A fast computation method for the response of hysteretic material structure based on power series approximation

Abstract

The application discloses a kind of hysteresis material structure response fast calculation method based on power series approximation, comprising the following steps: establishing the vibration response calculation equation with hysteresis damping model;Power series approximation is used to kernel function in convolution integral;The second derivative of convolution integral is calculated to obtain the relationship with response;Integral iterative calculation is carried out based on three-point interpolation format;And direct integration method is combined to calculate structural response.The application reduces the complexity of convolution calculation, improves the calculation efficiency;On the basis of ensuring calculation accuracy, the calculation efficiency is high, time-consuming is short, numerical stability is high, can simulate the response calculation in actual engineering structure, has great practical value.

Description

Technical Field

[0001] This invention relates to the field of calculating the structural response of hysteretic composite materials, and in particular to a fast calculation method for the structural response of hysteretic materials based on power series approximation. Background Technology

[0002] Damping plays a crucial role in structural dynamic analysis, and its mechanisms and models have long been a focus of attention for experts both domestically and internationally. Currently, viscous damping models are commonly used. While this model reduces the difficulty of solving the dynamic equations, it neglects the complexity of damping mechanisms in actual engineering materials and structures, and cannot accurately represent the damping mechanisms of hysteretic materials and structures. Generalized damping models, derived from viscoelastic constitutive relations, are a class of time-nonlocal damping models that consider stress-strain history, possess memory properties, and can represent the damping mechanisms of hysteretic composite materials and structures under various strain rates. Traditional methods for calculating structural dynamic response cannot calculate the response of time-delayed structural systems.

[0003] Because hysteresis damping is prevalent in modern composite structures, it is typically described using a generalized damping model with memory properties. In recent years, a few researchers have investigated the dynamic response of structures with generalized damping models exhibiting hysteresis. These studies generally employ extended classical response calculation methods combined with numerical integration methods involving convolution to obtain the dynamic response in the time domain. The dynamic response under the exponential kernel function is then compared with the analytical solution to analyze the accuracy of the response calculation results. However, the numerical calculation of convolution consumes a significant amount of time, greatly reducing the efficiency of response calculation.

[0004] The design of the dynamic response calculation method for generalized damping model structures faces two conflicting objectives: computational accuracy and computational efficiency. Improving computational efficiency while meeting the requirements for computational accuracy is an urgent problem to be solved. Summary of the Invention

[0005] Purpose of the invention: The purpose of this invention is to provide a fast calculation method for the structural response of hysteretic materials based on power series approximation.

[0006] Technical solution: The present invention includes the following steps:

[0007] (1) Establish the vibration response calculation equation with a hysteresis damping model;

[0008] (2) The kernel function within the convolution integral is approximated using a power series;

[0009] (3) The relationship between the second derivative of the convolution integral and the response is calculated;

[0010] (4) Perform integral iterative calculation based on the three-point interpolation scheme;

[0011] (5) Combine with the direct integration method to calculate the structural response.

[0012] Furthermore, step (1) establishes the vibration response calculation equation with a hysteresis damping model based on the dynamic model as follows:

[0013]

[0014] in, Represents the mass coefficient matrix. Represents the damping coefficient matrix. Represents the stiffness coefficient matrix. Represents the external load vector. For kernel function, , and Let them be acceleration, velocity, and displacement vectors, respectively.

[0015] Further, step (2) includes:

[0016] Determine the number of terms in a power series The kernel function within the convolution integral is approximated using a power series.

[0017]

[0018] in, and These are the coefficients of different powers of the power series.

[0019] The convolution integral can then be expressed as:

[0020]

[0021] in ;

[0022] Furthermore, the first derivative of the convolution integral in step (3) is expressed as:

[0023]

[0024] Among them, selected We can obtain:

[0025]

[0026] The second derivative of the convolution integral can be expressed as:

[0027]

[0028] in, The displacement at the current moment, This represents the displacement at the initial moment.

[0029] Furthermore, in step (4), the second derivative function is calculated using a three-point difference scheme as follows:

[0030]

[0031] in, , and They are respectively , and The integral value at time t.

[0032] Establish points Iterative calculation format:

[0033]

[0034] Furthermore, in step (4), based on the integral... The iterative calculation format allows us to obtain the iterative calculation formula for the convolution integral:

[0035]

[0036] Furthermore, step (5) includes numerically calculating the displacement and velocity at any given time using a dynamic response solution method. The displacement vector can be expressed as:

[0037]

[0038] in, , , and They are respectively , , and The displacement vector, velocity vector, acceleration vector, and acceleration vector at each moment. For algorithm parameters,

[0039] The velocity vector can be represented as:

[0040]

[0041] in, These are algorithm parameters, and they must meet certain conditions.

[0042] Furthermore, the parameters and The following conditions must be met:

[0043] , .

[0044] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages: The present invention establishes the functional relationship between the second derivative of the convolution integral term and the acceleration and velocity response with power series approximation, and constructs the iterative calculation formula using a three-point difference scheme, which reduces the complexity of convolution calculation and improves the calculation efficiency; while ensuring calculation accuracy, the present invention has high calculation efficiency, short time consumption, and high numerical stability, and can simulate the response calculation in actual engineering structures, which has great practical value. Attached Figure Description

[0045] Figure 1 This is a flowchart of the present invention;

[0046] Figure 2 This is a comparison chart of the calculation results of the present invention with those of the Newamrk method, the state-space method, and the Bathe method. Detailed Implementation

[0047] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0048] like Figure 1 As shown, the fast calculation method for the structural response of hysteretic materials based on power series approximation described in this invention includes the following steps:

[0049] (1) Set the initial parameters of the structural dynamic response and determine the simulation time. and the number of simulation time steps Calculate the time step ;

[0050] (2) Given the initial displacement According to the undamped dynamic equation: Calculate the initial acceleration ,in, Represents the mass coefficient matrix. Represents the damping coefficient matrix. This represents the stiffness coefficient matrix.

[0051] (3) Solve the Newmark method based on the dynamic response. displacement at time ,speed and acceleration response .

[0052] (4) Calculate using the rectangular integral scheme Convolution integral at time step: The initial integral value is 0.

[0053] (5) Solve for any time interval The displacement response, velocity response, and acceleration response; solving for the response at any given time. The displacement response, velocity response, and acceleration response can be expressed using the following formulas:

[0054] Acceleration response solution In the formula , ,

[0055] Velocity response solution Displacement response solution ,

[0056] in, , , and They are respectively , , and The displacement vector, velocity vector, acceleration vector, and acceleration vector at each moment. These are the algorithm parameters.

[0057] (6) Integral Cocoa can use iterative loop calculation .

[0058] (7) Amplitude given Repeat step (6) above to calculate the response at the next time step.

[0059] Taking the dynamic response calculation of a multi-degree-of-freedom composite material structure as an example, given the mass parameters, damping parameters and stiffness parameters, the simulation duration is determined according to step (1). and calculate step size Given initial values The response at the first time step is calculated according to steps (2) and (3); the responses at subsequent time steps are calculated according to steps (5), (6), and (7). This is compared with the traditional direct time-domain integration method. Figure 2 As shown, the response curves completely overlap, indicating that the method can accurately calculate the response.

[0060] Table 1 Comparison of calculation time for different calculation methods

[0061] 200 9 s 7 s 2.2 s 400 102 s 77 s 4.5 s 600 220 s 158 s 6.4 s 800 270 s 210 s 9.7 s 1000 360 s 287 s 13.6 s

[0062] Regarding computation time, the computation time for structural responses with various degrees of freedom was compared. Table 1 shows the computation time of three methods. The results indicate that, for different degrees of freedom, the computation time of this method is significantly lower than existing methods, and the difference becomes increasingly pronounced as the number of degrees of freedom increases. In comparison, the proposed method greatly reduces computation time while maintaining computational accuracy. Considering that actual structures often have hundreds or even tens of thousands of degrees of freedom, this method is more suitable for application in response calculation programs for real-world engineering structures.

Claims

1. A fast computation method of hysteretic material structural response based on power series approximation, characterized in that, Includes the following steps: (1) Establish the vibration response calculation equation with a hysteresis damping model; (2) The kernel function within the convolution integral is approximated using a power series; (3) The relationship between the second derivative of the convolution integral and the response is calculated; (4) Perform integral iterative calculation based on the three-point interpolation scheme; (5) Calculate the structural response by combining it with the direct integration method; The vibration response calculation equation with hysteresis damping model established in step (1) based on the dynamic model is as follows: ， wherein, denotes the mass coefficient matrix, denotes the damping coefficient matrix, denotes the stiffness coefficient matrix, denotes the external load vector, is the kernel function, , and are the acceleration, velocity and displacement vectors, respectively. Step (2) includes: Determining the number of power series terms Using a power series approximation for the kernel function within the convolution integral ， wherein and are the coefficients of the different powers of the power series; The convolution integral can then be expressed as: ， wherein ; The first derivative of the convolution integral in step (3) is expressed as: ， wherein the selected gives: ， The second derivative of the convolution integral is expressed as: ， in, The displacement at the current moment, This represents the displacement at the initial moment; The second derivative function is calculated using the three-point difference scheme in step (4): ， in, , and They are respectively , and The integral value at time t. Establish points Iterative calculation format: ， In step (4), based on the integral The iterative calculation format allows us to obtain the iterative calculation formula for the convolution integral: 。 2. The fast calculation method for the structural response of hysteretic materials based on power series approximation according to claim 1, characterized in that, Step (5) includes numerically calculating the displacement and velocity at any given time using a dynamic response solution method. The displacement vector can be expressed as: in, , , and They are respectively , , and The displacement vector, velocity vector, acceleration vector, and acceleration vector at each moment. For algorithm parameters, The velocity vector can be represented as: in, These are algorithm parameters, and they must meet certain conditions.

3. The fast calculation method for the structural response of hysteretic materials based on power series approximation according to claim 2, characterized in that, The parameters and The following conditions must be met: ， 。

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