Viscoelasticity coupling beam unit equivalent stiffness matrix and kinetic equation modeling method

Abstract

The invention relates to a viscoelastic coupling beam unit equivalent stiffness matrix and kinetic equation modeling method, which is characterized in that a force deformation behavior of a material depending on time is described based on a viscoelastic integral constitutive equation, and the equivalent stiffness matrix and kinetic equation of a viscoelastic beam unit are further established in combination with a weighted margin method. The problem that according to an existing method, due to elastic hypothesis, it is difficult to analyze the viscous influence on unit node response caused by the cumulative effect of the strain dynamic process under the dynamic load is solved, the accuracy of beam unit dynamic response calculation is improved, and a more universal discrete unit is provided for a finite element method; and the method has profound significance for improving the analysis and design level of high-end mechanical equipment and structures.

Description

Technical Field

[0001] The present application relates to the technical field of finite element modeling and analysis, and in particular to a method for modeling an equivalent stiffness matrix of a beam unit coupled with viscoelasticity and a dynamic equation. Background Art

[0002] As an important discrete unit in the finite element method, the beam unit has the advantages of simple geometric structure and fewer degrees of freedom. Based on the discrete structure or mechanical component of the beam unit, the unit stiffness matrix is ​​assembled, and the mass and damping matrix is ​​established to form the overall finite element dynamic equation, and finally the static and dynamic response of the structure or component is analyzed. This technology has been widely used in the flexible analysis of high-end equipment or complex structures in the fields of bridges, buildings, aerospace vehicles, high-speed railways, automobile industry, machine tools, precision instruments, etc.

[0003] The traditional finite element beam unit modeling method is based on the elastic assumption and uses the elastic constitutive equation to establish the stiffness matrix and transient displacement, ignoring the viscous part of the material's viscoelasticity, which makes it difficult to accurately analyze the impact of the cumulative effect of the loading and deformation history process on the beam unit node response. Summary of the invention

[0004] Based on this, it is necessary to provide a coupled viscoelastic beam unit equivalent stiffness matrix and dynamic equation modeling method that can solve the above technical problems.

[0005] 1. The present application provides a method for modeling an equivalent stiffness matrix and a dynamic equation of a beam element coupled with viscoelasticity, the method comprising:

[0006] Construct the equivalent stiffness matrix and dynamic equation of the viscoelastic beam unit; the equivalent stiffness matrix and dynamic equation are constructed based on the viscoelastic integral constitutive equation and the weighted residual method; the dynamic equation includes the corresponding relationship between node load, displacement, acceleration, and equivalent integral displacement;

[0007] Dynamic loads are applied to the nodes of the viscoelastic beam element, and the dynamic equations are solved to obtain the dynamic responses of the nodes of the viscoelastic beam element.

[0008] 2. The construction process of the equivalent stiffness matrix and dynamic equation of the above viscoelastic beam element includes:

[0009] The equilibrium equation is expressed as an equivalent integral using the weighted residual method to obtain the stiffness term of the dynamic equation.

[0010] According to the stiffness terms and geometric equations, the equivalent stiffness matrix and equivalent integrated displacement are determined;

[0011] The dynamic equations are determined based on the equivalent stiffness matrix, equivalent integrated displacement, mass matrix of the beam element and the nodal loads.

[0012] 3. The above process of determining the equivalent stiffness matrix and equivalent integral displacement based on the stiffness term and geometric equation includes:

[0013] Substitute the matrix expression of the geometric equation and the integral viscoelastic constitutive equation into the stiffness term to transform it into the stiffness term of coupled viscoelasticity; the stiffness term of coupled viscoelasticity includes the equivalent stiffness matrix and the equivalent integral displacement.

[0014] 4. The construction process of the above viscoelastic integral constitutive equation includes:

[0015] Obtaining the relaxation modulus of the viscoelastic material; the relaxation modulus is measured based on a stress relaxation experiment performed on the viscoelastic material in advance;

[0016] The time derivative of the relaxation modulus and the strain history are integrated to determine the viscoelastic integral constitutive equation.

[0017] 5. The transformation process of the above-mentioned coupled viscoelastic stiffness term includes:

[0018] According to the viscoelastic integral constitutive equation, the stiffness term is transformed into the product relationship between the equivalent stiffness matrix and the nodal displacement and the equivalent integral displacement, and the variation of the nodal displacement is reduced.

[0019] 6. In one embodiment, the process of solving the dynamic equation to obtain the dynamic response of the node of the viscoelastic beam unit includes:

[0020] The dynamic response of the node includes displacement, velocity and acceleration;

[0021] According to the dynamic equation, determine the effective stiffness matrix and the effective load at time t+Δt;

[0022] Determine the node displacement at time t+Δt based on the inverse matrix of the effective stiffness matrix and the effective load at time t+Δt;

[0023] The node velocity and acceleration at time t+Δt are determined by integration based on the node displacement at time t+Δt and the node displacement, velocity, and acceleration at time t.

[0024] 7. In one embodiment, the process of determining the effective stiffness matrix and the effective load at time t+Δt according to the dynamic equation includes:

[0025] According to the equivalent stiffness matrix and the mass matrix, the effective stiffness matrix is ​​determined;

[0026] Determine the effective load at time t+Δt based on the node load and the equivalent integrated displacement at time t+Δt.

[0027] This application introduces material viscoelasticity when constructing the dynamic equation of the beam unit, changes the stiffness term in the dynamic equation, and enables the equation to accurately reflect the viscosity effect of the material under dynamic load. This solves the problem that the existing modeling method is difficult to accurately analyze the viscosity effect of the cumulative effect on the unit node response under dynamic load due to the elastic assumption, thereby improving the accuracy of dynamic response calculation, especially under dynamic load conditions such as impact, and can more accurately predict the displacement, velocity, acceleration and other responses of the beam unit node. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 A schematic diagram of a modeling and solution process of a viscoelastic beam unit in one embodiment;

[0029] Figure 2 is a schematic diagram of a beam unit in one embodiment;

[0030] Figure 3 A schematic flow chart of a process for constructing an equivalent stiffness matrix and dynamic equations of a viscoelastic beam unit in an embodiment;

[0031] Figure 4 A schematic diagram of a process for solving the dynamic equations of a beam unit node in one embodiment;

[0032] Figure 5a is a graph showing the response of a sinusoidal load to a viscoelastic beam node displacement in one embodiment;

[0033] Figure 5b A diagram showing the complex load and viscoelastic beam node displacement response in one embodiment;

[0034] Figure 6a is a diagram showing the influence of viscoelastic parameters on the displacement response of beam nodes under sinusoidal loads in one embodiment;

[0035] Figure 6b Graph showing the influence of viscoelastic parameters on the displacement response of beam nodes under complex loads in one embodiment. DETAILED DESCRIPTION

[0036] In order to make the purpose, technical solution and advantages of the present application more clearly understood, the present application is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

[0037] In an exemplary embodiment, Figure 1 As shown, the present application provides a method for modeling an equivalent stiffness matrix and dynamic equation of a beam unit coupled with viscoelasticity, which is applied to a terminal and specifically includes the following steps:

[0038] S201, construct the equivalent stiffness matrix and dynamic equations of the viscoelastic beam element.

[0039] Among them, the beam element is a basic unit used for discrete structures or mechanical components in the finite element method. It is an idealized model of the actual beam structure, with a certain geometric shape (such as length, cross-sectional area, etc.) and material properties (such as elastic modulus, density, etc.). The schematic diagram of the beam element is as follows Figure 2 shown.

[0040] The constitutive equation of elastic materials usually describes the relationship between stress and strain, without time terms. On this basis, the constitutive equation of viscoelastic materials introduces the time term, and the integral form includes the integral of the strain history. In the finite element method, the weighted residual method is usually used to express the equivalent integral form of the equilibrium equation, and is used to subsequently substitute the constitutive equation and geometric equation to establish the equivalent stiffness matrix and dynamic equation of the viscoelastic beam unit.

[0041] S202, applying dynamic loads to the nodes of the viscoelastic beam unit, and solving the dynamic equations to obtain the dynamic responses of the nodes of the viscoelastic beam unit.

[0042] First, determine the type of dynamic load that may be applied in practice, such as periodic vibration loads during machining and impact loads during object collisions, and set the beam unit node load vector. Then, select a suitable integration algorithm, such as Newmark-β and generalized α method, and set the integration step and simulation time. Finally, set the initial values ​​of the displacement and velocity of the beam unit node, integrate and solve the dynamic equation, and obtain the dynamic response of the beam unit node at each moment during the entire simulation process, including displacement, velocity, and acceleration.

[0043] Based on the above embodiments, in an exemplary embodiment, see Figure 3 , the construction process of the equivalent stiffness matrix and dynamic equation of the viscoelastic beam element includes the following steps:

[0044] S301, performing equivalent integral representation on the equilibrium equation according to the weighted residual method to obtain the stiffness term of the dynamic equation.

[0045] Among them, the equilibrium equation is a differential equation that describes the relationship between stress, body force and inertia force at any point on the cross section of the beam unit. The weighted residual method is used to obtain the equivalent integral form of the differential equation, in which the product of stress and strain is the stiffness term of the dynamic equation. The stiffness term can be expressed in tensor form as:

[0046]

[0047] in, For strain, is stress, Since the directions do not interfere with each other even for a three-dimensional beam element, the integral can be simplified to one dimension.

[0048] S302, determining an equivalent stiffness matrix and an equivalent integral displacement according to the stiffness term and the geometric equation.

[0049] Among them, the geometric equation is a differential equation that describes the relationship between displacement and strain at any point on the cross section of the beam unit, and is usually expressed in matrix form in the finite element method. The stress variation data of the viscoelastic material under constant strain conditions is measured through stress relaxation experiments, so as to fit the relaxation modulus of the material, and integrate the time derivative of the relaxation modulus and the strain history to obtain the viscoelastic integral constitutive equation, which describes the relationship between stress, strain and time. The equation can be written as:

[0050]

[0051] Where Y is the relaxation modulus of the viscoelastic material, is the strain rate, and t is the time of the dynamic process.

[0052] Substitute the matrix expression of the geometric equation and the viscoelastic integral constitutive equation into the stiffness term. Due to the arbitrariness of the node displacement variation of the beam element, the node displacement variation in all terms containing the stiffness term in the dynamic equation is eliminated. Therefore, the original stiffness term can be transformed into the coupled viscoelastic stiffness term, which describes the product relationship between the equivalent stiffness matrix and the node displacement and the equivalent integral displacement, which can be written as:

[0053]

[0054]

[0055] Where B is the strain matrix of the beam element, is the displacement vector of the beam element node. and is the viscoelastic equivalent stiffness matrix of the beam element, are the equivalent integral displacements of the dynamic process, and their calculation formulas are as follows:

[0056]

[0057] The equivalent stiffness matrix is ​​the stiffness matrix obtained after the viscoelasticity of the coupled material is taken into account. Compared with the traditional elastic stiffness matrix, the equivalent stiffness matrix not only reflects the transient elasticity of the material, but also takes into account the material's characteristics similar to viscous fluid flow. The equivalent integral displacement is the integral of a displacement history introduced after considering the viscoelasticity of the material. Therefore, the stiffness term of the transformed coupled viscoelasticity can be used to analyze the coupling effect of the deformation history of the structure under dynamic load on the dynamic equation, and has the ability to describe the material memory characteristics.

[0058] S303, determining the dynamic equation according to the equivalent stiffness matrix, the equivalent integral displacement, the mass matrix of the beam element and the node load.

[0059] The stiffness term of the original beam element dynamic equation has been transformed into the product relationship between the equivalent stiffness matrix and the node displacement and the equivalent integral displacement. The remaining terms are not affected by viscoelasticity, and the damping term is ignored. Therefore, the dynamic equation of the beam element coupled with the viscoelasticity of the material can be written as:

[0060]

[0061] in, and are the mass matrix and nodal load vector of the beam element, and are the displacement and acceleration vectors of the beam element nodes, respectively.

[0062] Based on the above embodiments, in an exemplary embodiment, see Figure 4 , selecting a suitable integration method, solving the dynamic equation, and obtaining the dynamic response of the node of the viscoelastic beam unit includes:

[0063] S401, select an integration method and determine simulation and integration parameters.

[0064] Under dynamic loads, implicit methods such as the Newmark-β method are usually used to integrate and solve the dynamic equations of the beam element. The integral parameter in this method can be written as:

[0065]

[0066] in, and is the integral parameter of the Newmark-β method, and a suitable value can be selected by trial and error. is a constant that depends on the integration parameter, is the time step.

[0067] S402, determining the effective stiffness matrix and the effective load at time t+Δt according to the dynamic equation.

[0068] The effective stiffness matrix is ​​constructed by using the equivalent stiffness matrix and mass matrix multiplied by the transient displacement state in the stiffness term of coupled viscoelasticity, and the effective load at time t+Δt is determined based on the node load and the equivalent integrated displacement at time t+Δt, which can be written as:

[0069]

[0070] in, and are the effective stiffness matrix and effective load, respectively, is the velocity of the beam element node.

[0071] S403, integrating and calculating the node displacement, velocity and acceleration at time t+Δt according to the effective stiffness matrix and the effective load at time t+Δt.

[0072] First, the node displacement at time t+Δt is determined according to the product of the inverse matrix of the effective stiffness matrix and the effective load at time t+Δt. Then, the node velocity and acceleration at time t+Δt are obtained by integration based on the node displacement, velocity, and acceleration at time t, which can be written as:

[0073]

[0074] In a specific embodiment, the research object is a planar cantilever beam whose material is a polymer (phenolic tape), which is regarded as an Euler–Bernoulli beam unit. The geometric and material parameters of the cantilever beam are shown in Table 1.

[0075] Table 1 Geometric and material parameters of cantilever beam

[0076]

[0077] The stiffness term in the equivalent integral of the beam element dynamic equation is expressed as ;

[0078] The relaxation modulus of the material is determined experimentally and can be written as , where the coefficients can be written as , and then substitute into equation (2) to establish the integral constitutive equation, which can be written as In addition, if viscoelasticity is neglected, the elastic modulus of the material is 8.9 Gpa.

[0079] Based on the above viscoelastic integral constitutive equation and the geometric equation of the beam element, the above integral expression (1) can be further written as:

[0080]

[0081] The strain matrix can be written as .

[0082] According to the arbitrariness of variation, the above integral expression can be further written as:

[0083]

[0084] Therefore, the equivalent stiffness matrix is:

[0085]

[0086] The viscoelastic equivalent integrated displacement is:

[0087]

[0088] In addition, the stiffness matrix of the elastic beam is , and the node load vector is set to Therefore, the above items can be used to establish the dynamic equation of the viscoelastic beam unit, and then the Newmark-β method can be used to integrate and solve the dynamic equation.

[0089] The calculation results of the above embodiment show that, compared with the elastic beam element, the beam element coupled with the viscoelastic effect of the material established in the present application can more accurately memorize and analyze the influence of the cumulative effect of the process on the dynamic response under dynamic load conditions, as shown in Figures 5 and 6. Figure 5a , under high-frequency sinusoidal loads, the displacement amplitude of the nodes of the elastic beam unit is larger than that of the viscoelastic beam unit, but the fluctuation frequency and phase are similar to the load variation law. Figure 5b , under low-frequency loads, the displacement of the viscoelastic beam unit node shows viscosity compared with the load and elastic displacement, that is, the viscoelastic displacement lags behind the load change. As the load tends to be stable, the viscoelastic and elastic displacements tend to be equal.

[0090] Since the viscoelastic properties of materials depend on temperature and are usually temperature-dependent under actual working conditions, the effect of different relaxation times on the node displacements of viscoelastic beam elements is further discussed. Figure 6a , under high-frequency sinusoidal loads, appropriately reducing the relaxation time will cause a lag in the displacement response and increase the fluctuation amplitude. Figure 6b ,Under the action of low-frequency load, appropriately reducing the relaxation time increases the overall synchronization of the displacement response and the load change, but causes the high-frequency part to lag behind the load.

[0091] The above rules show that, compared with elastic displacement, viscoelastic displacement response depends not only on transient loads, but also on the historical process of loading and deformation, proving that solid materials have memory. The elastic assumption makes the solid only remember the undeformed initial configuration, resulting in a response that is independent of the historical process and has strong immediacy and synchronization. Therefore, the traditional elastic beam unit cannot analyze the cumulative effect of the loading history on the stress-strain response. Although it is applicable under long-period steady-state loads, it will produce large analysis deviations under dynamic load conditions such as impact. The examples prove that the viscoelastic beam unit established in this application can effectively solve this problem and has stronger universality.

[0092] It should be understood that, although the various steps in the flowcharts involved in the above-mentioned embodiments are displayed in sequence according to the indication of the arrows, these steps are not necessarily executed in sequence according to the order indicated by the arrows. Unless there is a clear explanation in this article, the execution of these steps does not have a strict order restriction, and these steps can be executed in other orders. Moreover, at least a part of the steps in the flowcharts involved in the above-mentioned embodiments can include multiple steps or multiple stages, and these steps or stages are not necessarily executed at the same time, but can be executed at different times, and the execution order of these steps or stages is not necessarily to be carried out in sequence, but can be executed in turn or alternately with other steps or at least a part of the steps or stages in other steps.

[0093] Based on the same inventive concept, the embodiment of the present application also provides a device for implementing the above-mentioned coupled viscoelastic beam unit equivalent stiffness matrix and dynamic equation modeling, including an equation building module and a response calculation module. The implementation solution provided by the device to solve the problem is similar to the implementation solution recorded in the above method, so it will not be repeated here.

[0094] In one embodiment, a computer program product is also provided, and when the computer program is executed by a processor, the above method can be implemented. The computer program product includes one or more computer instructions. When these computer instructions are loaded and executed on a computer, part or all of the above method can be implemented in whole or in part according to the process or function described in the embodiment of the present application.

[0095] According to some embodiments of the present application, a computer device is provided, including a memory and a processor, wherein a computer program is stored in the memory, and the above-mentioned embodiments are implemented when the processor executes the computer program. In addition, a non-temporary computer-readable storage medium including instructions is also provided, such as a memory including instructions, and the above-mentioned instructions can be executed by a processor of an electronic device to complete the above-mentioned method. For example, the non-temporary computer-readable storage medium can be a ROM, a random access memory (RAM), a CD-ROM, a magnetic tape, a floppy disk, an optical data storage device, etc.

[0096] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data used for analysis, stored data, displayed data, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties, and the collection, use and processing of relevant data must comply with relevant laws, regulations and standards of relevant countries and regions.

[0097] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0098] The above-described embodiments only express several implementation methods of the present application, and the descriptions thereof are relatively specific and detailed, but they cannot be understood as limiting the scope of the present application. It should be pointed out that, for a person of ordinary skill in the art, several variations and improvements can be made without departing from the concept of the present application, and these all belong to the protection scope of the present application. Therefore, the protection scope of the present application shall be subject to the attached claims.

Claims

1. A method for modeling an equivalent stiffness matrix and dynamic equations of a beam element coupled with viscoelasticity, characterized in that: The method comprises: Constructing an equivalent stiffness matrix and a dynamic equation of a viscoelastic beam unit; the equivalent stiffness matrix and the dynamic equation are constructed based on a viscoelastic integral constitutive equation and a weighted residual method; the dynamic equation includes a corresponding relationship among node load, displacement, acceleration, and equivalent integral displacement; Dynamic loads are applied to the nodes of the viscoelastic beam unit, and the dynamic equations are solved to obtain the dynamic responses of the nodes of the viscoelastic beam unit.

2. The method according to claim 1, characterized in that The construction process of the equivalent stiffness matrix and dynamic equation of the viscoelastic beam unit includes: The equilibrium equation is expressed by equivalent integral according to the weighted residual method to obtain the stiffness term of the dynamic equation; Determining the equivalent stiffness matrix and the equivalent integral displacement according to the initial stiffness term and the geometric equation; The dynamic equation is determined according to the equivalent stiffness matrix, the equivalent integrated displacement, the mass matrix of the beam element and the node load.

3. The method according to claim 2, characterized in that According to the stiffness term and the geometric equation, the process of determining the equivalent stiffness matrix and the equivalent integral displacement includes: Substituting the matrix expression of the geometric equation and the viscoelastic integral constitutive equation into the stiffness term, the stiffness term is converted into a coupled viscoelastic stiffness term; the coupled viscoelastic stiffness term includes the equivalent stiffness matrix and the equivalent integral displacement.

4. The method according to claim 3, characterized in that The construction process of the viscoelastic integral constitutive equation includes: Obtaining a relaxation modulus of a viscoelastic material; the relaxation modulus is measured based on a stress relaxation experiment performed on the viscoelastic material in advance; The time derivative of the relaxation modulus and the strain history are integrated to determine the viscoelastic integral constitutive equation.

5. The method according to claim 3, characterized in that: The transformation process of the stiffness term of the coupled viscoelasticity includes: According to the viscoelastic integral constitutive equation, the initial stiffness term is converted into the product relationship between the equivalent stiffness matrix and the node displacement and the equivalent integral displacement, and the variation of the node displacement is reduced.

6. The method according to claim 1, characterized in that The process of solving the dynamic equation to obtain the dynamic response of the node of the viscoelastic beam unit includes: The dynamic response of the node includes displacement, velocity and acceleration; According to the dynamic equation, determining the effective stiffness matrix and the effective load at time t+Δt; Determine the node displacement at time t+Δt according to the inverse matrix of the effective stiffness matrix and the effective load at time t+Δt; The node velocity and acceleration at time t+Δt are determined by integration according to the node displacement at time t+Δt and the node displacement, velocity and acceleration at time t.

7. The method according to claim 6, characterized in that Based on the dynamic equations, the process of determining the effective stiffness matrix and the effective load at time t+Δt includes: Determining the effective stiffness matrix according to the equivalent stiffness matrix and the mass matrix; The effective load at time t+Δt is determined according to the node load and the equivalent integrated displacement at time t+Δt.

8. A device for modeling an equivalent stiffness matrix and dynamic equations of a beam element coupled with viscoelasticity, characterized in that: The device comprises: An equation construction module, used to construct an equivalent stiffness matrix and a dynamic equation of a viscoelastic beam unit; the equivalent stiffness matrix and the dynamic equation are constructed based on a viscoelastic integral constitutive equation and a weighted residual method; the dynamic equation includes a correspondence between node loads, displacements, accelerations, and equivalent integral displacements; The response calculation module is used to apply dynamic loads to the nodes of the viscoelastic beam unit and solve the dynamic equations to obtain the dynamic responses of the nodes of the viscoelastic beam unit.

9. A computer program, characterized in that When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 7 are implemented.

10. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that: When the processor executes the computer program, the steps of the method according to any one of claims 1 to 7 are implemented.

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