Frequency domain TMM calculation method for pipeline containing elastic element

Abstract

The invention discloses an elastic element-containing pipeline frequency domain TMM calculation method, and relates to the technical field of pipeline system dynamics analysis. The method comprises the following steps: firstly, establishing a straight pipe local coordinate system and dividing system units, and constructing a pipeline element 14 equation model based on a Timoshenko beam theory; constructing a field transfer matrix and a point transfer matrix of the elastic element and the elastic support by referring to a four-end parameter method and combining impedance parameters, and integrating the field transfer matrix and the point transfer matrix into a system overall transfer matrix; and finally, establishing a complete frequency domain TMM calculation logic, and solving a pipeline frequency domain response through a node force balance condition, a continuous condition and a boundary condition. The method solves the problems that a traditional method is difficult to consider both precision and efficiency and poor in universality, can accurately reflect the dynamic characteristics of the elastic element, greatly improves the calculation efficiency, is adaptive to various elastic elements, and can provide reliable support for low-vibration design of pipelines in the fields of ocean engineering, shipbuilding and the like.

Description

Technical Field

[0001] This invention relates to the field of pipeline system dynamics analysis technology, specifically to a frequency domain TMM calculation method for pipelines containing elastic elements. It is applicable to complex pipeline systems containing elastic elements (such as flexible joints and bellows) and elastic supports in industries such as marine engineering, shipbuilding, and aerospace. It is particularly useful for the frequency domain dynamics analysis of chain-structure pipelines on ships, providing a core calculation method for low-vibration design and vibration control of pipeline systems. Background Technology

[0002] Piping systems are critical infrastructure for transferring fluid mass, energy, and momentum, and are widely used in marine engineering, shipbuilding, aerospace, and other fields. Under the combined influence of internal fluid excitations (such as pressure fluctuations) and external environmental loads (such as mechanical vibrations), piping systems are prone to flow-induced vibrations, leading to structural fatigue, noise radiation, and reduced system safety. Especially in scenarios with high stealth requirements, such as ships, vibration noise propagating through the hull can threaten overall stealth safety. Therefore, accurately predicting vibration characteristics is a core aspect of low-vibration design in the design, optimization, and fault diagnosis of piping systems.

[0003] Currently, the calculation methods for flow-induced vibration in pipeline systems are mainly divided into two categories:

[0004] (1) Three-dimensional full-time domain method: Based on computational fluid dynamics (CFD) to simulate the flow inside the pipe, and combined with the finite element method (FEM) to solve the pipe wall vibration. Although this method can provide high-precision dynamic data, it requires a lot of mesh generation and computational resources, which is inefficient and difficult to apply to complex pipeline systems in engineering practice.

[0005] (2) Three-dimensional time-frequency hybrid method: The computational burden is reduced by partial simplification, but the efficiency is still poor due to the complexity of the pipeline system (such as large length-to-diameter ratio and diverse component forms).

[0006] In recent years, the frequency domain transfer matrix method (TMM) has been introduced into the analysis of chain-structured pipelines due to its high efficiency. Based on a 14-equation model (such as Timoshenko beam theory), TMM simplifies the three-dimensional problem into a one-dimensional wave equation, significantly improving computational efficiency. However, existing TMM models are mainly designed for rigid pipelines and have significant limitations when dealing with elastic elements (such as flexible nozzles and bellows) and elastic supports.

[0007] (1) The model is limited to axial effects and fails to fully cover multi-directional fluid forces;

[0008] (2) There is a lack of a unified method for constructing the field transfer matrix of elastic elements, which makes it impossible to accurately describe the dynamic characteristics of elastic elements (such as impedance parameters);

[0009] (3) It relies on experimental databases or complex numerical simulations and has poor versatility.

[0010] None of the above methods can solve the flow-induced vibration problem of pipeline systems containing elastic elements while balancing accuracy and efficiency, highlighting the urgent need to develop high-precision and high-efficiency frequency domain TMM calculation methods. Summary of the Invention

[0011] (I) Purpose of the Invention

[0012] To address the shortcomings of existing technologies, such as the inability of traditional rigid pipeline TMM calculation methods to accurately describe the characteristics of elastic elements, the insufficient accuracy or inefficiency of simple elastic pipeline analysis methods, and the lack of a unified method for constructing the field transfer matrix of elastic elements and a complete frequency domain TMM calculation logic, this invention aims to provide a frequency domain TMM calculation method for pipelines containing elastic elements. This method establishes a field transfer matrix construction method for elastic elements and elastic supports, forming a complete frequency domain TMM calculation logic. This enables high-precision and efficient calculation of the frequency domain response of pipeline systems containing elastic elements, solving the problems of difficulty in balancing accuracy and efficiency and poor versatility in the analysis of complex elastic pipelines using traditional methods. This provides reliable technical support for the design optimization of pipeline systems containing elastic elements.

[0013] (II) Technical Solution

[0014] To achieve the above objectives, this invention provides a frequency domain TMM calculation method for pipelines containing elastic elements. Its core lies in the method for constructing the field transfer matrix of the elastic element and elastic support, and the corresponding frequency domain TMM calculation logic, specifically including:

[0015] (1) Basic modeling and coordinate system setting

[0016] 1. Establish a local coordinate system for the straight pipe.

[0017] Establish a local coordinate system for the straight pipe based on the right-hand rule (see...). Figure 2 The z-direction is along the pipe axis, and the yz plane is horizontal. To simplify the calculation, the effect of gravity is ignored. Each straight pipe element includes 14 variables: 6 forces, 6 velocities, sound pressure, and fluid vibration velocity. According to their vibration mode, they are divided into three categories: axial vibration, transverse vibration, and torsional vibration.

[0018] 2. System Unit Division

[0019] Based on the research on the piping system, it is first divided into different piping units, such as straight pipe units, bend units, elastic element units (such as flexible nozzles and bellows), elastic support units, and nodes. Secondly, according to the nodal force balance and continuity conditions, branch, external load, and coordinate transformation matrices, as well as common boundary constraint matrices, are established to provide a structural foundation for the subsequent construction of the transfer matrix. This step ensures that the piping system is decomposed into chain-like units, facilitating the application of TMM calculation logic.

[0020] 3. Establish a 14-equation model for piping components using Timoshenko beam theory.

[0021] A 14-equation model of the pipeline component is established using Timoshenko beam theory to analyze its vibration characteristics, which are uniformly expressed as Equation 1:

[0022]

[0023] In the formula: V f V is the average pulsating velocity of the fluid cross-section inside the pipe; P is the fluid pressure variable inside the pipe; v z The axial vibration velocity of the pipeline; v y f is the transverse vibration velocity of the pipe wall; y This refers to the transverse shear force of the pipe wall; The lateral torsional angular velocity variable of the straight pipe element; m x v represents the lateral bending moment variable of a straight pipe element. x f is the transverse vibration velocity of the pipe wall; x This refers to the transverse shear force of the pipe wall; The lateral torsional angular velocity variable of the straight pipe element; m y Let be the lateral bending moment variable of the straight pipe element; m is the torsional angular velocity. z Torque; axial force f in the pipeline z The expressions for A, B, and C differ depending on the piping component. For straight pipe components, the parameters are defined as follows:

[0024] A is a 14-dimensional identity matrix;

[0025]

[0026] In the above formula:

[0027] V0 is the steady flow velocity of the fluid;

[0028] This is the corrected fluid volume constant;

[0029] v is the Poisson's ratio of the pipeline;

[0030] E is the Young's modulus of the pipeline;

[0031] I p The moment of inertia of the pipeline;

[0032] κ is the shear coefficient. For thin-walled tubes, κ = (1+v) / (2+v);

[0033] G is the shear elastic modulus of the pipeline, G=E / (2(1+v));

[0034] T = ρ f I f +ρ p I p I f Let I be the axial moment of inertia of the fluid. f =πR 4 / 4;J p A is the polar moment of inertia of the pipe wall; p ρ is the cross-sectional area of ​​the pipe. p Pipeline density;

[0035] M = ρ p A p +ρ f A f A f ρ is the cross-sectional area of ​​the fluid; f The fluid density is given.

[0036] (2) Constructing the field transfer matrix of the elastic element

[0037] 1. Establish the force and velocity transmission characteristics at the input and output ends of the elastic element.

[0038] Drawing inspiration from the four-terminal parameter method, piping systems containing elastic elements (flexible connectors, bellows) are considered as linear mechanical systems (see...). Figure 6 Considering its dynamic transmission characteristics, the transmission characteristics of force and velocity at the input and output ends of the elastic element are obtained using the four-terminal parameter method (Equation 2), and then simplified into the form of a transmission matrix (Equation 3).

[0039]

[0040] In the formula, Z 11 Z 22 The mechanical impedance at the origin, Z 21 Z 12 To transmit mechanical impedance, the methods for obtaining mechanical impedance can be divided into: (a) building a separate test bench for the elastic element and measuring the origin mechanical impedance Z through vibration test. 11 Z 22 and the mechanical impedance Z 12 Z 21(a) The test frequency range is 5-1000Hz; (b) Establish a numerical calculation model of the elastic element, extract the resisting force and response speed, and obtain the mechanical impedance Z at the origin. 11 Z 22 and the mechanical impedance Z 12 Z 21 v1 and f1 are the input velocities and forces, and v2 and f2 are the output velocities and forces.

[0041]

[0042] f = Z b v, where f is the force; Z b ν is the input impedance of the flexible base, which can be obtained by experiment, numerical calculation, or a combination of both; v is the speed.

[0043] 2. Constructing the field transfer matrix of the elastic element and system integration

[0044]

[0045] (Formula 4), i = 1, 2, 3, representing the force and velocity transmission matrices at the input and output ends of the elastic element in the axial, lateral, and longitudinal directions, respectively; E is the identity matrix.

[0046] The field transfer matrix of the elastic element is constructed based on the impedance parameters (Formula 4). After the field matrix of the elastic element is constructed, the overall transfer matrix of the system is updated according to the nodal force balance condition and the continuity condition to ensure that the dynamic characteristics of the elastic element are integrated into the system dynamics model.

[0047] (3) Constructing the elastic support field transfer matrix

[0048] 1. Establish a pipeline support system model

[0049] Based on the actual flexible support form of the pipeline system, a pipeline support system model is established (see...). Figure 3 This includes the vibration-isolated pipe, elastic supports, and flexible base. Analogous to the process of establishing the transfer matrix at a branch pipe point, the supports and flexible base are considered as a branch pipe. At the pipe support node, the state vectors on the pipe are Ф... L and Ф R The state vector of the pipeline support is Ф U The state vector at the connection between the support and the flexible base is Ф. D .

[0050] 2. Establish the field transfer matrix of the elastic support.

[0051] Considering the variation of stiffness and damping of pipeline supports with frequency in practical engineering, and drawing analogy to the construction of the field transfer matrix of an elastic element, the field transfer matrix of the elastic support is as follows:

[0052]

[0053] The corresponding propagation relationship of the upper and lower state vectors is as follows

[0054] Φ U =U Z Φ D (Formula 6)

[0055] Considering the non-rigidity of the flexible base, the force and velocity balance equation in the z-direction at the flexible base is:

[0056] f = Z b v, in this formula, Z b The input impedance of the flexible base is given. The impedance parameters of the elastic support and the flexible base are obtained by: (a) using the finite element numerical calculation method to obtain the impedance parameters of the elastic support and the flexible base; (b) using experimental measurements to obtain the input impedance Z-impedance parameters of the elastic support and the flexible base; and (c) combining numerical calculation and experimental methods to obtain the impedance parameters of the elastic support and the flexible base.

[0057] The flexible base is treated as a special boundary condition, and its boundary matrix B b for

[0058]

[0059] 3. Establish the point transfer matrix at the pipe support in the flexible support system.

[0060] Analogous to the treatment of branch pipes, the pipe supports and flexible bases in the vibration isolation system are treated as special branch pipes, and a point transfer matrix is ​​established at the pipe supports in the elastic support system:

[0061]

[0062] In the formula, P1, P2, and P3 are the node state vector correlation matrices, and A2 and A3 are matrix decomposition parameters. 0 7×14 It is a 14-dimensional identity matrix.

[0063] Therefore, the transfer matrix of the pipeline elastic support point can be obtained (Formula 9), where Us includes the influence of pipeline support and flexible base.

[0064]

[0065] Based on impedance parameters, construct the elastic support field transfer matrix (Formula 5) and the point transfer matrix (Formula 9). After the elastic support field matrix is ​​constructed, combine it with the field matrix of the existing system (including straight pipes and elastic elements) according to the pipeline connection relationship. Based on the nodal force balance condition and continuity condition, update the overall transfer matrix of the pipeline system to realize the dynamic coupling between the elastic support and the system.

[0066] (4) Establish the frequency domain TMM calculation logic for pipelines containing elastic elements.

[0067] 1. Divide the piping system into piping units and nodes;

[0068] 2. Establish the field matrix of each component based on the fluid-structure interaction equation 14 model of each pipeline component;

[0069] 3. Establish the node matrix based on the nodal force balance and continuity conditions;

[0070] 4. Establish the overall transfer matrix of the pipeline system based on the pipeline structure. The specific integration process includes updating the overall transfer matrix of the pipeline system in stages (see steps 4, 7, and 10 of the calculation and verification process).

[0071] 5. Solve by combining the initial and final boundary conditions.

[0072] Φ s =U1·U2·…·U n ·Φ e (Formula 9)

[0073] Where, Φ s and Φ e These are the state vectors at the beginning and end of the pipeline system, U1, U2, ..., U... n This represents the field transfer matrix for each pipeline component (including straight pipes, elastic elements, elastic supports, etc.).

[0074] (III) Calculation and Verification Process

[0075] This invention addresses the vibration transmission characteristics and flow-induced vibrations of ship piping. Therefore, a TMM calculation method based on the frequency domain approach, suitable for fluid-structure interaction analysis of complex piping systems, is established. The specific calculation and verification process includes:

[0076] Step 1: Based on the pipeline system under study, divide it into straight pipe units, elastic element units, elastic support units and nodes. According to the node force balance conditions and continuity conditions, establish the branch, external load and coordinate transformation matrices as well as common boundary constraint matrices.

[0077] Step 2: For each straight pipe unit, establish 14 equations based on Timoshenko beam theory to construct the straight pipe field transfer matrix;

[0078] Step 3: Using the elastic element field transfer matrix construction method, combined with the elastic element impedance parameters obtained from experiments or numerical calculations, construct the elastic element field transfer matrix;

[0079] Step 4: Based on the connection relationship of the pipeline system (such as the series connection of straight pipes and elastic elements), update the overall transfer matrix of the system containing elastic elements (Formula 9) according to the nodal force balance condition and continuity condition.

[0080] Step 5: Substitute the obtained elastic element impedance parameters into the frequency domain TMM to calculate the pipeline vibration response. Compare the results with finite element or experimental data to verify the accuracy of the part containing the elastic element;

[0081] Step 6: Using the above-mentioned method for constructing the elastic support field transfer matrix, and combining the impedance parameters of the elastic support and flexible base obtained by experiments or numerical calculations, construct the elastic support field transfer matrix and the corresponding point transfer matrix.

[0082] Step 7: Based on Step 4, add elastic support units. According to the connection relationship (such as the position of support nodes), combine the field transfer matrix and point transfer matrix of the elastic support (the result of Step 6) with the field matrix of the existing system (including straight pipes and elastic elements). According to the nodal force balance condition and continuity condition, update the overall transfer matrix of the system containing elastic supports to realize the dynamic coupling between the elastic support and the system.

[0083] Step 8: Set the boundary conditions for the piping system (e.g., fixed end or free end), substitute the obtained impedance parameters including the elastic support into the frequency domain TMM calculation to determine the piping vibration response (apply a simple harmonic force with an amplitude of 1N in the middle section of the pipe), and compare the results with the finite element analysis or experimental data to verify the correctness of the elastic support component.

[0084] Step 9: Design and build the pipeline under test containing elastic elements and elastic supports. Repeat steps 3 and 6, and construct the field transfer matrix of the elastic elements and elastic supports by combining the elastic element impedance parameters obtained from experiments or numerical calculations.

[0085] Step 10: For the complete pipeline system (Step 9), based on the connection relationship, update the field transfer matrix of all straight pipes, elastic elements and elastic supports according to the nodal force balance condition and the continuity condition to obtain the final overall system transfer matrix.

[0086] Step 11: For the overall transfer matrix in Step 10, set the boundary conditions of the pipeline system, substitute all the impedance parameters including elastic elements and elastic supports into the frequency domain TMM, and calculate the frequency domain response of the pipeline including elastic elements and elastic supports.

[0087] Step 12: Design and build a test bench, use experimental methods to obtain the vibration response of the pipeline under test containing elastic elements and elastic elements, and compare the test results with the TMM calculation results in Step 11.

[0088] (iv) Beneficial effects

[0089] (1) The frequency domain TMM calculation method for pipelines with elastic elements proposed in this invention can accurately reflect the dynamic characteristics of elastic elements by integrating the impedance / stiffness parameters of elastic elements obtained by experiments or numerical calculations. This solves the problem that existing TMM models cannot fully cover the multi-directional force influence of elastic elements and significantly improves the accuracy of vibration calculation for pipeline systems with elastic elements.

[0090] (2) Based on the 14-equation model of Timoshenko beam theory, the three-dimensional structure is simplified into a one-dimensional beam element. Combined with the solution logic of frequency domain TMM, the large amount of meshing and node mapping work in the three-dimensional CFD-FEM hybrid method is avoided. While ensuring the accuracy of calculation, the computational complexity is greatly reduced, the consumption of computational resources is reduced, and the solution efficiency is significantly improved.

[0091] (3) This method can directly utilize the characteristic parameters (impedance / stiffness) of commonly used elastic elements in engineering. Through matrix integration, the requirement for calculation input is reduced, avoiding additional measurement costs. It has strong adaptability and can be widely applied to pipeline systems containing complex elastic elements such as flexible pipes, bellows, and vibration isolators.

[0092] (4) It can simultaneously calculate the vibration of measuring points on the pipeline and the base, providing comprehensive dynamic data support for the vibration control and low vibration design of the pipeline system, and further improving the theoretical system of flow-induced vibration of the pipeline system. Attached Figure Description

[0093] Figure 1 This is a flowchart of the method of the present invention.

[0094] Figure 2 This is the local coordinate system of the straight tube of the present invention.

[0095] Figure 3 This is a schematic diagram of the pipeline support system model of the present invention.

[0096] Figure 4 This is a schematic diagram of the transfer matrix method of the present invention.

[0097] Figure 5 This is a schematic diagram of the elastic element and elastic support pipeline of the present invention.

[0098] Figure 6 The TMM calculation model for the pipeline system with elastic elements of this invention

[0099] Figure 7The velocity impedance result of the pipeline system with elastic element of this invention obtained by numerical calculation.

[0100] Figure 8 Comparison of TMM calculation and finite element results for this invention with elastic elements

[0101] Figure 9 The TMM calculation model for the elastically supported pipeline system of this invention.

[0102] Figure 10 The velocity impedance results of the elastically supported pipeline system of this invention were obtained through numerical calculation.

[0103] Figure 11 Comparison of TMM calculation and finite element results for the elastically supported pipeline system of this invention.

[0104] Figure 12 Comparison of TMM calculation and experimental results for the pipeline system with elastic elements and elastic supports of the present invention. Detailed Implementation

[0105] The following detailed description of the frequency domain TMM calculation method for pipelines containing elastic elements, in conjunction with the accompanying drawings and test cases of typical ship piping systems, provides an explanation of the present invention.

[0106] I. Experimental System Setup and Basic Parameter Preparation

[0107] (I) Design of the test pipeline system

[0108] This embodiment takes a typical shipboard piping system containing elastic elements as the research object. Figure 5 The test bench was constructed, and its specific structure is as follows:

[0109] (1) Core components of the pipeline: The elastic element in this pipeline system is a DN65 flexible connector, with DN65 steel pipes (straight pipe units) of 1m in length connected in series at both ends to form a chain-type basic structure of "straight pipe 1-elastic element-straight pipe 2".

[0110] (2) Elastic support configuration: EA25 type vibration isolators are installed at the midpoint of straight pipe 1 and straight pipe 2 as elastic support units, and the flexible base is the fixed foundation structure of the test bench.

[0111] (3) Measurement point and excitation arrangement: Vibration response measurement points (measurement point 1 and measurement point 2) are set at a distance of 2 times the pipe diameter (i.e. 130 mm) from the flange end face of the two straight pipes respectively; a simple harmonic excitation force is applied to the right end of the pipeline system (the end of straight pipe 2) through the exciter, and the excitation frequency range is 5-1000Hz.

[0112] (4) Straight pipe structure and material parameters: DN65 steel pipe with an inner diameter of 65mm, a wall thickness of 5mm, and a material density ρ p=7880kg / m 3 Young's modulus E = 2.01 × 10⁻⁶ 11 Pa, Poisson's ratio v = 0.3; the fluid inside the pipe is water with density ρ f =1000kg / m 3 .

[0113] (II) Preprocessing of Key Parameters

[0114] (1) Calculation of basic parameters of straight pipe unit: Based on the above geometric and material parameters, combined with Timoshenko beam theory and the definition of the 14 equation model.

[0115] Pipe cross-sectional area: Where D = 65mm (inner diameter), t = 5mm (wall thickness);

[0116] Shear modulus:

[0117] Shear coefficient: (Values ​​for thin-walled tubes);

[0118] Pipe wall polar moment of inertia: J p =2I p ≈2.04×10 -7 m 4 ;

[0119] Fluid axial moment of inertia:

[0120] Comprehensive quality parameter: M = ρ p A p +ρ f A f =7880 × 9.42 × 10 -4 +1000×3.32×10 -3 ≈10.6kg / m;

[0121] Combined inertial parameters: T = ρ f I f +ρ p I p =1000×1.15×10 -6 +7880×1.02×10 -7 ≈1.95×10 - 3 kg·m.

[0122] Corrected fluid volume constant: The bulk modulus of the fluid is K = 2.1 × 10⁻⁶. 9 Pa, relative wall thickness of the pipeline Substituting the values, we get K'≈1.98×10⁻⁶.9 Pa.

[0123] (2) Obtaining the impedance parameters of the elastic element:

[0124] Impedance of elastic element: Constructing a separate test bench for elastic elements ( Figure 6 The mechanical impedance Z at the origin of the DN65 flexible connector was measured through a vibration test. 11 Z 22 and the mechanical impedance Z 12 Z 21 The test results are as follows Figure 7 It covers a frequency range of 5-1000Hz; the DN65 flexible connector has a symmetrical structure, satisfying the reciprocity principle, i.e., Z 11 =Z 22 Z 12 =Z 21 .

[0125] Impedance of elastic support and flexible base: The impedance parameters of the EA25 vibration isolator and flexible base were obtained using finite element numerical calculation or experimental methods (see...). Figure 10 ).

[0126] II. Implementation of Frequency Domain TMM Calculation

[0127] Step 1: Based on the structural composition of the test piping system, divide the piping into units and nodes:

[0128] Piping units: Straight pipe unit 1, straight pipe unit 2, elastic element unit, elastic support unit 1 (midpoint of straight pipe 1), elastic support unit 2 (midpoint of straight pipe 2);

[0129] Nodes: Node 1 (connection end between straight pipe 1 and elastic element), Node 2 (connection end between elastic element and straight pipe 2), Node 3 (installation end of elastic support at the midpoint of straight pipe 1), Node 4 (installation end of elastic support at the midpoint of straight pipe 2);

[0130] After division, a complete chain-like structural unit system is formed, laying the foundation for the construction of the transfer matrix.

[0131] Step 2: Construct the straight pipe field transfer matrix:

[0132] For straight pipe element 1 and straight pipe element 2, a 14-equation model is established based on Timoshenko beam theory. Substituting the straight pipe foundation parameters calculated in step 1, the field transfer matrix of the straight pipe is constructed:

[0133] (1) Define a 14-dimensional state vector:

[0134] (2) 14-Equation Model:

[0135] Where: A is a 14-dimensional identity matrix;

[0136]

[0137] (3) Deriving the field transfer matrix of the straight tube unit: Using the above 14 equations model and the frequency domain solution method, the field transfer matrix U of the straight tube unit 1 is obtained. st1 The field transfer matrix U of straight tube unit 2 st2 The matrix dimensions are all 14×14, reflecting the fluid-structure interaction vibration transmission characteristics of the straight tube element.

[0138] Step 3: Construction of the field transfer matrix of the elastic element

[0139] The method for constructing the field transfer matrix of the elastic element as described in the invention, combined with the impedance parameters of the elastic element obtained in step 1, is implemented as follows:

[0140] (1) Transmission characteristics based on the four-terminal parameter method: The elastic element is regarded as a linear mechanical system, and the force-velocity relationship between its input and output ends is as follows:

[0141] Convert to transfer matrix form: In the formula The mechanical impedance at the origin

[0142] To transmit mechanical impedance, this embodiment is obtained experimentally ( Figure 7 ).

[0143] (2) Constructing the field transfer matrix of the elastic element: Elastic element field transfer matrix U T It is a 14×14 dimensional matrix:

[0144]

[0145] in: i = 1, 2, 3, representing the force and velocity transmission matrices at the input and output ends of the elastic element in the axial, lateral, and longitudinal directions, respectively; E is the identity matrix, and the remaining submatrices are zero matrices.

[0146] Step 4: Integration of the overall transfer matrix of the system containing elastic elements

[0147] Based on the series connection relationship of the pipeline system (straight pipe 1 - node 1 - elastic element - node 2 - straight pipe 2), and according to the nodal force balance condition and continuity condition, matrix multiplication is performed in the chain structure sequence.

[0148] U sub =U st1 ·P1·U t ·P2·U st2

[0149] Where P1 and P2 are the point transfer matrices of node 1 and node 2, respectively, established based on nodal force balance and continuity conditions, both with a dimension of 14×14, ensuring Φ s =U1·U2·…·U n ·Φ e The transmission logic (Φ) s Φ e (These are the initial and final state vectors, respectively).

[0150] Step 5: Accuracy verification of the part containing elastic elements

[0151] (1) Set boundary conditions: Set the beginning of the pipeline (left end of straight pipe 1) as a fixed end (velocity-related components in the state vector);

[0152] (2) Response calculation: U sub Substitute the values ​​into the frequency domain TMM solution formula to calculate the vibration response (velocity amplitude) at measuring point 1 and measuring point 2;

[0153] (3) Result Verification: Compare the calculation results with the finite element method (FEM) calculation results, such as... Figure 8 The amplitude magnitude and peak frequency position deviations within the two 1000Hz frequency ranges shown are both less than 10%, verifying the accuracy of the elastic element field transfer matrix construction method and subsystem integration logic.

[0154] Step 6: Construction of the elastic support field transfer matrix and point transfer matrix

[0155] The method for constructing the elastic support field transfer matrix as described in the invention, combined with the impedance parameters of the elastic support and flexible base obtained in step 1, is implemented as follows:

[0156] Establish a model of a resilient support system (see) Figure 3 Treating the EA25 type vibration isolator and flexible base as special branch pipes, define the state vectors Φ at the upper and lower ends of the support. U (pipeside), Φ D (Base side), state vector Ф on both sides of the tube body L and Ф R ;

[0157] (2) Construct the elastic support field transfer matrix:

[0158] Elastic support field transfer matrix U Z for:

[0159]

[0160] The propagation relationship between the upper and lower state vectors is Φ. U =U Z Φ D .

[0161] (3) Constructing the boundary matrix of the flexible base: Considering the non-rigidity of the flexible base, its force-velocity balance equation in the z-direction is f = Z b v(Z b (Input impedance of flexible base), boundary matrix B b It is a 7×14 dimensional matrix:

[0162]

[0163] (4) Constructing the elastic support point transfer matrix: Analogous to the branch pipe processing method, define A2 and A3 are matrix factorization parameters, 0 7×14 Given a 14-dimensional identity matrix, the elastic support point transfer matrix is: Finally, the transfer matrix Φ of the elastic support point is obtained. s1 (Node 3), Φ s2 (Node 4):

[0164] Among them, P3-P8 are the node state vector correlation matrices, which ensure the dynamic coupling between the elastic support and the pipeline system.

[0165] Step 7: Update the overall transfer matrix with elastic support

[0166] In step 4, U sub Based on this, elastic support units are added, matrix integration is performed according to the support node positions, and the overall transmission including elastic support is updated according to the node force balance condition and continuity condition:

[0167] U total =U st1 ·P3·U s1 ·P1·U t ·P2·U s2 ·P4·U st2

[0168] Among them, P3 and P4 are the connection point transfer matrices between the flexible support nodes and the straight pipe units, and the integration process strictly follows the pipeline topology. Figure 9 This enables dynamic coupling between the elastic support and the subsystem containing elastic elements. total This is the updated 14×14 dimensional global transfer matrix.

[0169] Step 8: Verification of the correctness of the part with elastic support

[0170] (1) Boundary condition setting: Keep the boundary conditions of step 5 unchanged, and apply an additional harmonic excitation force with an amplitude of 1N in the middle section of the pipeline (node ​​2 position);

[0171] (2) Response calculation: Utotal Substituting into the frequency domain TMM solution formula Φ s =U1·U2·…·U n ·Φ e Calculate the vibration response at measuring point 1 and measuring point 2;

[0172] (3) Result Verification: Compare the calculation results with the finite element method results, such as... Figure 11 As shown, the response change trends of the two are completely consistent, and the resonance peak deviation is less than 12%, which verifies the correctness of the elastic support field transfer matrix, the point transfer matrix construction method and the system coupling logic.

[0173] Step 9: Reconstruction of the field transfer matrix of the complete pipeline under test

[0174] Design and construct a complete pipeline under test, including elastic elements and elastic supports. Figure 5 Repeat steps 3 and 6.

[0175] (2) Using the impedance parameters of the elastic elements and elastic supports in the complete pipeline that have been obtained, construct the field transfer matrix U' of the elastic element. t Elastic support field transfer matrix U′ Z1 、U′ Z2 and point transfer matrix, U′ S1 、U′ s2 The construction process is the same as steps 3 and 6.

[0176] Step 10: Integration of the overall transfer matrix of the complete piping system

[0177] Based on the complete pipeline topology (straight pipe 1 - node 3 - elastic support 1 - node 1 - elastic element - node 2 - elastic support 2 - node 4 - straight pipe)

[0178] U final =U st1 ·U′ s1 ·P′1·U′ t ·P′2·U′ s2 ·U st2

[0179] Where P1' and P2' are the point transfer matrices of the complete pipeline nodes, and based on the node force balance condition and the continuity condition, the final 14×14 dimensional complete pipeline system overall transfer matrix U is obtained. final .

[0180] Step 11: Calculation of the complete pipeline frequency domain response

[0181] (1) Set complete pipeline boundary conditions: The left end of straight pipe 1 is fixed (velocity-related component is 0), and a simple harmonic excitation force with an amplitude of 1N is applied to the right end of straight pipe 2 (consistent with the test conditions, refer to...). Figure 9 );

[0182] (2) Substitute parameters to solve: Substitute U final Substituting the impedance parameters of all elastic elements (pipes, supports) into the frequency domain TMM, combined with Φ s =U final ·Φ e Solve the frequency domain vibration response (vibration velocity amplitude) of measuring point 1 and measuring point 2 to obtain the TMM calculation results of the complete pipeline.

[0183] Step 12: Experimental Verification and Result Analysis

[0184] Test bench construction: The test bench is constructed according to the complete pipeline design scheme. A simple harmonic excitation force consistent with the calculated working conditions is applied by a vibrator, and vibration response data of measuring point 1 and measuring point 2 are collected by an acceleration sensor.

[0185] Results Comparison: Compare the vibration response data obtained from the experiment with the TMM calculation results from step 11, such as... Figure 12 As shown, in the frequency range of 5-1000Hz:

[0186] Amplitude magnitude: The deviation between the two is less than 15%, which meets the engineering accuracy requirements;

[0187] Trend of change: The positions of the resonance peaks and troughs are completely consistent, and the fluctuation characteristics in the mid-to-high frequency range are consistent;

[0188] Conclusion: The overall effectiveness and accuracy of the frequency domain TMM calculation method for pipelines with elastic elements proposed in this invention have been verified.

[0189] III. Summary of Implementation Results

[0190] This embodiment, following the calculation and verification process described in the invention, fully demonstrates the core advantages of the method of the present invention through experimental verification of a typical marine piping system containing elastic elements:

[0191] Accuracy: By accurately fusing the impedance parameters of elastic elements and using a unified transfer matrix construction method, the shortcomings of traditional TMM models in covering multi-directional forces are effectively solved, and the calculation results are in high agreement with the experimental data.

[0192] High efficiency: The 14-equation model based on Timoshenko beam theory simplifies the three-dimensional problem into a one-dimensional solution, resulting in high computational efficiency;

[0193] Versatility: It directly utilizes commonly used impedance / stiffness parameters in engineering, and is compatible with various elastic elements such as flexible joints and vibration isolators. It can be widely used in complex pipeline systems in fields such as marine engineering and shipbuilding.

[0194] The scope of protection of this invention is not limited to the above embodiments. All equivalent substitutions or modifications made based on the technical solutions and inventive concepts of this invention should be covered within the scope of protection of this invention.

Claims

1. A method for calculating the frequency domain TMM of a pipeline containing elastic elements, characterized in that, This includes basic modeling and coordinate system setting, construction of the field transfer matrix of elastic elements, construction of the field transfer matrix of elastic supports, and establishment of the frequency domain TMM calculation logic for pipelines containing elastic elements. The specific steps are as follows: (1) Basic modeling and coordinate system setting: 1.1 Establish a local coordinate system for the straight pipe according to the right-hand rule. The z-direction is along the pipe axis, and the yz plane is the horizontal plane. Ignore the influence of gravity. Each straight pipe micro-element contains 14 variables, which are divided into three categories according to the vibration mode: axial vibration, transverse vibration and torsional vibration. 1.2 Divide the unit into straight pipe units, elastic element units, elastic support units and nodes to form a chain unit structure. Based on the node force balance condition and continuity condition, establish the branch, external load and coordinate transformation matrix and common boundary constraint matrix. 1.3 A 14-equation model for the piping components is established using Timoshenko beam theory, uniformly expressed as Equation 1: y(z,t)=[V f ,P,v z ,f z ,v y ,f y ,θ x ,m x ,v x ,f x ,θ y ,m y ,θ z ,m z ] T A is a 14-dimensional identity matrix, and B and C are coefficient matrices of the corresponding dimensions. The definitions of each parameter are consistent with the specification. (2) Construct the field transfer matrix of the elastic element: 2.1 Drawing on the four-terminal parameter method, the elastic element is considered as a linear mechanical system, yielding formula 2 for the transmission characteristics of force and velocity at the input and output ends: And simplified to the transfer matrix form, Formula 3: Z 11 Z 22 The mechanical impedance at the origin, Z 21 Z 12 To transmit mechanical impedance, it is obtained through experimentation or numerical calculation; 2.2 Formula 4 for constructing the field transfer matrix of an elastic element based on impedance parameters: in These represent the axial, lateral, and longitudinal transmission matrices of the elastic element, respectively, with E being the identity matrix. 2.3 Based on the pipeline connection relationship, the field transfer matrix of the elastic element and the point matrix of the straight pipe unit and node are combined, and the overall system transfer matrix is ​​updated according to the node force balance condition and the continuity condition. (3) Construct the elastic support field transfer matrix: 3.1 Establish a pipeline support system model including the vibration-isolated pipeline, elastic support, and flexible base, and define the state vector Ф on the pipeline. L and Ф R The state vector Ф of the pipeline support U The state vector Ф at the connection between the support and the flexible base D ; 3.2 Formula 5 for constructing the field transfer matrix of the elastic support: The corresponding relationship between the upper and lower state vectors is given by formula 6: Φ U =U Z Φ D ; 3.3 Considering the non-rigid flexible base, its force and velocity balance equation in the z-direction is f = Z b v, Formula 7 for constructing the boundary matrix: Z b The input impedance of the flexible base; 3.4 Analogous to the branch pipeline treatment method, establish the point transfer matrix formula 8 for the pipeline support in the flexible support system: in P1, P2, and P3 are the node state vector correlation matrices, and A2 and A3 are matrix decomposition parameters. 0 7×14 Given a 14-dimensional identity matrix, we can obtain Formula 9 for the transfer matrix of the pipeline elastic support points: 3.5 Based on the pipeline connection relationship, the elastic support field transfer matrix and point transfer matrix are combined with the existing system matrix, and the overall pipeline system transfer matrix is ​​updated according to the nodal force balance condition and the continuity condition. (4) Establish the frequency domain TMM calculation logic for pipelines containing elastic elements: 4.1 Divide the piping system into piping units and nodes; 4.2 Establish the field matrix of each component based on the fluid-structure interaction equation 14 model of each pipeline component; 4.3 Establish the point matrix of the nodes based on the nodal force balance and continuity conditions; 4.4 Establish the overall transfer matrix of the pipeline system based on the pipeline structure. The integration process includes phased matrix multiplication. 4.5 Combining the initial and final boundary conditions, according to formula Φ s =U1·U2……U n ·Φ e Solve for Φ. s and Φ e These are the state vectors at the beginning and end of the pipeline system, U1, U2, ..., U... n This is the field transfer matrix for each pipeline component (including straight pipes, flexible fittings, flexible supports, etc.).

2. The method for calculating the frequency domain TMM of a pipeline containing an elastic element according to claim 1, characterized in that... The elastic element includes a flexible conduit and a bellows.

3. The method for calculating the frequency domain TMM of a pipeline containing an elastic element according to claim 1, characterized in that... The 14 variables in step (1) are: V f V is the average pulsating velocity of the fluid cross-section inside the pipe; P is the fluid pressure variable inside the pipe; v z The axial vibration velocity of the pipeline; v y f is the transverse vibration velocity of the pipe wall; y This refers to the transverse shear force of the pipe wall; The lateral torsional angular velocity variable of the straight pipe element; m x v represents the lateral bending moment variable of a straight pipe element. x f is the transverse vibration velocity of the pipe wall; x This refers to the transverse shear force of the pipe wall; The lateral torsional angular velocity variable of the straight pipe element; m y Let be the lateral bending moment variable of the straight pipe element; m is the torsional angular velocity. z Torque; axial force f in the pipeline z .

4. The method for calculating the frequency domain TMM of a pipeline containing an elastic element according to claim 1, characterized in that... The specific form of the coefficient matrix B in step (1) is as follows: in 5. The method for calculating the frequency domain TMM of a pipeline containing an elastic element according to claim 1, characterized in that... The specific form of the coefficient matrix C in step (1) is as follows: The parameters of each submatrix are determined based on the characteristics of the piping components.

6. The method for calculating the frequency domain TMM of a pipeline containing an elastic element according to claim 1, characterized in that, The methods for obtaining the mechanical impedance in step (2) can be divided into: (a) building a separate test bench for the elastic element and measuring the mechanical impedance Z at the origin through vibration test. 11 Z 22 and the mechanical impedance Z 12 Z 21 (a) The test frequency range is 5-1000Hz; (b) Establish a numerical calculation model of the elastic element, extract the resisting force and response speed, and obtain the mechanical impedance Z at the origin. 11 Z 22 and the mechanical impedance Z 12 Z 21 .

7. The method for calculating the frequency domain TMM of a pipeline containing an elastic element according to claim 1, characterized in that... The methods for obtaining the impedance parameters of the elastic support and flexible base in step (3) are as follows: (a) the impedance parameters of the elastic support and flexible base are obtained by using the finite element numerical calculation method; (b) the impedance parameters of the elastic support and flexible base are obtained by experimental measurement; and (c) the impedance parameters of the elastic support and flexible base are obtained by combining numerical calculation and experimental methods.

8. The method for calculating the frequency domain TMM of a pipeline containing an elastic element according to claim 1, characterized in that, It also includes the calculation and verification process, specifically: Step 1: Divide the elements into straight pipe units, elastic element units, elastic support units, and nodes. Based on the node force balance conditions and continuity conditions, establish the branch, external load, coordinate transformation matrix, and common boundary constraint matrices. Step 2: Construct the straight pipe field transfer matrix; Step 3: Construct the field transfer matrix of the elastic element; Step 4: Based on the connection relationship of the pipeline system (such as the series connection of straight pipes and elastic elements), update the overall transfer matrix of the system containing elastic elements according to the nodal force balance condition and the continuity condition. Step 5: Calculate the pipeline vibration response and compare it with finite element or experimental data for verification; Step 6: Construct the elastic support field transfer matrix and the corresponding point transfer matrix; Step 7: Combine the elastic support field transfer matrix and point transfer matrix with the existing system matrix, and update the overall transfer matrix containing the elastic support according to the nodal force balance condition and the continuity condition. Step 8: Calculate the pipeline vibration response and compare it with the finite element results or experimental data for verification; Step 9: Set up the complete pipeline under test, and construct the corresponding field transfer matrix using the obtained impedance parameters; Step 10: Update the field transfer matrix of all straight pipes, elastic elements and elastic supports according to the nodal force balance condition and the continuity condition to obtain the overall transfer matrix of the complete system. Step 11: Calculate the complete pipeline frequency domain response; Step 12: Set up the test bench to obtain test data and compare and verify it with the calculation results.

9. The method for calculating the frequency domain TMM of a pipeline containing an elastic element according to claim 8, characterized in that, The verification standard in step 5 is that the deviation of the amplitude and peak frequency position between the calculated results and the finite element or experimental data is less than 10%.

10. The method for calculating the frequency domain TMM of a pipeline containing an elastic element according to claim 8, characterized in that, The verification criteria in step 8 are: the response change trend of the calculation results is consistent with that of the finite element results, and the resonance peak deviation is less than 12%; the verification criteria in step 12 are: the amplitude deviation of the calculation results and the experimental data is less than 15%, and the positions of the resonance peak and valley are completely consistent.

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