Semi-analytical dynamic modeling method for liquid-filled series-parallel pipeline system based on beam-shell coupling

Abstract

The invention belongs to the technical field of mechanical dynamics, and discloses a beam-shell coupling-based semi-analytical dynamics modeling method for a liquid-filled series-parallel pipeline system. A parallel small-length-diameter-ratio pipeline system and a parallel large-length-diameter-ratio pipeline system are respectively equivalent to a beam sub-structure and a shell sub-structure, coupling connection between the structures is realized through a spatial distribution spring group, the coupling effect of the two structures is converted into corresponding coupling spring potential energy, a system coupling matrix is deduced, and a kinetic model of a parallel liquid filling pipeline is established; a series small-length-diameter-ratio pipeline system and a series large-length-diameter-ratio pipeline system are connected in series, small-length-diameter-ratio pipelines are equivalent to shell structures respectively, small-length-diameter-ratio pipe sections are modeled through shell units, pipe joint sections are modeled through variable cross-section beam units, coupling between pipe joints and adjacent pipelines is achieved through the virtual spring technology, a system coupling matrix is deduced, and the system coupling matrix is obtained. And establishing a kinetic model of the series liquid filling pipeline. Through comparison and verification with an inherent frequency result obtained by a finite element model and a hammering experiment, the established model has reliability in solving inherent characteristics.

Description

Technical Field

[0001] This invention relates to the field of mechanical dynamics technology, and in particular to a semi-analytical dynamic modeling method for a liquid-filled series-parallel pipeline system based on beam-shell coupling. Background Technology

[0002] The internal space of an aero-engine is compact, and the series-parallel piping system undertakes the task of transporting various media such as fuel, lubricating oil, and gas, making it a key component for ensuring stable engine operation. From a structural perspective, piping systems can be mainly divided into two categories: series piping systems and parallel piping systems. They are connected through pipe fittings and double-clamp structures, forming a complex spatial network transport system. However, while previous literature has extensively studied the dynamic characteristics of individual fluid-filled pipes, there is a gap in the academic research on semi-analytical dynamic modeling and vibration characteristic solutions for series-parallel piping with small and large length-to-diameter ratios. Summary of the Invention

[0003] In view of this, based on the theoretical foundation of beams and shells, a semi-analytical dynamic modeling method for liquid-filled series-parallel pipeline systems with significant geometric differences in small and large aspect ratios in aero-engines is proposed. The two types of pipelines are respectively represented as beam and shell substructures. When small and large aspect ratio pipelines are connected in parallel or series pipe joints are connected to a small aspect ratio pipeline system, an additional rotational displacement is introduced into the vibration displacement of the shell model to achieve coupling between the shell substructure and the beam substructure. The coupling connection between the structures is achieved through spatially distributed spring groups, converting the coupling effect into the potential energy of the corresponding coupling springs, and thus deriving the coupling matrix of the system. Considering the nonlinear characteristics of single / double clamp supports, virtual work generated by nonlinear external forces is introduced, and dynamic models of series-parallel liquid-filled pipelines with different aspect ratios are established. The reliability of the established model in solving inherent characteristics is verified by comparing the results of vibration tests with those of the finite element model. Conducting semi-analytical dynamic modeling of complex series-parallel piping systems for aero-engines not only helps to improve the dynamic modeling theory system of structural coupling mechanisms, but also provides important theoretical support for the optimized design of aero-engine piping systems.

[0004] The technical solution adopted in this invention is as follows: A semi-analytical dynamic modeling method for a liquid-filled series-parallel pipeline system based on beam-shell coupling, comprising the following steps:

[0005] S1. Considering the beam-shell coupling, a dynamic model of the parallel liquid-filled pipeline is performed. Based on Timoshenko beam theory, the kinetic energy, potential energy, and allowable displacement function of the liquid-filled straight beam pipeline with a large length-to-diameter ratio, as well as the potential energy generated by a single clamp, are obtained. Based on Mindlin-Reissner straight shell theory, the kinetic energy, potential energy, and allowable displacement function of the liquid-filled straight shell pipeline with a small length-to-diameter ratio, as well as the potential energy generated by a single clamp, are obtained. A spring is used to simulate the coupling of the straight beam and the curved beam with a double clamp, and the potential energy of the coupling spring is obtained. The kinetic energy, potential energy, and allowable displacement function of the straight beam and the curved beam, as well as the potential energy stored in the coupling spring, are substituted into the Lagrange energy equation. After discretization, the stiffness matrix, mass matrix, and damping matrix are obtained. After combination, the dynamic equation of the parallel liquid-filled pipeline is obtained.

[0006] S2. Considering the beam-shell coupling, a dynamic model of the series-filled pipeline is performed. Based on the Mindlin-Reissner bent shell theory, the kinetic energy, potential energy, and allowable displacement function of the small aspect ratio filled pipeline are obtained. The pipe joint section is modeled using beam elements. A virtual spring is used to couple the pipe joint and the small aspect ratio pipeline, and the coupling potential energy between the pipe joint and the adjacent pipeline is obtained. The kinetic energy, potential energy, allowable displacement function, and coupling potential energy of the straight shell, bent shell, and pipe joint are substituted into the Lagrange energy equation. After discretization, the stiffness matrix, mass matrix, and damping matrix are obtained, and the dynamic equation of the series-filled pipeline is obtained.

[0007] The kinetic energy, potential energy, allowable displacement function, and potential energy generated by the single clamp of the large aspect ratio liquid-filled straight beam pipeline are obtained as follows:

[0008] Determine the model parameters of the high aspect ratio liquid-filled straight beam pipeline; the high aspect ratio liquid-filled straight beam pipeline L / D>50, where L represents the pipeline length and D represents the pipeline diameter;

[0009] Based on Timoshenko beam theory, a dynamic model of a liquid-filled straight beam pipeline with a large length-to-diameter ratio is established; in the local coordinate system o of the straight beam pipeline section. i -x i y i z i Below, the displacement field of the straight beam pipe segment is defined as: u si v si and w si These represent the straight beam pipe segments along x. i y i and z i Axial displacement in the direction; θ si φ si and ψ si They represent the orbit around x. i y i and z i Angular displacement in direction; the geometric parameters of the dynamic model of a high aspect ratio liquid-filled straight beam pipeline include: li d represents the axial length of the i-th straight beam segment; i and D i These represent the inner and outer diameters of a high length-to-diameter ratio liquid-filled straight beam pipeline, respectively.

[0010] Potential energy U of a single straight beam pipe segment si Represented as:

[0011]

[0012] Among them, A pi I represents the cross-sectional area of ​​the straight beam pipe section. yi and I zi They are respectively straight beam pipe sections around o i y i and around o i z i The moment of inertia of the cross section of the shaft, E represents the elastic modulus of the straight beam pipe section, G represents the shear modulus of the straight beam pipe section, J i Represents the torsional moment of inertia. This is the shear correction factor;

[0013] Kinetic energy T of a single straight beam pipe segment si The specific expression is:

[0014]

[0015] Where, ρ p I represents the density of a liquid-filled straight beam pipe with a large length-to-diameter ratio. xi Represents a straight beam pipe segment around o i x i Moment of inertia of the shaft section;

[0016] Elastic potential energy generated by a single clamp Calculated using the following formula:

[0017]

[0018] The stiffness of each group of elements is as follows: This refers to the axial position of each group of springs, with a total of N springs arranged. g One single clamp;

[0019] Kinetic energy of fluid in a straight beam pipe section Represented as:

[0020]

[0021] Where, ρ f and A fi J represents the fluid density and cross-sectional area, respectively. fi It is the torsional moment of inertia of the fluid, I fiIt is the moment of inertia of the fluid cross section;

[0022] Allowable displacement function χ for straight beam pipe segment i (x, t), χ i = u si , v si , w si , θ si , φ si , ψ si , is represented as:

[0023]

[0024] in, , , , , and The vector representing the allowable displacement function of the straight beam pipe segment. , , , , and These represent the generalized displacement coordinates of the straight beam pipe segment, , , , , and Let represent the unknown coefficients of the allowable displacement function of the straight beam pipe segment, respectively. , , , , and The coefficient of the auxiliary term representing the allowable displacement function of the straight beam pipe segment. It is the truncation number. ;

[0025] The auxiliary term representing the allowable displacement function of the straight beam pipe segment is specifically expressed as follows:

[0026] .

[0027] The kinetic energy, potential energy, allowable displacement function, and potential energy generated by the single clamp of the small aspect ratio liquid-filled straight shell pipe are obtained as follows:

[0028] Determine the model parameters for a liquid-filled straight shell pipeline with a small length-to-diameter ratio; the length-to-diameter ratio of the liquid-filled straight shell pipeline is L / D≤50;

[0029] Based on the Mindlin-Reissner shell theory, a dynamic model of a liquid-filled straight shell pipe with a small length-to-diameter ratio is established. A cylindrical coordinate system o-xβz is established in the mid-plane of the liquid-filled straight shell pipe with a small length-to-diameter ratio, and the coordinates of any point on the mid-plane are represented by (x, β). l and R c Let u represent the length along the x-direction and the radius of the mid-surface along the z-direction of the liquid-filled straight-shell pipe with a small aspect ratio, respectively; zi v zi and w zi Let θ represent the axial displacements of the mid-surface of the i-th cylindrical shell in the x, β, and z directions, respectively. zi and φ zi They are around u zi and v zi The rotational displacement of the middle surface;

[0030] Displacement field at any point on a straight shell pipe section Represented as:

[0031]

[0032] The linear strain-displacement relationship corresponding to the straight shell tube section is shown below:

[0033]

[0034] Where, ε x and ε β Let γ be the normal strain along the x and β directions of the straight shell, respectively. βz γ xz and γ xβ The corresponding shear strain;

[0035] Furthermore, the linear stress-strain relationship corresponding to the straight shell section is expressed as follows:

[0036]

[0037] Where, σ x and σ β Let τ be the normal stress in the x-direction and β-direction, respectively. βz τ xz and τ xβ The corresponding shear stress is μ, where μ is Poisson's ratio.

[0038] Potential energy generated by elastic deformation of straight shell pipe section Represented as:

[0039]

[0040] in, Represents the shear coefficient;

[0041] In addition, the kinetic energy of the straight shell section The expression is:

[0042]

[0043] Where I0 = ρh, I1 = 1 / 12ρh 3 ;

[0044] In the fluid assumption, the fluid is considered an ideal fluid that is inviscid, irrotational, and incompressible, and flows steadily laterally relative to the pipe, neglecting the fluid's gravitational potential energy; the kinetic energy T of the fluid inside the shell... fi Represented as:

[0045]

[0046] in, For fluid velocity;

[0047] Potential energy generated when a single clamp supports a straight shell pipe section Represented as:

[0048]

[0049] in, , for the stiffness of a single clamp; x i N represents the axial position of each set of springs. g This refers to the number of single-unit clamps;

[0050] The allowable displacement function for a straight shell pipe section is:

[0051]

[0052] in, , , , , and The unknown coefficients represent the allowable displacement function of a straight shell pipe section; , , , and The unknown coefficients representing auxiliary terms; and Represents the axial half-wave number and circumferential wave number along the x and β directions, respectively; and Represents the cutoff numbers for the allowable axial displacement function and the allowable circumferential displacement function; The auxiliary term representing the allowable displacement function is expressed as follows:

[0053] .

[0054] The dynamic equations for the parallel liquid-filled pipelines are established as follows;

[0055] When a double-clamped hydraulic straight beam pipeline with a large length-to-diameter ratio and a hydraulic straight shell pipeline with a small length-to-diameter ratio are connected in parallel, the small length-to-diameter hydraulic straight shell pipeline is divided into Q units circumferentially. Each unit is connected to the large length-to-diameter ratio pipeline using spring sets, thus achieving the parallel connection of the large length-to-diameter hydraulic straight beam pipeline and the small length-to-diameter hydraulic straight shell pipeline. Each spring set includes translational stiffness and corresponding damping in three directions, and torsional stiffness and corresponding damping in three directions, expressed as follows: and χ = u, v, w, θ, φ, ψ; the mass of the double clamp is evenly divided into 4 parts with a mass of m. d The concentrated mass points are applied to the corresponding positions of each spring group on the two pipes; considering the soft nonlinear behavior exhibited by the double clamps under dynamic load, and These represent the nonlinear stiffness and nonlinear damping coefficient of the double clamp along the z-direction, respectively. and These represent the nonlinear stiffness and nonlinear damping coefficient of the double clamp along the y-direction, respectively.

[0056] A rotational angular displacement ψ is introduced; by establishing a deformation compatibility equation between the normal rotational angular displacement ψ and the translational displacement components u and v in local coordinates, the singularity of the stiffness matrix is ​​eliminated. The specific expression is as follows:

[0057]

[0058] Furthermore, the vibration displacement components of a liquid-filled straight-shell pipe with a small aspect ratio are transformed from the local cylindrical coordinate system to the global Cartesian coordinate system in order to construct the energy functional of the coupled spring assembly. The calculation formulas for the translational and rotational displacement components of the straight-shell structure transformed into the global coordinate system are as follows:

[0059]

[0060] The elastic potential energy U stored in the parallel coupled spring of the straight shell and straight beam structure is obtained. e It is calculated using the following formula:

[0061]

[0062] Where, x di The x represents the position of the double clamp spring assembly on a liquid-filled straight beam pipeline with a large length-to-diameter ratio. dj and β dj The position of the double clamp spring assembly on the liquid-filled straight pipe with a small length-to-diameter ratio is represented by Q, which is 8.

[0063] Nonlinear restoring force function of each set of springs in a double clamp in a beam-shell coupled parallel liquid-filled straight pipeline system Represented as:

[0064]

[0065] in, ;

[0066] The virtual work generated by the nonlinear external force of single and double clamps Represented as:

[0067]

[0068] Where, δ D It is the Diclave function, f Θ The nonlinear external force generated by a single clamp;

[0069] Substituting the kinetic energy, potential energy, and allowable displacement functions of the liquid-filled straight shell pipeline based on shell theory and the liquid-filled straight beam pipeline based on beam theory, along with the potential energy generated by the single clamp and the potential energy stored in the coupling spring, into the Lagrange equation, the specific expression is as follows:

[0070]

[0071] in, For generalized coordinates, T is the sum of all the kinetic energies mentioned above, and U is the sum of all the potential energies mentioned above. This is the sum of all the aforementioned ineffective efforts;

[0072] The dynamic equations for the parallel liquid-filled pipelines are then obtained as follows:

[0073]

[0074] in, K g Let K be the stiffness matrix of the clamp. f For the stiffness matrix affected by fluid, Let K be the coupling matrix of the beam-shell coupled parallel fluid-filled space piping system. p Here is the pipeline stiffness matrix. Stiffness matrix for clamp-supported pipelines with small length-to-diameter ratio; M f Add a mass matrix to the fluid, M p Let M be the pipeline quality matrix. c The mass matrix of the double clamp; , Let G represent the Rayleigh damping matrix of the beam-shell coupled parallel fluid-filled space piping system, and let G represent the fluid damping matrix.

[0075] The kinetic energy, potential energy, and allowable displacement functions of the small aspect ratio fluid-filled bend are as follows:

[0076] Determine the model parameters of the liquid-filled bend with a small length-to-diameter ratio; establish a circumferential coordinate system o-α at the midplane of the liquid-filled bend with a small length-to-diameter ratio. c β c z, Select the bent shell section α c and β c The directions are used as the two principal directions of the ring coordinate system, and the coordinates of any point on the intermediate surface are represented by (α). c , β c ) indicates that β c Let α0 be the circumferential angle of any point on the bend in the shell section from the reference point, and α0 be the central angle of the corresponding bend in the shell section, with a range of variation of [missing information]. ;R t R represents the bending radius of the bent shell section. c Let u be the radius of the mid-surface, h be the thickness of the fluid-filled bend with a small aspect ratio, and u be the radius of the mid-surface. wj v wj and w wj The bent shell section along α c β c and the linear displacement component in the z-direction, θ wj and φ wj They are orbiting α c and β c Angular displacement of the shaft;

[0077] Based on the Mindlin-Reissner shell theory, the displacement field at any point in a bent shell section can be uniformly expressed in the following form:

[0078]

[0079] The linear stress-strain relationship of the bent shell section is expressed as:

[0080]

[0081] Where, σ α and σ β They are structure α c Direction and β c Normal stress in the direction, τ zβ τ zα and τ βα ε represents the shear stress corresponding to the structure. α and ε β γ represents the normal strain of the structure. zβ γ zα and γ βα These are the structural shear strains;

[0082] Based on linear elasticity theory, the strain at any point of a structural element can be expressed as:

[0083]

[0084] Where, ε α,0 , ε β,0 γ zβ,0 γ zα,0 and γ βα,0 For the strain at the mid-surface of the structure, κ α κ β and κ βα This represents the change in curvature of the relative mid-surface of the structure;

[0085] The strain described above is described by the displacement of the mid-surface of the structure, specifically expressed as:

[0086]

[0087] in, and These represent the two radii of curvature corresponding to the bent shell section. , , and The Lamé coefficient is the coefficient corresponding to the bent shell section; , ;

[0088] For the bent shell section, the strain energy U of the shell is... εj It can be represented in the following form:

[0089]

[0090] Kinetic energy of the bent shell section The expression is:

[0091]

[0092] The bent shell section along α c Direction and β c The direction is expanded into a series, and the corresponding allowable displacement function is constructed, specifically expressed as:

[0093]

[0094] in, ; , , , and The unknown coefficients represent the allowable displacement function of the bent shell section; , , , and The coefficients of the unknowns in the auxiliary terms; For auxiliary terms in the Fourier series, specifically represented as:

[0095] .

[0096] The coupling potential energy between the pipe joint section and the adjacent small aspect ratio liquid-filled space pipe is established as follows;

[0097] The elastic potential energy generated by the virtual spring assembly between adjacent variable cross sections of the pipe joint It can be represented as:

[0098]

[0099] Kinetic energy T of the pipe joint vk and strain energy U vk It can be represented as:

[0100]

[0101] The pipe fitting is divided into P segments along the axial direction, and the length of each segment is p. k (k = 1, 2, ..., P) represents that u vk v vk and w vk These represent the pipe fittings at x k y k and z k Axial and lateral displacements in the direction, θ vk φ vk and ψ vk They represent the orbit around x. k y k and z k Directional torsion and angular displacement, A sk Let k be the cross-sectional area of ​​the pipe joint segment. For translational and rotational stiffness, ρ is the angle of rotation of the plane. v This represents the equivalent density of pipe fittings, where N represents the number of pipe fittings, and I... xk I yk and I zk These represent the pipe joint cross-section relative to the local coordinate system x. k y k and z k Moment of inertia of the shaft section, J k For each cross section of the pipe fitting relative to x k Torsional moment of inertia of the shaft.

[0102] When a pipe fitting is connected in series with a liquid-filled space pipeline with a small length-to-diameter ratio, the elastic potential energy U generated by the virtual spring assembly sb Specifically, it is expressed as follows:

[0103]

[0104] Where, x di x represents the position of the double-clamp spring assembly in the beam structure. dj ,β dj This indicates the location of the double clamp spring assembly within the shell structure;

[0105] The process of obtaining the dynamic equations for the series-connected liquid-filled pipeline is as follows;

[0106] The liquid-filled pipeline with a small length-to-diameter ratio is divided into multiple straight shell sections and bent shell sections. Based on the Mindlin-Reissner shell theory, the strain-displacement relationship of the straight shell section and the bent shell section is obtained, and the displacement of the middle surface of the straight shell section and the bent shell section is expressed by the modified Fourier series.

[0107] For space pipelines with small length-to-diameter ratios, it is necessary to analyze the force and displacement compatibility conditions at the interface between the straight shell section and the next adjacent curved shell section, and calculate the elastic potential energy stored in the coupling spring between the two structural units. Specifically, it can be expressed as:

[0108]

[0109] in, , represents the connection stiffness in each direction between the straight shell pipe section and the bent shell pipe section; the spatial transformation matrix is ​​represented by T, which is used to consider various connection combinations of adjacent pipe sections in the spatial pipeline.

[0110] The elastic potential energy generated at the interface between the bent shell section and the next adjacent straight shell section in a small aspect ratio space pipeline Represented as:

[0111]

[0112] Work done by external excitation load on each structural element Represented as:

[0113]

[0114] Among them, T z Let N be the spatial transformation matrix. n N represents the number of straight shell segments. m This represents the number of curved shell segments;

[0115] Considering the nonlinear support characteristics of the clamp, when the clamp supports a straight shell pipe section, the nonlinear force... Represented as:

[0116]

[0117] in, This represents the nonlinear stiffness of the single-clamp shell model. The coefficients of the Bouc-Wen model, These are hysteretic variables in the Bouc-Wen model;

[0118] Substituting the kinetic energy, potential energy, and allowable displacement functions of the straight shell section, the bent shell section, the single-joint clamp, and the pipe joint into the Lagrange energy equation, and considering the effect of Rayleigh damping, the dynamic equation of the space-filled pipeline system based on the beam-shell coupling theory is obtained, specifically expressed as:

[0119]

[0120] in, , , and The stiffness, mass, nonlinear stiffness, and damping matrix of a clamp-type series liquid-filled pipeline system with arbitrary small aspect ratio are represented. , Here is the boundary stiffness matrix. For the stiffness matrix affected by fluid, The stiffness matrix of a small aspect ratio pipe is derived from the stiffness matrix of a straight shell. and the stiffness matrix of the bent shell constitute, Stiffness matrix for pipes with small length-to-diameter ratio supported by clamps. To connect the stiffness matrix, The coupling stiffness matrix is ​​the series connection between the pipe joint and the adjacent small aspect ratio liquid-filled space pipeline. , The pipeline mass matrix consists of the mass matrices of straight shells and bent shells. and constitute, For the fluid mass matrix, Represents the external excitation force vector. Represents a nonlinear force vector; G represents the fluid damping matrix, C p It is the Rayleigh damping matrix of the pipeline system.

[0121] Compared with the prior art, the present invention has the following beneficial effects: The present invention addresses the dynamic characteristics of complex series-parallel piping systems in aero-engines, comprehensively considers fluid effects and structural coupling, and proposes a semi-analytical dynamic modeling method for liquid-filled series-parallel piping systems based on beam-shell coupling. The semi-analytical method is used to model complex series-parallel piping systems, thus improving the theoretical defects of traditional semi-analytical modeling methods in complex spatial configuration series-parallel piping systems.

[0122] This invention uses a beam-shell coupled series-parallel piping system as a verification case. The natural frequencies obtained by the semi-analytical model of this invention are compared with the natural frequencies obtained by the finite element model and the hammer impact test, and the errors are all less than 5%. The verification model has good computational efficiency and accuracy in predicting the inherent characteristics of beam-shell coupled series-parallel piping systems. Attached Figure Description

[0123] Figure 1 This is a schematic diagram of a straight beam pipe segment model;

[0124] Figure 2 This is a schematic diagram of a straight-shell pipe model;

[0125] Figure 3 This is a schematic diagram of a beam-shell coupled parallel piping model.

[0126] Figure 4 This is a schematic diagram of a bent shell pipe section model;

[0127] Figure 5 This is a schematic diagram of a series spatial pipeline model;

[0128] Figure 6 Finite element model of beam-shell parallel pipeline;

[0129] Figure 7 Wireframe diagram for hammer impact test of a series spatial pipeline model;

[0130] Figure 8 This is the overall frequency response function diagram for the series space pipeline test. Detailed Implementation

[0131] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0132] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0133] A semi-analytical dynamic modeling method for a liquid-filled series-parallel piping system based on beam-shell coupling includes the following steps:

[0134] S1. Considering the beam-shell coupling, a dynamic model of the parallel liquid-filled pipeline is performed. Based on Timoshenko beam theory, the kinetic energy, potential energy, and allowable displacement function of the liquid-filled straight beam pipeline with a large length-to-diameter ratio, as well as the potential energy generated by a single clamp, are obtained. Based on Mindlin-Reissner straight shell theory, the kinetic energy, potential energy, and allowable displacement function of the liquid-filled straight shell pipeline with a small length-to-diameter ratio, as well as the potential energy generated by a single clamp, are obtained. A spring is used to simulate the coupling of the straight beam and the curved beam with a double clamp, and the potential energy of the coupling spring is obtained. The kinetic energy, potential energy, and allowable displacement function of the straight beam and the curved beam, as well as the potential energy stored in the coupling spring, are substituted into the Lagrange energy equation. After discretization, the stiffness matrix, mass matrix, and damping matrix are obtained. After combination, the dynamic equation of the parallel liquid-filled pipeline is obtained.

[0135] S2. Considering the beam-shell coupling, a dynamic model of the series-filled pipeline is performed. Based on the Mindlin-Reissner bent shell theory, the kinetic energy, potential energy, and allowable displacement function of the small aspect ratio filled pipeline are obtained. The pipe joint section is modeled using beam elements. A virtual spring is used to couple the pipe joint and the small aspect ratio pipeline, and the coupling potential energy between the pipe joint and the adjacent pipeline is obtained. The kinetic energy, potential energy, allowable displacement function, and coupling potential energy of the straight shell, bent shell, and pipe joint are substituted into the Lagrange energy equation. After discretization, the stiffness matrix, mass matrix, and damping matrix are obtained, and the dynamic equation of the series-filled pipeline is obtained.

[0136] S3. Finite element modeling and simulation verification were performed on the beam-shell coupled parallel pipeline model; hammer impact test verification was performed on the beam-shell coupled series pipeline model; and the accuracy of the invention in the calculation of the beam-shell coupled liquid-filled series and parallel pipeline was verified by comparing the errors between different natural frequency results.

[0137] The kinetic energy, potential energy, allowable displacement function, and potential energy generated by the single clamp of the large aspect ratio liquid-filled straight beam pipeline are obtained as follows:

[0138] S11. Determine the parameters of the large length-to-diameter ratio pipeline model;

[0139] Furthermore, S1 specifically includes the following steps:

[0140] Figure 1 This is a schematic diagram of a straight beam pipe segment model; based on Timoshenko's straight beam theory, a dynamic model of a straight pipe segment with a large length-to-diameter ratio is established. In the local coordinate system o of the straight pipe segment... i -x i y i z i Below, the displacement field of the straight pipe section is defined as: u si v si and w si These represent the straight pipe sections along x. i y i and z i Axial and lateral displacements in the direction; θ si φ si and ψ si They represent the orbit around x. i y i and z i Torsional and angular displacements. The geometric parameters of the model include: l i d represents the axial length of the i-th straight pipe segment. i and D i These represent the inner and outer diameters of the pipe, respectively.

[0141] S12. Determine the parameters of the small length-to-diameter ratio pipeline model;

[0142] Figure 2 This is a schematic diagram of a straight shell pipe model. Based on the Mindlin-Reissner shell theory, a dynamic model of a straight pipe section with a small length-to-diameter ratio is established. A cylindrical coordinate system o-xβz is established at the mid-plane of the pipe, and the coordinates of any point on the mid-plane can be represented by (x, β). l and R c These represent the length of the pipeline along the x and z directions and the radius of the neutral layer, respectively. zi v zi and w zi Let θ represent the mid-surface displacements of the i-th cylindrical shell in the x, β, and z directions, respectively. zi and φ zi They are around u zi and v zi The rotational displacement of the mid-surface. Furthermore, virtual spring technology is used to simulate the boundary constraints at both ends of the pipeline, including three stiffness values. , and The translational spring, and two springs with stiffness values ​​of , Angle spring.

[0143] S13. Determine the model parameters of the beam-shell coupled parallel piping system;

[0144] Figure 3 This is a schematic diagram of a beam-shell coupled parallel pipeline model. A dynamic model of a liquid-filled parallel pipeline with a large and small length-to-diameter ratio (L / D ratio) is established using the beam-shell coupling concept. The large L / D ratio pipeline can be considered a beam structure, while the small L / D ratio pipeline is considered a straight shell structure. Double clamps are used as the connection structure to achieve structural coupling between the two pipelines. When the large and small L / D ratio pipelines are connected in parallel using double clamps, the small L / D ratio pipeline needs to be divided into Q units circumferentially. Each unit is connected to the large L / D ratio pipeline using spring sets, thus achieving parallel connection of the two pipelines. Each spring set includes translational stiffness and corresponding damping in three directions, and torsional stiffness and corresponding damping in three directions, represented as follows: and ,(χu, v, w, θ, φ, ψ). The mass of the double clamp is evenly divided into 4 masses of m. d Concentrated mass points are applied to the corresponding positions of each set of springs on the pipeline to achieve an accurate description of the mass effect of the double clamp. Furthermore, the soft nonlinear behavior exhibited by the double clamp under dynamic loads is considered. and These represent the nonlinear stiffness and nonlinear damping coefficient of the double clamp along the z-direction, respectively. and These represent the nonlinear stiffness and nonlinear damping coefficient of the double clamp along the y-direction, respectively.

[0145] Furthermore, in S11, the potential energy U of a single straight beam segment si Represented as:

[0146]

[0147] Among them, A pi I represents the cross-sectional area of ​​the straight beam pipe section. yi and I zi They are respectively straight beam pipe sections around o i y i and around o i z i The moment of inertia of the cross section of the shaft, E represents the elastic modulus of the straight beam pipe section, G represents the shear modulus of the straight beam pipe section, J i Represents the torsional moment of inertia. This is the shear correction factor;

[0148] Kinetic energy T of a single straight beam pipe segment siThe specific expression is:

[0149]

[0150] Where, ρ p I represents the density of a liquid-filled straight beam pipe with a large length-to-diameter ratio. xi Represents a straight beam pipe segment around o i x i Moment of inertia of the shaft section;

[0151] Elastic potential energy generated by a single clamp Calculated using the following formula:

[0152]

[0153] The stiffness of each group of elements is as follows: This refers to the axial position of each group of springs, with a total of N springs arranged. g One single clamp;

[0154] Kinetic energy of fluid in a straight beam pipe section Represented as:

[0155]

[0156] Where, ρ f and A fi J represents the fluid density and cross-sectional area, respectively. fi It is the torsional moment of inertia of the fluid, I fi It is the moment of inertia of the fluid cross section;

[0157] Allowable displacement function of straight beam pipe segment , is represented as:

[0158]

[0159] in, , , , , and The vector representing the allowable displacement function of the straight beam pipe segment. , , , , and These represent the generalized displacement coordinates of the straight beam pipe segment, , , , , and Let represent the unknown coefficients of the allowable displacement function of the straight beam pipe segment, respectively. , , , , and The coefficient of the auxiliary term representing the allowable displacement function of the straight beam pipe segment. It is the truncation number. ;

[0160] The auxiliary term representing the allowable displacement function of the straight beam pipe segment is specifically expressed as follows:

[0161] .

[0162] Furthermore, in S12, a dynamic model of a liquid-filled straight shell pipeline with a small length-to-diameter ratio is established based on the Mindlin-Reissner shell theory; a cylindrical coordinate system o-xβz is established in the plane of the liquid-filled straight shell pipeline with a small length-to-diameter ratio, and the coordinates of any point on the middle surface are represented by (x, β); l and R c Let u represent the length along the x-direction and the radius of the mid-surface along the z-direction of the liquid-filled straight-shell pipe with a small aspect ratio, respectively; zi v zi and w zi Let θ represent the axial displacements of the mid-surface of the i-th cylindrical shell in the x, β, and z directions, respectively. zi and φ zi They are around u zi and v zi The rotational displacement of the middle surface;

[0163] Displacement field at any point on a straight shell pipe section Represented as:

[0164]

[0165] The linear strain-displacement relationship corresponding to the straight shell tube section is shown below:

[0166]

[0167] Where, ε x and ε β Let γ be the normal strain along the x and β directions of the straight shell, respectively. βz γ xz and γ xβ The corresponding shear strain;

[0168] Furthermore, the linear stress-strain relationship corresponding to the straight shell section is expressed as follows:

[0169]

[0170] Where, σ x and σ βLet τ be the normal stress in the x-direction and β-direction, respectively. βz τ xz and τ xβ The corresponding shear stress is μ, where μ is Poisson's ratio.

[0171] Potential energy generated by elastic deformation of straight shell pipe section Represented as:

[0172]

[0173] in, Represents the shear coefficient;

[0174] In addition, the kinetic energy of the straight shell section The expression is:

[0175]

[0176] Where I0 = ρh, I1 = 1 / 12ρh 3 ;

[0177] In the fluid assumption, the fluid is considered an ideal fluid that is inviscid, irrotational, and incompressible, and flows steadily laterally relative to the pipe, neglecting the fluid's gravitational potential energy; the kinetic energy T of the fluid inside the shell... fi Represented as:

[0178]

[0179] in, For fluid velocity;

[0180] Potential energy generated when a single clamp supports a straight shell pipe section Represented as:

[0181]

[0182] in, , for the stiffness of a single clamp; x i N represents the axial position of each set of springs. g This refers to the number of single-unit clamps;

[0183] The allowable displacement function for a straight shell pipe section is:

[0184]

[0185] in, , , , , and The unknown coefficients represent the allowable displacement function of a straight shell pipe section; , , , and The unknown coefficients representing auxiliary terms; and Represents the axial half-wave number and circumferential wave number along the x and β directions, respectively; and Represents the cutoff numbers for the allowable axial displacement function and the allowable circumferential displacement function; The auxiliary term representing the allowable displacement function is expressed as follows:

[0186] .

[0187] Furthermore, in S13, when the double clamps connect the large-length-to-diameter (L / D) filled straight beam pipe and the small-length-to-diameter (L / D) filled straight shell pipe in parallel, the small-length-to-diameter (L / D) filled straight shell pipe is divided into Q units circumferentially. Each unit is connected to the large-length-to-diameter (L / D) pipe using spring sets, thus achieving the parallel connection of the large-length-to-diameter (L / D) filled straight beam pipe and the small-length-to-diameter (L / D) filled straight shell pipe. Each spring set includes translational stiffness and corresponding damping in three directions, and torsional stiffness and corresponding damping in three directions, expressed as follows: and χ = u, v, w, θ, φ, ψ; the mass of the double clamp is evenly divided into 4 parts with a mass of m. d The concentrated mass points are applied to the corresponding positions of each spring group on the two pipes; considering the soft nonlinear behavior exhibited by the double clamps under dynamic load, and These represent the nonlinear stiffness and nonlinear damping coefficient of the double clamp along the z-direction, respectively. and These represent the nonlinear stiffness and nonlinear damping coefficient of the double clamp along the y-direction, respectively.

[0188] A rotational angular displacement ψ is introduced; by establishing a deformation compatibility equation between the normal rotational angular displacement ψ and the translational displacement components u and v in local coordinates, the singularity of the stiffness matrix is ​​eliminated. The specific expression is as follows:

[0189]

[0190] Furthermore, the vibration displacement components of a liquid-filled straight-shell pipe with a small aspect ratio are transformed from the local cylindrical coordinate system to the global Cartesian coordinate system in order to construct the energy functional of the coupled spring assembly. The calculation formulas for the translational and rotational displacement components of the straight-shell structure transformed into the global coordinate system are as follows:

[0191]

[0192] The elastic potential energy U stored in the parallel coupled spring of the straight shell and straight beam structure is obtained. eIt is calculated using the following formula:

[0193]

[0194] Where, x di The x represents the position of the double clamp spring assembly on a liquid-filled straight beam pipeline with a large length-to-diameter ratio. dj and β dj The position of the double clamp spring assembly on the liquid-filled straight pipe with a small length-to-diameter ratio is represented by Q, which is 8.

[0195] Nonlinear restoring force function of each set of springs in a double clamp in a beam-shell coupled parallel liquid-filled straight pipeline system Represented as:

[0196]

[0197] in, ;

[0198] The virtual work generated by the nonlinear external force of single and double clamps Represented as:

[0199]

[0200] Where, δ D It is the Diclave function, f Θ The nonlinear external force generated by a single clamp;

[0201] Substituting the kinetic energy, potential energy, and allowable displacement functions of the liquid-filled straight shell pipeline based on shell theory and the liquid-filled straight beam pipeline based on beam theory, along with the potential energy generated by the single clamp and the potential energy stored in the coupling spring, into the Lagrange equation, the specific expression is as follows:

[0202]

[0203] in, For generalized coordinates, T is the sum of all the kinetic energies mentioned above, and U is the sum of all the potential energies mentioned above. This constitutes the aforementioned empty work;

[0204] The dynamic equations for the parallel liquid-filled pipelines are then obtained as follows:

[0205]

[0206] in, K g Let K be the stiffness matrix of the clamp. f For the stiffness matrix affected by fluid, Let K be the coupling matrix of the beam-shell coupled parallel fluid-filled space piping system. p Here is the pipeline stiffness matrix. Stiffness matrix for clamp-supported pipelines with small length-to-diameter ratio; M f Add a mass matrix to the fluid, M p Let M be the pipeline quality matrix. c The mass matrix of the double clamp; , Let G represent the Rayleigh damping matrix of the beam-shell coupled parallel fluid-filled space piping system, and let G represent the fluid damping matrix.

[0207] Furthermore, S2 specifically includes the following steps:

[0208] S21. Determine the model parameters for the liquid-filled bend pipeline with a small length-to-diameter ratio;

[0209] Figure 4 This is a schematic diagram of the bent shell pipe section model; a circumferential coordinate system o-α is established at the mid-plane of the liquid-filled bent pipe with a small length-to-diameter ratio. c β c z, Select the bent shell section α c and β c The directions are used as the two principal directions of the ring coordinate system, and the coordinates of any point on the intermediate surface are represented by (α). c ,β c ) indicates that β c Let α0 be the circumferential angle of any point on the bend in the shell section from the reference point, and α0 be the central angle of the corresponding bend in the shell section, with a range of variation of [missing information]. ;R t R represents the bending radius of the bent shell section. c Let u be the radius of the mid-surface, h be the thickness of the fluid-filled bend with a small aspect ratio, and u be the radius of the mid-surface. wj v wj and w wj The bent shell section along α c β c and the linear displacement component in the z-direction, θ wj and φ wj They are orbiting α c and β c Angular displacement of the shaft;

[0210] S22. Determine the coupling potential energy between the pipe joint section and the adjacent small length-to-diameter ratio liquid-filled space pipe;

[0211] S23. Determine the dynamic equations for the series-connected liquid-filled pipeline;

[0212] Furthermore, S21 specifically includes: based on the Mindlin-Reissner shell theory, the displacement field at any point in the bent shell section is uniformly expressed in the following form:

[0213]

[0214] The linear stress-strain relationship of the bent shell section is expressed as:

[0215]

[0216] Where, σ α and σ β They are structure α c Direction and β c Normal stress in the direction, τ zβ τ zα and τ βα ε represents the shear stress corresponding to the structure. α and ε β γ represents the normal strain of the structure. zβ γ zα and γ βα These are the structural shear strains;

[0217] Based on linear elasticity theory, the strain at any point of a structural element can be expressed as:

[0218]

[0219] Where, ε α,0 , ε β,0 γ zβ,0 γ zα,0 and γ βα,0 For the strain at the mid-surface of the structure, κ α κ β and κ βα This represents the change in curvature of the relative mid-surface of the structure;

[0220] The strain described above is described by the displacement of the mid-surface of the structure, specifically expressed as:

[0221]

[0222] in, and These represent the two radii of curvature corresponding to the bent shell section. , , and The Lamé coefficient is the coefficient corresponding to the bent shell section; , ;

[0223] For the bent shell section, the strain energy U of the shell is... εj It can be represented in the following form:

[0224]

[0225] Kinetic energy of the bent shell section The expression is:

[0226]

[0227] The bent shell section along α c Direction and β c The direction is expanded into a series, and the corresponding allowable displacement function is constructed, specifically expressed as:

[0228]

[0229] in, ; , , , and The unknown coefficients represent the allowable displacement function of the bent shell section; , , , and The coefficients of the unknowns in the auxiliary terms; For auxiliary terms in the Fourier series, specifically represented as:

[0230] .

[0231] Furthermore, S22 specifically includes:

[0232] The elastic potential energy generated by the virtual spring assembly between adjacent variable cross sections of the pipe joint It can be represented as:

[0233]

[0234] Kinetic energy T of the pipe joint vk and strain energy U vk It can be represented as:

[0235]

[0236] The pipe fitting is divided into P segments along the axial direction, and the length of each segment is denoted by _____. It means, u vk v vk and w vk These represent the pipe fittings at x k y k and z k Axial and lateral displacements in the direction, θ vk φ vk and ψ vk They represent the orbit around x. k y k and z k Directional torsion and angular displacement, A sk Let k be the cross-sectional area of ​​the pipe joint segment. For translational and rotational stiffness, ρ is the angle of rotation of the plane. v This represents the equivalent density of pipe fittings, where N represents the number of pipe fittings, and I... xk I yk and I zk These represent the pipe joint cross-section relative to the local coordinate system x. k y k and z k Moment of inertia of the shaft section, J k For each cross section of the pipe fitting relative to x k Torsional moment of inertia of the shaft.

[0237] When a pipe fitting is connected in series with a liquid-filled space pipeline with a small length-to-diameter ratio, the elastic potential energy U generated by the virtual spring assembly sb Specifically, it is expressed as follows:

[0238]

[0239] Where, x di x represents the position of the double-clamp spring assembly in the beam structure. dj ,β dj This indicates the location of the double clamp spring assembly within the shell structure;

[0240] Furthermore, S23 specifically includes:

[0241] Figure 5 This is a schematic diagram of a series-connected spatial pipeline model. For spatial pipelines with a small length-to-diameter ratio, it is necessary to analyze the force and displacement compatibility conditions at the interface between the straight shell section and the next adjacent curved shell section, and calculate the elastic potential energy stored in the coupling spring between the two structural units. Specifically, it can be expressed as:

[0242]

[0243] in, , represents the connection stiffness in each direction between the straight shell pipe section and the bent shell pipe section; the spatial transformation matrix is ​​represented by T, which is used to consider various connection combinations of adjacent pipe sections in the spatial pipeline.

[0244] The elastic potential energy generated at the interface between the bent shell section and the next adjacent straight shell section in a small aspect ratio space pipeline Represented as:

[0245]

[0246] Work done by external excitation load on each structural element Represented as:

[0247]

[0248] Among them, T z Let N be the spatial transformation matrix. n N represents the number of straight shell segments. m This represents the number of curved shell segments;

[0249] Considering the nonlinear support characteristics of the clamp, when the clamp supports a straight shell pipe section, the nonlinear force... Represented as:

[0250]

[0251] in, This represents the nonlinear stiffness of the single-clamp shell model. The coefficients of the Bouc-Wen model, These are hysteretic variables in the Bouc-Wen model;

[0252] Substituting the kinetic energy, potential energy, and allowable displacement functions of the straight shell section, the bent shell section, the single-joint clamp, and the pipe joint into the Lagrange energy equation, and considering the effect of Rayleigh damping, the dynamic equation of the space-filled pipeline system based on the beam-shell coupling theory is obtained, specifically expressed as:

[0253]

[0254] in, , , and The stiffness, mass, nonlinear stiffness, and damping matrix of a clamp-type series liquid-filled pipeline system with arbitrary small aspect ratio are represented. , Here is the boundary stiffness matrix. For the stiffness matrix affected by fluid, The stiffness matrix of a small aspect ratio pipe is derived from the stiffness matrix of a straight shell. and the stiffness matrix of the bent shell constitute, Stiffness matrix for pipes with small length-to-diameter ratio supported by clamps. To connect the stiffness matrix, The coupling stiffness matrix is ​​the series connection between the pipe joint and the adjacent small aspect ratio liquid-filled space pipeline. , The pipeline mass matrix consists of the mass matrices of straight shells and bent shells. and constitute, For the fluid mass matrix, Represents the external excitation force vector. Represents a nonlinear force vector; G represents the fluid damping matrix, C p It is the Rayleigh damping matrix of the pipeline system.

[0255] Furthermore, S3 specifically includes the following steps:

[0256] S31, Verification of the finite element model of beam-shell parallel pipeline;

[0257] Figure 6 For the finite element model of the beam-shell parallel pipeline, finite element models of the large and small length-to-diameter ratio (L / D ratio) pipeline systems were established using Shell281 and Beam188 combined elements. For the large L / D ratio pipeline, Beam188 elements were used for modeling. For the small L / D ratio pipeline model, it was divided into 16 segments circumferentially, containing a total of 13728 nodes and 4560 elements. Using Shell281 elements, a node was created at the center of the pipeline cross-section, and this node was sequentially connected to one ring of nodes on the outer ring of the pipeline cross-section using rigid constraints to ensure displacement compatibility between the shell elements and the nodes. Furthermore, Combine14 spring elements were used to constrain the degrees of freedom (including translational and rotational degrees of freedom) of the nodes to accurately simulate the double-clamp support characteristics of the actual parallel pipeline. The specific stiffness value of the double-clamp is: k. du = 5×10 5 N / m, k dv =5×10 6 N / m, k dw =5.5×10 6 N / m, k dθ =50 N·m / rad, k dφ =50 N·m / rad, k dψ =95 N·m / rad. The mass of the double clamp is equivalent to a concentrated mass point attached to the parallel pipe node, and is simulated using Mass21 elements.

[0258] S32, Verification by hammer impact test of beam-shell series pipeline;

[0259] Figure 7 The diagram shows the wireframe of the hammer impact test on the series spatial pipeline model. Modal hammer impact tests were further conducted on the series spatial pipeline. To compare and analyze the accuracy of different modeling methods, finite element models of the series pipeline were established using both solid element models and beam element models.

[0260] Furthermore, S31 specifically includes:

[0261] The natural frequencies calculated by the model of this invention were compared with the results of the finite element model. The comparison results for the first six natural frequencies of the pipeline system are detailed in Table 1. The maximum relative error between the natural frequencies calculated by the model of this invention and the finite element model was 2.56%, while the minimum relative error was only 0.77%. The error rate between the natural frequencies calculated by the model of this invention and the results of the finite element model was all less than 3%. This result fully verifies the calculation accuracy of the model of this invention in beam-shell parallel pipelines.

[0262] Table 1 Comparison of natural frequencies between the model of this invention and the finite element model

[0263]

[0264] Furthermore, S32 specifically includes:

[0265] Figure 8 The table shows the overall frequency response function of the series-connected spatial pipeline test. The natural frequencies measured by different models are detailed in Table 2. The study found that for a series-connected pipe with a small length-to-diameter ratio of 20 mm, the calculation results of the beam element model deviated significantly from the experimental values, with a large relative error, reaching a maximum of 17.29%. However, when using shell elements to model the pipe sections and beam elements to model the pipe joint sections, the maximum relative error was 8.16%. Compared with the finite element beam element model, the model of this invention showed a higher degree of agreement with the experimental results and the solid element model. This not only effectively improved the modeling accuracy but also considered computational efficiency. The solid element model took approximately 60 seconds to solve for the first four natural frequencies of the system, while the semi-analytical model established in this invention only required 20 seconds to complete the calculation with the same accuracy, resulting in a 66.7% improvement in computational efficiency.

[0266] Table 2 Comparison of natural frequencies of series tubes under fixed-end conditions

[0267]

Claims

1. A semi-analytical dynamic modeling method for fluid-filled series-parallel pipe systems based on beam-shell coupling, characterized in that, The method comprises the following steps: S1, considering the beam-shell coupling case, a parallel liquid-filled pipeline is modeled dynamically; based on the Timoshenko beam theory, kinetic energy, potential energy, displacement admissible function of a large length-diameter ratio liquid-filled straight beam pipeline and potential energy generated by a single coupling clamp are obtained; based on the Mindlin-Reissner straight shell theory, kinetic energy, potential energy, displacement admissible function of a small length-diameter ratio liquid-filled straight shell pipeline and potential energy generated by a single coupling clamp are obtained; a spring is used to simulate the coupling of a double coupling clamp to a straight beam and a bent beam and the potential energy stored in the coupling spring is obtained, the kinetic energy, potential energy, displacement admissible function of the straight beam and the bent beam and the potential energy stored in the coupling spring are substituted into the Lagrange energy equation, and after discretization, stiffness matrix, mass matrix and damping matrix are obtained, and after combination, the dynamic equation of the parallel liquid-filled pipeline is obtained; S2, considering the beam-shell coupling case, a series liquid-filled pipeline is modeled dynamically; based on the Mindlin-Reissner bent shell theory, kinetic energy, potential energy, displacement admissible function of a small length-diameter ratio liquid-filled bent pipeline are obtained; a beam element is used to model a pipe joint section, and a virtual spring is used to couple the pipe joint and the small length-diameter ratio pipeline and obtain the coupling potential energy between the pipe joint and the adjacent pipeline; the kinetic energy, potential energy, displacement admissible function of the straight shell, the bent shell and the pipe joint and the coupling potential energy are substituted into the Lagrange energy equation, and after discretization, stiffness matrix, mass matrix and damping matrix are obtained, and the dynamic equation of the series liquid-filled pipeline is obtained.

2. The semi-analytical dynamic modeling method of a series-parallel piping system based on beam-shell coupling according to claim 1, characterized in that, The kinetic energy, potential energy, displacement admissible function of the large length-diameter ratio liquid-filled straight beam pipeline and the potential energy generated by the single coupling clamp are obtained as follows: The model parameters of the large length-diameter ratio liquid-filled straight beam pipeline are determined; the large length-diameter ratio liquid-filled straight beam pipeline L / D>50, L represents the pipeline length, and D represents the pipeline diameter; Based on Timoshenko beam theory, the dynamic model of straight beam pipe with large length-diameter ratio is established. In the local coordinate system o i -x i y i z i The displacement field of straight beam pipe is defined as follows: u si , v si and w si represent the axial displacement of straight beam pipe along x i , y i and z i direction respectively; θ si , φ si and ψ si represent the rotational displacement of straight beam pipe around x i , y i and z i direction respectively. The geometric parameters of the dynamic model of the large aspect ratio liquid-filled straight beam pipeline include: l i represents the axial length of the i-th straight beam pipe segment; d i and D i respectively represent the inner diameter and the outer diameter of the large aspect ratio liquid-filled straight beam pipeline; Potential energy U of a single straight beam tube segment si is represented as: ； wherein A pi represents the cross-sectional area of the straight beam pipe section, I yi and I zi are the cross-sectional moments of inertia of the straight beam pipe section about the o i y i and o i z i axes, respectively, E represents the modulus of elasticity of the straight beam pipe section, G represents the shear modulus of the straight beam pipe section, J i represents the torsional moment of inertia, is a shear correction factor; Kinetic energy T of a single straight beam tube segment si The specific expression is: ； wherein p p represents the density of the large-aspect-ratio liquid-filled straight beam tube, I xi represents the cross-sectional moment of inertia of the straight beam tube segment about the o i x i axis; The elastic potential energy generated by the single coupling clamp Is calculated by the following formula: ； wherein the stiffness of each set of units is respectively is the axial position of each set of springs, arranged in total N g single-link clamps; Kinetic energy of fluid in straight beam pipe section is represented as: ； where p f and A fi represent the density and cross-sectional area of the fluid, respectively, J fi is the polar moment of inertia of the fluid, and I fi is the cross-sectional moment of inertia of the fluid. Displacement allowance function for straight beam pipe sections is expressed as: ； wherein , , , , and represent the displacement admissible function vectors of the straight beam pipe segment, , , , , and represent the generalized displacement coordinates of the straight beam pipe segment, , , , , and represent the unknown coefficients of the displacement admissible functions of the straight beam pipe segment, , , , , and represent the coefficients of the auxiliary terms of the displacement admissible functions of the straight beam pipe segment, is the truncation number, ; an auxiliary term representing the straight beam pipe segment displacement allowance function, specifically represented as: 。 3. The semi-analytical dynamic modeling method of a beam-shell coupled series-parallel piping system according to claim 1, wherein, The kinetic energy, potential energy, displacement admissible function of the small length-diameter ratio liquid-filled straight shell pipeline and the potential energy generated by the single coupling clamp are obtained as follows: The model parameters of the small length-diameter ratio liquid-filled straight shell pipeline are determined; the small length-diameter ratio liquid-filled straight shell pipeline L / D≤50; Based on Mindlin-Reissner shell theory, the dynamic model of small aspect ratio liquid-filled straight shell pipeline is established; the cylindrical coordinate system o-xβz is established at the small aspect ratio liquid-filled straight shell pipeline plane, and the coordinates of any point on the middle surface are represented by (x, β); l and R c respectively represent the length of the small aspect ratio liquid-filled straight shell pipeline along the x direction and the middle surface radius along the z direction; u zi , v zi and w zi are respectively the middle surface axial displacements of the i-th cylindrical shell in the x, β and z directions; θ zi and φ zi are respectively the middle surface rotational displacements around u zi and v zi ; displacement field at any point on the straight shell tube segment is represented as: ； The linear strain-displacement relationship corresponding to the straight shell pipe section is as follows: ； Where, ε x and ε β Let γ be the normal strain along the x and β directions of the straight shell, respectively. βz γ xz and γ xβ The corresponding shear strain; In addition, the linear stress-strain relationship corresponding to the straight shell pipe section is represented as: ； where σ x and σ β are normal stresses in the x and β directions, respectively, τ βz , τ xz , and τ xβ are corresponding shear stresses, and μ is the Poisson's ratio; Potential energy due to elastic deformation of straight shell segments is represented as: ； wherein represents the shear coefficient; Additionally, the kinetic energy of the straight shell tube section The expression is: ； where I0= p h, I1= 1 / 12 p h 3 ; In the fluid assumption, the fluid is regarded as an ideal fluid without viscosity, without rotation, and incompressible, and the fluid flows stably transversely relative to the pipeline, and the gravitational potential energy of the fluid is ignored; the kinetic energy T of the fluid in the shell is fi is represented as: ； wherein, is the fluid velocity; Potential energy generated when single coupling clamp supports straight shell tube section is represented as: ； wherein, Ks is the single coupling clamp stiffness; x i is the axial position of each group of springs, N g is the number of single coupling clamps; The displacement admissible function of the straight shell pipe section is: ； wherein , , , , and represent unknown coefficients of the straight shell tube segment displacement allowance function; , , , and represent unknown coefficients of the auxiliary term; and represent axial half wave numbers and circumferential wave numbers along the x direction and the β direction; and represent axial displacement allowance function truncation numbers and circumferential displacement allowance function truncation numbers; represents an auxiliary term of the displacement allowance function, and the specific expression is: 。 4. The semi-analytical dynamic modeling method of a series-parallel piping system based on beam-shell coupling according to claim 1, wherein, The dynamic equation of the parallel liquid-filled pipeline is established as follows; When the double clamp is connected in parallel with a large-length-ratio liquid-filled straight beam pipeline and a small-length-ratio liquid-filled straight shell pipeline, the small-length-ratio liquid-filled straight shell pipeline is evenly divided into Q units in the circumferential direction, and each unit is connected to the large-length-ratio pipeline by using a spring group, so as to realize the parallel connection of the large-length-ratio liquid-filled straight beam pipeline and the small-length-ratio liquid-filled straight shell pipeline; each group of spring groups includes translational stiffness and corresponding damping in three directions, and torsional stiffness and corresponding damping in three directions, and is represented as and , χ = u, v, w, θ, φ, ψ; the mass of the double clamp is evenly divided into four concentrated mass points with a mass of m d , which are respectively applied to the corresponding positions of each group of spring groups on the two pipelines; considering the soft nonlinear behavior of the double clamp under dynamic load, and represent the nonlinear stiffness and nonlinear damping coefficients of the double clamp in the z direction respectively, and represent the nonlinear stiffness and nonlinear damping coefficients of the double clamp in the y direction respectively; The rotation angle displacement ψ is introduced; the deformation compatibility equation between the normal rotation angle displacement ψ and the translation displacement components u, v in the local coordinate is established to achieve the purpose of eliminating the singularity of the stiffness matrix, and the specific expression is as follows: ； In addition, the vibration displacement components of the small length-diameter ratio liquid-filled straight shell pipeline are converted from the local column coordinate system to the global Cartesian coordinate system in order to construct the energy functional of the coupling spring group, and the calculation formula of the translation displacement and the rotation displacement components of the straight shell structure converted to the global coordinate system is as follows: ； The elastic potential energy U stored in the parallel coupling spring of the straight shell and straight beam structure is obtained e is calculated by the following formula: ； where x di represents the position of the double clamp spring group on the large length-diameter ratio liquid-filled straight shell pipeline, x dj and β dj represents the position of the double clamp spring group on the small length-diameter ratio liquid-filled straight shell pipeline, Q is taken as 8; Nonlinear restoring force function of each group spring in double clamp each group spring in beam-shell coupling parallel liquid-filled straight pipeline system is represented as: ； wherein ; Virtual work generated by non-linear external force of single and double clamps is expressed as: ； where δ D is the Dirichlet function, f Θ is the nonlinear external force generated by the single-link clamp The kinetic energy, potential energy and displacement admissible function of the liquid-filled straight shell pipeline based on the shell theory and the kinetic energy, potential energy and displacement admissible function of the liquid-filled straight beam pipeline based on the beam theory, the potential energy generated by the single coupling clamp and the potential energy stored in the coupling spring are substituted into the Lagrange equation, and the specific representation is as follows: ； wherein is the generalized coordinate, T is the sum of all kinetic energies, and U is the sum of all potential energies, is the virtual work. Then the dynamic equation of the parallel liquid-filled pipeline is as follows: ； wherein, , K g is the fluid influence stiffness matrix, f is the beam-shell coupling and parallel liquid-filled space piping coupling matrix, K p is the piping stiffness matrix, is the stiffness matrix of the clamp supporting small aspect ratio piping; , M f is the fluid added mass matrix, M p is the piping mass matrix, M c is the double-clamp mass matrix; , denotes the Rayleigh damping matrix of the beam-shell coupling and parallel liquid-filled space piping system, G denotes the fluid damping matrix.​ 5. The semi-analytical dynamic modeling method of series-parallel piping systems based on beam-shell coupling according to claim 1, characterized in that, The kinetic energy, potential energy, displacement admissible function of the small length-diameter ratio liquid-filled bent pipeline are as follows: Determine the model parameters of the small length-diameter ratio liquid-filled elbow pipe; Establish a ring coordinate system o-α at the plane of the small length-diameter ratio liquid-filled elbow pipe c β c z, select the bend shell pipe segment α c and β c direction as the two main directions of the ring coordinate system, the coordinates of any point on the middle surface are represented by (α c , β c ), β c is the circumferential angle of any point on the bend shell pipe segment to the reference point, α0 is the central angle of the bend shell pipe segment corresponding to the central angle, and the change interval is ; R t represents the bending radius of the bend shell pipe segment, R c is the radius of the middle surface, h is the thickness of the small length-diameter ratio liquid-filled elbow pipe, u wj , v wj and w wj are the linear displacement components of the bend shell pipe segment along α c , β c and z directions, θ wj and φ wj are the rotation displacement angles around α c and β c axes respectively; According to the Mindlin-Reissner shell theory, the displacement field of an arbitrary point in the curved shell pipe section is expressed in the following form: ； The linear stress-strain relationship of the curved shell pipe section is expressed in the following form: ； where σ α and σ β are the normal stresses in the directions α c and β c respectively, τ zβ , τ zα and τ βα are the shear stresses in the structure, ε α and ε β are the normal strains in the structure, and γ zβ , γ zα and γ βα are the shear strains in the structure, respectively. According to the linear elastic theory, the strain of an arbitrary point in the structural unit is expressed in the following form: ； where ε α,0 , ε β,0 , γ zβ,0 , γ zα,0 and γ βα,0 are the middle surface strains in the structure, κ α , κ β and κ βα represent the relative changes in curvature of the middle surface; The strain is described by the displacement of the intermediate surface of the structure, and is specifically expressed in the following form: ； wherein and denotes the two radii of curvature corresponding to the bend section, , , and are the Lame coefficients corresponding to the bend section; , ; For the bend shell section, the strain energy U of the shell εj is expressed in the form of ； Kinetic energy of a bent tube section The expression is: ； The curved shell segment is expanded along the α c direction and the β c direction, and the corresponding displacement admissible function is constructed, which is specifically represented as: ； wherein ; , , , and represent unknown coefficients of the displacement allowance function of the U-bend pipe section; , , , and represent unknown coefficients of the auxiliary term; is an auxiliary term of the Fourier series, and is specifically represented as: 。 6. The semi-analytical dynamic modeling method of series-parallel piping systems based on beam-shell coupling according to claim 1, characterized in that, The coupling potential energy of the pipe joint section and the adjacent small-length-to-diameter ratio liquid-filled space pipeline is established as follows: The elastic potential energy generated by the virtual spring set between adjacent variable cross sections of the pipe joint may be represented as: ； The kinetic energy T of the pipe joint vk and the strain energy U vk may be expressed as: ； where P is the number of segments, and u vk , v vk and w vk are the axial and transverse displacements of the pipe joint in the x k , y k and z k directions, respectively, θ vk , φ vk and ψ vk are the torsional and angular displacements of the pipe joint about the x k , y k and z k directions, respectively, A sk is the cross-sectional area of the kth segment of the pipe joint, is the translational and angular stiffness in each direction, is the angle of planar rotation, ρ v is the equivalent density of the pipe joint, N is the number of pipe joints, I xk , I yk and I zk are the moments of inertia of the cross-section of the pipe joint with respect to the local coordinate axes x k , y k and z k , respectively, and J k is the torsional moment of inertia of the cross-section of the pipe joint with respect to the x k axis. When the pipe joint is connected in series with the small length-diameter ratio liquid-filled space pipeline, the elastic potential energy U generated by the virtual spring set sb Specifically represented as: ； where x di represents the position of the double clamp spring set in the beam structure, x dj , β dj represents the position of the double clamp spring set in the shell structure.

7. The semi-analytical dynamic modeling method of series-parallel piping systems based on beam-shell coupling according to claim 1, characterized in that, The process of obtaining the dynamic equation of the series liquid-filled pipeline is as follows: The small-length-to-diameter ratio liquid-filled space pipeline is divided into a plurality of straight shell pipe sections and curved shell pipe sections; based on the Mindlin-Reissner shell theory, the strain-displacement relationship of the straight shell pipe section and the curved shell pipe section is obtained, and the intermediate surface displacement of the straight shell pipe section and the curved shell pipe section is expressed by using a modified Fourier series; For small length-diameter ratio space pipeline, the force and displacement compatibility conditions at the connection interface between the straight shell pipe segment and the next adjacent elbow shell pipe segment are analyzed to calculate the elastic potential energy stored in the coupling spring between the two structural units , which is specifically expressed as: ； wherein, represents the connection stiffness in each direction between straight and elbow pipe segments; the spatial transformation matrix is represented as T and is used to account for various connection combinations of adjacent pipe segments in a spatial pipeline. The elastic potential energy generated at the connecting interface between the adjacent straight shell pipe section and the next section of the curved shell pipe section in a small length-diameter ratio space pipeline is represented as: ； Work done by external excitation loads on each structural unit is represented as: ； where T z is the spatial transformation matrix, N n represents the number of straight shell segments, N m represents the number of curved shell segments; Considering the nonlinear support characteristics of the clamp, when the clamp supports the straight shell tube segment, the nonlinear force is expressed as: ； wherein, represents the nonlinear stiffness of the single-link clevis pair shell model, is a coefficient of the Bouc-Wen model, is a hysteretic variable of the Bouc-Wen model; The kinetic energy, potential energy and displacement admissible function of the straight shell pipe section, the curved shell pipe section, the single joint clamp and the pipe joint are substituted into the Lagrange energy equation, and the influence of Rayleigh damping is considered, so that the dynamic equation of the space liquid-filled pipeline system based on the beam-shell coupling theory is obtained, and is specifically expressed in the following form: ； where, , , and represent the stiffness, mass, nonlinear stiffness and damping matrices of the clamp-any small length-to-diameter ratio in-line fluid-filled piping system; , is the boundary stiffness matrix, is the fluid-affected stiffness matrix, is the small length-to-diameter ratio piping stiffness matrix composed of the straight shell stiffness matrix and the curved shell stiffness matrix , is the stiffness matrix when the clamp supports the small length-to-diameter ratio piping, is the connection stiffness matrix, is the coupling stiffness matrix of the pipe joint and the adjacent small length-to-diameter ratio in-line fluid-filled piping; , is the piping mass matrix composed of the straight shell and curved shell mass matrices and , is the fluid mass matrix, represents the external excitation force vector, represents the nonlinear force vector; , G represents the fluid damping matrix, C p is the Rayleigh damping matrix of the piping system.

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