A structural dynamic analysis method and device containing a boundary resistance device

Abstract

The present application relates to a structural dynamic analysis method and device containing a boundary resistance device, which calculates the equilibrium damping matrix and the equilibrium load matrix based on the displacement, velocity and acceleration of the structural system at the previous moment in a set time step; then combines the boundary nonlinear dynamic equation of resistance balance to solve the velocity of the nonlinear dimension; then obtains the velocity of the linear dimension based on the solution formula of the velocity of the linear dimension; and obtains the displacement, velocity and acceleration of the current moment based on the velocity of the nonlinear dimension and the velocity of the linear dimension. The above utilizes the characteristics of the boundary resistance and velocity correlation and the inherent characteristics of the boundary nonlinear dynamic equation, uses the dynamic equation with only nonlinear dimension and based on boundary resistance balance, and finally calculates the displacement, velocity and acceleration of the structural system at the current moment. The dimension of the nonlinear dynamic equation is greatly reduced, the iterative solution consumes less time, and solves the problem of non-convergence of nonlinear iteration of strong boundary resistance.

Description

Technical Field

[0001] The present application relates to the field of structural dynamic analysis of bridges containing boundary resistance devices, and in particular to a structural dynamic analysis method and device containing boundary resistance devices. Background Art

[0002] Bridge structures currently often incorporate frictional oscillating bearings and dampers to improve their dynamic performance. The resistance and deformation of these devices exhibit nonlinear characteristics, and the resistance they provide is equivalent to changing the structural boundary constraints. Dynamic analysis of structures incorporating boundary resistance devices is often referred to as boundary nonlinear dynamic analysis.

[0003] The boundary nonlinear dynamic equation used in boundary nonlinear dynamic analysis is usually expressed as follows:

[0004]

[0005] It can be seen that the boundary nonlinear dynamic equation is a second-order equation, which includes the displacement term u and the velocity term and acceleration terms is the boundary nonlinear resistance, which is related to displacement and velocity. Displacement, velocity, acceleration, boundary resistance and external load are all related to time. For simplicity, the following expressions do not have a time t. This second-order ordinary differential equation is difficult to solve, and scholars usually need to reduce the order of the above dynamic equation. The order reduction method is usually combined with the Newmark-Beta method, that is, converting the velocity term and acceleration term into displacement term. The reduced-order boundary nonlinear dynamic equation is shown in Formula 2.

[0006]

[0007] In the formula To convert stiffness and external loads. In the reduced-order nonlinear equation, the displacement term is an unknown, and the resistance term contains a displacement term. Although the above nonlinear dynamic equations have been reduced in order, due to the strong nonlinear characteristics of the boundary nonlinear resistance term, the dimensions of the above nonlinear equations generally reach tens of thousands, making the solution very slow and difficult to converge. The main reason for the non-convergence is that the boundary resistance Jacobian matrix contains unknown terms and is not clear, and the dimensions of the equations are coupled. Summary of the Invention

[0008] The embodiments of the present application provide a structural dynamic analysis method and device including a boundary resistance device to solve the problem in related technologies that due to the strong nonlinear characteristics of the boundary nonlinear resistance term, the dimension of the boundary nonlinear dynamic equation used for analysis generally reaches tens of thousands, the solution speed is very slow, and it is difficult to converge.

[0009] In a first aspect, a method for dynamic analysis of a structure including a boundary resistance device is provided, comprising:

[0010] Calculate the equilibrium damping matrix and equilibrium load matrix based on the displacement, velocity and acceleration of the structural system at the previous moment in the set time step;

[0011] According to the boundary nonlinear dynamic equation based on resistance balance, and in combination with the equilibrium damping matrix and the equilibrium load matrix, the nonlinear dimension velocity is solved; then, according to the solution formula of the nonlinear dimension velocity and the linear dimension velocity, the linear dimension velocity is obtained;

[0012] According to the velocity of the nonlinear dimension and the velocity of the linear dimension, the displacement, velocity and acceleration of the structural system at the current moment are obtained.

[0013] In some embodiments, calculating the equilibrium damping matrix and the equilibrium load matrix based on the displacement, velocity, and acceleration of the structural system at a previous moment in a set time step includes the following steps:

[0014] According to formula 3, combined with the displacement, velocity and acceleration of the structural system at the previous moment in the set time step, the equivalent damping matrix and equivalent load matrix of the structural system are calculated;

[0015] Extracting and splitting the equivalent damping matrix to obtain equivalent damping matrices of different dimensions; splitting the equivalent load array to obtain equivalent load arrays of different dimensions;

[0016] According to the equivalent damping matrices of different dimensions and the equivalent load matrix of the same dimension, and in combination with Formula 4 and Formula 5, a balanced damping matrix and a balanced load matrix are obtained.

[0017] In some embodiments, the displacement, velocity and acceleration of the structural system at the previous moment are u0, and

[0018] The formula three is:

[0019]

[0020] Among them, C eq is the equivalent damping matrix, Q eq is the equivalent load array, h is the set time step; the u0, and are the displacement, velocity and acceleration at the previous moment respectively; Q is the load matrix; C is the damping matrix; K is the stiffness matrix; and M is the mass matrix.

[0021] In some embodiments, extracting and splitting the equivalent damping matrix to obtain equivalent damping matrices of different dimensions; splitting the equivalent load array to obtain equivalent load arrays of different dimensions specifically includes the following steps:

[0022] From the equivalent damping matrix C eq Extract and split the equivalent damping matrix of l rows and l columns The equivalent damping matrix with m rows and m columns The equivalent damping matrix with l rows and m columns The equivalent damping matrix with m rows and l columns

[0023] From the equivalent load array Q eq Extract the equivalent load array of l rows and m rows of equivalent load arrays

[0024] The l is the dimension of the boundary resistance, corresponding to the nonlinear dimension; m is the dimension of the non-boundary resistance, corresponding to the linear dimension, and the total dimension is n, n=l+m.

[0025] In some embodiments, the formula 4 is:

[0026]

[0027] The formula five is:

[0028]

[0029] in, is the transformation damping matrix with m rows and l columns; is the transformation damping matrix with l rows and l columns; is the velocity transformation matrix with m rows; is the equilibrium damping matrix corresponding to the boundary resistance dimension l, is the equilibrium load matrix corresponding to the boundary resistance dimension l.

[0030] In some embodiments, the boundary nonlinear dynamic equation based on resistance balance is: in, is the equilibrium damping matrix corresponding to the boundary resistance dimension l, D l is the equilibrium resistance array corresponding to the boundary resistance dimension l, is the equilibrium load array corresponding to the boundary resistance dimension l; is the velocity of the nonlinear dimension l;

[0031] The solution formula for the speed of the linear dimension is:

[0032] in, is the speed in linear dimension.

[0033] In some embodiments, the displacement, velocity, and acceleration of the structural system at the current moment are obtained based on the velocity of the nonlinear dimension and the velocity of the linear dimension, including the following steps:

[0034] By combining the obtained nonlinear dimension velocity and linear dimension velocity, the velocity of the total dimension can be determined to obtain the velocity of the structural system at the current moment;

[0035] Then calculate the displacement, velocity and acceleration at the current moment according to formula 8;

[0036] The formula eight is:

[0037]

[0038] in, is the velocity at the current moment; u is the displacement at the current moment; is the acceleration at the current moment.

[0039] In a second aspect, a structural dynamic analysis device including a boundary resistance device is provided, comprising:

[0040] The first module is used to calculate the equilibrium damping matrix and the equilibrium load matrix according to the displacement, velocity and acceleration of the structural system at the previous moment in the set time step;

[0041] The second module is used to solve the nonlinear dimension velocity based on the boundary nonlinear dynamic equation based on the resistance balance and in combination with the balanced damping matrix and the balanced load matrix; and then obtain the linear dimension velocity based on the solution formula of the nonlinear dimension velocity and the linear dimension velocity;

[0042] The third module is used to obtain the displacement, velocity and acceleration at the current moment according to the velocity of the nonlinear dimension and the velocity of the linear dimension.

[0043] The beneficial effects of the technical solution provided by this application include:

[0044] The embodiment of the present application provides a structural dynamic analysis method and device containing a boundary resistance device, which calculates the balanced damping matrix and the balanced load matrix based on the displacement, velocity and acceleration of the structural system at the previous moment in a set time step; solves the velocity of the nonlinear dimension based on the boundary nonlinear dynamic equation based on resistance balance and in combination with the balanced damping matrix and the balanced load matrix; then obtains the velocity of the linear dimension based on the solution formula of the velocity of the linear dimension; obtains the displacement, velocity and acceleration of the current moment based on the velocity of the nonlinear dimension and the velocity of the linear dimension. The above utilizes the characteristics of the boundary resistance and velocity correlation and the inherent characteristics of the boundary nonlinear dynamic equation, and uses the dynamic equation with only nonlinear dimension and based on boundary resistance balance to finally calculate the displacement, velocity and acceleration of the structural system at the current moment. The dimension of the nonlinear dynamic equation is greatly reduced, the time consumption of the nonlinear iterative solution is very small, the calculation solution is greatly improved, and the problem of non-convergence of nonlinear iteration of strong boundary resistance is solved. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0046] Figure 1 A schematic flow chart of a structural dynamic analysis method including a boundary resistance device provided in an embodiment of the present application. DETAILED DESCRIPTION

[0047] To make the purpose, technical solutions, and advantages of the embodiments of this application more clear, the technical solutions in the embodiments of this application will be clearly and completely described below in conjunction with the drawings in the embodiments of this application. Obviously, the described embodiments are part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of this application.

[0048] The boundary nonlinear dynamic equation used in boundary nonlinear dynamic analysis is usually expressed as follows:

[0049]

[0050] It can be seen that the boundary nonlinear dynamic equation is a second-order equation, which includes the displacement term u and the velocity term and acceleration terms is the nonlinear boundary resistance, which is related to displacement and velocity. Displacement, velocity, acceleration, boundary resistance, and external load are all time-dependent. This second-order ordinary differential equation is difficult to solve, and researchers typically need to reduce the order of the above dynamic equation. This reduction method is often combined with the Newmark-Beta method, which converts the velocity and acceleration terms into displacement terms. The reduced-order boundary nonlinear dynamic equation is shown in Equation 2.

[0051]

[0052] In the formula To convert stiffness and external loads. In the reduced-order nonlinear equation, the displacement term is an unknown, and the resistance term contains a displacement term. Although the above nonlinear dynamic equations have been reduced in order, due to the strong nonlinear characteristics of the boundary nonlinear resistance term, the dimensions of these nonlinear equations generally reach tens of thousands, making the solution very slow and difficult to converge. The main reason for the non-convergence is that the boundary resistance Jacobian matrix contains unknown terms and is not clear, and the dimensions of the equations are coupled.

[0053] Therefore, in order to solve the problem that existing methods are difficult to achieve slow solution and non-convergence of boundary nonlinear dynamic equations, the embodiments of the present application provide a structural dynamic analysis method, device, equipment and medium containing a boundary resistance device, so as to solve the problem in related technologies that due to the strong nonlinear characteristics of the boundary nonlinear resistance term, the dimension when using the boundary nonlinear dynamic equation for analysis generally reaches tens of thousands, the solution speed is very slow, and it is not easy to converge.

[0054] See also Figure 1 In a first aspect, a method for dynamic analysis of a structure including a boundary resistance device is provided, comprising:

[0055] Step 100: Calculate the equilibrium damping matrix and the equilibrium load matrix based on the displacement, velocity, and acceleration of the structural system at the previous moment in the set time step;

[0056] Step 101: Solve the nonlinear dimension velocity based on the boundary nonlinear dynamic equation based on resistance balance and in combination with the equilibrium damping matrix and the equilibrium load matrix; then, obtain the linear dimension velocity based on the solution formula of the nonlinear dimension velocity and the linear dimension velocity; the nonlinear dimension corresponds to the boundary resistance device;

[0057] Step 102: derive the displacement, velocity, and acceleration of the structural system at the current moment according to the velocity of the nonlinear dimension and the velocity of the linear dimension.

[0058] By utilizing the characteristics of boundary resistance being related to velocity and the inherent characteristics of boundary nonlinear dynamic equations, a boundary nonlinear dynamic equation with only nonlinear dimensions and based on boundary resistance balance was established and used. Compared with conventional technologies, the technical advantages of the present invention are: (1) The dimension of the nonlinear dynamic equation is greatly reduced, the time consumption of nonlinear iterative solution is very small, the computational solution is greatly improved, and the problem of non-convergence of strong resistance nonlinear iteration is solved. For example, the dimension of the structure is usually hundreds of thousands, while the dimension of nonlinear boundary resistance is only dozens. Compared with conventional equations, the computational solution time is increased by more than 20 times. (2) The present invention is the first to propose a method based on resistance balance. The boundary nonlinear resistance is mainly related to velocity, and displacement is secondary. Conventional equations are established based on displacement balance. Nonlinear iterative solution is not direct and requires data conversion, resulting in poor iterative solution performance.

[0059] The above utilizes the characteristics of boundary resistance and velocity correlation and the inherent characteristics of the boundary nonlinear dynamic equation. Using the dynamic equation with only nonlinear resistance dimension and based on boundary resistance balance, the displacement, velocity and acceleration of the structural system at the current moment are finally calculated. The dimension of the nonlinear dynamic equation is greatly reduced, the iterative solution is time-saving, and the problem of non-convergence of nonlinear iteration with strong boundary resistance is solved.

[0060] In some preferred embodiments, step 100 specifically includes the following steps:

[0061] Step 1001: According to Formula 3, and in combination with the displacement, velocity and acceleration of the structural system at the previous moment in the set time step, calculate the equivalent damping matrix and equivalent load matrix of the structural system; the displacement, velocity and acceleration of the structural system at the previous moment are u0, and

[0062] Formula 3 is:

[0063]

[0064] Among them, C eq is the equivalent damping matrix, Q eq is the equivalent load array, h is the set time step; u0, and are the displacement, velocity and acceleration of the previous moment respectively; Q is the load matrix; C is the damping matrix; K is the stiffness matrix; M is the mass matrix.

[0065] Step 1002: Extract and split the equivalent damping matrix to obtain equivalent damping matrices of different dimensions; split the equivalent load array to obtain equivalent load arrays of different dimensions; specifically:

[0066] From the equivalent damping matrix C eqExtract and split the equivalent damping matrix of l rows and l columns The equivalent damping matrix with m rows and m columns The equivalent damping matrix with l rows and m columns The equivalent damping matrix with m rows and l columns That is, from C eq Extract the elements at the intersection of rows and columns corresponding to the resistance dimension to form an equivalent damping matrix with l rows and l columns From C eq Delete the elements at the intersection of the row and column corresponding to the resistance dimension, and the remaining elements form an equivalent damping matrix with m rows and m columns From C eq Extract the elements of the row position corresponding to the resistance dimension and delete the elements of the column position corresponding to the resistance dimension to form an equivalent damping matrix with l rows and m columns From C eq Extract the elements of the column position corresponding to the resistance dimension and delete the elements of the row position corresponding to the resistance dimension to form an equivalent damping matrix with m rows and l columns.

[0067] From the equivalent load array Q eq Extract the equivalent load array of l rows and m rows of equivalent load arrays That is, from Q eq Extract the row elements corresponding to the resistance dimension to form an equivalent load array of l rows From Q eq Extract the row elements corresponding to the non-resistance dimension to form an m-row equivalent load array

[0068] l is the dimension of boundary resistance, corresponding to the nonlinear dimension; m is the dimension of non-boundary resistance, corresponding to the linear dimension, so the total dimension is n, n=l+m.

[0069] Step 1003: Based on the equivalent damping matrix of different dimensions and the equivalent load matrix of the same dimension, and in combination with Formula 4 and Formula 5, the equilibrium damping matrix and the equilibrium load matrix are obtained. That is, the transformation matrix based on resistance balance is determined, the transformation damping matrix with m rows and l columns The transformation damping matrix with l rows and l columns and the velocity transformation matrix with m rows

[0070] Formula 4 is:

[0071]

[0072] Formula 5 is:

[0073]

[0074] in, is the transformation damping matrix with m rows and l columns; is the transformation damping matrix with l rows and l columns; is the velocity transformation matrix with m rows; is the equilibrium damping matrix corresponding to the boundary resistance dimension l, is the equilibrium load matrix corresponding to the boundary resistance dimension l.

[0075] In some preferred embodiments, in step 101: the boundary nonlinear dynamic equation based on resistance balance, i.e., Formula 6, is used to perform nonlinear numerical iterative solution to determine the speed of the nonlinear dimension, and Formula 7 can be used to determine the speed of the non-resistance dimension.

[0076] The boundary nonlinear dynamic equation based on resistance balance is: Corresponding to formula 6; where, is the equilibrium damping matrix corresponding to the boundary resistance dimension l, D l is the equilibrium resistance array corresponding to the boundary resistance dimension l, is the equilibrium load array corresponding to the boundary resistance dimension l; is the velocity of the nonlinear dimension l;

[0077] The solution formula for the speed of linear dimension is: Corresponding formula 7.

[0078] in, is the speed in linear dimension.

[0079] In some preferred embodiments, step 102 includes:

[0080] By combining the obtained nonlinear dimension speed and linear dimension speed, the speed of the total dimension can be determined to obtain the speed at the current moment;

[0081] Then calculate the displacement, velocity and acceleration at the current moment according to formula 8;

[0082] Formula 8 is:

[0083]

[0084] in, is the velocity at the current moment; u is the displacement at the current moment; is the acceleration at the current moment.

[0085] In a second aspect, a structural dynamic analysis device including a boundary resistance device is provided, comprising:

[0086] The first module is used to calculate the equilibrium damping matrix and the equilibrium load matrix according to the displacement, velocity and acceleration of the structural system at the previous moment in the set time step;

[0087] The second module is used to solve the nonlinear dimension velocity based on the boundary nonlinear dynamic equation based on the resistance balance and in combination with the equilibrium damping matrix and the equilibrium load matrix; then, the linear dimension velocity is obtained based on the solution formula of the nonlinear dimension velocity and the linear dimension velocity;

[0088] The third module is used to obtain the displacement, velocity and acceleration at the current moment according to the velocity of the nonlinear dimension and the velocity of the linear dimension.

[0089] According to a third aspect, an electronic device is provided, comprising:

[0090] Processor, memory, communication interface;

[0091] The memory is used to store executable instructions that can be executed by the processor;

[0092] The processor is configured to perform a structural dynamic analysis method including a boundary resistance device by executing executable instructions.

[0093] In a fourth aspect, a readable storage medium is provided, on which a computer program is stored, and when the computer program is executed by a processor, a structural dynamic analysis method containing a boundary resistance device is implemented.

[0094] In the above description, the memory can be either independent or integrated with the processor. Optionally, when the memory is a device independent of the processor, the electronic device may further include a bus for connecting the aforementioned devices. The electronic device is used to implement the technical solution in any of the aforementioned method embodiments, and its implementation principles and technical effects are similar and will not be further described here.

[0095] A readable storage medium stores a computer program, which, when executed by a processor, implements the technical solution provided by any of the aforementioned embodiments.

[0096] Those skilled in the art will appreciate that all or part of the steps in the above-described method embodiments can be implemented using hardware associated with program instructions. The aforementioned program can be stored in a computer-readable storage medium. When executed, the program performs the steps of the above-described method embodiments. The aforementioned storage medium includes various media capable of storing program code, such as ROM, RAM, magnetic disks, or optical disks.

[0097] In the description of this application, it should be noted that the terms "upper" and "lower" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on this application. Unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or an indirect connection through an intermediate medium, or it can be internal communication between two elements. For those of ordinary skill in the art, the specific meanings of the above terms in this application can be understood according to the specific circumstances.

[0098] It should be noted that, in this application, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device comprising a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device. In the absence of further restrictions, an element defined by the sentence "comprising a ..." does not exclude the presence of other identical elements in the process, method, article or device comprising the element.

[0099] The foregoing is merely a list of specific embodiments of the present application, intended to enable those skilled in the art to understand or implement the present application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, the present application is not limited to the embodiments shown herein, but is intended to conform to the broadest scope consistent with the principles and novel features of the present application.

Claims

1. A structural dynamic analysis method containing a boundary resistance device, characterized in that: It includes: Calculate the equilibrium damping matrix and equilibrium load matrix based on the displacement, velocity and acceleration of the structural system at the previous moment in the set time step; According to Formula 3, combined with the displacement, velocity, and acceleration of the structural system at the previous moment in the set time step, the equivalent damping matrix and equivalent load matrix of the structural system are calculated; Formula 3 is: ； in, is the equivalent damping matrix, is the equivalent load array, h is the set time step; 、 and are the displacement, velocity and acceleration at the previous moment respectively; Q is the load matrix; C is the damping matrix; K is the stiffness matrix; M is the mass matrix; Extract and split the equivalent damping matrix to obtain equivalent damping matrices of different dimensions; split the equivalent load array to obtain equivalent load arrays of different dimensions; Extract and split OK, Equivalent damping matrix of the column , m rows, m columns equivalent damping matrix 、 The equivalent damping matrix of m rows and m columns , m lines, Equivalent damping matrix of the column ; From the equivalent load array Extracted from Equivalent load array for rows and m rows of equivalent load arrays ; is the dimension of boundary resistance, corresponding to nonlinear dimension; m is the dimension of non-boundary resistance, corresponding to linear dimension, then the total dimension is n, n= +m; According to the equivalent damping matrices of different dimensions and the equivalent load arrays of different dimensions, and in combination with Formula 4 and Formula 5, the equilibrium damping matrix and the equilibrium load matrix are obtained; the formula 4 is: ； The formula five is: ； in, For m rows, l The transformation damping matrix of the column; for l OK, l The transformation damping matrix of the column; is the velocity transformation matrix with m rows; is the corresponding boundary resistance dimension The equilibrium damping matrix, is the corresponding boundary resistance dimension The balanced load matrix; According to the boundary nonlinear dynamic equation based on resistance balance, combined with the balanced damping matrix and the balanced load matrix, the nonlinear dimension velocity is solved; then, according to the solution formula of the nonlinear dimension velocity and the linear dimension velocity, the linear dimension velocity is obtained; the boundary nonlinear dynamic equation based on resistance balance is: ;in, is the corresponding boundary resistance dimension The equilibrium damping matrix, is the corresponding boundary resistance dimension The balanced resistance array, is the corresponding boundary resistance dimension balanced load array; Nonlinear dimension The speed of the linear dimension is: = ;in, is the speed in linear dimension; According to the velocity of the nonlinear dimension and the velocity of the linear dimension, the displacement, velocity and acceleration of the structural system at the current moment are obtained.

2. The structural dynamic analysis method including a boundary resistance device according to claim 1, characterized in that: According to the velocity of the nonlinear dimension and the velocity of the linear dimension, the displacement, velocity and acceleration of the structural system at the current moment are obtained, including the following steps: By combining the obtained nonlinear dimension velocity and linear dimension velocity, the velocity of the total dimension can be determined to obtain the velocity of the structural system at the current moment; Then calculate the displacement, velocity and acceleration at the current moment according to formula 8; The formula eight is: ； in, is the speed at the current moment; is the displacement at the current moment; is the acceleration at the current moment.

3. A structural dynamic analysis device including a boundary resistance device, characterized in that: It includes: The first module is used to calculate the equilibrium damping matrix and the equilibrium load matrix according to the displacement, velocity and acceleration of the structural system at the previous moment in the set time step; According to Formula 3, combined with the displacement, velocity, and acceleration of the structural system at the previous moment in the set time step, the equivalent damping matrix and equivalent load matrix of the structural system are calculated; Formula 3 is: ； in, is the equivalent damping matrix, is the equivalent load array, h is the set time step; 、 and are the displacement, velocity and acceleration at the previous moment respectively; Q is the load matrix; C is the damping matrix; K is the stiffness matrix; M is the mass matrix; Extract and split the equivalent damping matrix to obtain equivalent damping matrices of different dimensions; split the equivalent load array to obtain equivalent load arrays of different dimensions; Extract and split OK, Equivalent damping matrix of the column , m rows, m columns equivalent damping matrix 、 The equivalent damping matrix of m rows and m columns , m lines, Equivalent damping matrix of the column ; From the equivalent load array Extracted from Equivalent load array for rows and m rows of equivalent load arrays ; is the dimension of boundary resistance, corresponding to nonlinear dimension; m is the dimension of non-boundary resistance, corresponding to linear dimension, then the total dimension is n, n= +m; According to the equivalent damping matrices of different dimensions and the equivalent load arrays of different dimensions, and in combination with Formula 4 and Formula 5, the equilibrium damping matrix and the equilibrium load matrix are obtained; the formula 4 is: ； The formula five is: ； in, For m rows, l The transformation damping matrix of the column; for l OK, l The transformation damping matrix of the column; is the velocity transformation matrix with m rows; is the corresponding boundary resistance dimension The equilibrium damping matrix, is the corresponding boundary resistance dimension The balanced load matrix; The second module is used to solve the nonlinear dimension velocity based on the boundary nonlinear dynamic equation based on resistance balance, combined with the balanced damping matrix and the balanced load matrix; then, the linear dimension velocity is obtained based on the solution formula of the nonlinear dimension velocity and the linear dimension velocity; the boundary nonlinear dynamic equation based on resistance balance is: ;in, is the corresponding boundary resistance dimension The equilibrium damping matrix, is the corresponding boundary resistance dimension The balanced resistance array, is the corresponding boundary resistance dimension balanced load array; Nonlinear dimension The speed of the linear dimension is: = ;in, is the speed in linear dimension; The third module is used to obtain the displacement, velocity and acceleration at the current moment according to the velocity of the nonlinear dimension and the velocity of the linear dimension.

Images