A frequency domain-based wave function method for predicting fluid-structure coupling in liquid-filled pipelines

Abstract

The present invention relates to a method for predicting fluid-solid coupling of liquid-filled pipelines based on a wave function method within the frequency domain, comprising the following steps: S1, starting from the vibration control equation of the liquid-filled pipeline, wherein the pipeline vibration includes pipeline axial vibration, pipeline lateral vibration, and pipeline torsional vibration; S2, introducing dimensionless quantities; S3, performing a one-dimensional Fourier transform on the changed equation in the time domain and space domain; S4, obtaining homogeneous solutions and non-homogeneous solutions, and forming a general solution to the equation by superimposing and changing the homogeneous and non-homogeneous parts. The solution method includes three directions: pipeline axial vibration, lateral vibration, and torsional vibration. At the same time, the present invention is based on an accurate analytical calculation method, is easy to implement through programming, and has the advantages of excellent parametric modeling, a simple modeling process, and high computational efficiency. It can realize rapid prediction of fluid-solid coupling of liquid-filled pipelines through programming, and the prediction evaluation results can provide a reference for the vibration reduction and noise reduction design of related pipelines.

Description

Technical Field

[0001] The present invention relates to the technical field of pipeline fluid-solid coupling, in particular to a method for predicting fluid-solid coupling of a liquid-filled pipeline based on a wave function method within a frequency domain. Background Art

[0002] Liquid-filled pipelines are common in fuel and cooling systems on ships and aircraft. Vibration noise from mechanical equipment such as pumps and valves can propagate along the pipe walls. Excessive, long-term vibration and noise can cause fatigue damage to both the equipment and the pipelines. Therefore, to prevent resonance between the piping system's natural frequency and the equipment's excitation frequency, a prediction of fluid-structure interaction is necessary during the piping system design phase for reference by piping designers.

[0003] Common methods for existing pipeline calculations include Anasys and Abaqus. Both are based on the Finite Element Method (FEM) to simulate and analyze complex engineering problems. They discretize continuous physical regions into a finite number of units and predict and analyze the performance of the entire structure or system by solving the interactions between the units. Users can choose the appropriate software based on project requirements. Both provide users with accurate and efficient analysis tools to assist in product development and optimization. Traditional pipeline vibration and noise calculations require a lot of manual experience and require remodeling for different pipeline models. This is cumbersome, time-consuming, labor-intensive, and has low accuracy.

[0004] To this end, we propose a fluid-solid coupling prediction method for liquid-filled pipelines based on the wave function method in the frequency domain. Summary of the Invention

[0005] In response to the shortcomings of the above-mentioned existing production technology, the applicant provides a fluid-solid coupling prediction method for liquid-filled pipelines based on the wave function method in the frequency domain, which can obtain the general solution for the axial vibration, lateral vibration and torsional vibration of the pipeline.

[0006] At the same time, the present invention is based on a precise analytical calculation method, is easy to implement through programming, and has the advantages of excellent parametric modeling, simple modeling process, and high calculation efficiency.

[0007] The technical solutions adopted in the present invention are as follows:

[0008] A method for predicting fluid-solid coupling in a liquid-filled pipeline based on a wave function method in the frequency domain comprises the following steps:

[0009] S1. Starting from the vibration control equation of the liquid-filled pipeline, the pipeline vibration includes the pipeline axial vibration, pipeline lateral vibration and pipeline torsional vibration;

[0010] S2, at a certain point in the middle z=z0, subject to a single-point excitation force Fe iωt, taking the pipe displacement and the sound pressure in the pipe as unknown quantities, a dimensionless quantity is introduced;

[0011] S3. Perform one-dimensional Fourier transform on the transformed equation in time domain and space domain;

[0012] S4. Obtain homogeneous and inhomogeneous solutions, and form the general solution of the equation by superimposing and varying the homogeneous and inhomogeneous parts.

[0013] Furthermore, the axial vibration equation of the liquid-filled straight tube is:

[0014]

[0015] Where f z is the axial force on the pipe wall, u z is the axial displacement of the tube wall, u f is the axial displacement of the fluid in the pipe, and p is the acoustic pressure of the fluid in the pipe. The above four items are the four degrees of freedom of axial vibration of the liquid-filled pipe. R, h and A p are the inner diameter, wall thickness and cross-sectional area of ​​the pipeline respectively; ρ f and ρ p are the fluid density in the tube and the wall density respectively; E is the Young's modulus of the tube wall material; μ is the Poisson's ratio, K * is the bulk modulus of the fluid in the pipe; the pipe axis is the z direction.

[0016] Furthermore, a section of infinitely long liquid-filled pipe is pre-adopted in S2, and a single-point excitation force Fe is applied at a certain point z=z0 in the middle. iωt , taking the pipe displacement and the sound pressure in the pipe as unknown quantities, introduce the dimensionless quantity:

[0017] Transform the equations into

[0018]

[0019] Performing one-dimensional Fourier transform in both time and space domains on both sides of the equations yields:

[0020]

[0021] In the above formula, is the dimensionless wave number.

[0022] Furthermore, the homogeneous solution is the eigenvector at different frequencies, corresponding to the characteristic wave number. For the homogeneous equation, move the term containing the wave number to the right side of the equal sign and leave the rest on the left side to obtain the characteristic equation:

[0023]

[0024] The above formula (4) is a generalized eigenvalue problem. Solving this problem can obtain the eigenvalue and the corresponding eigenvector. The homogeneous solution of the pipeline axial vibration is:

[0025]

[0026] In the above formula, c n is the unknown coefficient.

[0027] Furthermore, applying Cram's rule to solve the nonhomogeneous equations (3) yields the nonhomogeneous solution

[0028]

[0029] In the above formula, Q(λ) is the determinant of the coefficient matrix.

[0030] Furthermore, performing an inverse Fourier transform on the solution of equation (6) above will yield the particular solution of the equation:

[0031]

[0032] Applying the residue theorem, we get:

[0033]

[0034] In the above formula, λ k is the characteristic wave number of equation (3) in the upper half plane of the complex plane. Adding equations (5) and (8) yields the following general solution:

[0035]

[0036] Furthermore, the general solution for the transverse vibration of the pipeline is:

[0037]

[0038] Furthermore, the general solution of the pipeline torsional vibration is:

[0039]

[0040] The beneficial effects of the present invention are as follows:

[0041] This paper proposes a frequency-domain wave function-based method for predicting fluid-structure interaction in liquid-filled pipelines. Starting from the vibration control equation for liquid-filled pipelines, dimensionless parameters are introduced and the transformed equation is subjected to a one-dimensional Fourier transform in both the time and spatial domains. The general solution of this transformed equation is expressed as a superposition of homogeneous and inhomogeneous components. This solution addresses three aspects of pipeline vibration: axial, lateral, and torsional.

[0042] At the same time, the present invention is based on a precise analytical calculation method, is easy to implement through programming, has the characteristics of excellent parametric modeling, simple modeling process, and high calculation efficiency. It can realize the rapid prediction of fluid-solid coupling of liquid-filled pipelines through programming, and the prediction evaluation results can provide a reference for the vibration and noise reduction design of related pipelines. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a pipeline subjected to concentrated excitation force in the present invention.

[0044] Figure 2 This is the pipe displacement response diagram when lateral excitation is applied to a single pipe in the present invention.

[0045] Figure 3 This is a comparison diagram of the fluid velocity in the pipe when axial excitation is applied to a single pipe in the present invention. DETAILED DESCRIPTION

[0046] The specific embodiments of the present invention will be described below with reference to the accompanying drawings.

[0047] The present invention proposes a method for predicting fluid-solid coupling of liquid-filled pipelines based on the wave function method in the frequency domain. Starting from the vibration control equation of the liquid-filled pipeline, dimensionless parameters are introduced, and the changed equation is subjected to a one-dimensional Fourier transform in the time domain and space domain. The general solution of the above-mentioned changed equation is written as a superposition of homogeneous and non-homogeneous parts. The solution method includes three directions: axial vibration, lateral vibration, and torsional vibration of the pipeline. Through the present invention, rapid prediction of fluid-solid coupling of liquid-filled pipelines can be achieved through programming, and the prediction evaluation results can provide a reference for the design of vibration and noise reduction of related pipelines.

[0048] In this embodiment, the axial vibration equation of the liquid-filled straight tube is:

[0049]

[0050] Where f z is the axial force on the pipe wall, u z is the axial displacement of the tube wall, u f is the axial displacement of the fluid in the pipe, and p is the acoustic pressure of the fluid in the pipe. The above four items are the four degrees of freedom of axial vibration of the liquid-filled pipe. R, h and A p are the inner diameter, wall thickness and cross-sectional area of ​​the pipeline respectively; ρ f and ρ p are the fluid density in the tube and the wall density respectively; E is the Young's modulus of the tube wall material; μ is the Poisson's ratio, K * is the bulk modulus of the fluid in the pipe; the pipe axis is the z direction.

[0051] In this embodiment, a section of infinitely long liquid-filled pipe is used, and a single-point excitation force Fe is applied at a certain point z=z0 in the middle. i ωt, taking the pipe displacement and the sound pressure in the pipe as unknown quantities, the dimensionless quantity is introduced Transform the equations into

[0052]

[0053] Perform one-dimensional Fourier transform on both sides of the equations in time and space domains to obtain

[0054]

[0055] In the above formula, is the dimensionless wave number. For simplicity and clarity, the “~” symbol above the dimensionless quantity is omitted below. The solution of the above system of equations contains homogeneous and inhomogeneous parts. The homogeneous solution is the eigenvector at different frequencies, corresponding to the characteristic wave number. To align the equation, move the term containing the wave number to the right side of the equal sign and leave the rest on the left side of the equal sign to obtain the characteristic equation

[0056]

[0057] The above equation (4) is a generalized eigenvalue problem. Solving this problem can obtain the eigenvalue and the corresponding eigenvector. The homogeneous solution of the pipeline axial vibration is:

[0058]

[0059] In the above formula, c n are unknown coefficients. Applying Cram’s rule to solve the nonhomogeneous equations (25), we obtain the solution

[0060]

[0061] In the above formula, Q(λ) is the determinant of the coefficient matrix. Performing inverse Fourier transform on the solution of formula (6) will yield the particular solution of the equation

[0062]

[0063] Applying the residue theorem, we get

[0064]

[0065] In the above formula, λ k is the characteristic wave number of equation (25) in the upper half plane of the complex plane. Adding equations (27) and (30) yields a general solution of the form

[0066]

[0067] For the sake of simplicity, the above formula only gives the solution for semi-infinite domain.

[0068] Similarly, the control equation for the lateral vibration of the pipeline is:

[0069]

[0070] For the transverse vibration equation of the pipe in the xoz plane, the dimensionless quantity is introduced The equations are dimensionless and Fourier transformed to obtain the equations

[0071]

[0072] In the above formula, The generalized eigenvalue problem corresponding to equation (11) is

[0073]

[0074] Solving the above eigenvalue problem and equation (33) can obtain the homogeneous and nonhomogeneous solutions of the partial differential equations respectively, and superimposing the two can obtain the general solution

[0075]

[0076] The same method can be used to obtain the solution of the pipeline in the yoz plane. Since the equations of motion in the yoz plane are different from those in the xoz plane in terms of signs, the wave numbers and eigenvectors obtained are different. Therefore, the general solution in the yoz plane is expressed as

[0077]

[0078] According to the torsional vibration equation of the pipeline, the non-homogeneous equation and characteristic equation after Fourier transformation can be directly written

[0079]

[0080]

[0081] In the above formula, Combining the solutions of the above two equations, we can get the general solution of the differential equation as follows:

[0082]

[0083] Combining the above solutions, we can get the total wave function solution of the straight tube. The homogeneous part and the inhomogeneous part of the solution are the superposition of 14 and 7 wave components respectively.

[0084]

[0085] but

[0086]

[0087] The liquid-filled straight pipe has a total of 6 structural displacement degrees of freedom and 1 acoustic pressure degree of freedom. The vector v is transformed to be divided into two groups, one of which is the variables corresponding to the 7 degrees of freedom, and the other is the corresponding derived variables. The two groups of variables are sorted in a certain order to obtain

[0088]

[0089] Through such a transformation, the eigenvectors on the right side of Equation (41) will also be divided into two groups.

[0090] The present invention is based on an accurate analytical calculation method, is easy to implement through programming, and has the advantages of excellent parametric modeling, simple modeling process, and high calculation efficiency.

[0091] like Figure 2 and Figure 3 As shown, the calculation results of the wave function calculation pipeline proposed in the patent are compared as follows:

[0092] By comparing with the traditional finite element FEM, the pipeline calculation method proposed in the patent of this invention basically overlaps with the calculation results of the finite element software, indicating that the method proposed in the patent of this invention is correct and consistent.

[0093] The above description is an explanation of the present invention, not a limitation of the present invention. The scope of the present invention is defined in the claims. Any modifications may be made within the scope of protection of the present invention.

Claims

1. A method for predicting fluid-solid coupling in liquid-filled pipelines based on the wave function method in the frequency domain, characterized in that: The steps include: S1. Starting from the vibration control equation of the liquid-filled pipeline, the pipeline vibration includes the pipeline axial vibration, pipeline lateral vibration and pipeline torsional vibration; S2. At a point in the middle Inspired by a single point , taking the pipe displacement and the sound pressure in the pipe as unknown quantities, a dimensionless quantity is introduced; S3. Perform one-dimensional Fourier transform on the transformed equation in time domain and space domain; S4. Obtaining a homogeneous solution: The homogeneous solution is the eigenvector at different frequencies, corresponding to the characteristic wavenumber. The equation obtained in S3 is adjusted by moving the term containing the wavenumber to the right side of the equal sign and leaving the rest on the left side to obtain the characteristic equation, which is a generalized eigenvalue problem. Solving this problem can obtain the eigenvalue and the corresponding eigenvector, and obtain the homogeneous solution for the pipeline axial vibration; Obtaining nonhomogeneous solutions: Apply Cram's rule to solve the nonhomogeneous equations (3) and obtain nonhomogeneous solutions; The general solution of the equation is formed by superimposing and changing the homogeneous and inhomogeneous parts; The general solution for the transverse vibration of the pipeline is: (13) The general solution of the pipeline torsional vibration is: (17)。 2. The method for predicting fluid-solid coupling in a liquid-filled pipeline based on a wave function method in the frequency domain according to claim 1, characterized in that: The axial vibration equation of the liquid-filled straight tube is: (1) In the formula is the axial force on the pipe wall, is the axial displacement of the tube wall, is the axial displacement of the fluid in the tube, is the sound pressure of the fluid in the pipe; the above four items are the four degrees of freedom of axial vibration of the liquid-filled pipeline, R, h and are the inner diameter, wall thickness and cross-sectional area of ​​the pipeline respectively; and are the fluid density in the tube and the wall density respectively; E is the Young's modulus of the tube wall material; is Poisson's ratio, is the bulk modulus of the fluid in the pipe; the pipe axis is the z direction.

3. The method for predicting fluid-solid coupling in a liquid-filled pipeline based on a wave function method in the frequency domain according to claim 2, characterized in that: In the S2, a fluid-filled pipe of infinite length is pre-adopted, at a certain point in the middle. Inspired by a single point , taking the pipe displacement and the sound pressure in the pipe as unknown quantities, introduce the dimensionless quantity: , , , , , , changing the equations into (2) Performing one-dimensional Fourier transform in both time and space domains on both sides of the equations yields: (3) In the above formula, , , , is the dimensionless wave number.

4. The method for predicting fluid-solid coupling in a liquid-filled pipeline based on a wave function method in the frequency domain according to claim 1, characterized in that: The characteristic equation in S4 is: (4) The above formula (4) is a generalized eigenvalue problem. Solving this problem can obtain the eigenvalue and the corresponding eigenvector. The homogeneous solution is: (5) In the above formula, is the unknown coefficient.

5. The method for predicting fluid-solid coupling in a liquid-filled pipeline based on a wave function method in the frequency domain according to claim 1, characterized in that: Apply Cram's rule to solve the nonhomogeneous equations (3) and obtain the nonhomogeneous solution (6) In the above formula, is the determinant of the coefficient matrix.

6. The method for predicting fluid-solid coupling in a liquid-filled pipeline based on a wave function method in the frequency domain according to claim 5, characterized in that: Performing inverse Fourier transform on the solution of equation (6) above will yield the specific solution of the equation: (7) Applying the residue theorem, we get: (8) In the above formula, is the characteristic wave number of equation (3) in the upper half plane of the complex plane. Adding equations (5) and (8) yields the following general solution: (9)。

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