain rule. Finally, update each parameter to be identified:

[0165]

[0166] During iterative updates, an adaptive learning rate can be used to accelerate model convergence.

[0167] The number of iterations or stopping conditions can be determined by setting a maximum number of iterations or setting a threshold.

[0168] The preload estimation module executes a preload estimation program written in Python, which uses Newton's iteration method to numerically solve the equations obtained above. Newton's iteration method is a numerical method based on Taylor expansion used to solve nonlinear equations of the form f(x) = 0. Its core idea is to gradually approximate the root of the equation through linear approximation. The steps of this method are as follows:

[0169] 1. Initial conditions set:

[0170] Choose an initial value x0 (close to the true root);

[0171] Set a convergence threshold ∈ (e.g., 10) -6 );

[0172] Set the maximum number of iterations N. max (To prevent infinite loops.)

[0173] 2. Iteration:

[0174]

[0175] In the above formula, x n This is the approximate root of the equation obtained in the nth iteration.

[0176] 3. Termination conditions (stop if any one of them is met):

[0177] The function value is small enough: |f(x) n )∣<∈;

[0178] The iteration step size is small enough: |x n+1 -x n |<∈;

[0179] By following the steps above to write a preload estimation program and substituting the measured natural vibration frequency into it, an accurate estimate of the bolt preload can be obtained, thereby measuring the tightness of the fastening connection device.

[0180] Obviously, those skilled in the art can make various modifications and variations to the embodiments of the present invention without departing from the spirit and scope of the invention. Therefore, if these modifications and variations fall within the scope of the claims of the present invention and their equivalents, the present invention also intends to include these modifications and variations.

Claims

1. A theoretical method for estimating the tightness of a rotating bolt based on vibration, characterized in that, Includes the following steps: Step 1: Establish a physical model of the bolt; Step two: Measure the natural vibration frequency of the bolt; Step 3: Parameter identification, to obtain the equation between the natural vibration frequency of the bolt and the preload, which contains specific boundary stiffness parameter values; Step four: Solve the equation to obtain a numerical solution, and estimate the bolt preload based on the measured natural vibration frequency.

2. The theoretical method as described in claim 1, characterized in that, In step one, the bolt is physically modeled using Euler-Bernoulli beam theory. The vibration equation of the Euler-Bernoulli beam is: In the above formula, E is the elastic modulus, I is the moment of inertia of the cross section, U is the deflection of the beam, ρ is the density of the beam, A is the cross-sectional area, x is the abscissa, and N is the axial tension of the beam. After considering rotational and axial vibration excitation, the vibration equation becomes: In the above formula, R is the distance between the bolt shaft and the rotation shaft, and Ω is the rotational angular velocity; For ease of solution, the vibration equation needs to be dimensionless: Where U0 is the centrifugal static displacement, T is the characteristic time, and Ω cr The critical speed is given by the following expressions: The dimensionless vibration equation becomes: To solve for the vibration response using the method of separation of variables, let: η(x,t)=φ(x)·Υ(t) After separating the variables and rearranging, we obtain the intrinsic equations: The eigenvalue equation is an ordinary differential equation, which yields four characteristic roots, including two real roots and two imaginary roots. The four characteristic roots are as follows: Based on the four characteristic roots mentioned above, the general form of the general solution to the differential equation can be written: η(ξ)=C1cosh(λ1ξ)+C2sinh(λ1ξ)+C3cos(λ2ξ)+C4sin(λ2ξ) In the above formula, C1-C4 are four coefficients to be determined; Substitute the boundary conditions again: -η″′(0,τ)+pη′(0,τ)=K u n(0,t) η″(0,τ)=K θ n(0,t) -η″′(1,τ)+pη′(1,τ)=-K u n(1,t) η″(1,τ)=-K θ n(1,t) Among them, K u For boundary tensile stiffness, K θ Boundary torsional stiffness, boundary tensile stiffness, and boundary torsional stiffness are two system parameters that need to be identified. Solving the system of equations simultaneously, we obtain the matrix equation: AC = 0 Among them, s1=sinh(λ1), s2=cosh(λ1), s3=sinh(λ2), s4=cosh(λ2), c1=sin(λ1), c2=cos(λ1), c3=sin(λ2), c4=cos(λ2); For the matrix equation to have a non-zero solution, the determinant of A must be equal to 0, resulting in the following equation: |A|=0 In the formula, λ1 and λ2 are both functions of the dimensionless preload p and the natural vibration frequency w, thus obtaining an equation for the dimensionless preload p and the natural vibration frequency w.

3. The theoretical method as described in claim 2, characterized in that, Step two specifically includes: An accelerometer was used to measure the vibration response of the bolt. The natural vibration frequency of the bolt is obtained by performing an FFT on the measured vibration response.

4. The theoretical method as described in claim 3, characterized in that, Step three specifically includes: Assume the system model is as follows: f(ω,p,θ)=0 In the above formula, θ is a vector composed of parameters that need to be identified, p is the dimensionless preload, and w is the natural vibration frequency. The sum of squared prediction errors is used as the loss function: Use the chain rule to calculate the gradient of the loss function. Finally, update each parameter to be identified: During the iterative update process, an adaptive learning rate is used to accelerate model convergence. The number of iterations or stopping conditions can be determined by setting a maximum number of iterations or setting a threshold.

5. The theoretical method as described in claim 4, characterized in that, Step four involves using Newton's iteration method to numerically solve the equation, gradually approximating the root through linear approximation. This includes: Initial conditions are set by selecting an initial value x0 close to the true root; a convergence threshold ∈ is set; and the maximum number of iterations N is set. max ; Iteration, In the above formula, x n This is the approximate root of the equation obtained in the nth iteration; Termination condition: The game will stop if any of the following conditions are met: The function value is small enough: |f(x) n )∣<∈; The iteration step size is small enough: |x n+1 -x n ∣<∈.

6. A theoretical device for estimating the tightness of a rotating bolt, characterized in that, The theoretical device employs any one of the theoretical methods described in claims 1-5, and includes a measurement module, a wireless transmission module, a parameter identification module, and a preload estimation module. The measurement module is used to measure the natural vibration frequency of the bolt; The parameter identification module is used to identify the parameters of the two boundary stiffnesses of the bolt model and obtain the equation between the natural vibration frequency and the preload of the bolt containing specific boundary stiffness parameter values. The wireless transmission module is used to transmit the natural vibration frequency measured by the measurement module to the parameter identification module and the preload estimation module; The preload estimation module uses the natural vibration frequency as input and estimates the bolt preload by solving the equation between the natural vibration frequency and the preload.

7. The theoretical apparatus as described in claim 6, characterized in that, The measurement module includes an accelerometer and an MCU with signal connections; An accelerometer is used to measure the vibration response of a bolt; The MCU is used to preprocess and save the vibration data measured by the accelerometer.

8. The theoretical method as described in claim 7, characterized in that, The wireless transmission module is a Bluetooth module.

9. The theoretical method as described in claim 6, characterized in that, The parameter identification module uses the gradient descent method to identify the parameters of the boundary stiffness.

10. The theoretical method as described in claim 6, characterized in that, The preload estimation module uses Newton's iterative method to solve the equations with exact numerical solutions.

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