Free vibration semi-analytical calculation method for liquid filling functional gradient pipeline on elastic foundation

Abstract

The invention discloses a free vibration semi-analytical calculation method for a liquid filling functional gradient pipeline on an elastic foundation, and relates to the technical field of solid mechanics and fluid-solid coupling vibration analysis. The method comprises the following steps: acquiring materials, geometry and boundary parameters of a pipeline, a fluid and a foundation, establishing a fluid-solid coupling model based on a three-dimensional elastic theory, simulating foundation support by adopting a two-parameter Pasternak model, and deducing a control equation and boundary conditions of a pipeline structural domain and a fluid domain; discretizing a pipeline reference surface and a fluid boundary through a two-dimensional high-order spectrum unit, and converting an equation into a standard coordinate system in combination with proportional boundary coordinate transformation; proportional boundary finite element control equations of a structural domain and a fluid domain are deduced respectively and converted into a first-order differential equation set, then a fine integration method and an eigenvalue decomposition method are adopted for solving, and then based on a fluid-solid coupling condition coupling equation, the equation is converted into a generalized eigenvalue problem for solving the inherent frequency and the vibration mode. According to the method, only precision, efficiency and discrete boundaries are considered, and support is provided for pipeline design optimization and safety evaluation.

Description

Technical Field

[0001] This invention relates to the field of solid mechanics and fluid-structure interaction vibration analysis, and in particular to a semi-analytical calculation method for analyzing the free vibration characteristics of a fluid-filled functional gradient pipe structure on an elastic foundation. Background Technology

[0002] Functionally graded pipelines (FGRs), with their continuously adjustable material properties along the thickness direction, are widely used in engineering scenarios such as fuel transportation in aerospace propulsion systems, deep-water oil pipelines in ships, and coolant pipelines in nuclear power plants. These scenarios typically involve a multi-physics coupled environment of "pipeline-fluid-foundation"—the pipeline must continuously transport fluid media such as fuel, crude oil, and coolant, while the external structure relies on an elastic foundation for support and fixation. Their free vibration characteristics directly determine the system's operational safety and service life. For example, in the aerospace field, excessive pipeline vibration can lead to unstable fluid transport, causing fluctuations in engine thrust; in marine engineering, the superposition of pipeline vibration and wave loads can cause weld cracking, leading to the risk of crude oil leakage; in nuclear power plant scenarios, abnormal vibration of coolant pipelines can even affect the heat exchange efficiency of the nuclear reactor, creating potential safety hazards. Therefore, accurately analyzing the free vibration characteristics of fluid-filled FGRs on elastic foundations is a core requirement for ensuring the reliable operation of such engineering systems.

[0003] Current technical solutions for vibration analysis of fluid-filled pipelines on elastic foundations have significant limitations. Firstly, traditional finite element methods require separate three-dimensional volumetric meshing for the pipeline's structural and fluid domains. When dealing with problems such as continuous thickness variations in functionally graded materials (FJCTs) and dynamic fluid-structure coupling, mesh generation is not only difficult and computationally intensive, but also prone to numerical discretization errors due to mesh distortion. This is particularly problematic in complex geometric scenarios such as multi-cavity pipelines with varying diameters, where computational efficiency falls short of the need for rapid comparison of multiple solutions during engineering design. Secondly, analytical methods based on classical shell theory or first-order shear deformation theory require simplified assumptions regarding the pipeline's three-dimensional stress state and fluid dynamic pressure distribution. These assumptions fail to accurately capture the impact of FJCT non-uniformity and the dual-parameter support effect (compression and shear) of the elastic foundation on vibration characteristics, leading to significant discrepancies between calculation results and actual engineering conditions. This makes them unsuitable for high-precision applications such as aerospace and marine engineering.

[0004] While the Scaled Boundary Finite Element Method (SBFEM) offers advantages such as discrete boundary conditions and radial analytical solutions, combining the flexibility of numerical methods with the accuracy of analytical methods, its traditional application relies on a fixed scaling center. When simulating structures with large aspect ratios and regular geometric shapes, such as pipes, it is prone to introducing geometric discretization errors due to the mismatch between the scaling center and the pipe axis. Furthermore, it has not developed a dedicated computational framework for multi-field coupling scenarios involving functionally graded materials (FJCTs), fluid filling, and elastic foundations. This makes it difficult to efficiently integrate the power-law distribution characteristics of FJCTs along their thickness and accurately integrate the reaction force calculations of the Pasternak two-parameter foundation model, resulting in significant shortcomings in the adaptability of this method in practical engineering applications.

[0005] Existing technical solutions cannot simultaneously meet the triple requirements of accurate modeling of multi-field coupling, efficient calculation of complex scenarios, and deep adaptation to engineering needs. There is an urgent need for a semi-analytical calculation technique that can closely fit the actual engineering environment and avoid the limitations of pure theoretical derivation, so as to fill the gap between numerical methods and engineering applications. Summary of the Invention

[0006] To address the aforementioned technical problems, this application discloses a semi-analytical calculation method for the free vibration of a fluid-filled functional gradient pipeline on an elastic foundation, specifically including:

[0007] Parameters of the elastic foundation, functionally graded pipe and internal filling fluid are obtained. The parameters include material properties, geometric dimensions and boundary conditions. The material properties include the elastic modulus, mass density and volume fraction index of the ceramic phase and the metallic phase of the pipe. The boundary conditions include the free surface of the fluid, the bottom and the fluid-solid interface constraints.

[0008] A fluid-structure interaction model in Cartesian coordinate system is established. Based on three-dimensional elasticity theory, the geometric and physical equations of the channel structure domain are derived. The two-parameter Pasternak model is used to simulate the foundation support effect. The control equations and boundary conditions of the fluid domain expressed in terms of hydrodynamic pressure are derived.

[0009] The pipe structure reference surface and the fluid domain boundary are discretized using two-dimensional high-order spectral units. The basic equations of the structural domain and the fluid domain are transformed to the proportional boundary coordinate system through shape function interpolation and coordinate transformation.

[0010] The proportional boundary finite element control equations of the structural domain and fluid domain are derived separately, and then transformed into a set of first-order variable coefficient differential equations. The precise integration method and eigenvalue decomposition method are used to solve them respectively, and the dynamic stiffness equation of the structural domain and the dynamic equation of the fluid domain are obtained.

[0011] Based on the force balance and kinematic compatibility conditions at the fluid-structure interaction interface, the fluid dynamic pressure is used as an additional load to couple the equations of the structural domain and the fluid domain, forming the system dynamic control equations.

[0012] The system dynamics control equations are transformed into a generalized eigenvalue problem, and the natural frequencies and corresponding mode shapes of the pipeline are obtained by solving the problem, thus completing the free vibration characteristic analysis.

[0013] Preferably, the functionally graded pipe is composed of a ceramic phase and a metallic phase, and the material properties change continuously along the pipe thickness direction according to a power law function. The volume fractions of the ceramic phase and the metallic phase respectively satisfy: , ,in, , These are volume fraction functions for the ceramic phase and the metallic phase, respectively. For the dimensionless thickness of the functionally graded material pipe, The thickness is measured from the inner surface. It is a volume fraction index used to control the variation and distribution of material along the thickness direction.

[0014] Preferably, the effective material properties of the elastic modulus and mass density of the functionally graded pipe are expressed as follows: , ,in, , These are the elastic moduli of the ceramic phase and the metallic phase, respectively. , These are the mass densities of the ceramic phase and the metallic phase, respectively.

[0015] The elastic modulus and mass density at any cross-section along the thickness direction satisfy: , .

[0016] Preferably, based on the small deformation assumption, the strain-displacement relationship of the pipeline structural domain is as follows: Among them, displacement field Differential operators Defined as: Using Hooke's law in matrix form, the stress-strain relationship is expressed as: Among them, the elasticity matrix , It is Poisson's ratio.

[0017] Preferably, the foundation reaction force simulated by the two-parameter Pasternak model is expressed as follows: ,in, Let be the radial displacement vector of the functionally graded material pipe; For the Laplace operator; , These are the compression modulus and shear modulus of the elastic foundation, respectively.

[0018] In a Cartesian coordinate system, the fundamental reaction force at any point within the solution domain is expressed through a global displacement field as: ,in, This represents the fundamental reaction force considering the global node degrees of freedom. , These represent the compressive stiffness matrix and the shear stiffness matrix of the foundation, respectively.

[0019] Preferably, the governing equations of the fluid domain are the three-dimensional Laplace equations: ,in, The pressure is dynamic water pressure; the boundary conditions of the fluid domain include the fluid-structure interface, the free surface, and the bottom surface. The fluid-structure interface is... The free surface is , ,in, For fluid density, Let be the acceleration of the container wall along the outward normal. For outward unit normals.

[0020] Preferably, when discretizing the reference surface of the pipeline structure, a scaled boundary finite element coordinate system is established. The element interpolation expression is:

[0021]

[0022] in, , Let the coordinates be those of the inner surface. , Let the coordinates be those of the outer surface. The shape function matrix of the higher-order spectral elements;

[0023] The coordinate transformation relationship is as follows: ,in, It is a Jacobian matrix.

[0024] Preferably, after deriving the finite element governing equations for the scaled boundary of the structural domain, the transformed first-order variable-coefficient differential equation is: ,in, , Let be the pipe displacement vector. For the internal force vectors of the fluid domain nodes, The coefficient matrix of the structural domain;

[0025] After solving using the precise integration method, the equilibrium equations of the structured domain are obtained: ,in, , , , A submatrix of the stiffness matrix. , These are the displacement vectors of the inner and outer surfaces, respectively. , These are the equivalent nodal forces on the inner and outer surfaces, respectively. The force exerted by the fluid at the interface.

[0026] Preferably, after deriving the finite element governing equations for the proportional boundary of the fluid domain, the transformed first-order variable-coefficient differential equation is: in, , For the internal force vectors of the fluid domain nodes, This is the fluid domain coefficient matrix;

[0027] After solving using the eigenvalue decomposition method, the fluid domain equilibrium equations are obtained: ,in, Here is the stiffness matrix of the fluid domain. This is the mass matrix of the fluid-structure interaction.

[0028] Preferably, the expression for the generalized eigenvalue problem is:

[0029]

[0030] in, , , , For the stiffness matrix of the structural domain, , , , For the structural domain quality matrix submatrix, Add a mass matrix to the fluid. For the natural frequency, , These are the displacement vectors of the outer and inner surface nodes, respectively.

[0031] Compared with the prior art, the technical solution of this application has the following technical effects:

[0032] This invention can accurately adapt to actual engineering scenarios involving multi-physics coupling of pipelines, fluids, and foundations. It constructs a pipeline structure model through three-dimensional elasticity theory, avoiding the simplification assumptions of traditional shell theory. At the same time, it combines a two-parameter Pasternak model to fully reproduce the compression and shear support effects of elastic foundations. It can also accurately capture the power-law distribution characteristics of functionally graded materials along the thickness, eliminating errors caused by geometric discretization and material simplification from a theoretical perspective. This makes the calculation results more consistent with engineering practice and provides a reliable theoretical basis for the analysis of pipeline vibration characteristics.

[0033] This invention discretizes the pipeline reference surface and the fluid domain boundary using two-dimensional high-order spectral elements, eliminating the need for three-dimensional volume meshing of the structural and fluid domains as required by the traditional finite element method. This significantly reduces the system's degrees of freedom and the difficulty of mesh generation, lowers the computational scale, and avoids numerical fluctuations caused by mesh distortion. It can quickly complete multi-scheme comparison calculations, better meeting the computational efficiency requirements of the engineering design stage.

[0034] This invention constructs a unified semi-analytical computational framework that combines the scaled boundary finite element method with the refined integral method and eigenvalue decomposition method. It can adapt to the complex geometry of pipelines through scaled boundary coordinate transformation, improve computational accuracy through analytical solutions, and flexibly integrate multi-physics field effects such as functionally graded materials, fluid-structure interaction, and elastic foundations. It does not require adjustment of the core algorithm for different scenarios and is applicable to pipeline vibration analysis needs in multiple fields such as aerospace, marine engineering, and nuclear power.

[0035] This invention can directly output the natural frequencies and mode shapes of pipelines, providing a clear direction for pipeline structural design optimization. It can also adjust material gradient distribution, foundation support parameters, or pipeline geometry to control vibration amplitude. At the same time, it provides data support for vibration monitoring and safety assessment during pipeline operation, helping to avoid risks such as unstable fluid transport and structural cracking caused by abnormal vibration, and ensuring the long-term reliable operation of the engineering system.

[0036] The above description is only an overview of the technical solution of this application. In order to better understand the technical means of this application and implement it in accordance with the contents of the specification, and to make the above and other objects, features and advantages of this application more obvious and understandable, the preferred embodiments of this application are described in detail below with reference to the accompanying drawings.

[0037] The above and other objects, advantages and features of this application will become more apparent to those skilled in the art from the following detailed description of specific embodiments in conjunction with the accompanying drawings. Attached Figure Description

[0038] To more clearly illustrate the technical solutions in the embodiments of this application 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 some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. In all drawings, similar elements or parts are generally identified by similar reference numerals. In the drawings, the elements or parts are not necessarily drawn to scale.

[0039] Based on the description of the figures and their corresponding technical content in the document, the titles of the figures are as follows:

[0040] Figure 1This is a flowchart illustrating the numerical simulation method for free vibration of a fluid-filled functional gradient pipeline on an elastic foundation according to the present invention.

[0041] Figure 2 This is a schematic diagram of the structure of the fluid-filled functional gradient pipeline on the elastic foundation in an embodiment of the present invention;

[0042] Figure 3 This is a schematic diagram of the discretization of the inner surface of the pipe based on similar surface technology used in this invention;

[0043] Figure 4 This is a schematic diagram of the boundary discretization of the fluid domain proportional boundary coordinate system used in this invention;

[0044] Figure 5 This is a schematic diagram of a functional gradient pipeline based on elasticity, which is the second example of the present invention. Detailed Implementation

[0045] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. In the following description, specific details such as specific configurations and components are provided merely to help fully understand the embodiments of this application. Therefore, those skilled in the art should understand that various changes and modifications can be made to the embodiments described herein without departing from the scope and spirit of this application. In addition, for clarity and brevity, descriptions of known functions and structures are omitted in the embodiments.

[0046] It should be understood that the phrase "an embodiment" or "this embodiment" throughout the specification means that a specific feature, structure, or characteristic related to the embodiment is included in at least one embodiment of this application. Therefore, "an embodiment" or "this embodiment" appearing throughout the specification does not necessarily refer to the same embodiment. Furthermore, these specific features, structures, or characteristics can be combined in any suitable manner in one or more embodiments.

[0047] Furthermore, reference numerals and / or letters may be repeated in different examples within this application. Such repetition is for the purpose of simplification and clarity and does not in itself indicate a relationship between the various embodiments and / or settings discussed.

[0048] In this article, the term "and / or" is merely a description of the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can mean: A exists alone, B exists alone, and A and B exist simultaneously. The term " / and" in this article describes another type of relationship between related objects, indicating that two relationships can exist. For example, A / and B can mean: A exists alone, and A and B exist alone. In addition, the character " / " in this article generally indicates that the related objects before and after it are in an "or" relationship.

[0049] In this article, the term "at least one" is merely a description of the relationship between related objects, indicating that there can be three relationships. For example, "at least one of A and B" can mean: A exists alone, A and B exist simultaneously, or B exists alone.

[0050] It should also be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion.

[0051] Example 1

[0052] This embodiment mainly describes a semi-analytical calculation method for the free vibration of a fluid-filled functional gradient pipe on an elastic foundation, such as... Figure 1 As shown, it specifically includes:

[0053] Parameters of the elastic foundation, functionally graded pipe and internal filling fluid are obtained. The parameters include material properties, geometric dimensions and boundary conditions. The material properties include the elastic modulus, mass density and volume fraction index of the ceramic phase and the metallic phase of the pipe. The boundary conditions include the free surface of the fluid, the bottom and the fluid-solid interface constraints.

[0054] A fluid-structure interaction model in Cartesian coordinate system is established. Based on three-dimensional elasticity theory, the geometric and physical equations of the channel structure domain are derived. The two-parameter Pasternak model is used to simulate the foundation support effect. The control equations and boundary conditions of the fluid domain expressed in terms of hydrodynamic pressure are derived.

[0055] A fluid-structure interaction model of a fluid-filled functionally graded pipe on an elastic foundation is established in Cartesian coordinates. Based on three-dimensional elasticity theory, considering the gradient variation of material properties along the thickness direction, the geometric and physical equations of the pipe's structural domain are obtained. The internal fluid is considered as an inviscid, incompressible, and irrotational ideal fluid. Considering the fluid-structure interaction effect, the governing equations of the fluid domain expressed in terms of hydrodynamic pressure and the boundary conditions at the free surface, bottom, and interface are derived. Simultaneously, a two-parameter Pasternak model is used to simulate the supporting effect of the elastic foundation on the pipe.

[0056] The pipe structure reference surface and the fluid domain boundary are discretized using two-dimensional high-order spectral units. The basic equations of the structural domain and the fluid domain are transformed to the proportional boundary coordinate system through shape function interpolation and coordinate transformation.

[0057] The reference surface of the pipeline structure is discretized using two-dimensional high-order spectral elements. Based on similarity surface technology, the fundamental equations of the pipeline structure are transformed to the scaled boundary coordinate system through shape function interpolation and coordinate transformation of the elements. Combining the principle of virtual work and integrating the reaction force of the elastic foundation, the scaled boundary finite element governing equations of the pipeline structure are derived. By introducing equivalent nodal force vectors, the governing equations are transformed into a system of first-order variable-coefficient differential equations along the pipeline thickness direction, which are then solved using the precise integration method. Finally, the dynamic stiffness equation of the structural domain integrating the elastic foundation effect is obtained.

[0058] The proportional boundary finite element control equations of the structural domain and fluid domain are derived separately, and then transformed into a set of first-order variable coefficient differential equations. The precise integration method and eigenvalue decomposition method are used to solve them respectively, and the dynamic stiffness equation of the structural domain and the dynamic equation of the fluid domain are obtained.

[0059] The boundary of the fluid domain is discretized using two-dimensional high-order spectral elements. Interpolation and coordinate transformation of the elements are performed using shape functions to transform the fundamental equations and boundary conditions of the fluid domain into a scaled boundary coordinate system. The weighted residual method is used to establish the scaled boundary finite element governing equations of the fluid domain. By introducing intermediate variables, the governing equations are transformed into a system of first-order variable-coefficient differential equations, which are then solved using the eigenvalue decomposition method. Finally, the fluid domain dynamic equations describing the relationship between hydrodynamic pressure and the normal displacement at the interface are obtained.

[0060] Based on the force balance and kinematic compatibility conditions at the fluid-structure interaction interface, the fluid dynamic pressure is used as an additional load to couple the equations of the structural domain and the fluid domain, forming the system dynamic control equations.

[0061] Based on the correspondence between the nodes of the fluid domain and the structural domain at the coupling interface, the obtained dynamic equations of the structural domain and the fluid domain are expressed in blocks. Based on the force balance condition and kinematic compatibility condition at the fluid-structure interaction interface, the fluid dynamic pressure is introduced as an additional load into the structural equations, and the dynamic control equations of the entire system are obtained through coupling.

[0062] The system dynamics control equations are transformed into a generalized eigenvalue problem, and the natural frequencies and corresponding mode shapes of the pipeline are obtained by solving the problem, thus completing the free vibration characteristic analysis.

[0063] For the free vibration problem, the governing equations of the coupled system are expressed as a generalized eigenvalue problem. By solving this eigenvalue problem, the natural frequencies and corresponding mode shapes of the fluid-filled functionally graded pipeline on the elastic foundation can be directly obtained. Based on these calculation results, the free vibration characteristics of the system can be comprehensively analyzed, and its dynamic response law can be determined.

[0064] Furthermore, when establishing the structural model of the fluid-filled functionally graded material (FGM) pipe-elastic foundation system in the Cartesian coordinate system (x,y,z), it is necessary to construct mathematical models for the FGM pipe structural domain, the Pasternak elastic foundation, and the fluid domain respectively. The basic equations and boundary conditions of each domain are derived theoretically, as follows:

[0065] FGM pipes are composed of a ceramic phase (reinforcing phase) and a metallic phase (matrix phase). Their material properties change continuously along the pipe thickness direction according to a power-law function. The volume fractions of the ceramic and metallic phases satisfy the following conditions:

[0066]

[0067]

[0068] in, , These are volume fraction functions for the ceramic phase and the metallic phase, respectively. The dimensionless thickness of the functionally graded pipeline ( (This represents the thickness measured from the inner surface); n is the volume fraction index, used to control the variation and distribution of the material along the thickness direction.

[0069] The effective material properties of elastic modulus and mass density can be expressed as:

[0070]

[0071]

[0072] in, , These are the elastic moduli of the ceramic phase and the metallic phase, respectively. , These are the mass densities of the ceramic phase and the metallic phase, respectively.

[0073] The elastic modulus and mass density at any cross-section in the thickness direction can be obtained:

[0074]

[0075]

[0076] Based on the assumption of small deformation, the relationship between strain and displacement of the pipeline is as follows:

[0077]

[0078] Among them, displacement field Differential operators The definitions are as follows:

[0079]

[0080]

[0081] Using Hooke's law in matrix form, the stress-strain relationship can be expressed as:

[0082]

[0083] Among them, the elasticity matrix for:

[0084]

[0085] The elasticity matrix of the functionally graded pipeline can be rewritten as:

[0086]

[0087]

[0088]

[0089] Similarly, the density distribution of the functionally graded pipeline can be rewritten as:

[0090]

[0091] For the fluid domain, assuming the fluid is an incompressible, inviscid, irrotational, and ideal fluid with small wave steepness, its irrotational harmonic motion satisfies linear wave theory. Therefore, the fluid governing equations expressed in terms of the hydrodynamic pressure variable p satisfy the three-dimensional Laplace equation:

[0092] exist middle

[0093] exist middle

[0094] exist middle

[0095] exist middle

[0096] in, Indicates hydrodynamic pressure, Represents the fluid domain. Indicates the outward unit normal. Indicates fluid density, This represents the acceleration of the container wall along the outward normal. Represents gravitational acceleration. , and These represent the fluid-structure interface, the free surface, and the bottom surface, respectively.

[0097] The supporting effect of the foundation is simulated using a two-parameter Pasternak foundation model. The foundation reaction force acting on the bottom surface of the pipeline structure can be expressed as:

[0098]

[0099] in, This represents the radial displacement vector of the functionally graded pipeline. For the Laplace operator; , These are the compression modulus and shear modulus of the elastic foundation, respectively.

[0100] In the Cartesian coordinate system, the fundamental reaction force at any point in the solution domain is expressed by the global displacement field as:

[0101]

[0102] in, This represents the fundamental reaction force considering the global node degrees of freedom. and These represent the compressive stiffness matrix and the shear stiffness matrix of the foundation, respectively. The transformation matrix is ​​defined as:

[0103]

[0104] in, It is a circular coordinate system.

[0105] Furthermore, the inner surface of the pipeline structural domain is discretized using two-dimensional high-order spectral elements to establish a scaled boundary finite element coordinate system. The expression for interpolating each element using shape functions is as follows:

[0106]

[0107]

[0108]

[0109] in, and Represents the coordinates of the inner surface. and This represents the coordinate difference between the inner and outer surfaces; , , It is the coordinate vector of the nodes on the inner surface. , , For the outer surface, the subscript k indicates the number of element nodes; It is the shape function matrix of higher-order spectral elements.

[0110] The coordinate transformation relationship of the basic equations of the pipe in Cartesian coordinates is as follows:

[0111]

[0112] Jacobian matrix determinant Represented as:

[0113]

[0114] in,

[0115]

[0116]

[0117]

[0118] According to the Taylor series expansion, the reciprocal of the Jacobian determinant can be expressed as:

[0119]

[0120]

[0121]

[0122]

[0123]

[0124]

[0125]

[0126] ,

[0127] , ,

[0128] ,

[0129] Based on the concept of isoparametric transformation, the displacement field within the element can be represented using the same nodal interpolation shape function as the coordinate mapping; therefore, the local approximation of the displacement field at any point in the solution domain is as follows:

[0130]

[0131] The strain field and stress field can be reformulated as:

[0132]

[0133]

[0134] Among them, the strain matrix of the FGM pipe , for:

[0135]

[0136]

[0137] in

[0138]

[0139]

[0140] Applying the principle of virtual work, the governing equations of the proportional boundary finite element method for a pipeline structure considering ground reaction forces are expressed as follows:

[0141]

[0142] in, Represents the pipe displacement vector Regarding coordinates The second-order partial derivative, superscript " " represents the transpose of a matrix. , , All are coefficient matrices, and their specific expressions are as follows:

[0143]

[0144]

[0145]

[0146] The equilibrium boundary conditions for the inner and outer surfaces are:

[0147]

[0148]

[0149]

[0150] in, Represents the nodal internal force vector. and These represent the coordinates of the inner and outer surfaces, respectively. and These represent the equivalent nodal forces acting on the inner and outer surfaces, respectively.

[0151] Furthermore, based on the displacement vectors of each node in the structural domain... With nodal internal force vector The proportional boundary finite element governing equations of the FGM pipeline are transformed into first-order variable coefficient differential equations as follows:

[0152]

[0153]

[0154] in, Indicate intermediate variables Regarding coordinates The first-order partial derivative, The coefficient matrix of the structural domain is expressed as follows:

[0155]

[0156] Among them, the superscript " " represents the inverse of a matrix.

[0157] This uses polynomial fitting to determine the expansion coefficients. In particular, by utilizing quadratic polynomials at appropriately chosen discrete points Perform coefficient matrix approximation, for example

[0158]

[0159]

[0160] Among them, matrix , , It is independent of The constant coefficient matrix can be obtained by discrete positions. Fitting least squares To determine, we obtain:

[0161]

[0162]

[0163] in, Represents the exponent matrix, This represents the coefficients to be determined.

[0164] Substituting the boundary conditions yields:

[0165]

[0166]

[0167]

[0168] By performing matrix exponentiation using the precise integration method, the equilibrium equations of the structure domain are obtained:

[0169]

[0170] in, This represents the stiffness matrix of the FGM pipe. , , , They represent The submatrix is ​​expressed as follows:

[0171]

[0172] in

[0173]

[0174] After rewriting the above equations as force-displacement relationships, we can obtain...

[0175]

[0176] in, The force exerted by a fluid at an interface can be expressed as:

[0177]

[0178] in, This represents the hydrodynamic pressure acting on the interface. It is the unit outward normal vector. It can be represented as:

[0179]

[0180] Furthermore, the fluid domain boundary is discretized using two-dimensional high-order spectral elements to establish a scaled boundary finite element coordinate system. The expression for interpolating each element using shape functions is as follows:

[0181]

[0182]

[0183]

[0184] in, and Indicates the coordinates of the scaling center. and Represents the coordinates of the fluid surface. , and Represents the coordinates of discrete nodes on the fluid surface. For the shape function of the higher-order spectral unit in the fluid domain.

[0185] The fundamental equations of fluid dynamics in Cartesian coordinates are transformed to obtain the following:

[0186]

[0187]

[0188]

[0189] The finite element governing equations for the proportional boundary of the fluid domain, established using the weighted residual method, are as follows:

[0190]

[0191] The boundary conditions are:

[0192]

[0193] The coefficient matrix is ​​defined as follows:

[0194]

[0195]

[0196]

[0197]

[0198] in

[0199]

[0200] The first-order reconstruction of the second-order equation is achieved by introducing a new variable, as shown below.

[0201]

[0202] in

[0203]

[0204]

[0205] The solution yields:

[0206]

[0207] in, and It is the integration constant.

[0208] Given that the eigenvalues ​​have negative real parts, for a finite fluid domain, the integration constant is... It must be zero. This allows the elimination of constants. And the following relationship is generated at the boundary of the fluid domain:

[0209]

[0210] The equilibrium equations for the entire fluid domain are obtained as follows:

[0211]

[0212] Furthermore, by dividing the fluid domain into different types of nodes, the resulting fluid domain dynamic equilibrium equations are expressed in blocks:

[0213]

[0214] Among them, the subscript " "" indicates a common node at the fluid-structure interface, subscript " "" indicates a node on the free surface of the fluid, subscript " " " indicates a node on the bottom surface.

[0215] From the first and third terms of the above equation, we can obtain the following relationship:

[0216]

[0217]

[0218] get:

[0219]

[0220] Similarly, by distinguishing between structural nodes (subscript "I") and non-interface nodes (subscript "N") on the fluid-structure interface, we can obtain:

[0221]

[0222] Further results were obtained:

[0223]

[0224] in

[0225]

[0226] Furthermore, using the initial and boundary conditions, the free vibration frequencies of the fluid-structure interaction system are obtained by solving the generalized eigenvalue problem:

[0227]

[0228] In this embodiment, a structural model of a fluid-filled functionally graded pipeline on an elastic foundation is established. Based on elasticity theory and the Laplace equation, the fundamental equations of the structural and fluid domains are established. Higher-order spectral element shape functions are used to interpolate and approximate the structural and fluid domains respectively. The proportional boundary finite element control equations of the structural and fluid domains are derived through coordinate transformation combined with the virtual work principle and the weighted residual method. The equations are solved using the precise integration method and the eigenvalue solving method, further obtaining the dynamic equilibrium equations of the structural and fluid domains. Based on the coupling of equilibrium and compatibility conditions, the free vibration control equations of the fluid-filled functionally graded pipeline are obtained. This invention reduces the computational cost of the mechanical properties of fluid-filled functionally graded pipelines, thereby improving computational efficiency.

[0229] Based on Example 1, this example details a semi-analytical calculation method for the free vibration of a fluid-filled functionally graded pipeline on an elastic foundation. It verifies the numerical simulation of the free vibration characteristics of this invention in fluid-structure interaction of an isotropic pipeline through the analysis of the free vibration characteristics. Specifically:

[0230] like Figure 2 As shown, given a pipe container model that is unfilled, half-filled, and fully filled with liquid, the geometric parameters of the container structure are radius and length. Container height ,thickness The material parameter is the elastic modulus. Poisson's ratio ,density The liquid inside the container is assumed to be water, and its density is... .

[0231] like Figure 3As shown, this is the discretization method of the high-order spectral elements for the pipeline structure reference surface, including three element types: 4th, 5th, and 6th order. The upper half of the figure presents the geometric shape of the pipeline structure (a curved functionally graded pipeline), and the reference surface used for scaled boundary coordinate transformation is marked by similar surfaces. The lower half shows the details of the high-order spectral elements of different orders: each element has a different number of nodes distributed in the local coordinate system (η-ζ) (the 4th order element has the fewest nodes, and the 6th order element has the most nodes). This type of high-order spectral element is the core tool for discretizing the pipeline reference surface in this invention—through shape function interpolation, the complex curved surface geometry of the pipeline can be transformed to the standard scaled boundary coordinate system, which not only preserves the actual geometric shape of the pipeline, but also utilizes the high precision characteristics of high-order elements to avoid the geometric approximation errors of traditional low-order elements when discretizing complex curved surfaces.

[0232] like Figure 4 As shown, the scaled boundary finite element discretization logic of the three-dimensional pipeline structure includes similarity centers and higher-order spectral elements. The cube in the figure represents the geometric model of the three-dimensional functional gradient pipeline. The similarity center is the reference point for the coordinate transformation of the scaled boundary. The geometry of each part of the pipeline can be obtained by scaling outward from this center. The magnified area on the right is the higher-order spectral element on the pipeline boundary (node ​​distribution in the local coordinate system η-ζ). This invention only needs to discretize the outer boundary of the pipeline (instead of the three-dimensional volume mesh of the traditional method) to complete the modeling of the pipeline structural domain: by utilizing the scaling characteristics of the similarity center and combining the high-precision interpolation of the higher-order spectral elements, it can accurately reproduce the three-dimensional geometry of the pipeline, significantly reduce the discrete degrees of freedom, and avoid the mesh distortion problem caused by volume meshing.

[0233] like Figure 5 As shown, a support model of a functionally graded pipeline on an elastic foundation (Pasternak two-parameter foundation) is presented. The curved surface structure in the figure is the functionally graded pipeline (the geometric parameters of the pipeline are labeled a, b, h, and R). The spring layer and shear layer below the pipeline correspond to the compressive support and shear support effects of the Pasternak model, respectively. The spring layer simulates the normal reaction force of the foundation on the pipeline (corresponding to the foundation compressive modulus), while the shear layer simulates the shear force transmission inside the foundation (corresponding to the foundation shear modulus).

[0234] The free vibration frequencies of the fluid-structure interaction system were obtained by solving the generalized eigenvalue problem. The first eight natural frequencies were selected and compared with the results calculated by the finite element method in existing technologies such as Mazuche et al. (1996), Ergin and Ugurlu (2004), and Marimuthu et al. (2015). Table 1 shows the comparison of the first eight natural frequencies of the fluid filling fluid in the isotropic pipe with no filling, half filling, and complete filling with liquid obtained by the present invention and other methods. As can be seen from Table 1, the results calculated by the present invention basically converge at the 7th element order, and the converged results are basically consistent with the finite element method calculation results of Mazuche et al. (1996), Ergin and Ugurlu (2004), and Marimuthu et al. (2015), indicating that the present invention can achieve high accuracy with fewer meshes and lower computational cost.

[0235] Table 1 Comparison of the first eight natural frequencies of isotropic pipe-filled fluid calculated by this invention and other methods.

[0236] To further verify the efficiency and accuracy of the provided numerical analysis method, the free vibration characteristics of a functionally graded pipeline on an elastic foundation were analyzed. Table 2 shows the convergence of the natural frequencies of the functionally graded pipeline on the elastic foundation. The geometric parameters are a=l=10h=0.2. The material parameters correspond to the bottom layer aluminum and the top layer alumina, respectively. The relevant foundation parameters and calculation results were normalized to... and The natural frequency of the structure is normalized to Three types of foundation support conditions were considered: no foundation support, Winkler foundation support, and Pasternak foundation support. The computational model used 4th to 7th order spectral elements with an 8×2 mesh, and its results were compared with those based on FSDT.

[0237] Table 2 Comparison of the first eight natural frequencies of a functionally graded pipeline based on elasticity calculated by this invention and other methods.

[0238] As shown in Table 2, the calculation results obtained in this paper are highly consistent with the comparison reference, which not only demonstrates the accuracy of the calculation and analysis of the free vibration of the functionally graded pipe structure placed on the basis of two-parameter elasticity, but also shows its robustness under different foundation scenarios.

[0239] This embodiment details the numerical simulation of the free vibration characteristics of a fluid-filled functionally graded pipeline on an elastic foundation, demonstrating its ability to accurately reproduce vibration patterns under multi-physics coupling. By combining three-dimensional elasticity theory with the scaled boundary finite element method, it avoids the accuracy deficiencies of traditional simplified models. It accurately captures the influence of functionally graded material inhomogeneity, hydrodynamic pressure effects, and the dual-parameter support of the elastic foundation on vibration characteristics. Furthermore, the calculation can be completed using only discrete boundaries, effectively balancing accuracy and efficiency. The simulation results provide a reliable basis for pipeline structure vibration control and parameter optimization, avoiding engineering design risks caused by simulation deviations, and are adaptable to pipeline vibration analysis needs in various scenarios.

[0240] The above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. For those skilled in the art, the present invention can have various modifications and variations. Any changes, modifications, substitutions, integrations, and parameter changes made to these embodiments within the spirit and principles of the present invention, without departing from the principles and spirit of the present invention, through conventional substitutions or to achieve the same function, fall within the scope of protection of the present invention.

Claims

1. A semi-analytical calculation method for the free vibration of a fluid-filled functional gradient pipeline on an elastic foundation, characterized in that, include: Parameters of the elastic foundation, functionally graded pipe and internal filling fluid are obtained. The parameters include material properties, geometric dimensions and boundary conditions. The material properties include the elastic modulus, mass density and volume fraction index of the ceramic phase and the metallic phase of the pipe. The boundary conditions include the free surface of the fluid, the bottom and the fluid-solid interface constraints. A fluid-structure interaction model in Cartesian coordinate system is established. Based on three-dimensional elasticity theory, the geometric and physical equations of the channel structure domain are derived. The two-parameter Pasternak model is used to simulate the foundation support effect. The control equations and boundary conditions of the fluid domain expressed in terms of hydrodynamic pressure are derived. The pipe structure reference surface and the fluid domain boundary are discretized using two-dimensional high-order spectral units. The basic equations of the structural domain and the fluid domain are transformed to the proportional boundary coordinate system through shape function interpolation and coordinate transformation. The proportional boundary finite element control equations of the structural domain and fluid domain are derived separately, and then transformed into a set of first-order variable coefficient differential equations. The precise integration method and eigenvalue decomposition method are used to solve them respectively, and the dynamic stiffness equation of the structural domain and the dynamic equation of the fluid domain are obtained. Based on the force balance and kinematic compatibility conditions at the fluid-structure interaction interface, the fluid dynamic pressure is used as an additional load to couple the equations of the structural domain and the fluid domain, forming the system dynamic control equations. The system dynamics control equations are transformed into a generalized eigenvalue problem, and the natural frequencies and corresponding mode shapes of the pipeline are obtained by solving the problem, thus completing the free vibration characteristic analysis.

2. The semi-analytical calculation method for free vibration of a fluid-filled functional gradient pipeline on an elastic foundation according to claim 1, characterized in that, The functionally graded conduit is composed of a ceramic phase and a metallic phase, and the material properties change continuously along the thickness direction of the conduit according to a power law function. The volume fractions of the ceramic phase and the metallic phase satisfy the following: , ,in, , These are volume fraction functions for the ceramic phase and the metallic phase, respectively. For the dimensionless thickness of the functionally graded material pipe, The thickness is measured from the inner surface. It is a volume fraction index used to control the variation and distribution of material along the thickness direction.

3. The semi-analytical calculation method for free vibration of a fluid-filled functional gradient pipeline on an elastic foundation according to claim 2, characterized in that, The effective material properties of the elastic modulus and mass density of the functionally graded pipe are expressed as follows: , ,in, , These are the elastic moduli of the ceramic phase and the metallic phase, respectively. , These are the mass densities of the ceramic phase and the metallic phase, respectively. The elastic modulus and mass density at any cross-section along the thickness direction satisfy: , .

4. The semi-analytical calculation method for free vibration of a fluid-filled functional gradient pipeline on an elastic foundation according to claim 1, characterized in that, Based on the assumption of small deformation, the strain-displacement relationship of the pipeline structural domain is as follows: Among them, displacement field Differential operators Defined as: Using Hooke's law in matrix form, the stress-strain relationship is expressed as: Among them, the elasticity matrix , It is Poisson's ratio.

5. The semi-analytical calculation method for free vibration of a fluid-filled functional gradient pipeline on an elastic foundation according to claim 1, characterized in that, The ground reaction force simulated by the two-parameter Pasternak model is expressed as follows: ,in, Let be the radial displacement vector of the functionally graded material pipe; For the Laplace operator; , These are the compression modulus and shear modulus of the elastic foundation, respectively. In a Cartesian coordinate system, the fundamental reaction force at any point within the solution domain is expressed through a global displacement field as: ,in, This represents the fundamental reaction force considering the global node degrees of freedom. , These represent the compressive stiffness matrix and the shear stiffness matrix of the foundation, respectively.

6. The semi-analytical calculation method for free vibration of a fluid-filled functional gradient pipeline on an elastic foundation according to claim 1, characterized in that, The governing equations of the fluid domain are the three-dimensional Laplace equations: ,in, The pressure is dynamic water pressure; the boundary conditions of the fluid domain include the fluid-structure interface, the free surface, and the bottom surface. The fluid-structure interface is... The free surface is The bottom surface is ,in, For fluid density, Let be the acceleration of the container wall along the outward normal. For outward unit normals.

7. The semi-analytical calculation method for free vibration of a fluid-filled functional gradient pipeline on an elastic foundation according to claim 1, characterized in that, When discretizing the reference surface of the pipeline structure, establish a scaled boundary finite element coordinate system. The element interpolation expression is: in, , Let the coordinates be those of the inner surface. , Let the coordinates be those of the outer surface. The shape function matrix of the higher-order spectral elements; The coordinate transformation relationship is as follows: ,in, It is a Jacobian matrix.

8. The semi-analytical calculation method for free vibration of a fluid-filled functional gradient pipeline on an elastic foundation according to claim 1, characterized in that, After deriving the finite element governing equations for the scaled boundary of the structural domain, the transformed first-order variable-coefficient differential equation is: ,in, , Let be the pipe displacement vector. For the internal force vectors of the fluid domain nodes, The coefficient matrix of the structural domain; After solving using the precise integration method, the equilibrium equations of the structured domain are obtained: ,in, , , , A submatrix of the stiffness matrix. , These are the displacement vectors of the inner and outer surfaces, respectively. , These are the equivalent nodal forces on the inner and outer surfaces, respectively. The force exerted by the fluid at the interface.

9. The semi-analytical calculation method for free vibration of a fluid-filled functional gradient pipeline on an elastic foundation according to claim 1, characterized in that, After deriving the finite element governing equations for the proportional boundary of the fluid domain, the transformed first-order variable-coefficient differential equation is: in, , For the internal force vectors of the fluid domain nodes, This is the fluid domain coefficient matrix; After solving using the eigenvalue decomposition method, the fluid domain equilibrium equations are obtained: ,in, Here is the stiffness matrix of the fluid domain. This is the mass matrix of the fluid-structure interaction.

10. The semi-analytical calculation method for free vibration of a fluid-filled functional gradient pipeline on an elastic foundation according to claim 1, characterized in that, The expression for the generalized eigenvalue problem is: in, , , , For the stiffness matrix of the structural domain, , , , For the structural domain quality matrix submatrix, Add a mass matrix to the fluid. For the natural frequency, , These are the displacement vectors of the outer and inner surface nodes, respectively.

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