Time-varying finite element analysis method for variable length axial moving chord system with complex boundary conditions

Abstract

The application discloses a time-varying finite element analysis method for a variable-length axial movement chord line system with complex boundary conditions, a mathematical model of the system is established according to system dynamics characteristics and mechanical principles; a finite element model of a three-node nonlinear high-precision unit is established; according to the Hamilton principle, a consistent mass matrix, a consistent damping matrix and a consistent stiffness matrix of the unit are derived; a Newmark_beta method is used to process the boundary conditions to obtain the velocity and acceleration of the displacement excitation after preprocessing; a 0-1 replacement method is used to determine the dynamic equation of the system without boundary processing; the Newmark_beta method is used for step-by-step solving, and the state between calculation steps is obtained by using a modified Akima cubic Hermite interpolation algorithm. Finally, the lateral vibration of the single-rope winding type hoist lifting rope caused by the boundary displacement disturbance under the condition of arbitrary speed tension is obtained, and the technical problem that the existing variable-length axial movement system cannot consider the transient fluctuation response caused by the boundary excitation can be solved.

Description

Technical Field

[0001] This invention belongs to the field of finite element analysis technology, specifically a time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions. Background Technology

[0002] Axial moving chord systems, as typical transmission systems, are characterized by high efficiency and large load-bearing capacity, making them suitable for various engineering applications, including but not limited to elevator lifting systems, cableway transmission systems, cable car transportation systems, and rope-driven robot systems. These systems typically employ drum winding for propulsion. However, in actual engineering, the drum inevitably suffers from mechanical manufacturing errors and wear after prolonged operation, resulting in the drum not being a perfectly round shape. Furthermore, most ultra-long-distance winding lifting systems use multi-layer winding, which inevitably leads to inter-layer transitions in the winding rope and displacement impact excitation when the rope passes through the transition zone. The external disturbances analyzed above will all cause additional lateral vibrations in the axial moving chord, which can significantly reduce transmission efficiency and even cause safety accidents. This makes the analysis of lateral vibrations in axial movement particularly important. The disturbances analyzed above all involve forced displacement excitation at the rope endpoints, i.e., they belong to the first type of boundary condition.

[0003] Traditional research techniques rely on Hamilton's principle to establish dynamic equations, which are then solved using numerical methods. However, introducing the first type of boundary condition is often a challenging problem in practical numerical solutions. For example, the traditional Galerkin method introduces the first type of boundary condition using two trial functions: a non-homogeneous part satisfying the first type of boundary condition and a homogeneous part satisfying the zero boundary condition. With a limited number of trial functions, numerical studies may exhibit a phenomenon where the first type of boundary condition perturbation instantaneously acts on the entire computational domain, failing to reflect its wave characteristics. This is impractical, especially in cases of impact-type excitations occurring during interlayer transitions in wound ropes. Methods capable of accurately characterizing the first type of boundary condition are primarily based on various methods developed from the d'Alembert traveling wave method. However, due to the assumption of linear equations, these methods cannot be used for the more practical study of lateral vibrations in length-variable tension, geometrically nonlinear axially moving systems. Summary of the Invention

[0004] In view of this, the purpose of this invention is to provide a time-varying finite element analysis method for a variable-length axially moving string system with complex boundary conditions, aiming to solve the technical problem that the existing technology cannot consider the transient fluctuation response caused by boundary excitation in variable-length axial motion systems.

[0005] To achieve the above objectives, the present invention provides the following technical solution:

[0006] A time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions includes the following steps:

[0007] Step 1: Establish a dynamic model based on the dynamic characteristics and mechanical principles of the variable-length axially moving string system;

[0008] Step 2: Construct the element matrix and end mass matrix

[0009] A variable-length axially movable chord system includes a rope and end masses;

[0010] The rope is divided into multiple elements using the variable finite element method. The kinetic energy, potential energy, and virtual work done by external forces in each element are determined. A finite element model of a three-node nonlinear high-precision element is established. Based on Hamilton's principle, the uniform mass matrix, uniform damping matrix, and uniform stiffness matrix of the element are determined.

[0011] Determine the mass matrix, damping matrix, and stiffness matrix of the end mass;

[0012] Step 3: Assemble the matrices of all elements and determine the dynamic equations of the system without boundary treatment;

[0013] Step 4: Use the Newmark_beta method to preprocess the displacement excitation e(t) to obtain the velocity and acceleration of the preprocessed displacement excitation;

[0014] Step 5: Perform 0-1 substitution method to obtain the finite element equations with boundary displacement excitation;

[0015] Step 6: Solve the finite element equations step by step using the Newmark_beta method, and obtain the state and calculation residual vector between calculation steps using the modified Akima cubic Hermite interpolation algorithm;

[0016] Step 7: Determine if the 2-norm of the calculated residual vector in the current iteration step is less than the set threshold: if yes, proceed to step 8; if no, proceed to step 6.

[0017] Step 8: Determine if the calculation is complete: if yes, the calculation is complete; if no, proceed to step 2.

[0018] Furthermore, in step one, a fixed coordinate system XOY is established with the upper end of the rope as the origin. The dynamic model of the variable-length axially moving string system is as follows:

[0019]

[0020] Where: ρ, E, A, c, p and f(X, t) are the linear density of the rope, Young's modulus, equivalent cross-sectional area, damping, linear or nonlinear indicator and distributed load of the chord, respectively; X is the distance to the fixed coordinate system; t represents time.

[0021] V represents the rate of increase of the target speed over time; Indicates the acceleration of the target increase;

[0022] T represents the static tension of the rope;

[0023] w represents the deflection of the rope;

[0024] This represents the derivative of the rope's deflection with respect to time.

[0025] This represents the second derivative of the rope's deflection with respect to time.

[0026] w′ represents the derivative of the rope's deflection with respect to spatial coordinates;

[0027] w′′ represents the second derivative of the rope's deflection with respect to spatial coordinates;

[0028] The second derivative of the rope's deflection with respect to time and space coordinates is given.

[0029] m e c e k e These are the equivalent concentrated mass at the end, equivalent damping, and equivalent stiffness, respectively.

[0030] l(t) is the time-varying length of the rope;

[0031] e(t) represents the boundary condition at the upper end of the rope.

[0032] Furthermore, in step two, the virtual work done by the kinetic energy, potential energy, and external force of each unit are as follows:

[0033]

[0034]

[0035]

[0036] Among them, E ki E represents the kinetic energy of the i-th unit; pi δW represents the potential energy of the i-th unit. i The virtual work done by the external force on the i-th element is represented by ε; i represents the i-th element; ε is the Lagrange strain; l i (t) represents the time-varying length of the i-th unit; w i Indicates the deflection change of the i-th unit; xi Indicates the position of the i-th unit; This is the matter derivative operator.

[0037] Furthermore, the uniform mass matrix, uniform damping matrix, and uniform stiffness matrix of the element are as follows:

[0038]

[0039]

[0040]

[0041] in: and , respectively, are the element mass matrix, element stiffness matrix, and element damping matrix; g and N are the nodal displacement vector and shape function, respectively; This represents the first derivative of a shape function vector with respect to time. N' represents the second derivative of the shape function vector with respect to time; N' represents the first derivative of the shape function vector with respect to spatial coordinates; N' T N represents the transpose of the first derivative of the shape function vector with respect to spatial coordinates; T This represents the transpose of a vector of shape functions; It is the time-varying length of the cell; The target velocity is the ratio of the element length to the total length.

[0042] Furthermore, the mass matrix, damping matrix, and stiffness matrix of the end mass are as follows:

[0043]

[0044]

[0045]

[0046] Where: m cage k cage and c cage These are the mass matrix, stiffness matrix, and damping matrix of the end masses, respectively; me (t) represents the length of the end mass; SUM represents the summation function; E and E represent the density, stiffness coefficient, damping coefficient, and Young's modulus of the end mass, respectively.

[0047] Furthermore, in step three, the dynamic equation of the system without boundary treatment is:

[0048]

[0049] Where: M, C, and F are the mass matrix, damping matrix, and force vector of the model without boundary treatment, respectively; K linear and K nonlinear These are the linear stiffness matrix and nonlinear stiffness matrix of the model without boundary treatment, respectively; q(t) is the nodal displacement vector; The derivative of the nodal displacement vector with respect to time; The second derivative of the nodal displacement vector with respect to time.

[0050] Furthermore, in step four, the velocity and acceleration of the preprocessed displacement excitation are respectively:

[0051]

[0052] in: This indicates the velocity of the displacement excitation after preprocessing; The acceleration of the displacement excitation after preprocessing; γ and β are parameters of the Newmark_beta method; Δt is the calculation step size.

[0053] Furthermore, in step five, the finite element equations for introducing boundary displacement excitation are as follows:

[0054]

[0055] Where: m gh c represents the components of the system's consistent quality matrix; gh k represents the components of the system's uniform damping matrix. gh The components representing the uniform stiffness matrix of the system include nonlinear parts; g = 2, 3, ..., n; h = 2, 3, ..., n; q i The components of the displacement vector representing all discrete degrees of freedom of the system; This represents the components of the velocity vector for all discrete degrees of freedom of the system after boundary treatment; Represents the components of the acceleration vector for all discrete degrees of freedom of the system after boundary treatment; i = 1, 2, ..., n;

[0056] Rewriting it in a unified, indirect form, we obtain the finite element equations after handling the boundary conditions:

[0057]

[0058] in, This represents the system consistency quality matrix after boundary condition processing. This represents the system uniform damping matrix after boundary condition processing; This represents the system load vector after boundary condition processing; This represents the displacement vector of all degrees of freedom of the system after boundary condition processing; This represents the velocity vectors of all degrees of freedom of the system after boundary condition processing; This represents the acceleration vector of all degrees of freedom of the system after boundary condition processing; Represents the linear components of the system's uniform stiffness matrix after boundary condition processing; This represents the nonlinear component of the system uniform stiffness matrix after boundary condition processing.

[0059] Furthermore, in step six, the incremental balance equation established at the end of the calculation step using the Newmark_beta method is as follows:

[0060]

[0061] Incremental expressions for velocity and acceleration:

[0062]

[0063] The equilibrium equations are obtained as follows:

[0064]

[0065] in: This represents the equivalent static stiffness term in the Newmark_beta method calculation step; This represents the equivalent static load term in the Newmark_beta method calculation step; and:

[0066]

[0067] Where: Rs is the calculated residual vector of the current iteration step; and:

[0068]

[0069] Where: Rs0 is the residual vector calculated in the previous iteration step;

[0070] The displacement, velocity, and acceleration at the end of the step are calculated as follows:

[0071]

[0072] in: and These represent the displacement, velocity, and acceleration at the end of the calculation step, respectively. and These represent the displacement increment, velocity increment, and acceleration increment at the end of the calculation step, respectively.

[0073] The beneficial effects of this invention are as follows:

[0074] The axially moving string system is essentially a time-varying length continuous gyroscope system. Current finite element methods all use the material point method under the Lagrange description, which involves the judgment of birth and death elements and cannot introduce Coriolis forces for complex modal analysis. For this type of variable-length continuous gyroscope system with first-type boundary conditions, this invention presents a time-varying finite element analysis method for variable-length axially moving string systems with complex boundary conditions. The method establishes a mathematical model of the system based on its dynamic characteristics and mechanical principles; divides the system into elements, establishing a finite element model with three-node nonlinear high-precision elements; derives the uniform mass matrix, uniform damping matrix, and uniform stiffness matrix of the elements based on Hamilton's principle; processes the boundary conditions using the Newmark_beta method to obtain the pre-processed velocity and acceleration of the displacement excitation; uses the 0-1 permutation method to obtain the finite element equations with boundary displacement excitation; solves stepwise using the Newmark_beta method, and obtains the state between calculation steps using the modified Akima cubic Hermite interpolation algorithm. Finally, the lateral vibration of the hoisting rope of a single-rope wound hoist caused by boundary displacement disturbance under arbitrary velocity and tension conditions is obtained. This invention can accurately handle the essential boundary conditions of a single-rope wound hoist with variable-length axial movement, obtaining the lateral vibration of the axially moving chord system caused by boundary displacement disturbance under arbitrary speed and tension, reflecting the transient fluctuation characteristics caused by boundary excitation. It satisfies the adaptability and accuracy requirements for solving the dynamic equations of a single-rope wound hoist. Attached Figure Description

[0075] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration:

[0076] Figure 1 This is a schematic diagram of a typical single-rope winding hoist.

[0077] Figure 2 This is a flowchart of the time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions according to the present invention. Detailed Implementation

[0078] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0079] like Figure 1 The diagram shows a schematic of a single-rope winding hoist. It is assumed that the drum can be well controlled, and the object being lifted can be considered as the end mass. A fixed coordinate system OXY is established with the upper end of the rope as the origin, e(t) is the boundary condition at the upper end of the rope, the time-varying length of the rope is l(t), i is the element number, and d... i(t) represents the starting position of the i-th unit, o i xy is an inertial coordinate system established with the initial position as the origin, l i (t) represents the time-varying length of the i-th unit, w i (x i ,t) is the distance from the origin of the inertial coordinate system x i The deflection at the position, where X is the distance to the fixed coordinate system, and the mass of the end mass is m. e The equivalent damping of the end mass is c. e The equivalent stiffness of the end mass is k. e .

[0080] The following describes the specific implementation of the time-varying finite element analysis method for the variable-length axial moving chord system with complex boundary conditions of the present invention, using the single-rope winding hoist as an example.

[0081] The time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions in this embodiment includes the following steps:

[0082] Step 1: Establish a dynamic model based on the dynamic characteristics and mechanical principles of the variable-length axially moving string system.

[0083] Specifically, in this embodiment, a fixed coordinate system XOY is established with the upper end of the rope as the origin. The dynamic model of the variable-length axially moving string system is as follows:

[0084]

[0085] Where: ρ, E, A, c, p and f(X, t) are the linear density of the rope, Young's modulus, equivalent cross-sectional area, damping, linear or nonlinear indicator and distributed load of the chord, respectively; X is the distance to the fixed coordinate system; t represents time.

[0086] V represents the rate of increase of the target speed over time; express……;

[0087] T represents the static tension of the rope;

[0088] w represents the deflection of the rope;

[0089] This represents the derivative of the rope's deflection with respect to time.

[0090] This represents the second derivative of the rope's deflection with respect to time.

[0091] w′ represents the derivative of the rope's deflection with respect to spatial coordinates;

[0092] w″ represents the second derivative of the rope's deflection with respect to spatial coordinates;

[0093] The second derivative of the rope's deflection with respect to time and space coordinates is given.

[0094] m e c e k e These are the equivalent concentrated mass at the end, equivalent damping, and equivalent stiffness, respectively.

[0095] l(t) is the time-varying length of the rope;

[0096] e(t) represents the boundary condition at the upper end of the rope.

[0097] Step 2: Construct the element matrix and end mass matrix. Specifically, the variable-length axially movable string system includes the rope and end masses. The element matrix and end mass matrix are constructed for the rope and end masses respectively below.

[0098] (1) Constructing the unit matrix

[0099] The rope was divided into multiple elements using the variable finite element method. The kinetic energy, potential energy, and virtual work done by external forces in each element were determined, and a finite element model of a three-node nonlinear high-precision element was established. Specifically, the kinetic energy, potential energy, and virtual work done by external forces in each element are as follows:

[0100]

[0101]

[0102]

[0103] Among them, E ki E represents the kinetic energy of the i-th unit; pi δW represents the potential energy of the i-th unit. i The virtual work done by the external force on the i-th element is represented by ε; i represents the i-th element; ε is the Lagrange strain; l i (t) represents the time-varying length of the i-th unit; w i Indicates the deflection change of the i-th unit; x i Indicates the position of the i-th unit; This represents the differential operator for the matter derivative.

[0104] Based on Hamilton's principle, the uniform mass matrix, uniform damping matrix, and uniform stiffness matrix of the element are determined. Specifically, Hamilton's principle is as follows:

[0105]

[0106] Where t0 and t1 represent the upper and lower limits of arbitrary time integration in Hamilton's principle, respectively;

[0107] The uniform mass matrix, uniform damping matrix, and uniform stiffness matrix of the element are as follows:

[0108]

[0109]

[0110]

[0111] in: and , respectively, are the element mass matrix, element stiffness matrix, and element damping matrix; g and N are the nodal displacement vector and shape function, respectively; This represents the first derivative of a shape function vector with respect to time. N' represents the second derivative of the shape function vector with respect to time; N' represents the first derivative of the shape function vector with respect to spatial coordinates; N' T N represents the transpose of the first derivative of the shape function vector with respect to spatial coordinates; T Represents the transpose of a vector of shape functions; It is the time-varying length of the cell; The target velocity is the ratio of the element length to the total length.

[0112] (2) Construct the end mass matrix

[0113] In this embodiment, the mass matrix, damping matrix, and stiffness matrix of the end mass are:

[0114]

[0115]

[0116]

[0117] Where: m cage k cage and c cage These are the mass matrix, stiffness matrix, and damping matrix of the end masses, respectively; me (t) represents the length of the end mass; SUM represents the summation function; E and E represent the density, stiffness coefficient, damping coefficient, and Young's modulus of the end mass, respectively.

[0118] Step 3: Assemble the matrices of all elements to determine the dynamic equations of the system without boundary treatment. In this embodiment, the dynamic equations of the system without boundary treatment are:

[0119]

[0120] Where: M, C, and F are the mass matrix, damping matrix, and force vector of the model without boundary treatment, respectively; Klinear and K nonlinear These are the linear stiffness matrix and the nonlinear stiffness matrix of the model without boundary treatment, respectively. The first derivative of the displacement vector with respect to time of all degrees of freedom of all discrete nodes of the system is the velocity vector of the system. It represents the second derivative of the displacement vector with respect to time of all degrees of freedom of all discrete nodes in the system, i.e., the acceleration vector of the system.

[0121] Step 4: Use the Newmark_beta method to preprocess the displacement excitation e(t) to obtain the velocity and acceleration of the preprocessed displacement excitation. In this embodiment, the velocity and acceleration of the preprocessed displacement excitation are as follows:

[0122]

[0123] in: This indicates the velocity of the displacement excitation after preprocessing; The acceleration of the displacement excitation after preprocessing; γ and β are parameters of the Newmark_beta method; Δt is the calculation step size.

[0124] Step 5: Perform 0-1 substitution method processing to obtain the finite element equations with boundary displacement excitation. In this embodiment, the finite element equations with boundary displacement excitation are:

[0125]

[0126] Where: m gh c represents the components of the system's consistent quality matrix; gh k represents the components of the system's uniform damping matrix. gh The components representing the uniform stiffness matrix of the system include a nonlinear part; g = 2, 3, ..., n; h = 2, 3, ..., n; g i The components of the displacement vector representing all discrete degrees of freedom of the system; This represents the components of the velocity vector for all discrete degrees of freedom of the system after boundary treatment; Represents the components of the acceleration vector for all discrete degrees of freedom of the system after boundary treatment; i = 1, 2, ..., n;

[0127] make Rewriting it in a unified, indirect form, we obtain the finite element equations after handling the boundary conditions:

[0128]

[0129] in, This represents the system consistency quality matrix after boundary condition processing. This represents the system uniform damping matrix after boundary condition processing; This represents the system load vector after boundary condition processing; This represents the displacement vector of all degrees of freedom of the system after boundary condition processing; This represents the velocity vectors of all degrees of freedom of the system after boundary condition processing; This represents the acceleration vector of all degrees of freedom of the system after boundary condition processing; Represents the linear components of the system's uniform stiffness matrix after boundary condition processing; This represents the nonlinear component of the system uniform stiffness matrix after boundary condition processing.

[0130] Step Six: Solve the finite element equations step-by-step using the Newmark_beta method, and obtain the state and residual vector between computational steps using the modified Akima cubic Hermite interpolation algorithm. Using the Newmark_beta method, the incremental equilibrium equation established at the final time of the computational step is:

[0131]

[0132] Incremental expressions for velocity and acceleration:

[0133]

[0134] The equilibrium equations are obtained as follows:

[0135]

[0136] in: This represents the equivalent static stiffness term in the Newmark_beta method calculation step; This represents the equivalent static load term in the Newmark_beta method calculation step; and:

[0137]

[0138] Where Rs is the calculated residual vector of the current iteration step.

[0139] Thus, we can obtain its temporary displacement, velocity, and acceleration increments. Furthermore, we can obtain the displacement, velocity, and acceleration at the end of the calculation step as follows:

[0140]

[0141] in: and These represent the displacement, velocity, and acceleration at the end of the calculation step, respectively. and These represent the displacement increment, velocity increment, and acceleration increment at the end of the calculation step, respectively.

[0142] The displacement, velocity, and acceleration at the start of the current time step are obtained from the previous calculation step using modified Akima cubic Hermite interpolation based on the two-grid data of this calculation step. The nonlinear stiffness matrix of the current time step is updated, and the computational residual vector of the current time step is obtained simultaneously.

[0143]

[0144] Where Rs0 is the residual vector calculated in the previous iteration step.

[0145] Step 7: Determine if the 2-norm of the calculated residual vector Rs in the current iteration step is less than the set threshold: if yes, proceed to step 8; if no, proceed to step 6.

[0146] Step 8: Determine if the calculation is complete: if yes, the analysis ends; if no, proceed to Step 2.

[0147] The embodiments described above are merely preferred embodiments for fully illustrating the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions, characterized in that: Includes the following steps: Step 1: Establish a dynamic model based on the dynamic characteristics and mechanical principles of the variable-length axially moving string system; Step 2: Construct the element matrix and end mass matrix A variable-length axially movable chord system includes a rope and end masses; The rope is divided into multiple elements using the variable finite element method. The kinetic energy, potential energy, and virtual work done by external forces in each element are determined, and a finite element model of a three-node nonlinear high-precision element is established. Based on Hamilton's principle, determine the uniform mass matrix, uniform damping matrix, and uniform stiffness matrix of the element; Determine the mass matrix, damping matrix, and stiffness matrix of the end mass; Step 3: Assemble the matrices of all elements and determine the dynamic equations of the system without boundary treatment; Step 4: Use the Newmark_beta method for displacement excitation. Preprocessing is performed to obtain the velocity and acceleration of the displacement excitation after preprocessing; Step 5: Perform 0-1 substitution method to obtain the finite element equations with boundary displacement excitation; Step 6: Solve the finite element equations step by step using the Newmark_beta method, and obtain the state and computational residual vector between computational steps using the modified Akima cubic Hermite interpolation algorithm; Step 7: Determine if the 2-norm of the calculated residual vector in the current iteration step is less than the set threshold: if yes, proceed to step 8; if no, proceed to step 6. Step 8: Determine if the calculation is complete: if yes, the calculation is complete; if no, proceed to step 2.

2. The time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions according to claim 1, characterized in that: Step one specifically involves establishing a fixed coordinate system XOY with the upper end of the rope as the origin, and the dynamic model of the variable-length axially moving string system is as follows: in: , , , , and These are the linear density of the rope, Young's modulus, equivalent cross-sectional area, damping, linear or nonlinear characteristics, and distributed load of the chord. This represents the distance to a fixed coordinate system. Indicates time; This indicates the rate at which the target speed increases over time. This indicates the target's acceleration. Indicates the static tension of the rope; Indicates the deflection of the rope; This represents the derivative of the rope's deflection with respect to time. This represents the second derivative of the rope's deflection with respect to time. The derivative of the rope's deflection with respect to spatial coordinates; This represents the second derivative of the rope's deflection with respect to spatial coordinates. The second derivative of the rope's deflection with respect to time and space coordinates is given. , , These are the equivalent concentrated mass at the end, equivalent damping, and equivalent stiffness, respectively. The time-varying length of the rope; This indicates the boundary conditions at the upper end of the rope.

3. The time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions according to claim 2, characterized in that: In step two, the virtual work done by the kinetic energy, potential energy, and external force of each unit are as follows: in, Indicates the first The kinetic energy of the unit; Indicates the first The potential energy of the unit; Indicates the first The virtual work done by the unit external force; Indicates the first unit; Lagrange strain; Indicates the first The time-varying length of the cell; Indicates the first Changes in the deflection of the unit; Indicates the first The location of the unit; This represents the differential operator for the matter derivative.

4. The time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions according to claim 3, characterized in that: The uniform mass matrix, uniform damping matrix, and uniform stiffness matrix of the element are as follows: in: , and These are the element mass matrix, element stiffness matrix, and element damping matrix, respectively. and These are the nodal displacement vector and the shape function, respectively. This represents the first derivative of a shape function vector with respect to time. This represents the second derivative of the shape function vector with respect to time. This represents the first derivative of a shape function vector with respect to spatial coordinates; This represents the transpose of the first derivative of the shape function vector with respect to spatial coordinates; Represents the transpose of a vector of shape functions; It is the time-varying length of the cell; The target velocity is the ratio of the element length to the total length.

5. The time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions according to claim 1, characterized in that: The mass matrix, damping matrix, and stiffness matrix of the end mass are as follows: in: , and These are the mass matrix, stiffness matrix, and damping matrix of the end masses, respectively; The length of the end mass; This represents the summation function; , , and These are the density, stiffness coefficient, damping coefficient, and Young's modulus of the end mass, respectively.

6. The time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions according to claim 1, characterized in that: In step three, the dynamic equation of the system without boundary treatment is: in: , and These are the mass matrix, damping matrix, and force vector of the model without boundary treatment; and These are the linear stiffness matrix and the nonlinear stiffness matrix of the model without boundary treatment, respectively; The displacement vector representing all degrees of freedom of all discrete nodes in the system; The first derivative of the displacement vector with respect to time of all degrees of freedom of all discrete nodes of the system is the velocity vector of the system. It represents the second derivative of the displacement vector with respect to time of all degrees of freedom of all discrete nodes in the system, i.e., the acceleration vector of the system.

7. The time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions according to claim 1, characterized in that: In step four, the velocity and acceleration of the preprocessed displacement excitation are as follows: in: This indicates the velocity of the displacement excitation after preprocessing; The acceleration of the displacement excitation after preprocessing; and These are all parameters of the Newmark_beta method; This is the step size for the calculation.

8. The time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions according to claim 7, characterized in that: In step five, the finite element equations for introducing boundary displacement excitation are as follows: in: The components of the system's consistent quality matrix are represented. Represents the components of the system's uniform damping matrix; The components representing the uniform stiffness matrix of the system include nonlinear components; ; ; The components of the displacement vector representing all discrete degrees of freedom of the system; This represents the components of the velocity vector for all discrete degrees of freedom of the system after boundary treatment; This represents the components of the acceleration vector for all discrete degrees of freedom of the system after boundary treatment; ; Rewriting it in a unified, indirect form, we obtain the finite element equations after handling the boundary conditions: in, This represents the system consistency quality matrix after boundary condition processing; This represents the system uniform damping matrix after boundary condition processing; This represents the system load vector after boundary condition processing; This represents the displacement vector of all degrees of freedom of the system after boundary condition processing; This represents the velocity vectors of all degrees of freedom of the system after boundary condition processing; This represents the acceleration vector of all degrees of freedom of the system after boundary condition processing; Represents the linear components of the system's uniform stiffness matrix after boundary condition processing; This represents the nonlinear component of the system uniform stiffness matrix after boundary condition processing.

9. The time-varying finite element analysis method for a variable-length axially moving chord system with complex boundary conditions according to claim 8, characterized in that: In step six, using the Newmark_beta method, the incremental balance equation established at the end of the calculation step is: Incremental expressions for velocity and acceleration: The equilibrium equations are obtained as follows: in: This represents the equivalent static stiffness term in the Newmark_beta method calculation step; This represents the equivalent static load term in the Newmark_beta method calculation step; and: in: Let be the residual vector calculated for the current iteration step; and: in: This is the residual vector calculated in the previous iteration step; The displacement, velocity, and acceleration at the end of the step are calculated as follows: in: , and These represent the displacement, velocity, and acceleration at the end of the calculation step, respectively. , and These represent the displacement increment, velocity increment, and acceleration increment at the end of the calculation step, respectively.

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