A calculation method for vibration and stress fatigue analysis of nonlinear boundary hydraulic pipelines

Abstract

This invention discloses a computational method for vibration and stress fatigue analysis of hydraulic pipelines under nonlinear boundary conditions. The method first establishes the dynamic control equations of the hydraulic pipeline under nonlinear boundary conditions, then solves and verifies the system using the harmonic balance method and the Runge-Kutta method to obtain the response and stress distribution of the entire pipeline. Based on the calculation results, the tensile and bending stresses at arbitrary locations in the pipeline are further solved. This identifies the location with the highest stress, and fatigue life is predicted based on the maximum stress. This method more closely approximates the actual working conditions of hydraulic pipelines, improves the accuracy of stress analysis, and contributes to in-depth research on the fatigue failure mechanisms of hydraulic pipelines, providing theoretical support for engineering design and fault early warning.

Description

Technical Field

[0002] This invention relates to the field of pipeline engineering technology, specifically to a method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines. Background Technology

[0004] Hydraulic pipelines, as a key component of hydraulic systems, are widely used in automobiles, aerospace, ships, and various engineering machinery. Their structural stability and operational reliability have a significant impact on the overall system performance. In actual operation, hydraulic pipelines often endure pulsating loads from internal high-pressure fluids as well as external mechanical or environmental excitations, resulting in complex vibration responses. Hydraulic pipelines exposed to severe vibration for extended periods are prone to fatigue damage, potentially leading to ruptures, leaks, and other serious malfunctions, posing significant risks to equipment operation and personnel safety, causing severe economic losses, and even triggering catastrophic accidents. Currently, research on the vibration characteristics of hydraulic pipelines is largely based on ideal boundary conditions such as simply supported, fixed, or cantilevered structures. While this modeling approach simplifies the analysis process to some extent, it fails to accurately reflect the boundary constraint characteristics of hydraulic pipelines in actual engineering applications. In practical applications, the boundaries of hydraulic pipelines are often influenced by complex support structures, flexible connectors, and environmental excitations, exhibiting strong nonlinear characteristics, leading to significant shortcomings in the accuracy and applicability of existing methods. Therefore, it is necessary to conduct stress and fatigue analysis research that better reflects engineering realities, based on the physical nature of hydraulic pipeline vibration and combined with nonlinear boundary conditions.

[0005] To address the aforementioned issues, there is an urgent need to propose a vibration and stress fatigue analysis and calculation method for hydraulic pipelines under nonlinear boundary conditions. This method aims to improve the accuracy of the analysis results and enhance their guiding value for engineering applications, thereby providing theoretical basis and technical support for the structural optimization design and fault early warning of related equipment. Summary of the Invention:

[0006] To more closely approximate the actual boundary conditions of hydraulic pipelines and improve the accuracy of fatigue dynamic stress in hydraulic pipelines for prediction and optimization design of hydraulic pipeline vibration fatigue life, this paper proposes a nonlinear boundary hydraulic pipeline vibration and stress fatigue calculation method.

[0007] Specifically, the following steps are included:

[0008] S1. Based on the structural characteristics, material parameters, geometric dimensions, and internal and external load conditions of the hydraulic pipeline, establish the dynamic control differential equations of the hydraulic pipeline under nonlinear boundary constraints. The control equations and boundary conditions of the hydraulic pipeline are derived using the generalized Hamilton's principle.

[0009] S2. Solve for the frequency and modes of the hydraulic pipeline.

[0010] S21. Ignoring the effects of fluid flow, damping, and nonlinear terms in the governing equations, the governing equations and boundary conditions of the linearized system are obtained. The natural frequencies and modes of the pipe can then be solved.

[0011] S22. The modal functions obtained by solving the linearized equations are used as trial functions and weight functions in the Galerkin truncation method to solve for the frequency and modes of the nonlinear boundary hydraulic pipeline.

[0012] S3. First, the Galerkin truncation method (GTM) is used to discretize the governing equations in space into a system of ordinary differential equations. Then, the harmonic balance method (HBM) and the Runge-Kutta method (RK) are used to solve the system of equations to obtain the amplitude-frequency response.

[0013] S31. To verify the correctness and stability of the frequency domain solution, the fourth-order Runge-Kutta method is used to perform time-domain integration on the original nonlinear control equations to obtain the system's time response under given initial conditions and loads.

[0014] S32. To solve for the steady-state response of the system under periodic excitation, the harmonic balance method is introduced. The system response is expressed as a combination of a finite number of sine and cosine functions, which are then substituted into the governing equations to obtain the amplitude-frequency response characteristic curve of the system. The solution is then compared and verified with the fourth-order Runge-Kutta result to ensure its reliability.

[0015] S4. Based on stress theory, calculate the tensile stress and bending stress on the neutral surface of the hydraulic pipeline. Considering that the tensile vibration stress is uniformly distributed on the pipeline cross-section, while the bending vibration stress occurs at the top and bottom of the pipeline cross-section, accurately assess the stress state and select the bottom of the pipeline as the analysis location.

[0016] S41. Solve for the tensile stress on the neutral surface of a hydraulic pipeline.

[0017] S42. Solve for the bending stress of the hydraulic pipeline.

[0018] S43. The values ​​of tensile vibration stress and bending vibration stress are added together to evaluate the fatigue stress of the pipe.

[0019] S5. By analyzing the stress distribution and fatigue life calculation results of the entire pipeline, the region with the greatest stress or the shortest life is determined, providing a basis for pipeline structure improvement, support optimization, or maintenance strategies.

[0020] Preferably, the total energy of the hydraulic pipeline in S11 is specifically: kinetic energy and strain energy. Let m1 and v1 be the mass and velocity of the infinitesimal elements in the pipe, and m2 and v2 be the mass and velocity of the fluid within the infinitesimal elements in the pipe. The governing equations and boundary conditions for the pipe are derived based on the generalized Hamiltonian principle and variational principles. Since the modal expansion method is difficult to directly handle nonlinear and time-dependent boundary conditions, the δ(x) function is used to realize its spatial positioning. The linear support reaction forces at both ends of the hydraulic pipeline are retained in the boundary conditions, while the nonlinear support reaction forces are regarded as external excitation terms and introduced into the control equation.

[0021] Preferably, after obtaining the linearized system equations in S2, its natural frequencies are calculated, specifically as follows:

[0022] , Assume the solution is: , to obtain C n The coefficient matrix is ​​as follows: To ensure that the coefficients C in the modal solution in If not all elements are zero, the determinant of the matrix is ​​zero. Also, let C... 1n =1, thus obtaining the modal functions of the pipeline. Using the modal functions of a linear static pipeline as the trial function and weight function, the corresponding linear derived system is: The free vibration displacement of the conveying pipeline can be set as: ,according to

[0023] The eigenvalue ω can be calculated as the natural frequency of the pipe's lateral free vibration. For a given eigenvalue, the complex eigenvector Q can be determined. k Therefore, the modal function of the lateral free vibration of the fluid transport pipeline can be expressed as: .

[0024] Preferably, in S31, the governing equations are truncated into ordinary differential equations before being solved. The specific calculation is as follows: The solution to the governing equations is set as: In the formula, N is an integer greater than or equal to 1, and ϕ n (x) is the modal function of a linear hydraulic pipeline, q n (t) represents the generalized displacement due to the lateral vibration of the hydraulic pipeline. Multiply both sides of the equation by the trial function φ. k (x), and integrate from 0 to L. Finally, the Runge-Kutta method is used for numerical solution.

[0025] Preferably, in S32, the governing equations can be rewritten in standard form using the harmonic balance method, as specifically calculated below: The solution is set as follows: By organizing the coefficients corresponding to each harmonic, we can obtain: The unknown coefficient z in the formula can be solved using Newton's iteration method. However, due to the singularity of the iteration matrix, the iteration method may fail at inflection points. This problem can be solved using a prediction correction method, namely the pseudo-arc length method. The basic idea of ​​this method is to introduce an undetermined coefficient ω, i.e.: This yields the following nonlinear algebraic equation: Where z0 and ω0 represent the values ​​of z and ω in the previous iteration step, respectively. ∆s represents the increment of the arc length. The corresponding Jacobian matrix J is... The corresponding unit tangent vector is: The prediction process can be represented as: In the formula, s is the arc length. The initial value problem in the formula can be solved using the classical Euler method, and its calculation formula is: , where y p For predicting the solution, y0 is the initial value. The calculation is performed using the iterative format shown below: .

[0026] Preferably, the displacement and velocity at any point in S41 can be obtained using the harmonic balance method in S32, and the tensile stress equation is derived based on Lagrange strain. The calculation method is as follows:

[0027] Preferably, creep effects can be neglected when calculating bending stress in S42. Based on the small deflection theory and the relationship between strain and displacement, the dynamic stress-strain expression caused by pipe vibration can be derived: The calculation method is as follows: Where D is the outer diameter of the hydraulic pipe and E is the elastic modulus. Trial function, The frequency of the excitation.

[0028] Preferably, the service life calculation method in S5 is as follows: The relationship between stress intensity and crack propagation rate is established: , where a is the crack size, N is the number of stress cycles, and m and C are material constants. The range of stress intensity factor variation: Crack propagation life: Where a0 represents the initial crack, a c This represents the critical crack. f(a) is a standard geometric function with finite boundaries: Where b is the pipe circumference. Under constant amplitude stress, N can be expressed as:

[0029]

[0030] After obtaining the number of stress cycles N, the service life is determined: .

[0031] As can be seen from the above technical solution, this invention discloses a method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines. Compared with existing methods, it has the following beneficial effects:

[0032] (1) This invention considers the nonlinear boundary conditions commonly encountered in the actual operation of hydraulic pipelines, such as flexible support, elastic clamping and multi-point connection, and breaks through the limitations of traditional ideal boundary models such as simply supported, fixed support and cantilever, so as to more realistically reflect the actual working state of hydraulic pipelines and improve the modeling accuracy.

[0033] (2) By solving the amplitude-frequency response of the system under nonlinear boundary conditions and combining it with the point stress calculation method, the distribution law of tensile stress and bending stress in the pipeline can be accurately identified, providing a scientific basis for the identification of stress concentration points and structural safety assessment.

[0034] (3) This invention uses the Galerkin method to spatially discretize the entire pipeline and obtain the response and stress distribution of the entire pipeline. It can identify the stress distribution and lifetime distribution of the entire pipeline structure and generate stress field diagrams or lifetime diagrams.

[0035] (4) This invention calculates the maximum stress response under real working conditions and then evaluates the life of hydraulic pipelines based on fatigue theory, making the prediction results more reliable and of engineering reference value, which helps to extend the service life of pipelines and formulate reasonable maintenance cycles.

[0036] (5) The vibration response and stress results obtained by the present invention can be used to optimize the support arrangement and damping structure design of hydraulic pipelines, providing effective technical support for the structural optimization and safety design of hydraulic systems. Attached Figure Description

[0038] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort, wherein:

[0039] Figure 1 shows a mathematical model for analyzing vibration and stress fatigue in nonlinear boundary hydraulic pipelines according to the present invention.

[0040] Figure 2 This is a schematic flowchart of a method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines according to the present invention.

[0041] Figure 3 This is a comparison diagram of numerical and analytical methods for the vibration and stress fatigue analysis calculation method of a nonlinear boundary hydraulic pipeline according to the present invention.

[0042] Figure 4 This invention provides a stress distribution diagram along the cross-section of a hydraulic pipeline for a nonlinear boundary hydraulic pipeline vibration and stress fatigue analysis calculation method.

[0043] Figure 5 This invention provides a method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines, including a diagram showing the total stress distribution in a hydraulic pipeline. Specific implementation methods

[0044] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. It should be understood that the described embodiments are only some embodiments of the present invention, and 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.

[0045] In the following description, when referring to the accompanying drawings, the same numbers in different drawings denote the same or similar elements unless otherwise indicated. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application. Rather, they are merely examples of apparatuses and methods consistent with some aspects of this application as detailed in the appended claims.

[0046] This invention proposes a method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines, such as... Figure 1 As shown, it includes the following steps:

[0047] S1. Based on the structural characteristics, material parameters, geometric dimensions, and internal and external load conditions of the hydraulic pipeline, establish the dynamic control differential equations of the hydraulic pipeline under nonlinear boundary constraints. The control equations and boundary conditions of the hydraulic pipeline are derived using the generalized Hamilton's principle.

[0048] S2. Solve for the natural frequencies and modes of the hydraulic pipeline.

[0049] S21. Ignoring the effects of fluid flow, damping, and nonlinear terms in the governing equations, the governing equations and boundary conditions of the linearized system are obtained. The natural frequencies and modes of the pipe can then be solved.

[0050] S22. The modal functions obtained by solving the linearized equations are used as trial functions and weight functions in the Galerkin truncation method to solve for the frequency and modes of the nonlinear boundary hydraulic pipeline.

[0051] S3. First, the Galerkin truncation method (GTM) is used to discretize the governing equations in space into a system of ordinary differential equations. Then, the harmonic balance method (HBM) and the Runge-Kutta method (RK) are used to solve the system of equations to obtain the amplitude-frequency response.

[0052] S31. To verify the correctness and stability of the frequency domain solution, the fourth-order Runge-Kutta method is used to perform time-domain integration on the original nonlinear control equations to obtain the system's time response under given initial conditions and loads.

[0053] S32. To solve for the steady-state response of the system under periodic excitation, the harmonic balance method is introduced. The system response is expressed as a combination of a finite number of sine and cosine functions, which are then substituted into the governing equations to obtain the amplitude-frequency response characteristic curve of the system. The solution is then compared and verified with the fourth-order Runge-Kutta result to ensure its reliability.

[0054] S4. Based on stress theory, calculate the tensile stress and bending stress on the neutral surface of the hydraulic pipeline. Considering that the tensile vibration stress is uniformly distributed on the pipeline cross-section, while the bending vibration stress occurs at the top and bottom of the pipeline cross-section, accurately assess the stress state and select the bottom of the pipeline as the analysis location.

[0055] S41. Solve for the tensile stress on the neutral surface of a hydraulic pipeline.

[0056] S42. Solve for the bending stress of the hydraulic pipeline.

[0057] S43. The values ​​of tensile vibration stress and bending vibration stress are added together to evaluate the fatigue stress of the pipe.

[0058] S5. By analyzing the stress distribution and fatigue life calculation results of the entire pipeline, the region with the greatest stress or the shortest life is determined, providing a basis for pipeline structure improvement, support optimization, or maintenance strategies.

[0059] To further implement the above technical solution, the specific calculation of the hydraulic pipeline mathematical model in S1 is as follows:

[0060] Kinetic and strain energy of the pipeline:

[0061] Where ρ p and A p These represent the pipe material density and cross-sectional area of ​​the conveying pipe, ρ. f and A f Let be the density and cross-sectional area of ​​the liquid inside the pipe. u(x, t) is the lateral vibration displacement of the pipe, and w(x, t) is the axial displacement of the pipe. The axial strain of the pipe element is: The elastic constitutive relation of the pipeline is given by the following equation: , where μ is a viscoelastic parameter derived from the Kelvin material derivative. Work done by the supporting reaction force: According to the generalized Hamiltonian principle: The governing equations and boundary conditions of the pipeline are obtained by using variational principles:

[0062]

[0063]

[0064] The linear support reactions at both ends of the hydraulic pipeline are retained in the boundary conditions, while the nonlinear support reactions are treated as external excitations and introduced into the governing equations, with their spatial location achieved using the δ(x) function. The governing equations and corresponding boundary conditions can then be reformulated as follows:

[0065] To further implement the above technical solution, after obtaining the linearized system equations in S2, its natural frequencies are calculated, specifically as follows:

[0066] Assume the solution is:

[0067]

[0068] Get C i n The coefficient matrix is ​​as follows: ,in:

[0069] To make the coefficients C in the modal solution in If not all elements are zero, the determinant of the matrix is ​​zero. Also, let C... 1n =1, so the modal function of the pipe can be obtained:

[0070] Using the modal functions of a linear static pipeline as the trial function and weight function, the corresponding linear derived system is: The free vibration displacement of the conveying pipeline can be set as: ,according to

[0071] The eigenvalue ω can be determined as the natural frequency of the pipe's lateral free vibration. For a given eigenvalue, the complex eigenvector Q can be determined. k Therefore, the modal function of the lateral free vibration of the fluid transport pipeline can be expressed as: .

[0072] To further implement the above technical solution, the governing equations in S31 are first truncated using the GTM method, and the amplitude-frequency response is solved using the fourth-order Runge-Kutta method. The specific calculations are as follows:

[0073] First, based on the Galerkin truncation method, the governing equations of the nonlinear boundary hydraulic pipeline are discretized into a set of coupled ordinary differential equations. Assume the solution to the governing equations is: In the formula, N is an integer greater than or equal to 1, and ϕ n (x) is the modal function of a linear hydraulic pipeline, q n (t) represents the generalized displacement due to lateral vibration of the hydraulic pipeline. The weighting function ψ in the Galerkin method... m Let (x) be the modal function ϕ. n The form (x) can be simplified by integrating x over the range from 0 to L as follows:

[0074] in:

[0075] The amplitude-frequency response can be obtained by solving the above ordinary differential equation using the fourth-order Runge-Kutta method.

[0076] To further implement the above technical solution, the governing equations in S32 can be rewritten in standard form using the harmonic balance method, as calculated below: The solution is set as follows:

[0077] The third harmonic assumption is as follows: By organizing the coefficients corresponding to each harmonic, we can obtain: The unknown coefficient z in the formula can be solved using Newton's iteration method. However, due to the singularity of the iteration matrix, the iteration method may fail at inflection points. This problem can be solved using a prediction correction method, namely the pseudo-arc length method. The basic idea of ​​this method is to introduce an undetermined coefficient ω, i.e.: The following nonlinear algebraic equation is obtained:

[0078]

[0079] Where z0 and ω0 represent the values ​​of z and ω in the previous iteration step, respectively. ∆s represents the increment of the arc length. The corresponding Jacobian matrix J is...

[0080] The corresponding unit tangent vector is: The prediction process can be represented as: In the formula, s is the arc length. The initial value problem in the formula can be solved using the classical Euler method, and its calculation formula is: , where y pFor predicting the solution, y0 is the initial value. The calculation is performed using the iterative format shown below:

[0081] To further implement the above technical solution, the displacement and velocity at any point in S41 can be obtained using the harmonic balance method in S32, and the tensile stress equation is derived based on Lagrange strain. The calculation method is as follows:

[0082] To further implement the above technical solution, creep effects can be neglected when calculating bending stress in S42. Based on the small deflection theory and the relationship between strain and displacement, the dynamic stress-strain expression caused by pipe vibration can be derived. The calculation method is as follows: Where D is the outer diameter of the hydraulic pipe and E is the elastic modulus. Trial function, The frequency of the excitation.

[0083] To further implement the above technical solution, the service life calculation method in S5 is as follows: The relationship between stress intensity and crack propagation rate is established using Paris theory:

[0084] , where a is the crack size, N is the number of stress cycles, and m and C are material constants. The range of stress intensity factor variation: Crack propagation life: Where a0 represents the initial crack, a c This represents the critical crack. f(a) is a standard geometric function with finite boundaries: Where b is the circumference of the pipe. Since b is much larger than a, f a Treated as a constant of 1, N can be expressed as follows under constant amplitude stress: After obtaining the number of stress cycles N, the service life is determined:

[0085] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application, and should all be included within the protection scope of this application.

Claims

1. A method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines, characterized in that, Includes the following steps: S1. Based on the structural characteristics, material parameters, geometric dimensions, and internal and external load conditions of the hydraulic pipeline, establish the dynamic control differential equations of the hydraulic pipeline under nonlinear boundary constraints. The control equations and boundary conditions of the hydraulic pipeline are derived using the generalized Hamiltonian principle. S2. Solve for the frequency and modes of the hydraulic pipeline. S3. First, use the Galerkin truncation method to discretize the control equations in space into a system of ordinary differential equations. Then, use the harmonic balance method (HBM) and the Runge-Kutta method to solve this system of equations, obtaining the amplitude-frequency response. S4. Based on stress theory, calculate the tensile stress and bending stress on the neutral surface of the hydraulic pipeline. Considering that the tensile vibration stress is uniformly distributed across the pipeline cross-section, while the bending vibration stress occurs at the top and bottom of the pipeline cross-section, accurately assess the stress state and select the bottom of the pipeline as the analysis location. S5. Through analysis of the stress distribution and fatigue life calculation results of the entire pipeline, determine the region with the maximum stress or the shortest life, providing a basis for pipeline structure improvement, support optimization, or maintenance strategies.

2. The method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines according to claim 1, characterized in that, Based on the generalized Hamilton's principle and variational principles, the pipeline control equations and boundary conditions are derived. Since the modal expansion method is difficult to directly handle nonlinear and time-dependent boundary conditions, the δ(x) function is used to realize its spatial positioning. The linear support reaction forces at both ends of the hydraulic pipeline are retained in the boundary conditions, while the nonlinear support reaction forces are regarded as external excitation terms and introduced into the control equation.

3. The method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines according to claim 1, characterized in that, Ignoring the effects of flow velocity, external excitation, and nonlinear terms, after obtaining the linearized system equations in S2, its natural frequencies and modes are calculated.

4. The method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines according to claim 1, characterized in that, In S32, the amplitude-frequency response of the system is obtained according to the harmonic balance method and the pseudo-arc length method, and then verified using the Runge-Kutta method.

5. The method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines according to claim 1, characterized in that, The displacement and velocity at any point in S41 can be obtained by the harmonic balance method in S32, and the tensile stress equation is obtained based on Lagrange strain.

6. The method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines according to claim 1, characterized in that, When calculating bending stress in S42, creep effects can be ignored. Based on the small deflection theory and the relationship between strain and displacement, the dynamic stress-strain expression caused by pipeline vibration can be derived, and the calculation method is as follows: Where D is the outer diameter of the hydraulic pipe and E is the elastic modulus. Trial function, The frequency of the excitation.

7. The method for analyzing and calculating vibration and stress fatigue in nonlinear boundary hydraulic pipelines according to claim 1, characterized in that, The service life calculation method in S5 is as follows: The relationship between stress intensity and crack propagation rate is established using Paris theory: Where a is the crack size, N is the number of stress cycles, and m and C are material constants. The range of stress intensity factor variation: Crack propagation life: Where a0 represents the initial crack, a c This represents the critical crack. f(a) is a standard geometric function with finite boundaries: Where b is the pipe circumference, and under constant amplitude stress, N can be expressed as: After obtaining the number of stress cycles N, the service life is determined: .

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