Modeling Method for Equivalent Stiffness Matrix and Dynamic Equation of Coupled Viscoelastic Beam Elements

Abstract

This application relates to a method for modeling the equivalent stiffness matrix and dynamic equation of a beam element coupled with viscoelasticity. This method describes the time-dependent stress-deformation behavior of materials based on the integral-type constitutive equation of viscoelasticity, and further combines the weighted residual method to establish the equivalent stiffness matrix and dynamic equation of the viscoelastic beam element. This solves the problem that the existing methods are difficult to analyze the cumulative effect of the strain dynamic history under dynamic loads on the viscous influence of the element node response due to the elastic assumption, improves the accuracy of the calculation of the dynamic response of the beam element, provides a more general discrete element for the finite element method, and has far-reaching significance for improving the analysis and design level of high-end mechanical equipment and structures.

Description

Technical Field

[0001] This application relates to the technical field of finite element modeling and analysis, and particularly to a method for modeling the equivalent stiffness matrix and dynamic equation of a beam element coupled with viscoelasticity. Background Art

[0002] As an important discrete element in the finite element method, the beam element has the advantages of simple geometric structure and fewer degrees of freedom. Based on the beam element to discretize structures or mechanical components, assemble the element stiffness matrix, and establish the mass and damping matrices, so as to form the overall finite element dynamic equation, and finally analyze the static and dynamic responses of the structure or component. This technology has been widely used in the flexible analysis of high-end equipment or complex structures in fields such as bridges, buildings, aerospace aircraft, high-speed rails, automotive industry, machine tools, and precision instruments.

[0003] The traditional finite element beam element modeling method is based on the elastic hypothesis, uses the elastic constitutive equation to establish the stiffness matrix and transient displacement, and ignores the viscous part of material viscoelasticity, thus making it difficult to accurately analyze the influence of the cumulative effect of the loading and deformation history process on the node response of the beam element. Summary of the Invention

[0004] Based on this, in view of the above technical problems, it is necessary to provide a method for modeling the equivalent stiffness matrix and dynamic equation of a beam element coupled with viscoelasticity that can solve the above problems.

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

[0006] Construct the equivalent stiffness matrix and dynamic equation of the viscoelastic beam element; 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 relationships between node loads, displacements, accelerations, and equivalent integral displacements;

[0007] Apply dynamic loads to the nodes of the viscoelastic beam element, and solve the dynamic equation to obtain the dynamic response of the nodes of the viscoelastic beam element.

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

[0009] Perform equivalent integral representation on the equilibrium equation according to the weighted residual method to obtain the stiffness term of the dynamic equation;

[0010] Determine the equivalent stiffness matrix and equivalent integral displacement according to the stiffness term and the geometric equation;

[0011] Determine the dynamic equation according to the equivalent stiffness matrix, equivalent integral displacement, mass matrix of the beam element, and node load.

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

[0013] Substitute the matrix expression of the geometric equation and the viscoelastic integral constitutive equation into the stiffness term to transform it into a stiffness term coupled with viscoelasticity; the stiffness term coupled with 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] Obtain the relaxation modulus of the viscoelastic material; the relaxation modulus is measured based on a prior stress relaxation experiment on the viscoelastic material.

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

[0017] 5. The transformation process of the above stiffness term coupled with viscoelasticity includes:

[0018] According to the viscoelastic integral constitutive equation, transform the stiffness term into a product relationship between the equivalent stiffness matrix and the nodal displacement and the equivalent integral displacement, and cancel out the variation of the nodal displacement.

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

[0020] The dynamic response of the nodes 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] According to the inverse matrix of the effective stiffness matrix and the effective load at time t + Δt, determine the nodal displacement at time t + Δt.

[0023] According to the nodal displacement at time t + Δt and the nodal displacement, velocity, and acceleration at time t, determine the nodal velocity and acceleration at time t + Δt by integration.

[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] Determine the effective stiffness matrix according to the equivalent stiffness matrix and the mass matrix.

[0026] Determine the effective load at time t + Δt according to the nodal load and the equivalent integral displacement at time t + Δt.

[0027] When constructing the dynamic equation of the beam element, the viscoelasticity of the material is introduced in this application, which changes the stiffness term in the dynamic equation, enabling the equation to accurately reflect the viscous effect of the material under dynamic loads. This solves the problem that the existing modeling methods are difficult to accurately analyze the viscous effect of the cumulative effect on the node response of the element under dynamic loads due to the elastic assumption, thereby improving the accuracy of the dynamic response calculation. Especially under dynamic load conditions such as impact, it can more accurately predict the responses such as displacement, velocity, and acceleration of the beam element nodes. Description of the Drawings

[0028] Figure 1 It is a schematic diagram of the modeling and solution process of the viscoelastic beam element in an embodiment;

[0029] Figure 2 It is a schematic diagram of the beam element in an embodiment;

[0030] Figure 3 It is a schematic diagram of the process of constructing the equivalent stiffness matrix and dynamic equation of the viscoelastic beam element in an embodiment;

[0031] Figure 4 It is a schematic diagram of the solution process of the dynamic equation of the beam element node in an embodiment;

[0032] Figure 5a It is a diagram of the sine load and the displacement response of the viscoelastic beam node in an embodiment;

[0033] Figure 5b It is a diagram of the complex load and the displacement response of the viscoelastic beam node in an embodiment;

[0034] Figure 6a It is a diagram of the influence of viscoelastic parameters on the displacement response of the beam node under sine load in an embodiment;

[0035] Figure 6b It is a diagram of the influence of viscoelastic parameters on the displacement response of the beam node under complex load in an embodiment. Detailed Description of the Embodiment

[0036] In order to make the objectives, technical solutions, and advantages of this application clearer, the following further elaborates on this application in combination with the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not used to limit this application.

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

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

[0039] Among them, the beam element is a basic element used to discretize structures or mechanical components in the finite element method. It is an idealized model of an actual beam structure, with certain geometric shapes (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 shown in Figure 2 shown.

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

[0041] S202, Apply dynamic loads to the nodes of the viscoelastic beam element and solve the dynamic equation to obtain the dynamic response of the nodes of the viscoelastic beam element.

[0042] Among them, first determine the types of dynamic loads that may be encountered in practice, such as periodic vibration loads during the machining process and impact loads during object collisions, etc., and set the beam element node load vector. Then, select a suitable integration algorithm, such as the Newmark-β method and the generalized α method, and set the integration step size and simulation time. Finally, set the initial values of the displacements and velocities of the beam element nodes, integrate and solve the dynamic equation to obtain the dynamic response of the beam element nodes at each moment during the entire simulation process, including displacements, velocities, and accelerations.

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

[0044] S301, Perform equivalent integral representation of 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 the stress, body force, and inertial force at any point on the cross-section of the beam element. The weighted residual method is used to obtain the equivalent integral form of this differential equation, and the product term of stress and strain is the stiffness term of the dynamic equation. This stiffness term can be denoted in tensor form as:

[0046]

[0047] Among them, is the strain, is the stress, Since even for three-dimensional beam elements, there is no interference among different directions, this integral can be simplified to one-dimensional.

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

[0049] Among them, the geometric equation is a differential equation describing the relationship between the displacement and strain of any point on the cross-section of the beam element, and is usually expressed in matrix form in the finite element method. By conducting stress relaxation experiments, the data of the stress change with time under constant strain conditions of the viscoelastic material are measured, and then the relaxation modulus of this material is obtained by fitting. Integrating the time derivative of the relaxation modulus and the strain history, the integral-type constitutive equation of viscoelasticity is obtained, which describes the relationship among stress, strain, and time. This equation can be written as:

[0050]

[0051] Among them, 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 integral-type constitutive equation of viscoelasticity into the stiffness term. And due to the arbitrariness of the variation of the beam element nodal displacement, the variation of the nodal displacement in all terms containing the stiffness term in the dynamic equation is cancelled out. Therefore, the original stiffness term can be transformed into a stiffness term coupled with viscoelasticity, which describes the product relationship between the equivalent stiffness matrix and the nodal displacement and the equivalent integral displacement, and can be written as:

[0053]

[0054]

[0055] Among them, B is the strain matrix of the beam element, is the displacement vector of the beam element nodes. and are the viscoelastic equivalent stiffness matrices of the beam element, is the equivalent integral displacement of the dynamic process. Their calculation formulas are as follows:

[0056]

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

[0058] S303. Determine the dynamic equation according to the equivalent stiffness matrix, equivalent integral displacement, mass matrix of the beam element, and nodal 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 nodal displacement and 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 considering the influence of material viscoelasticity can be written as:

[0060]

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

[0062] Based on the above embodiments, in an exemplary embodiment, please refer to Figure 4 , the process of selecting a suitable integration method to solve the dynamic equation and obtaining the dynamic response of the nodes of the viscoelastic beam element includes:

[0063] S401. Select an integration method and determine the simulation and integration parameters.

[0064] Under the action of dynamic load, the Newmark-β method, an implicit method, is usually used to solve the dynamic equation of the beam element. The integration parameters in this method can be written as:

[0065]

[0066] Wherein, and are the integration parameters of the Newmark-β method, and appropriate values can be selected by the trial method. is a constant dependent on the integration parameters. is the time step.

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

[0068] Construct the effective stiffness matrix using the equivalent stiffness matrix and mass matrix in the stiffness term coupling viscoelasticity multiplied by the transient displacement state, and determine the effective load at time t+Δt according to the nodal load and the equivalent integral displacement at time t+Δt, which can be written as:

[0069]

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

[0071] S403, according to the effective stiffness matrix and the effective load at time t+Δt, integrally calculate the node displacements, velocities and accelerations at time t+Δt.

[0072] First, determine the node displacements at time t+Δt according to the product of the inverse matrix of the effective stiffness matrix and the effective load at time t+Δt, and then, combined with the node displacements, velocities and accelerations at time t, obtain the node velocities and accelerations at time t+Δt through integration, which can be written as:

[0073]

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

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

[0076]

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

[0078] Determine the relaxation modulus of the material according to experiments, which can be written as , and the coefficient therein can be written as , and then substitute it into Equation (2) to establish an integral constitutive equation, which can be written as . In addition, if viscoelasticity is ignored, the elastic modulus of this 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] Among them, 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 integral displacement is as follows:

[0087]

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

[0089] The calculation results of the above embodiments show that, compared with the elastic beam element, the beam element established in this application that couples the viscoelastic effect of the material 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 FIGS. 5 and 6. Refer to Figure 5a , under the action of high-frequency sinusoidal load, the displacement amplitude of the nodes of the elastic beam element is greater than that of the viscoelastic beam element, but the fluctuation frequency and phase are similar to the load change law. Refer to Figure 5b , under the action of low-frequency load, the displacement of the nodes of the viscoelastic beam element 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 characteristic parameters of the material vary depending on temperature and usually have variable temperature under actual working conditions, the influence of different relaxation times on the nodal displacement of the viscoelastic beam element is further discussed. Refer to Figure 6a , under the action of high-frequency sinusoidal load, appropriately reducing the relaxation time causes the displacement response to lag and increases the fluctuation amplitude. Refer to 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 the elastic displacement, the viscoelastic displacement response not only depends on the transient load, but is also affected by the historical process of loading and deformation, proving that solid materials have memory. The elastic assumption makes the solid only memorize the undeformed initial configuration, resulting in the response not depending on the historical process, with strong immediacy and synchronization. Therefore, the traditional elastic beam element cannot analyze the cumulative influence of the loading history on the stress-strain response, although it is applicable under long-term steady-state loads, but there will be a large analysis deviation under dynamic load conditions such as impact. The embodiments prove that the viscoelastic beam element established in this application can effectively solve this problem and has stronger universality.

[0092] It should be understood that although the steps in the flowcharts involved in the above-described embodiments are shown sequentially according to the indications of the arrows, these steps are not necessarily executed sequentially in the order indicated by the arrows. Unless explicitly stated herein, there is no strict order restriction for the execution of these steps, and these steps can be executed in other orders. Moreover, at least some of the steps in the flowcharts involved in the above-described embodiments may include multiple steps or multiple stages. These steps or stages are not necessarily executed at the same time, but can be executed at different times. The execution order of these steps or stages is not necessarily sequential either, but can be executed alternately or in turn with at least some of the steps or stages in other steps or other steps.

[0093] Based on the same inventive concept, an embodiment of the present application further provides a device for implementing the equivalent stiffness matrix and dynamic equation modeling of a beam unit involving coupling viscoelasticity as described above, including an equation construction module and a response calculation module. The implementation solution provided by this device to solve the problem is similar to the implementation solution described in the above method, so it will not be elaborated here.

[0094] In one embodiment, a computer program product is also provided. 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 according to the process or function described in the embodiments of the present application.

[0095] According to some embodiments of the present application, a computer device is provided, including a memory and a processor. A computer program is stored in the memory, and when the processor executes the computer program, the above embodiments are implemented. In addition, a non-transitory computer-readable storage medium including instructions is also provided, such as a memory including instructions. The above instructions can be executed by the processor of an electronic device to complete the above method. For example, the non-transitory computer-readable storage medium may be a ROM, a random access memory (RAM), a CD-ROM, a magnetic tape, a floppy disk, and 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 for analysis, stored data, displayed data, etc.) involved in the present 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 need to comply with relevant laws, regulations, and standards of relevant countries and regions.

[0097] The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity of description, 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, it should be considered as the scope recorded in this specification.

[0098] The above-described embodiments merely represent several implementation manners of the present application. The description is relatively specific and detailed, but it should not be construed as a limitation on the patent scope of the present application. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several modifications and improvements can still be made, 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 appended 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: 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; Integrating the time derivative of the relaxation modulus and the strain history to determine a viscoelastic integral constitutive equation; The equilibrium equation is expressed as an equivalent integral using the weighted residual method to obtain the stiffness term of the dynamic equation. Substituting the matrix expression of the geometric equation and the viscoelastic integral constitutive equation into the stiffness term, converting it into a coupled viscoelastic stiffness term, and determining the equivalent stiffness matrix and equivalent integral displacement of the viscoelastic beam unit; Constructing a dynamic equation according to the equivalent stiffness matrix, the equivalent integral displacement, the mass matrix of the beam element and the node load; 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 stiffness term of the coupled viscoelasticity includes the equivalent stiffness matrix and the equivalent integrated displacement.

3. The method according to claim 1, characterized in that The transformation process of the stiffness term of the coupled viscoelasticity includes: According to the viscoelastic integral constitutive equation, the 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.

4. 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.

5. The method according to claim 4, 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.

6. The method according to claim 1, characterized in that 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.

7. The method according to claim 1, characterized in that The stiffness term of the dynamic equation is the product term of stress and strain.

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 building module is used to obtain 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; Integrating the time derivative of the relaxation modulus and the strain history to determine a viscoelastic integral constitutive equation; The equilibrium equation is expressed as an equivalent integral using the weighted residual method to obtain the stiffness term of the dynamic equation. Substituting the matrix expression of the geometric equation and the viscoelastic integral constitutive equation into the stiffness term, converting it into a coupled viscoelastic stiffness term, and determining the equivalent stiffness matrix and equivalent integral displacement of the viscoelastic beam unit; Constructing a dynamic equation according to the equivalent stiffness matrix, the equivalent integral displacement, the mass matrix of the beam element and the node load; 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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