A semi-analytical dynamic modeling method for an L-shaped fluid conveying pipeline

Abstract

The present invention provides a semi-analytical dynamic modeling method for an L-shaped fluid delivery pipeline, comprising: based on a semi-analytical method, decomposing the L-shaped pipeline into a straight-bend-straight model, and establishing a dynamic model of the L-shaped fluid delivery pipeline system; performing model verification on the established dynamic model of the L-shaped fluid delivery pipeline system to obtain a verified model; based on the verified model, simulating the influence of fluid parameters on the natural frequency of the L-shaped fluid delivery pipeline, and verifying the correctness of the dynamic model of the L-shaped fluid delivery pipeline system. The present invention uses ANSYS software and experimental tests to obtain the modal vibration mode and natural frequency of the L-shaped pipeline clamp system, and compares them with the semi-analytical analysis results to verify the effectiveness of the method. The influence of flow rate and pressure on the natural frequency is also studied, and compared with the finite element method, the change trend is consistent. Numerical calculation results show that the semi-analytical method has high accuracy in calculating the natural frequency of the L-shaped fluid delivery pipeline.

Description

Technical Field

[0001] The present invention relates to the field of mechanical dynamics technology, and in particular to a semi-analytical dynamics modeling method for an L-shaped fluid delivery pipeline. Background Art

[0002] Pipelines are crucial components connecting aircraft engine lubricating oil systems, fuel systems, regulating systems, starting systems, and other accessories. When fluid flows through these pipelines, vibration and resonance within the pipeline system can induce further failures, directly impacting aircraft engine testing progress. Therefore, studying the dynamics of fluid transport within pipelines is a crucial issue. In academic research, aviation hydraulics typically simplifies pipelines into a fluid transport model and a clamp support model.

[0003] Due to the internal geometry constraints of aircraft engines, straight or curved pipes alone are no longer sufficient for engineering purposes. Therefore, the dynamic characteristics of combined straight and curved fluid delivery pipelines warrant investigation. Among these, the L-shaped pipeline model is closer to engineering practice, and the research results are more practical. Currently, extensive research has been conducted on the dynamic analysis of fluid delivery pipelines. To better analyze the impact of fluid-structure interaction (FSI) on the inherent characteristics of fluid delivery pipelines, researchers have employed methods such as the transfer matrix method, semi-analytical methods, and the finite element method to establish pipeline system models. The transfer matrix method is suitable for chained pipeline structures and has a wide range of applications, but it often suffers from numerical instability. The finite element method has advantages in handling complex pipeline configurations and boundary conditions, but suffers from low computational efficiency. The semi-analytical method is often used to address nonlinear vibration problems in fluid delivery pipelines, offering high solution efficiency. However, while it is suitable for single pipes, it is difficult to apply to complex piping systems. Consequently, research on the application of semi-analytical methods to L-shaped pipeline modeling is limited. Summary of the Invention

[0004] In response to the technical problems raised above, a semi-analytical dynamic modeling method for an L-shaped fluid delivery pipeline is provided. The method of the present invention applies a semi-analytical method to model the L-shaped fluid delivery pipeline, and has high solution efficiency while ensuring calculation accuracy.

[0005] The technical means adopted in the present invention are as follows:

[0006] A semi-analytical dynamic modeling method for an L-shaped fluid delivery pipeline includes:

[0007] Based on the semi-analytical method, the L-shaped pipeline is decomposed into a straight-bend-straight model, and the dynamic model of the L-shaped fluid transmission pipeline system is established;

[0008] The established L-shaped fluid transmission pipeline system dynamics model is verified to obtain a verified model;

[0009] Based on the verified model, the influence of fluid parameters on the natural frequency of the L-shaped fluid pipeline is simulated to verify the correctness of the dynamic model of the L-shaped fluid pipeline system.

[0010] Furthermore, the semi-analytical method is used to establish a dynamic model of the L-shaped fluid delivery pipeline system, including:

[0011] Establish the pipe bending control equation;

[0012] Establish the straight pipe control equation;

[0013] The control equations of the curved pipe and the straight pipe are combined to establish the dynamic equations of the L-shaped fluid conveying pipeline system.

[0014] Furthermore, the establishment of the bend control equation specifically includes:

[0015] Use u respectively x 、u y and u z The displacements in the x1, y1, and z1 directions of any point in the elbow are expressed as follows:

[0016]

[0017] u y (x,z,t)=v(x,t)-zθ(x,t),

[0018] u z (x,y,t)=w(x,t)+yθ(x,t),

[0019] Among them, θ, φ and is the cross-sectional rotation about the three coordinate axes at a point; the relationship between rotation and displacement is as follows:

[0020]

[0021] Lagrange strain theory is used to describe geometric nonlinearity and the nonlinear relationship between strain and displacement:

[0022]

[0023] According to the modified couple stress theory, the partial part of the couple stress tensor m and the symmetric curvature tensor γ are expressed as follows:

[0024]

[0025]

[0026] m=2l 2 μγ

[0027] Where u is the displacement vector, θ is the rotation vector, and l is the material length scale parameter for measuring the coupling stress effect;

[0028] The strain energy of the curved pipe is expressed as follows:

[0029]

[0030] The fluid velocity at the centerline of the curved pipe is expressed as follows:

[0031]

[0032] Among them, A p is the cross-sectional area of the pipe, A f is the cross-sectional area of the fluid, ρ f is the fluid density, ρ p is the pipeline density, U p is the work done by the axial force P, U f The work done for the fluid pressure;

[0033] Applying the extended Hamiltonian principle, the governing equation of the elbow supported by the clamp is obtained. The integral time is from t1 to t2, and the expression is as follows:

[0034]

[0035]

[0036] Among them, δT is the virtual kinetic energy, δU is the virtual potential energy, and δW is the virtual work.

[0037] Furthermore, the establishment of the straight pipe control equation specifically includes:

[0038] According to Hamilton's principle, the fluid-solid coupling vibration equation of straight pipe fluid transportation is obtained:

[0039]

[0040] Furthermore, the control equations for the curved pipe and the straight pipe are combined to establish the dynamic equations for the L-shaped fluid delivery pipeline system, which specifically include:

[0041] In order to obtain an approximate solution in a finite-dimensional function space, the continuous system is discretized using the Galerkin discretization technique, and the deformation of the curved pipe is expressed as:

[0042]

[0043]

[0044]

[0045]

[0046]

[0047] v r (x)=C1sinβ2x+C2cosβ2x+C3sinhβ2x+C4coshβ2x

[0048] w r (x)=C5sinβ3x+C6cosβ3x+C7sinhβ3x+C8coshβ3x

[0049] Assume that the boundary conditions at both ends of the elastic clamp supporting the pipe are:

[0050] v r ″(0)=0,EI1v r ″′(0)=-K v v r (0)

[0051] v r ″(l)=0,EI1v r ″′(l)=K v v r (l)

[0052] Therefore, the following equation can be obtained through calculation:

[0053] D0=sinβ2l-cosβ2l-2G sinhβ2l+coshβ2l

[0054] D2=sinβ2l-sinhβ2l

[0055] D3=-2G sinβ2l-cosβ2l+coshβ2l

[0056] D4=sinβ2l-sinhβ2l

[0057]

[0058] The deformation of a straight pipe can be expressed as:

[0059]

[0060]

[0061]

[0062]

[0063]

[0064] v′ r(x) = C9sinβ5x + C 10 cosβ5x+C 11 sinhβ5x+C 12 coshβ5x

[0065] w′ r (x) = C 13 sinβ6x+C 14 cosβ6x+C 15 sinhβ6x+C 16 coshβ6x

[0066] Assume that the modal function of the lateral vibration of a straight pipe is in the form of:

[0067] C9=C 11

[0068]

[0069]

[0070] Combining straight and curved pipes, the dynamic equation is simplified to:

[0071]

[0072] Furthermore, the model verification of the established L-shaped fluid delivery pipeline system dynamics model specifically includes:

[0073] The modal verification is based on the hammer test of the L-shaped fluid pipeline. The test instruments required for the modal verification include a three-axis accelerometer, a force hammer and a 12-channel LMS system.

[0074] Furthermore, based on the verified model, simulating the influence of fluid parameters on the natural frequency of the L-shaped fluid delivery pipeline to verify the correctness of the L-shaped fluid delivery pipeline system dynamic model specifically includes:

[0075] Through multiple hammer tests, the modal vibration shape and natural frequency of the L-shaped fluid transmission pipeline system were obtained;

[0076] The semi-analytical method is used to obtain the modal vibration shape and natural frequency of the fluid transmission pipeline;

[0077] The correctness of the model is verified by comparing the modal vibration shapes and frequency results of the two.

[0078] Compared with the prior art, the present invention has the following advantages:

[0079] 1. The semi-analytical dynamic modeling method for an L-shaped fluid delivery pipeline provided by the present invention is the first to apply the semi-analytical method to model an L-shaped fluid delivery pipeline, and has a high solution efficiency while ensuring calculation accuracy.

[0080] 2. The semi-analytical dynamic modeling method for L-shaped fluid delivery pipelines provided by this invention first performs segmented modeling. Based on the Hamiltonian principle, a dynamic model for a curved fluid delivery pipeline system is derived. This model is then derived for a straight pipe, followed by a straight-and-bend combination. Numerical analysis and pipeline hammer tests verify the accuracy of the model. Furthermore, the effects of fluid velocity and pressure on the pipeline's natural frequency are studied, and the critical velocity and critical pressure of the L-shaped fluid delivery pipeline are determined. This method has practical engineering significance and provides support for subsequent vibration analysis of L-shaped fluid delivery pipelines.

[0081] Based on the above reasons, the present invention can be widely promoted in the fields of mechanical dynamics and the like. BRIEF DESCRIPTION OF THE DRAWINGS

[0082] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.

[0083] Figure 1 The figure is a flow chart of the inherent characteristic analysis method of the L-shaped fluid delivery pipeline of the present invention.

[0084] Figure 2 Schematic diagram of the model of the L-shaped fluid delivery pipeline of the present invention.

[0085] Figure 3 Schematic diagram of the comparison between the numerical and experimental results of the frequency response function of the L-shaped fluid delivery pipeline system of the present invention.

[0086] Figure 4 Schematic diagram of the modal test results of the L-shaped fluid delivery pipeline of the present invention.

[0087] Figure 5 This is a schematic diagram of the comparison results between the natural frequency change curves of each order under different fluid velocities of the L-shaped fluid delivery pipeline of the present invention and the finite element method.

[0088] Figure 6 Schematic diagram of the comparison between the natural frequency change curves of each order under different fluid pressures of the L-shaped fluid delivery pipeline of the present invention and the finite element method. DETAILED DESCRIPTION

[0089] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts should fall within the scope of protection of the present invention.

[0090] It should be noted that the terms "first", "second", etc. in the description and claims of the present invention and the above-mentioned drawings are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that the numbers used in this way can be interchanged where appropriate, so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions. For example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units clearly listed, but may include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.

[0091] like Figure 1 、 2 As shown, the present invention provides a semi-analytical dynamic modeling method for an L-shaped fluid delivery pipeline, comprising:

[0092] S1. Based on the semi-analytical method, the L-shaped pipeline is decomposed into a straight-bend-straight model, and the dynamic model of the L-shaped fluid transmission pipeline system is established;

[0093] In specific implementation, as a preferred embodiment of the present invention, the specific implementation process of step S1 is as follows:

[0094] S11, establishing the pipe bending control equation;

[0095] Use u respectively x 、u y and u z The displacements in the x1, y1, and z1 directions of any point in the elbow are expressed as follows:

[0096]

[0097] u y (x,z,t)=v(x,t)-zθ(x,t),

[0098] u z (x,y,t)=w(x,t)+yθ(x,t),

[0099] Among them, θ, φ and is the cross-sectional rotation about the three coordinate axes at a point; the relationship between rotation and displacement is as follows:

[0100]

[0101] In order to obtain the static nonlinear deformation of the bent tube, the Lagrangian strain theory is used to describe the geometric nonlinearity and the nonlinear relationship between strain and displacement:

[0102]

[0103] According to the modified couple stress theory, the partial part of the couple stress tensor m and the symmetric curvature tensor Υ are expressed as follows:

[0104]

[0105]

[0106] m=2l 2 μγ

[0107] Where u is the displacement vector, θ is the rotation vector, and l is the material length scale parameter for measuring the coupling stress effect;

[0108] The strain energy of the curved pipe is expressed as follows:

[0109]

[0110] The fluid velocity at the centerline of the curved pipe is expressed as follows:

[0111]

[0112] Among them, A p is the cross-sectional area of the pipe, A f is the cross-sectional area of the fluid, ρ f is the fluid density, ρ p is the pipeline density, U p is the work done by the axial force P, U f The work done for the fluid pressure;

[0113] Applying the extended Hamiltonian principle, the governing equation for the elbow supported by the clamp is obtained. The integral time is from t1 to t2, and the expression is as follows:

[0114]

[0115]

[0116] Among them, δT is the virtual kinetic energy, δU is the virtual potential energy, and δW is the virtual work.

[0117] S12, establishing the straight pipe control equation;

[0118] According to Hamilton's principle, the fluid-solid coupling vibration equation of straight pipe fluid transportation is obtained:

[0119]

[0120] S13. Combine the curved pipe control equation and the straight pipe control equation to establish the L-shaped fluid delivery pipeline system dynamic equation.

[0121] In order to obtain an approximate solution in a finite-dimensional function space, the continuous system is discretized using the Galerkin discretization technique, and the deformation of the curved pipe is expressed as:

[0122]

[0123]

[0124]

[0125]

[0126]

[0127] v r (x)=C1sinβ2x+C2cosβ2x+C3sinhβ2x+C4coshβ2x

[0128] w r (x)=C5sinβ3x+C6cosβ3x+C7sinhβ3x+C8coshβ3x

[0129] Assume that the boundary conditions at both ends of the elastic clamp supporting the pipe are:

[0130] v r ″(0)=0,EI1v r ″′(0)=-K v v r (0)

[0131] v r ″(l)=0,EI1v r ″′(l)=K v v r (l)

[0132] Therefore, the following equation can be obtained through calculation:

[0133] D0=sinβ2l-cosβ2l-2G sinhβ2l+coshβ2l

[0134] D2=sinβ2l-sinhβ2l

[0135] D3=-2G sinβ2l-cosβ2l+coshβ2l

[0136] D4=sinβ2l-sinhβ2l

[0137]

[0138] Since there is also elastic support in the longitudinal direction, the solution method of the modal function is the same as that of the transverse modal function. The governing equation of the elbow can be simplified as:

[0139]

[0140] The deformation of a straight pipe can be expressed as:

[0141]

[0142]

[0143]

[0144]

[0145]

[0146] v′ r (x) = C9sinβ5x + C 10 cosβ5x+C 11 sinhβ5x+C 12 coshβ5x

[0147] w′ r (x) = C 13 sinβ6x+C 14 cosβ6x+C 15 sinhβ6x+C 16 coshβ6x

[0148] Assume that the modal function of the lateral vibration of a straight pipe is in the form of:

[0149] C9=C 11

[0150]

[0151]

[0152] Combining straight and curved pipes, the dynamic equation is simplified to:

[0153]

[0154] S2. verifying the established L-shaped fluid transmission pipeline system dynamics model to obtain a verified model;

[0155] In practice, as a preferred embodiment of the present invention, modal verification based on hammer testing of an L-shaped fluid delivery pipeline was performed. The test instruments required for modal verification include a three-axis accelerometer (PCB 356A01), a force hammer (PCB 086C01), and a 12-channel LMS system (SC-XS12-A). The parameters of the fluid delivery pipeline system are shown in Table 1. The clamp consists of two bands and a metal rubber. Using a self-designed fixture, the clamp stiffness, K, was obtained through multiple measurements. y =3.53×10 6 N / m, K z =4.98×10 7 N / m, K θy =58.84Nm / rad, K θz =27.37Nm / rad.

[0156] Table 1 Parameters of the fluid delivery pipeline system.

[0157]

[0158] S3. Based on the verified model, simulate the influence of fluid parameters on the natural frequency of the L-type fluid pipeline to verify the correctness of the dynamic model of the L-type fluid pipeline system.

[0159] To ensure the accuracy of the test results, multiple hammer tests were conducted to obtain the frequency response function of the L-shaped piping system supported by clamps at both ends, as shown in the figure. Figure 3 As shown in Table 2(a) and Table 2(b), there are three natural frequencies in the 0-1000 Hz plane and out of the plane respectively. The natural frequency of the pipeline is identified by the peak value of the frequency response function.

[0160] Table 2 (a) Comparison of the natural frequencies of the L-shaped pipes supported by clamps at both ends

[0161]

[0162] Table 2(b) Comparison of the natural frequencies of the L-shaped pipeline supported by clamps at both ends

[0163]

[0164]

[0165] The comparison of the experimental modal vibration shape and the simulated modal vibration shape shows good agreement, which proves the feasibility of the semi-analytical hypothetical modal method. Based on the above dynamic model, Figure 5To simulate the effect of flow velocity on the natural frequencies of each order, a comparison chart with the finite element method is provided. By comparing the two, the trend of the natural frequencies decreasing with increasing flow velocity is consistent, which once again verifies the correctness of the established model. Figure 6 A comparison chart of the simulation of the effect of fluid pressure on each order of natural frequency and the finite element method.

[0166] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A semi-analytical dynamic modeling method for an L-shaped fluid delivery pipeline, characterized in that: include: Based on the semi-analytical method, the L-shaped pipeline is decomposed into a straight-bend-straight model, and the dynamic model of the L-shaped fluid transmission pipeline system is established, including: Establish the pipe bending control equation; Establish the straight pipe control equation; The control equations of the curved pipe and the straight pipe are combined to establish the dynamic equations of the L-shaped fluid delivery pipeline system, which include: In order to obtain an approximate solution in a finite-dimensional function space, the continuous system is discretized using the Galerkin discretization technique, and the deformation of the curved pipe is expressed as: v r (x)=C1sinβ2x+C2cosβ2x+C3sinhβ2x+C4coshβ2x w r (x)=C5sinβ3x+C6cosβ3x+C7sinhβ3x+C8coshβ3x Assume that the boundary conditions at both ends of the elastic clamp supporting the pipe are: in r ″(0)=0,EI1v r ″′(0)=-K v in r (0) in r ″(l)=0,EI1v r ″′(l)=K v in r (l) Therefore, the following equation can be obtained through calculation: D0=sinβ2l-cosβ2l-2Gsinhβ2l+coshβ2l D2=sinβ2l-sinhβ2l D3=-2Gsinβ2l-cosβ2l+coshβ2l D4=sinβ2l-sinhβ2l The deformation of a straight pipe can be expressed as: v r ′(x)=C9sinβ5x+C 10 cosβ5x+C 11 sinhβ5x+C 12 coshβ5x w r ′(x)=C 13 sinβ6x+C 14 cosβ6x+C 15 sinhβ6x+C 16 coshβ6x Assume that the modal function of the lateral vibration of a straight pipe is in the form of: C9=C 11 Combining straight and curved pipes, the dynamic equation is simplified to: The established L-shaped fluid transmission pipeline system dynamics model is verified to obtain a verified model; Based on the verified model, the influence of fluid parameters on the natural frequency of the L-shaped fluid pipeline is simulated to verify the correctness of the dynamic model of the L-shaped fluid pipeline system.

2. The semi-analytical dynamic modeling method of an L-shaped fluid delivery pipeline according to claim 1, characterized in that: The establishment of the bend control equation specifically includes: The displacements of any point in the elbow in the x1, y1 and z1 directions are represented by ux, uy and uz respectively, as follows: u y (x,z,t)=v(x,t)-zθ(x,t), u z (x,y,t)=w(x,t)+yθ(x,t), Among them, θ, φ and is the cross-sectional rotation about the three coordinate axes at a point; the relationship between rotation and displacement is as follows: Lagrange strain theory is used to describe geometric nonlinearity and the nonlinear relationship between strain and displacement: According to the modified couple stress theory, the partial part of the couple stress tensor m and the symmetric curvature tensor γ are expressed as follows: m=2l 2 mg Where u is the displacement vector, θ is the rotation vector, and l is the material length scale parameter for measuring the coupling stress effect; The strain energy of the curved pipe is expressed as follows: The fluid velocity at the centerline of the curved pipe is expressed as follows: Among them, A p is the cross-sectional area of the pipe, A f is the cross-sectional area of the fluid, ρ f is the fluid density, ρ p is the pipeline density, U p is the work done by the axial force P, U f The work done for the fluid pressure; Applying the extended Hamiltonian principle, the governing equation of the elbow supported by the clamp is obtained. The integral time is from t1 to t2, and the expression is as follows: Among them, δT is the virtual kinetic energy, δU is the virtual potential energy, and δW is the virtual work.

3. The semi-analytical dynamic modeling method of an L-shaped fluid delivery pipeline according to claim 1, characterized in that: The establishment of the straight pipe control equation specifically includes: According to Hamilton's principle, the fluid-solid coupling vibration equation of straight pipe fluid transportation is obtained:

4. The semi-analytical dynamic modeling method of an L-shaped fluid delivery pipeline according to claim 1, characterized in that: The model verification of the established L-shaped fluid transmission pipeline system dynamic model specifically includes: The modal verification is based on the hammer test of the L-shaped fluid pipeline. The test instruments required for the modal verification include a three-axis accelerometer, a force hammer and a 12-channel LMS system.

5. The semi-analytical dynamic modeling method of an L-shaped fluid delivery pipeline according to claim 1, characterized in that: The method of simulating the influence of fluid parameters on the natural frequency of the L-shaped fluid transmission pipeline based on the verified model and verifying the correctness of the dynamic model of the L-shaped fluid transmission pipeline system specifically includes: Through multiple hammer tests, the modal vibration shape and natural frequency of the L-shaped fluid transmission pipeline system were obtained; The semi-analytical method is used to obtain the modal vibration shape and natural frequency of the fluid transmission pipeline; The correctness of the model is verified by comparing the modal vibration shapes and frequency results of the two.

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