Method, system and medium for predicting vibration of an l-shaped fluid conveying pipe with nonlinear support considering stress stiffening effect

Abstract

A method, system, and medium for predicting the vibration of an L-shaped flow pipeline with nonlinear support, considering the stress stiffening effect, are disclosed. The method includes: determining the natural frequencies and dynamic response at any point on the L-shaped pipeline under pulsating internal flow, considering the stress stiffening effect; calculating the static load of the steady internal flow on the L-shaped pipeline and obtaining the static equilibrium configuration using the Newton-Raphson iterative method; calculating the axial force of the pipeline structure based on the static equilibrium configuration and using the axial force to accurately calculate the additional lateral stiffness caused by the stress stiffening effect; and finally, transferring the additional lateral stiffness to the solution of the inherent characteristics and time-domain response. This invention achieves accurate calculation of the inherent characteristics and time-domain response analysis of an L-shaped flow pipeline with nonlinear support, considering the stress stiffening effect under pulsating flow. When considering different pipeline configurations, only the coordinate transformation matrix and the corresponding fluid static load vector need to be changed. Furthermore, it can solve complex nonlinear supports.

Description

Technical Field

[0001] This invention relates to the field of mechanical dynamics, and in particular to a method, system, and medium for predicting the vibration of an L-shaped conveying pipeline with nonlinear support, considering the stress stiffening effect. Background Technology

[0002] Pipelines are widely used in aerospace, petroleum, and nuclear industries, among others. Due to space constraints and equipment layout limitations, L-shaped pipelines are extensively used in practical engineering. However, influenced by pumps and other equipment, the fluid within the pipe often exhibits periodic pulsation characteristics, and fluid-structure interaction can easily trigger destructive parametric resonances. Simultaneously, to meet installation and vibration isolation requirements, numerous constraints such as clamps are introduced in engineering projects, resulting in complex nonlinear boundaries and intermediate elastic support characteristics in the pipeline system. Therefore, accurately predicting the nonlinear dynamic behavior of L-shaped pipelines with complex supports under pulsating flow is of significant engineering importance for ensuring the safe operation of the system.

[0003] Chinese invention patent application CN202310959736 is currently the closest technical solution to this invention. This solution addresses parallel flow transmission pipeline systems with clamp-supported nonlinear structures and employs a semi-analytical method to establish a dynamic model. However, this solution has the following shortcomings: First, the modeling object is limited to parallel straight pipe configurations, failing to address coordinate transformations caused by the curvature of bends in L-shaped pipes and the change in the direction of fluid centrifugal force, thus it cannot be directly used for the dynamic analysis of L-shaped flow transmission pipelines; second, the semi-analytical method relies on pre-derived modal functions, which are cumbersome to derive when dealing with complex nonlinear boundaries, limiting its versatility; third, this solution does not solve for the static equilibrium configuration of the fluid-structure interaction system, failing to transfer the axial force generated by the steady-state fluid centrifugal force and its resulting stress stiffening effect to subsequent solutions, leading to an incomplete description of the system stiffness matrix and insufficient accuracy in calculating natural frequencies.

[0004] The static equilibrium configuration not only determines the initial configuration of the pipeline, but also generates a significant axial force due to the centrifugal force of the fluid in equilibrium steady state under the constraints at both ends. This axial force induces a "stress stiffening" effect similar to that of a prestressed beam. Existing technologies have failed to effectively transfer and couple the stiffening effect induced by the axial force in static equilibrium to the time-domain solution stage of the dynamic response, directly reducing the accuracy of the calculation of the system's natural frequencies.

[0005] In summary, existing methods still have technical gaps in synchronously handling complex nonlinear boundary constraints and static-dynamic coupling effects, making it difficult to accurately predict the actual vibration behavior of pipelines under pulsating flow excitation. Summary of the Invention

[0006] To address the aforementioned shortcomings in the dynamic modeling of L-shaped flow conveying pipelines with nonlinear supports in existing technologies, this invention provides a method, system, and medium for predicting the vibration response of L-shaped pipelines with nonlinear supports based on finite element simulation. This method, based on finite element simulation technology, discretizes the physical pipeline structure and incorporates the influence of static equilibrium configuration on stress stiffening effect under steady-state flow-induced loads. This enables high-precision prediction of the nonlinear vibration response of the pipeline structure under pulsating internal flow excitation, providing a reliable analytical tool for pipeline system safety assessment and support parameter optimization.

[0007] To achieve the above objectives, the technical solution of the present invention includes:

[0008] A vibration prediction method for an L-shaped flow transmission pipeline with nonlinear support, considering stress stiffening effect, includes the following steps:

[0009] Step S1: Establish the initial structural dynamic equations of the L-shaped flow transmission pipeline with nonlinear supports:

[0010] Based on Euler beam theory and using the finite element method, the physical L-shaped pipe is discretized into multiple finite elements. Each element is described by a two-node Euler beam element, and each node has three degrees of freedom: axial displacement, lateral displacement, and cross-sectional rotation. Based on the pipe's geometric parameters (including straight pipe length, bend radius, pipe inner diameter, and wall thickness), material parameters (including Young's modulus, Poisson's ratio, and structural density), and the physical parameters of the fluid inside the pipe (including fluid density and velocity), a set of initial structural dynamic equations for the L-shaped conveying pipe with nonlinear supports is established.

[0011] The initial set of structural dynamic equations is expressed as follows:

[0012]

[0013] Where q represents the generalized displacement vector, a point on it represents the first derivative with respect to time, and two points on it represent the second derivative with respect to time. K L K NL F represents the linear and nonlinear stiffness matrices of the pipeline structure. ext It is a vector containing the externally distributed forces. M is the mass matrix. C is the Rayleigh damping matrix. , The additional damping and stiffness terms are due to the fluid-structure interaction effect.

[0014] For the bends in L-shaped pipes, multiple straight beam elements are used to approximate the circular arc bending spatial configuration. The length of each straight beam element is determined by dividing the bend radius and bending angle equally. The physical spatial mapping between the local coordinate system and the global coordinate system is achieved through a coordinate transformation matrix between adjacent elements.

[0015] Step S2: Solve for the static equilibrium spatial configuration of the L-shaped pipe:

[0016] By removing time-varying dynamic terms (including inertial terms, damping terms, and time-varying external excitation terms) from the initial set of structural dynamic equations, the static equilibrium equations describing the physical pipe structure under steady-state fluid loads are obtained.

[0017] The static equilibrium equation is as follows:

[0018]

[0019] Among them, the generalized static load vector of the system Depend on Projecting onto the system's generalized coordinates yields:

[0020]

[0021] Where H T Let T be the shape function matrix based on the Hermite interpolation function, where the superscript T denotes the transpose of the matrix. The cell length is given by 'e', ​​the superscript 'e' indicates the cell number, and N is the number of cells. The expression is:

[0022]

[0023] Where n represents the unit normal vector radially outward from the center of curvature of the bend. The average flow velocity of the fluid inside the pipe. The cross-sectional area of ​​the fluid inside the pipe. Let R be the fluid density and R be the radius of the bend. The Newton-Raphson numerical iterative method is used to solve the static equilibrium equations to obtain the static equilibrium spatial configuration of the physical pipe structure under steady-state flow-induced loads. During the iteration process, the convergence criterion adopts dual control of the displacement increment norm and the static residual norm. When the displacement increment norm between iteration steps is less than a first preset tolerance and the residual norm of the current step is less than a second preset tolerance, the static equilibrium spatial configuration is considered to have converged, thus ensuring the physical accuracy of the static solution.

[0024] Step S3: Calculate the stress stiffening effect caused by axial force and generate the element axial force additional stiffness matrix:

[0025] Based on the static equilibrium spatial configuration, the axial force borne by each finite element of the physical pipe structure is calculated. The stress stiffening effect caused by this axial force is then calculated. The actual physical axial force value is multiplied by a geometric matrix formed by the integral of the derivative of the Hermite interpolation function with respect to x to generate an element axial force-added stiffness matrix (i.e., the element geometric stiffness matrix) characterizing the contribution of this effect to the lateral stiffness of the element.

[0026] Step S4: Correct the system stiffness matrix and update the coordinate transformation matrix:

[0027] The element axial force additional stiffness matrices of each finite element are assembled into a global additional stiffness matrix K after coordinate transformation. g This is superimposed on the overall stiffness matrix of the physical pipe structure to correct the stiffness distribution of the system;

[0028] Simultaneously, the coordinate transformation matrix of each finite element is updated based on the element space rotation angle increment generated by the static equilibrium spatial configuration.

[0029] Step S5: Construct the complete time-domain dynamic equations and solve for the nonlinear vibration response:

[0030] Using the updated coordinate transformation matrix, the mass matrix, damping matrix, stiffness matrix, and external force vectors are transformed to the global coordinate system and assembled to construct the complete time-domain dynamic equations of the structure:

[0031] The fluid-structure interaction additional term (additional damping) in the time-domain dynamic equations of the structure Additional stiffness The fluid load term is updated in real time based on the instantaneous flow velocity at each moment to simulate the parametric excitation of the physical piping system.

[0032] Finally, the nonlinear vibration response of the pipeline was solved using the Newmark-β scheme combined with the Newton-Raphson iterative method.

[0033] Preferably, the static solution is obtained by using the Newton-Raphson iterative method. Subsequently, in order to establish the mapping relationship from the global static potential type to the local element internal forces, from Extract the static displacement vector of the node corresponding to the e-th element. Based on this displacement vector, the actual spatial length of the element after deformation. It can be expressed as a geometric function of nodal coordinates and static displacements. Therefore, the element axial force induced by the steady-state load can be calculated in reverse. :

[0034]

[0035] In the formula S p Let E be the cross-sectional area of ​​the pipe structure, and E be the Young's modulus of the pipe structure. Let be the initial length of the e-th unit.

[0036] Construct the element axial force additional stiffness matrix k using element axial force. g eAnd superimposed on the original stiffness matrix:

[0037]

[0038] Where k g e Add a stiffness matrix to the element axial force. Let be the shape function matrix based on the Hermite interpolation function, and the dot above the symbol indicates the derivative with respect to x.

[0039] According to static solution The spatial rotation increment of each finite element is used to update the matrix. : In the formula, T0 is the initial coordinate transformation matrix, and matrix T disp ( ) is represented as:

[0040]

[0041] in This represents the change in element angle, with the subscript 'e' indicating the corresponding element number.

[0042] The dynamic equations are constructed using the updated coordinate transformation matrix.

[0043]

[0044]

[0045]

[0046]

[0047] This invention also provides a vibration prediction system for an L-shaped flow transmission pipeline with nonlinear support, considering the stress stiffening effect, for implementing the above method, characterized in that it includes:

[0048] The physical model discretization module is used to discretize the physical L-shaped pipe into multiple finite elements based on beam theory, and to establish an initial set of structural dynamic equations based on the pipe structural parameters and the fluid parameters inside the pipe.

[0049] The static equilibrium solution module is used to remove time-varying terms from the equation set to construct the static equilibrium equations, and to solve for the static equilibrium spatial configuration of the pipeline structure using a numerical iteration method.

[0050] The stiffening effect calculation module is used to calculate the actual axial internal force of the unit based on the static equilibrium spatial configuration and generate the unit axial force additional stiffness matrix characterizing the stress stiffening effect.

[0051] The matrix correction and coordinate update module is used to superimpose the additional stiffness matrix onto the total stiffness matrix and update the coordinate transformation matrix of the unit according to the static equilibrium configuration.

[0052] The time-domain dynamic response solution module is used to construct the complete time-domain equation using the corrected total stiffness matrix and the updated coordinate transformation matrix, and to solve the nonlinear vibration response of the pipeline structure using the step-by-step integration method.

[0053] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the above-described method.

[0054] The beneficial effects of this invention are as follows:

[0055] 1. This invention considers the influence of stiffness hardening effect caused by static equilibrium configuration on the inherent characteristics and nonlinear vibration of L-shaped pipes under pulsating flow.

[0056] 2. This invention analyzes the influence of bend radius and nonlinear support stiffness on the dynamic behavior of L-shaped pipes, providing a reference for the structural design, parameter optimization, and engineering application of elastically supported L-shaped flow transmission pipes in practical engineering.

[0057] 3. The natural frequency and vibration response of the nonlinear supported L-shaped conveying pipe obtained by the present invention are consistent with the results obtained from the literature and COMSOL multiphysics simulation software, which proves the accuracy of the present invention. Attached Figure Description

[0058] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0059] Figure 1 This is a flowchart of a vibration prediction method for an L-shaped transmission pipeline with nonlinear support, considering the stress stiffening effect, according to an embodiment of the present invention.

[0060] Figure 2 This is a schematic diagram of a nonlinear supported L-shaped pipe according to an embodiment of the present invention. The red curve in the diagram represents the form of pulsating flow. Indicates the average flow velocity amplitude. The amplitude of the pulsating flow velocity. The frequency of the pulse is t, and time is t.

[0061] Figure 3This invention relates to the effect of average flow velocity on the first four natural frequencies of a pipeline system. (a) shows the variation curves of the first four natural frequencies of the pipeline under different average flow velocities without considering the stress stiffening effect, and compares them with the calculation results of the Absolute Nodal Coordinates (ANCF) method. (b) shows the variation curves of the first four natural frequencies of the pipeline under different average flow velocities with considering the stress stiffening effect.

[0062] Figure 4 This is a comparison chart of the time-domain response of a certain point in the L-shaped pipeline and the calculation results of the COMSOL physical simulation software in an embodiment of the present invention. Detailed Implementation

[0063] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments. However, exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that the invention will be more comprehensive and complete, and will fully convey the concept of the exemplary embodiments to those skilled in the art. The same reference numerals in the figures denote the same or similar structures, and therefore their detailed description will be omitted.

[0064] This embodiment uses, as follows Figure 2 Taking the nonlinear supported L-shaped flow conveying pipe as an example, the finite element modeling and solution process of the present invention is described in detail. The geometric and material parameters of the nonlinear supported L-shaped flow conveying pipe are shown in Table 1.

[0065]

[0066] Figure 1 The flowchart below shows the solution process for a finite element model correction method for an L-shaped pipe with nonlinear support under parametric excitation according to the present invention. The method is implemented according to the following steps:

[0067] Step S1: Construct the dynamic equations for an L-shaped pipe with nonlinear support under parametric excitation based on the geometric and material parameters of the elastically supported flow conveying pipe.

[0068]

[0069] Where q represents the generalized displacement vector, a point on it represents the first derivative with respect to time, and two points on it represent the second derivative with respect to time. K L K NL F represents the linear and nonlinear stiffness matrices of the pipeline structure. ext It is a vector containing the externally distributed force. C is the Rayleigh damping matrix. , These represent the additional damping and stiffness terms introduced by the fluid-structure interaction effect. M is the mass matrix.

[0070] Step S2, the static equation is:

[0071]

[0072] Where F is the generalized static load vector of the system, derived from... Projecting onto the system's generalized coordinates yields: ,

[0073] Where H T Let T be the shape function matrix based on the Hermite interpolation function, where the superscript T denotes the transpose of the matrix. The cell length is given by 'e', ​​the superscript 'e' indicates the cell number, and N is the number of cells. The expression is:

[0074]

[0075] Where n represents the unit normal vector radially outward from the center of curvature of the bend. The average flow velocity of the fluid inside the pipe. The cross-sectional area of ​​the fluid inside the pipe. R is the fluid density, and R is the radius of the bend.

[0076] Step S3: Use the Newton-Raphson iterative method to obtain the static solution. ;

[0077] Step S4, Use Inverse calculation of axial force in pipe structure unit:

[0078]

[0079] in From The static displacement vector of the node corresponding to the e-th element is extracted. Based on this displacement vector, the actual spatial length of the element after deformation is calculated. It can be expressed as a geometric function of nodal coordinates and static displacements. . Let E be the element axial force, E be the Young's modulus of the pipe structure, and S be the axial force. p Let E be the cross-sectional area of ​​the pipe structure, and E be the Young's modulus of the pipe structure. Let be the initial length of the e-th unit.

[0080] An additional stiffness matrix for element axial force is constructed using element axial force and then superimposed onto the original stiffness matrix for correction.

[0081]

[0082] Where H is the shape function matrix based on the Hermite interpolation function, the dot on the symbol indicates differentiation with respect to x, the superscript T indicates the transpose of the corresponding matrix, and k g eAdd a stiffness matrix to the axial force of the element.

[0083] Step S5, Use Update coordinate transformation matrix Right now:

[0084]

[0085] Where T disp ( It can be written as:

[0086]

[0087] in The angle caused by static deformation.

[0088] The dynamic equations are constructed using the updated coordinate transformation matrix, and the stiffness matrix is ​​corrected using an additional stiffness matrix.

[0089]

[0090]

[0091]

[0092]

[0093] The kinematic equations of the L-shaped fluid-structure interaction pipe based on the finite element method are obtained as follows:

[0094]

[0095] Finally, the number of grids N in the system was set to 100, and the natural frequencies were solved using the mass matrix and stiffness matrix. Figure 3 Table (a) shows a comparison between the method and the absolute nodal coordinate method without considering stiffening effects. The results show that the two methods have good consistency, demonstrating the accuracy of the mass matrix and stiffness matrix of this invention. Table 2 shows a comparison with COMSOL numerical simulation software when considering stiffness hardening, proving the accuracy of the stiffness matrix proposed in this invention. Figure 3 Figure (b) shows the natural frequencies of the system considering the stiffening effect. The results in the figure are consistent with... Figure 3 (a) The results obtained without considering the stiffening effect show significant differences, demonstrating the practical significance of this invention. The nonlinear vibration response of the pipeline was solved using the Newmark-β scheme combined with the Newton-Raphson iterative method. A simple harmonic concentrated force of 30 N and 10 Hz was applied at the horizontal nonlinear support. The flow velocity was set to 0. The accuracy of the pipeline system's time-domain response was verified using COMSOL numerical simulation software, with the horizontal intermediate support selected as the observation point. Figure 4The results show that the two are in good agreement, verifying that the present invention still has good accuracy in time-domain solution.

[0096]

[0097] Based on the above analysis results, this invention achieves good consistency with the absolute node coordinate method and COMSOL physical simulation software. Therefore, this invention realizes the calculation of the inherent characteristics and time-domain response analysis of L-shaped flow transmission pipes with nonlinear supports when considering stress stiffening effects.

[0098] In order to better understand the technical means of this application and to implement it according to the description, and to make the above and other objects, features and advantages of this application more apparent and understandable. It should be understood that the above general description is merely exemplary and explanatory, and does not limit the invention.

[0099] Embodiments of the invention will readily conceive of those skilled in the art upon consideration of the invention disclosed in the specification. This application is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. The specification and embodiments are to be considered exemplary only, and the true scope of the invention is indicated by the claims.

[0100] It should be understood that the present invention is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of the invention is limited only by the appended claims.

Claims

1. A method for predicting vibration of an L-shaped fluid-conveying pipe with nonlinear supports considering the effect of stress stiffening, characterized in that, Includes the following steps: Step 1: Based on Euler beam theory, the L-shaped pipe is discretized into multiple finite elements using the finite element method. According to the geometric parameters, material parameters and physical parameters of the fluid inside the pipe, the initial structural dynamic equations of the L-shaped conveying pipe with nonlinear support are established. The initial structural dynamic equations are used to characterize the force equilibrium state of the physical pipe structure under pulsating internal flow excitation. Step 2: Remove the time-varying dynamic terms from the initial structural dynamic equations to obtain the static equilibrium equations describing the physical pipe structure under steady-state fluid loads. Solve the static equilibrium equations using the Newton-Raphson iterative method to obtain the static equilibrium spatial configuration of the physical pipe structure under steady-state flow-induced loads. Step 3: Based on the static equilibrium spatial configuration, back-calculate the axial force of each finite element of the physical pipe structure, and calculate the stress stiffening effect caused by the axial force based on the axial force, and generate the element axial force additional stiffness matrix to characterize the contribution of the effect to the lateral stiffness of the element. Step 4: Assemble the element axial force additional stiffness matrix of each finite element into a global additional stiffness matrix, and superimpose it into the total stiffness matrix of the physical pipe structure to correct the stiffness distribution of the system. At the same time, update the coordinate transformation matrix of each finite element according to the element spatial rotation angle increment generated by the static equilibrium spatial configuration. Step 5: Using the updated coordinate transformation matrix and the corrected total stiffness matrix, construct the complete time-domain dynamic equations of the structure, and combine the Newmark-β stepwise integration scheme and the Newton-Raphson iterative method to solve the nonlinear vibration response of the physical pipe structure under pulsating internal flow excitation in the time domain.

2. A method for predicting vibrations of an L-shaped fluid conveying pipe with nonlinear supports considering stress stiffening effect as claimed in claim 1 wherein, In step 1, the initial structural dynamics equations are constructed based on fluid-structure interaction physical fields. The additional damping and stiffness terms in the equations are determined based on the interaction between the instantaneous velocity of the pulsatile flow inside the pipe and the pipe wall. The generalized force vector in the equations is determined based on the constraint reaction force applied to the pipe structure by external physical excitation and nonlinear support constraints.

3. The vibration prediction method for an L-shaped flow transmission pipeline with nonlinear support, considering stress stiffening effect, as described in claim 2, is characterized in that... The initial set of structural dynamic equations is as follows: In the formula, q represents the generalized displacement vector, where one point represents the first derivative with respect to time, and two points represent the second derivative with respect to time; K L K NL F represents the linear and nonlinear stiffness matrices of the pipeline structure. ext It is a vector containing the externally distributed force, and C is the Rayleigh damping matrix. , The additional damping and stiffness terms are due to the fluid-structure interaction effect, and M is the mass matrix.

4. A method for predicting vibration of an L-shaped fluid conveying pipe with nonlinear supports considering stress stiffening effect according to claim 1, characterized in that, The steady-state fluid load in step 2 is specifically the centrifugal force generated by the fluid when it flows through the bend. The physical value of the centrifugal force is determined based on the fluid density, the cross-sectional area of ​​the pipe, the average flow velocity in the pipe, and the radius of the bend. Its direction of action is radially outward along the center of curvature of the bend and is applied as a static load to each finite element node.

5. The vibration prediction method for an L-shaped flow transmission pipeline with nonlinear support, considering stress stiffening effect, as described in claim 1, is characterized in that... The static equilibrium equation in step 2 is: In the formula, For generalized static load vector, H T Let T be the shape function matrix based on the Hermite interpolation function, where the superscript T denotes the transpose of the matrix. The element length is given by 'e', ​​where 'e' indicates the element number and 'N' represents the number of elements. The expression is: In the formula, n represents the unit normal vector radially outward from the center of curvature of the bend. The average flow velocity of the fluid inside the pipe. The cross-sectional area of ​​the fluid inside the pipe. R is the fluid density, and R is the radius of the bend.

6. A method for predicting vibrations of an L-shaped fluid-conveying pipeline with nonlinear supports considering stress stiffening effect as claimed in claim 1 wherein, In step 3, the axial force of the unit is calculated in reverse. The specific steps are as follows: extract the axial elongation or compression of each finite element under static equilibrium spatial configuration, combine the Young's modulus of the pipe material and the cross-sectional area of ​​the pipe, and use Hooke's law to calculate the actual physical axial force value of the element.

7. A method for predicting vibrations of an L-shaped fluid-conveying pipeline with nonlinear supports considering the effect of stress stiffening according to claim 6, characterized in that, The stress stiffening effect in step 3 specifically includes the following steps: Step 3.1: Obtain the static solution using the Newton-Raphson iterative method. Afterwards, from Extract the static displacement vector of the node corresponding to the e-th element. Based on this static displacement vector, the actual spatial length l1 of the e-th element after deformation is... e Represented as a geometric function of nodal coordinates and static displacements ; From this, the unit axial force induced by the steady load is back-calculated : where S p is the cross-sectional area of the pipe structure, E is the Young's modulus of the pipe structure, is the initial length of the e-th element; Step 3.2: Construct the geometric stiffness matrix using the element axial force and superimpose to the original stiffness matrix to get the element axial force additional stiffness matrix k g e wherein is the shape function matrix based on Hermite interpolation functions, the dot above the symbol indicates differentiation with respect to x.

8. A method for predicting vibrations of an L-shaped fluid conveying pipe with nonlinear supports considering stress stiffening effect as claimed in claim 1 wherein, Step 4, which involves updating the coordinate transformation matrix, specifically includes the following steps: Step 4.1: Based on the static solution The spatial rotation increment of each finite element is used to update the matrix. : In the formula, T0 is the initial coordinate transformation matrix, and matrix T disp ( ) is represented as: in, This represents the change in element angle, with the subscript 'e' indicating the corresponding element number.

9. A method for predicting vibration of an L-shaped fluid conveying pipe with nonlinear supports considering stress stiffening effect as claimed in claim 1 wherein, The step 5 constructs the complete structural time-domain dynamic equation, specifically using the updated coordinate transformation matrix The mass matrix M, the damping matrix , the stiffness matrix and the external force vector F ext are assembled and converted; The final kinematic equations for the L-shaped fluid-structure interaction pipe based on the finite element method are as follows: In the formula, K g is the system global additional stiffness matrix considering the stress stiffening effect. Finally, the nonlinear vibration response of the pipeline under pulsating fluid is solved by using the Newmark-β format combined with the Newton-Raphson iteration method.

10. A method for predicting vibrations of an L-shaped fluid conveying pipe with nonlinear supports considering stress stiffening effect as claimed in claim 1 wherein, In step 1, for the bend section of the L-shaped pipe, multiple straight beam units are used to approximate its circular arc bending spatial configuration. The length of each straight beam unit is determined according to the bend radius and bending angle. The physical spatial mapping between the local coordinate system and the global coordinate system is realized between adjacent units through the coordinate transformation matrix.

11. A vibration prediction system for an L-shaped transmission pipeline with nonlinear support, considering stress stiffening effects, for implementing the method described in any one of claims 1 to 10, characterized in that, include: The physical model discretization module is used to discretize the physical L-shaped pipe into multiple finite elements based on beam theory, and to establish an initial set of structural dynamic equations based on the pipe structural parameters and the fluid parameters inside the pipe. The static equilibrium solution module is used to remove time-varying terms from the equation set to construct the static equilibrium equations, and to solve for the static equilibrium spatial configuration of the pipeline structure using a numerical iteration method. The stiffening effect calculation module is used to calculate the actual axial internal force of the unit based on the static equilibrium spatial configuration and generate the unit axial force additional stiffness matrix characterizing the stress stiffening effect. The matrix correction and coordinate update module is used to superimpose the additional stiffness matrix onto the total stiffness matrix and update the coordinate transformation matrix of the unit according to the static equilibrium configuration. The time-domain dynamic response solution module is used to construct the complete time-domain equation using the corrected total stiffness matrix and the updated coordinate transformation matrix, and to solve the nonlinear vibration response of the pipeline structure using the step-by-step integration method.

12. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps of the method as described in any one of claims 1 to 10.

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