Simulation method for dynamics behavior of rigid-flexible-thermo coupled thin plate system

Abstract

The application discloses a simulation method for dynamics behavior of a rigid-flexible-thermal coupling thin plate system, divides the thin plate system into grids according to set grid parameters, disperses the thin plate system into thin plate units according to the grids, assembles rigid-flexible coupling dynamics equations of each thin plate unit into rigid-flexible coupling dynamics equations of the thin plate system, assembles heat conduction equations of each thin plate unit into heat conduction equations of the thin plate system, combines the dynamics equations and the heat conduction equations into an equation group, obtains the dynamics equations of the rigid-flexible-thermal coupling thin plate system, uses a generalized- α difference method to perform difference on the dynamics equations of the rigid-flexible-thermal coupling thin plate system, uses a Newton-Raphson method to iteratively solve the difference format, obtains position coordinate arrays and temperature coordinate arrays of nodes of the thin plate system, performs coordinate conversion on the position coordinate arrays of the nodes of the thin plate system, and draws a transverse deformation-time curve of end points of the thin plate system and a deformation diagram of the thin plate system. The application realizes bidirectional coupling, is more in line with actual engineering backgrounds, and improves calculation precision.

Description

Technical Field

[0001] This invention belongs to the field of multibody system dynamics modeling, specifically a simulation method for the dynamic behavior of rigid-flexible-thermal coupled thin-plate systems. Background Technology

[0002] To adapt to the complex space environment, spacecraft are increasingly characterized by their large size, lightweight design, and precision. Many spacecraft components, such as large-aperture antennas or solar panels, can be considered as thin-plate structures. During their orbital flight, most spacecraft periodically enter and exit areas of sunlight, partial shadow, and shadow. Because the solar radiation heat flux experienced by the spacecraft varies significantly in these different regions, unstable vibrations, known as thermally induced vibrations, occur within the system. Clearly, these thermally induced vibrations degrade the spacecraft's reliability, attitude control, and other operational performance characteristics, and may even affect its lifespan. Therefore, establishing a rigid-flexible coupled dynamics model considering thermal field coupling is of great significance. Summary of the Invention

[0003] The purpose of this invention is to propose a simulation method for the dynamic response of a rigid-flexible-thermal coupled thin plate system.

[0004] The technical solution to achieve the purpose of this invention is: a simulation method for the dynamic behavior of a rigid-flexible-thermal coupled thin plate system, comprising the following steps:

[0005] Step 1: Set the geometric parameters, material parameters, mesh parameters, and temperature parameters of the thin plate system;

[0006] Step 2: Divide the thin plate system into a grid according to the set grid parameters, obtain the rigid-flexible coupling dynamic equation of each thin plate element through the kinetic energy theorem and the principle of virtual work, and assemble the rigid-flexible coupling dynamic equation of each thin plate element into the rigid-flexible coupling dynamic equation of the thin plate system; obtain the heat conduction equation of each thin plate element through the law of conservation of energy and the theory of heat conduction, and assemble the heat conduction equation of each thin plate element into the heat conduction equation of the thin plate system.

[0007] Step 3: Consider the relationship between the rigid-flexible coupling dynamic equation and the heat conduction equation of the thin plate system, and combine the dynamic equation and the heat conduction equation into a set of equations to obtain the dynamic equation of the rigid-flexible-thermal coupling thin plate system.

[0008] Step 4: Use the generalized α method to perform finite difference on the dynamic equations of the rigid-flexible-thermal coupled thin plate system, and use the Newton-Raphson method to iteratively solve the finite difference scheme to obtain the position coordinate array and temperature coordinate array of the nodes of the thin plate system.

[0009] Step 5: Perform coordinate transformation on the coordinate array of the node positions of the thin plate system, and draw a graph based on the transformation results to obtain the lateral deformation-time curve of the end point of the thin plate system and the deformation diagram of the thin plate system.

[0010] A simulation system for the dynamic response of a rigid-flexible-thermal coupled thin plate system is provided. Based on the aforementioned simulation method for the dynamic response of a rigid-flexible-thermal coupled thin plate system, the system realizes the simulation of the dynamic behavior of the thin plate system under the action of a thermal field.

[0011] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it simulates the dynamic behavior of the thin plate system under the action of a thermal field based on the dynamic response simulation method of the rigid-flexible-thermal coupled thin plate system.

[0012] A computer-readable storage medium storing a computer program, which, when executed by a processor, simulates the dynamic behavior of a thin-plate system under the action of a thermal field based on the aforementioned dynamic response simulation method for a rigid-flexible-thermal coupled thin-plate system.

[0013] Compared with the prior art, the significant advantages of this invention are as follows: (1) This invention can be developed into a visualization application based on MATLAB APP DESIGNER. By setting relevant parameters and running it, numerical simulation of the thin plate system in the thermal field can be realized, and the displacement field and temperature field of each node of the thin plate system can be obtained in real time. Compared with traditional programs, this invention is simple and convenient. (2) This invention realizes a unified description of the displacement field and temperature field. The neutral layer temperature, temperature gradient, and generalized position coordinates of each node of the thin plate system can all be solved using reduced thin plate elements under the same mesh, which simplifies the iterative steps in the dynamic solution process and improves the calculation accuracy. (3) This invention considers the interaction between the temperature field and the displacement field, realizes bidirectional coupling, and is more in line with the actual engineering background. Attached Figure Description

[0014] Figure 1 A flowchart illustrating the simulation method for the dynamic behavior of a rigid-flexible-thermal coupled thin-plate system.

[0015] Figure 2 This is a flexible thin-plate rectangular element that integrates temperature based on the absolute node coordinate method.

[0016] Figure 3 This is a schematic diagram of a flexible thin plate undergoing large-scale motion under the influence of heat flux density.

[0017] Figure 4 The iterative flowchart for solving the generalized -α method.

[0018] Figure 5 Initialization diagram of the APP's visual interface.

[0019] Figure 6 The image shows the results of running the example.

[0020] Figure 7 The time history diagram is for an example. Detailed Implementation

[0021] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments.

[0022] like Figure 1 As shown, the simulation method for the dynamic behavior of a rigid-flexible-thermal coupled thin plate system includes the following steps:

[0023] Step 1: Set the geometric parameters, material parameters, mesh parameters, and temperature parameters of the thin plate system.

[0024] In practical engineering, physical models such as solar sails and solar panels can be simplified into thin-plate systems consisting of a rectangular single-layer sheet. Since the operating environment of these physical models needs to consider temperature variations, parameters affecting temperature are added to the thin-plate system; these are called temperature parameters.

[0025] (1) Geometric parameters

[0026] The geometric parameters of the thin plate system are: length L, width W, and thickness h.

[0027] (2) Material parameters

[0028] Material parameters of the thin plate system: density ρ, elastic modulus E.

[0029] (3) Mesh parameters

[0030] Mesh parameters for the thin plate system: number of meshes n = n1 × n2, mesh length l e The grid width is w e .

[0031] (4) Temperature parameters

[0032] Temperature parameters of the thin plate system: specific heat capacity c, thermal conductivity k, coefficient of thermal expansion α, solar radiation heat flux s, surface absorptivity α up ,α low The reference temperature is T0.

[0033] Step 2: Divide the thin plate system into a grid according to the set grid parameters. Based on the grid, use the absolute nodal coordinate method to discretize the thin plate system into many thin plate elements. Obtain the rigid-flexible coupling dynamic equation of each thin plate element through the kinetic energy theorem and the principle of virtual work. Assemble the rigid-flexible coupling dynamic equation of each thin plate element into the rigid-flexible coupling dynamic equation of the thin plate system. Based on the thin plate elements discretized according to the grid, obtain the heat conduction equation of each thin plate element through the law of conservation of energy and the theory of heat conduction. Assemble the heat conduction equation of each thin plate element into the heat conduction equation of the thin plate system.

[0034] (1) Dynamic equations of rigid-flexible coupling based on thin plate system

[0035] Establishing the inertial coordinate system of the thin plate system as follows Figure 2 As shown, the length direction of the thin plate system is taken as the x-axis, the width direction as the y-axis, and the thickness direction as the z-axis, with the origin O being the intersection of the length, width, and thickness centerlines. The thin plate system is divided into n = n1 × n2 rectangular grids according to the grid parameters. The thin plate system is discretized using the absolute node coordinate method, with each rectangular grid considered as a thin plate element, resulting in n = n1 × n2 thin plate elements. n1 and n2 are the number of grids divided along the x and y directions, respectively. The thin plate system has a length of L and a width of W, and each thin plate element has a length of l. e Width is w e , where l e =L / n1, w e =W / n2.

[0036] For any point P on the thin plate system, in the inertial frame o-xyz, its position vector can be expressed as:

[0037] r = [r1 r2 r3] T =S·q (1)

[0038] Where r is the position vector of point P; r1, r2, r3 are the position components of point P in the o-xyz directions of the inertial coordinate system; S is the shape function matrix of the absolute nodal coordinate method, and q is the nodal position coordinate array. The expressions for S and q are as follows:

[0039] S = [S1I] 3×3 S2I 3×3 S3I 3×3 …S 12 I 3×3 ] 3×12 (2)

[0040]

[0041] Where S1 S2…S 12To use the absolute nodal coordinate method to discretize the shape function components in the shape function matrix of a thin plate system; I 3×3 It is a 3rd order identity matrix; For example Figure 3 The coordinate array of the node positions of the four boundary nodes A, B, C, and D on the thin plate element where point P is located.

[0042] Shape function components S1 S2…S 12 The specific expression is as follows:

[0043]

[0044] In equation (4), ξ = x e / l e ,η=y e / w e (0≤ξ,η≤1), with the length direction of the thin plate element as x e Direction, width y of the thin plate unit e Direction, with node A at the lower left corner of the thin plate element as the zero point. e x e ,y e These are the relative position coordinates of P in the coordinate system of the thin plate element when the thin plate element is undeformed.

[0045] According to the kinetic energy theorem, the kinetic energy of the thin plate element can be expressed as T. e :

[0046]

[0047] Where ρ is the density of the thin plate system; The derivative of the position vector of point P with respect to time; For the node velocity array; M e Let be the mass matrix of the thin plate element. The expression is as follows:

[0048]

[0049] Where h is the thickness of the thin plate system.

[0050] Based on the thin-plate assumption, the normal vectors of each element in the thin-plate system are perpendicular to the neutral layer. In this model, the neutral layer is the plane formed by the midpoints of the thickness direction of the thin-plate system. Therefore, the total strain at any point P on the thin-plate element is:

[0051] ε=ε0-zκ (7) where ε is the total strain at point P; according to the knowledge of mechanics of materials, the transition layer with zero stress in the thin plate system is taken as the neutral layer, ε0 is the strain vector of the neutral layer; z is the distance from point P to the neutral layer; κ is the curvature vector.

[0052] Based on Hooke's law, and according to the total strain ε at point P obtained from equation (7), the constitutive equation at point P can be expressed as:

[0053] σ=Eε (8) where σ is the total stress at point P; E is the elastic coefficient matrix of the thin plate system.

[0054] According to the principle of virtual work, the elastic force matrix of a thin plate element can be expressed as:

[0055]

[0056] Where F εe F is the elastic force matrix related to the strain terms of the thin plate element and the neutral layer at point P in the thin plate system; κe This is the elastic force matrix related to the curvature term of the thin plate element where point P is located in the thin plate system.

[0057] Mass matrix M of thin plate system unit e The elastic force matrix F related to the strain term of the neutral layer εe The elastic force matrix F related to the curvature term κe By assembling them according to the traditional finite element method based on the mesh, the overall mass matrix M of the thin plate system and the elastic force matrix F related to the strain term of the neutral layer can be obtained. ε The elastic force matrix F related to the curvature term κ The final dynamic equations based on the rigid-flexible coupling of the thin-plate system are:

[0058]

[0059] (2) Heat conduction equation based on thin plate system

[0060] According to the law of conservation of energy and the theory of heat conduction, the fundamental equation satisfied by the temperature field variable T can be expressed as:

[0061]

[0062] Where q x ,q y ,q z Let be the heat flux density in the x, y, z directions inside the thin plate system; Q be the heat source density inside the thin plate system; ρ be the density of the thin plate system; and c be the specific heat capacity of the thin plate system.

[0063] According to Fourier's law of heat transfer, the direction of heat flux density inside a thin-plate system is opposite to the temperature gradient, that is:

[0064]

[0065] Where k is the thermal conductivity of the thin plate system.

[0066] Substituting equation (12) into equation (11), we can obtain the heat conduction equation satisfied by the transient temperature field T of the thin plate system:

[0067]

[0068] Where v is the volume of the thin-plate system.

[0069] Adding solar radiation heat flow to the heat conduction equation (13), considering the distinction between the upper and lower plates, and taking the minimum value of the functional, we can obtain the variational equation of heat conduction in Cartesian coordinates for the temperature field variable:

[0070]

[0071] Where A S q represents the area of ​​the upper and lower surfaces of the thin plate system. up q represents the heat flux density on the upper surface of the thin-plate system. low The heat flux density on the lower surface of the thin-plate system; its form is as follows:

[0072]

[0073] Where α up α is the heat absorption coefficient of the upper surface; low Γ is the heat absorption coefficient of the lower surface; s is the solar radiation heat flux vector; |s| is the numerical value of the solar radiation heat flux; up Γ is the solar radiation angle coefficient of the upper surface; low is the solar radiation angle coefficient of the lower surface.

[0074] The temperature T at any point P in a thin plate system can be expressed as:

[0075] T = T0 + T N +zT M (16) Where T0 is the reference temperature of point P; T N The temperature of point P in the neutral layer is called the neutral layer temperature; T M Let be the temperature gradient at point P in the z-direction.

[0076] Based on the mesh discretized thin plate elements in the rigid-flexible coupling dynamic equation of the thin plate system in step 1 (1), the absolute nodal coordinate method is used to discretize the temperature of the thin plate system. Taking point P as an example, the non-zero elements in the first row of the shape function matrix S are taken to obtain T. N ,T M :

[0077]

[0078] Where I 1×1 It is a first-order identity matrix; q Nq is the array of temperature coordinates for the neutral layer at the nodes; M It is a coordinate array of nodal temperature gradients.

[0079] Substituting equations (16) and (17) into equation (14), we can obtain the heat conduction equations for the neutral layer temperature and temperature gradient of the thin plate element, respectively:

[0080]

[0081] in:

[0082]

[0083]

[0084]

[0085]

[0086]

[0087] Where C Ne For the thermal capacity array related to the temperature of the thin plate unit and the neutral layer, C Me For thin-plate elements and temperature gradient-related heat capacity arrays; K Ne For the thermally conductive array of thin-plate unit and neutral layer temperature, K Me For thin-plate elements and temperature gradient-dependent thermal arrays; Q Ne For the heat source array related to the temperature of the thin plate unit and the neutral layer, Q Me This is a heat source array related to thin plate elements and temperature gradients.

[0088] Similarly, the thermal capacity array C related to the neutral layer temperature in the thin plate unit is... Ne Temperature gradient-related heat capacity matrix C Me The temperature-dependent thermal conductivity array K of the neutral layer Ne Temperature gradient related thermal conductivity array K Me Neutral layer temperature-related heat source array Q Ne Temperature gradient related heat source array Q Me By assembling the mesh according to the traditional finite element method, the thermal capacity matrix C related to the neutral layer temperature in the thin plate system can be obtained. N Temperature gradient-related heat capacity matrix C M The temperature-dependent thermal conductivity array K of the neutral layer N Temperature gradient related thermal conductivity array K M Neutral layer temperature-related heat source array Q N Temperature gradient related heat source array Q MThe final heat conduction equation based on the thin-plate system is:

[0089]

[0090] Where q T =[q T N ,q T M ] T For nodal temperature coordinate arrays; For the heat capacity array of the thin plate system; For heat conduction arrays in thin-plate systems; It is a heat source array for a thin plate system.

[0091] Step 3: Consider the relationship between the dynamic equations and the heat conduction equations of the thin plate system, combine the dynamic equations and the heat conduction equations into a set of equations to obtain the dynamic equations of the rigid-flexible-thermal coupled thin plate system.

[0092] (1) Coupling terms of the nodal position coordinate array in the heat conduction equation of the thin plate system

[0093] The temperature variation of the thin-plate system is mainly affected by the solar radiation heat flux *s*. For example... Figure 2 As shown, when a thin plate system undergoes large deformation, the normal vector of the thin plate element at any point P on the system will change, which in turn affects the solar radiation angle coefficient.

[0094] Let the solar radiation heat flux be s, which represents the direction of solar radiation; let the normal vector of the upper surface at point P in the thin plate system be n, and the normal vector of the lower surface be -n.

[0095]

[0096] Where r is x Let r be the slope of the position vector of point P in the x-direction; y Let be the slope of the position vector at point P in the y-direction; n be the normal vector at point P; and ||n|| be the numerical value of the normal vector at point P. Solar radiation angle coefficient of the thin plate system surface. When Γ < 0, the upper surface of the thin plate receives solar heat radiation, denoted as Γ. up Conversely, the lower surface of the thin plate receives solar heat radiation, denoted as Γ. low After obtaining the solar radiation angle coefficient, the solar radiation of the system is:

[0097]

[0098] Where α up α is the heat absorption coefficient of the upper surface; low Γ represents the heat absorption coefficient of the lower surface; |s| represents the numerical value of solar radiation heat flux; upΓ is the solar radiation angle coefficient of the upper surface; low is the solar radiation angle coefficient of the lower surface.

[0099] Equation (27) reveals that the solar radiation angle coefficient of the heat source array in the heat conduction equation of the thin plate system is closely related to the slope of the position vector of point P in the x and y directions in the rigid-flexible coupling dynamic equation. Surface solar radiation angle coefficient This is a coupling term in the nodal position coordinate array of the heat conduction equation for a thin-plate system.

[0100] (2) Coupling terms of nodal temperature coordinate arrays in the rigid-flexible coupling dynamic equations of thin plate systems

[0101] As the temperature rises, thermal stress is generated within the thin plate system, leading to deformation and thus a change in the elastic properties of the system. When the temperature of the thin plate system changes, the constitutive equation for any point P on the system is:

[0102] σ=σ ε +σ T =Eε+Eε T (28) Where E is the elastic coefficient matrix of the thin plate system; σ is the total stress at point P considering the temperature effect. ε Let σ be the elastic stress at point P. T Let ε be the temperature stress at point P; and ε be the elastic strain at point P. T The thermal strain at point P caused by the temperature change.

[0103] Since temperature changes only affect the axial direction of the structure, and not the tangential direction, the thermal strain ε caused by temperature changes is therefore... T as follows:

[0104]

[0105] Where α is the coefficient of thermal expansion, and ΔT is the temperature change. ΔT takes the following form:

[0106] ΔT=T N +zT M (30) Where T N The neutral layer temperature; T M denoted as the temperature gradient along the z-direction; z represents the position coordinates of point P along the thickness direction.

[0107] Substituting equations (30) and (29) into equation (28) yields:

[0108] σ=Eε-EαΔT(31)

[0109] According to the principle of virtual work, the elastic force under temperature load can be expressed as:

[0110]

[0111] in This represents the elastic force under temperature load corresponding to the neutral layer temperature. This represents the elastic force under temperature load corresponding to the temperature gradient.

[0112] According to equation (32), it can be seen that the elastic force of temperature load is closely related to the temperature coordinate array. This is the coupling term of the nodal temperature coordinate array in the rigid-flexible coupling dynamic equation of a thin-plate system.

[0113] (3) Based on the rigid-flexible-thermal coupling dynamic equation of thin plate system

[0114] Substituting the solar radiation angle factor and the temperature load elastic force into the dynamic equations and heat transfer equations of the thin plate system, respectively, we finally obtain the rigid-flexible-thermal coupling dynamic equations based on the thin plate system:

[0115]

[0116] Where Φ(q,t) is the constraint equation of the dynamics, Φ(q T ,t) are the constraint equations of the heat conduction equation; Let be the partial derivative of the constraint equations of the dynamic equations with respect to the position coordinate column matrix q. The constraint equations for the heat conduction equations are applied to the temperature coordinate array q. T The partial derivative of .

[0117] Step 4: Use the generalized α method to perform finite difference on the dynamic equations of the rigid-flexible-thermal coupled thin plate system, and use the Newton-Raphson method to iteratively solve the finite difference scheme to obtain the position and temperature coordinate arrays of the nodes of the thin plate system.

[0118] Combining the nodal position coordinate array and the nodal temperature coordinate array, the thin-plate element with coupled temperature is represented as: r h =[r1 r2 r3 T] N T M The nodal coordinate array q of the coupled temperature all =[q,q N ,q M ].

[0119]

[0120] Step 3 yields the rigid-flexible-thermal coupled dynamic equations of the thin-plate system. A difference scheme is constructed using the generalized α method, and the Newton-Raphson method is employed for iterative processing. The specific steps are as follows:

[0121] (1) A difference scheme for constructing the dynamic equations of a rigid-flexible-thermal coupled thin plate system based on the generalized α method.

[0122] Nodal position coordinate array q0 and nodal temperature coordinate array under initial conditions of the thin plate system The nodal coordinate array q that forms the coupling temperature under initial conditions all0 =[q0,q T0 Introduce the parameters generalized acceleration a and Lagrange multiplier λ0.

[0123]

[0124] After time t = n × hh has elapsed, the nodal coordinate array obtained from the dynamic equations and heat conduction equations of the previous moment is used to estimate the q of the next time step. n+1 , an+1:

[0125]

[0126] Where hh is the time step, and α m ,α f β and γ are parameters related to the spectral radius ρ. They take the following form:

[0127]

[0128] The spectral radius ρ determines the distribution range of the energy dissipation frequency of the algorithm, where ρ∈[0,1]. Equation (34) is a difference scheme constructed based on the generalized-α method for the rigid-flexible-thermal coupling dynamic equations of the thin plate system.

[0129] (2) Solving the position coordinate matrix and temperature coordinate matrix of the nodes of the thin plate system based on the Newton-Raphson method.

[0130] The q constructed in step 4(1) based on the generalized-α method difference scheme n+1 , Substituting into equation (33), we can obtain:

[0131]

[0132] Among them G n+1 G1 is the difference matrix of the rigid-flexible-thermal coupled kinetic equations of the thin-plate system after the (n+1)th time step, without considering thermodynamic constraints. n+1 G2 is the difference matrix of the rigid-flexible coupling dynamic equations of the thin-plate system after the (n+1)th time step. n+1 This is the difference matrix of the heat conduction equation of the thin plate system after the (n+1)th time step.

[0133] Using the iterative method proposed by Arnold, the solution is obtained iteratively:

[0134]

[0135] in, This is the correction result of the kth iteration after the (n+1)th time step; This is the difference matrix of the rigid-flexible-thermal coupling dynamic equations of the thin plate system in the kth iteration after the (n+1)th time step; Indicates the kth time During iteration, for q all Jacobian matrix of partial derivatives.

[0136] Introducing correction factors The corrected iteration result after k+1 iterations is as follows:

[0137]

[0138] in The node coordinate array is the coupling temperature of the (k+1)th iteration after the (n+1)th time step; It is the Lagrange multiplier for the (k+1)th iteration after the (n+1)th time step.

[0139] If after k+1 iterations, the following conditions are met: This indicates that the accuracy of the iteration result meets the requirements, where tol is the allowable error. The obtained... Use this as the initial condition for the next time step, and update parameter a. n+1 Start the loop for the next time step, until the time expires.

[0140]

[0141] The specific iterative flowchart is as follows: Figure 4 By iteratively solving equation (39), after satisfying the allowable error, the final nodal temperature coordinate array q in the thin plate system is obtained. T And the column matrix q of node position coordinates. q and q T The position coordinate array and temperature coordinate array of the nodes of the thin plate system are obtained by solving the rigid-flexible-thermal coupling dynamic equation of the thin plate system based on the Newton-Raphson method.

[0142] Step 5: Perform coordinate transformation on the position coordinate array of the thin plate system, and draw a graph based on the transformation results to obtain the lateral deformation-time curve of the end point of the thin plate system and the deformation diagram of the thin plate system.

[0143] Step 4 yielded the nodal temperature coordinate array q of the thin-plate system at different times. TThe nodal position coordinate array q in the inertial coordinate system results in a large amount of simulation data, which is not intuitive and cannot directly reflect the changes in the dynamic behavior of the thin plate system under the influence of temperature.

[0144] To intuitively obtain the deformation of the system, it is necessary to perform coordinate transformation on the obtained node position coordinate matrix:

[0145]

[0146] Where u, v, w represent the deformations of the thin plate system in the x, y, z directions of the inertial coordinate system; the initial coordinates of point P are [x0, y0, 0]. T The generalized coordinates of point P are [r1 r2 r3] T θ is the angle through which the thin plate system rotates; cosθ and sinθ are the values ​​of r in the nodal position coordinates q0. x and r y .

[0147] Plotting the graph with time t and lateral deformation w as coordinate regions, the lateral deformation of the post-processed thin plate system is obtained as follows: Figure 6 .

[0148] Using the generalized coordinates of point P [r1 r2 r3] T Using the coordinate region as an example, plotting is performed at regular intervals to obtain the deformation history diagram of the thin plate system after post-processing, as shown below. Figure 7 .

[0149] This invention also proposes a simulation system for the dynamic response of a rigid-flexible-thermal coupled thin plate system. Based on the aforementioned simulation method for the dynamic response of a rigid-flexible-thermal coupled thin plate system, the dynamic behavior of the thin plate system under the action of a thermal field is simulated.

[0150] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it simulates the dynamic behavior of the thin plate system under the action of a thermal field based on the dynamic response simulation method of the rigid-flexible-thermal coupled thin plate system.

[0151] A computer-readable storage medium storing a computer program, which, when executed by a processor, simulates the dynamic behavior of a thin-plate system under the action of a thermal field based on the aforementioned dynamic response simulation method for a rigid-flexible-thermal coupled thin-plate system.

[0152] Example

[0153] To verify the effectiveness and accuracy of the present invention, this embodiment addresses the following: Figure 2 The rotating thin plate shown was studied to simulate the operation of solar panels in space.

[0154] The geometric, material, and temperature parameters of the thin-plate system required in step 1 are shown in Table 1. The initial solar radiation angle is 0°.

[0155] Table 1 Geometric parameters, material parameters, and temperature parameters of the thin plate system

[0156]

[0157] In this example, the thin plate system selects a mesh size of 30 = 10 × 3; the thin plate element length is l. e The value is 0.45m, and the grid width is w. e It is 0.5m.

[0158] Step 2: Based on the mesh generated in Step 1, the thin plate system is discretized into many thin plate elements using the absolute nodal coordinate method. The rigid-flexible coupling dynamic equations for each thin plate element are obtained using the kinetic energy theorem and the principle of virtual work. The rigid-flexible coupling dynamic equations of each thin plate element are then assembled into the rigid-flexible coupling dynamic equations of the thin plate system. Based on the thin plate elements discretized according to the mesh, the heat conduction equations for each thin plate element are obtained using the law of conservation of energy and heat conduction theory. The heat conduction equations of each thin plate element are then assembled into the heat conduction equations of the thin plate system. After obtaining the rigid-flexible coupling dynamic equations and heat conduction equations of the thin plate system, proceed to Step 3.

[0159] Step 3: Consider the relationship between the dynamic equation and the heat conduction equation of the thin plate system, combine the dynamic equation and the heat conduction equation into a set of equations to obtain the dynamic equation of the rigid-flexible-thermal coupled thin plate system, and then proceed to step 4.

[0160] Step 4: Use the generalized α method to perform finite difference on the dynamic equations of the rigid-flexible-thermal coupled thin plate system, and use the Newton-Raphson method to iteratively solve the finite difference scheme to obtain the position coordinate array and temperature coordinate array of the thin plate system nodes, then proceed to step 5.

[0161] Step 5: Perform coordinate transformation on the position coordinate matrix of the thin plate system, and plot the transformed result to obtain the lateral deformation-time curve at the end point of the thin plate system, such as... Figure 6 Deformation diagrams of thin plate systems, such as Figure 7 .

[0162] As shown in Figure 6, the displacement of point P in the z-direction increases to approximately -0.11 m, and then it begins to vibrate with -0.11 m as the equilibrium position. This demonstrates that the influence of temperature on the dynamic behavior of the thin plate cannot be simply ignored. The period of the thin plate vibration is approximately 0.5 s, and the amplitude is approximately 0.01 m. This type of vibration is called thermally induced vibration.

[0163] Based on previous research, this invention considers the dynamic behavior of flexible thin plates in a temperature field, as well as the influence of large-scale motion on thermodynamic boundary conditions. It outputs temperature and displacement change diagrams at the end points and configuration diagrams of the thin plate for use by those skilled in the art. The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as the combinations of these technical features do not contradict each other, they should be considered within the scope of this specification.

[0164] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these modifications and improvements all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. A simulation method for the dynamic behavior of a rigid-flexible-thermal coupled thin-plate system, characterized in that, Includes the following steps: Step 1: Set the geometric parameters, material parameters, mesh parameters, and temperature parameters of the thin plate system; Step 2: Divide the thin plate system into a grid according to the set grid parameters, obtain the rigid-flexible coupling dynamic equation of each thin plate element through the kinetic energy theorem and the principle of virtual work, and assemble the rigid-flexible coupling dynamic equation of each thin plate element into the rigid-flexible coupling dynamic equation of the thin plate system; obtain the heat conduction equation of each thin plate element through the law of conservation of energy and the theory of heat conduction, and assemble the heat conduction equation of each thin plate element into the heat conduction equation of the thin plate system. Step 3: Considering the relationship between the rigid-flexible coupling dynamic equations and the heat conduction equations of the thin plate system, combine the dynamic equations and the heat conduction equations into a system of equations to obtain the dynamic equations of the rigid-flexible-thermal coupled thin plate system; specifically: (1) Coupling terms of the nodal position coordinate array in the heat conduction equation of the thin plate system Let the solar radiation heat flow be It represents the direction of solar radiation; the normal vector of the upper surface at point P in the thin-plate system is... The normal vector of the lower surface is ,but: (26)； Among them Let be the slope of the position vector of point P in the x-direction; Let be the slope of the position vector of point P in the y-direction; Let be the normal vector at point P; Let P be the value of the normal vector at point P; Solar radiation angle coefficient of thin plate system surface ,when At that time, the upper surface of the thin plate is subjected to solar heat radiation, denoted as Conversely, the lower surface of the thin plate receives solar heat radiation, denoted as... After obtaining the solar radiation angle coefficient, the solar radiation of the system is: (27)； in The heat absorption coefficient of the upper surface; The heat absorption coefficient of the lower surface; This represents the numerical value of solar radiation heat flux; The solar radiation angle coefficient of the upper surface; The solar radiation angle coefficient of the lower surface; Equation (27) reveals that the solar radiation angle coefficient in the heat source array of the heat conduction equation in the thin plate system is closely related to the slope of the position vector of point P in the x and y directions in the rigid-flexible coupling dynamic equation. The surface solar radiation angle coefficient... This is a coupling term in the nodal position coordinate array of the heat conduction equation for a thin-plate system; (2) Coupling terms of nodal temperature coordinate array in the rigid-flexible coupling dynamic equation of thin plate system When the temperature of the thin plate system changes, the constitutive equation for any point P on the thin plate system is: (28)； in The elastic coefficient matrix of the thin plate system; To account for the total stress at point P under the influence of temperature, Let P be the elastic stress. The temperature stress at point P; Let P be the elastic strain. The thermal strain at point P caused by the temperature change; Since temperature changes only affect the axial changes of the structure, and are unaffected by changes in the tangential direction, the thermal strain caused by temperature changes... as follows: (29)； in The coefficient of thermal expansion is... Temperature changes take the following form: (30)； in The temperature of the neutral layer; denoted as the temperature gradient along the z-direction; z represents the position coordinates of point P along the thickness direction. Substituting equations (30) and (29) into equation (28), we get: (31)； The elastic force under temperature load, derived from the principle of virtual work, is expressed as: (32)； in This represents the elastic force under temperature load corresponding to the neutral layer temperature. The elastic force under temperature load corresponding to the temperature gradient; According to equation (32), it can be seen that the elastic force of temperature load is closely related to the temperature coordinate array. , This refers to the coupling term of the nodal temperature coordinate array in the rigid-flexible coupling dynamic equation of a thin-plate system. (3) Rigid-flexible-thermal coupling dynamic equations based on thin plate systems Substituting the solar radiation angle factor and the temperature load elastic force into the dynamic equations and heat transfer equations of the thin plate system, respectively, we finally obtain the rigid-flexible-thermal coupling dynamic equations based on the thin plate system: (33)； in These are the constraint equations for the dynamics. These are the constraint equations for the heat conduction equation; The constraint equations of the dynamic equations are given by the position coordinate array. The partial derivative, The constraint equations for the heat conduction equation are arranged in terms of the temperature coordinate matrix. The partial derivative; Step 4: Use the generalized α method to perform finite difference on the dynamic equations of the rigid-flexible-thermal coupled thin plate system, and use the Newton-Raphson method to iteratively solve the finite difference scheme to obtain the position coordinate array and temperature coordinate array of the nodes of the thin plate system. Step 5: Perform coordinate transformation on the coordinate array of the node positions of the thin plate system, and draw a graph based on the transformation results to obtain the lateral deformation-time curve of the end point of the thin plate system and the deformation diagram of the thin plate system.

2. The simulation method for the dynamic behavior of a rigid-flexible-thermal coupled thin-plate system according to claim 1, characterized in that, Step 1: Set the geometric parameters, material parameters, mesh parameters, and temperature parameters of the thin plate system, specifically as follows: (1) Geometric parameters Geometric parameters of the thin plate system: length L, width W, thickness h; (2) Material parameters Material parameters of the thin plate system: density ρ, elastic modulus E; (3) Mesh parameters Mesh parameters for the thin plate system: number of meshes The grid length is The grid width is ; (4) Temperature parameters Temperature parameters of the thin plate system: specific heat capacity c, thermal conductivity k, coefficient of thermal expansion. The solar radiation heat flux is s, and the surface absorptivity is The reference temperature is T0.

3. The simulation method for the dynamic behavior of a rigid-flexible-thermal coupled thin-plate system according to claim 1, characterized in that, Step 2: Divide the thin plate system into a grid according to the set grid parameters. Obtain the rigid-flexible coupling dynamic equations for each thin plate element using the kinetic energy theorem and the principle of virtual work. Assemble the rigid-flexible coupling dynamic equations of each thin plate element into the rigid-flexible coupling dynamic equations of the thin plate system. Obtain the heat conduction equations for each thin plate element using the law of conservation of energy and the theory of heat conduction. Assemble the heat conduction equations of each thin plate element into the heat conduction equations of the thin plate system. Specifically: (1) Dynamic equations of rigid-flexible coupling based on thin plate system An inertial coordinate system for the thin plate system is established with its length as the x-axis, width as the y-axis, and thickness as the z-axis, and the origin O at the intersection of the length, width, and thickness midlines. The thin plate system is then divided into sections according to the mesh parameters. The thin-plate system is discretized using a rectangular grid and the absolute nodal coordinate method. Each rectangular grid is considered as a thin-plate element. A thin plate unit, and Each is a thin plate system along and The number of grids is divided in two directions. The thin plate system has a length of L and a width of W. The length of each thin plate element is... , width is ,in , ; For any point P in a thin plate system, in an inertial frame Below, its position vector is represented as: (1)； in Let P be the position vector; Point P in the inertial coordinate system Position components in each direction; The shape function matrix is ​​the absolute nodal coordinate method. For the coordinate array of the node positions, and The expressions are as follows: (2)； (3)； in To use the absolute nodal coordinate method to discretize the shape function components in the shape function matrix of the thin plate system; It is a 3rd order identity matrix; This is a column matrix of the nodal positions of the four boundary nodes A, B, C, and D on the thin plate element containing point P; shape function components. The specific expression is as follows: (4)； In formula (4) Taking the length direction of the thin plate unit as Direction, width of the thin plate unit Direction, with node A at the lower left corner of the thin plate element as the zero point. , These are the relative position coordinates of P in the coordinate system of the thin plate element when the thin plate element is undeformed; According to the work-energy theorem, the kinetic energy of the thin plate element can be expressed as: : (5)； in The density of the thin plate system; The derivative of the position vector of point P with respect to time; For node velocity arrays; The mass matrix of the thin plate element is expressed as follows: (6)； in The thickness of the thin plate system; Based on the thin-plate assumption, the normal vector of each element in the thin-plate system is perpendicular to the neutral layer, which is a plane composed of the midpoints of the thickness direction of the thin-plate system. Then, the total strain at any point P on the thin-plate element is: (7) ； in The total strain at point P; The strain vector of the neutral layer; The distance of point P from the neutral layer; It is a curvature vector; Based on Hooke's law, the total strain at point P is obtained according to equation (7). The constitutive equation for point P is expressed as: (8)； in The total stress at point P; The elastic coefficient matrix of the thin plate system; Based on the principle of virtual work, the elastic force matrix of a thin plate element can be expressed as: (9)； in This is the elastic force matrix related to the strain terms of the thin plate element and the neutral layer at point P in the thin plate system. This is the elastic force matrix related to the curvature term of the thin plate element where point P is located in the thin plate system; Mass matrix of thin plate system units Elastic force matrix related to neutral layer strain term The elastic force matrix related to the curvature term The overall mass matrix of the thin plate system is obtained by assembling according to the mesh. Elastic force matrix related to neutral layer strain term The elastic force matrix related to the curvature term Finally, the rigid-flexible coupling dynamic equations based on the thin plate system are obtained: (10)； (2) Heat conduction equation based on thin plate system According to the law of conservation of energy and the theory of heat conduction, the temperature field variable... The fundamental equation that satisfies this is expressed as: (11)； in For the interior of the thin plate system Heat flux density in the direction; The density of heat sources within the thin-plate system; The density of the thin-plate system; The specific heat capacity of the thin plate system; According to Fourier's law of heat transfer, the direction of heat flux density inside a thin-plate system is opposite to the temperature gradient, that is: (12)； in The thermal conductivity of the thin plate system; Substituting equation (12) into equation (11), we obtain the transient temperature field of the thin plate system. The heat conduction equation that is satisfied is: (13)； in For the volume of the thin plate system; Adding solar radiation heat flow to the heat conduction equation (13), considering the distinction between the upper and lower plates, and taking the minimum value of the functional, we obtain the heat conduction variational equation of the temperature field variable in Cartesian coordinates: (14)； in For the area of ​​the upper and lower surfaces of the thin plate system; The heat flux density on the upper surface of the thin plate system. The heat flux density on the lower surface of the thin plate system is expressed as follows: (15)； in The heat absorption coefficient of the upper surface; The heat absorption coefficient of the lower surface; This is the solar radiation heat flux vector; This represents the numerical value of solar radiation heat flux; The solar radiation angle coefficient of the upper surface; The solar radiation angle coefficient of the lower surface; Temperature at any point P in the thin plate system Represented as: (16)； in The reference temperature for point P; The temperature of point P in the neutral layer is called the neutral layer temperature. Let P be the temperature gradient in the z-direction. The absolute nodal coordinate method is used to discretize the temperature of the thin plate system. For point P, the shape function matrix is ​​taken. The non-zero elements in the first row of the array are then obtained. , : (17)； in It is a first-order identity matrix; This is a coordinate array of the neutral layer temperature at the nodes; For the nodal temperature gradient coordinate array; Substituting equations (16) and (17) into equation (14), we obtain the heat conduction equations for the neutral layer temperature and temperature gradient of the thin plate element, respectively: (18)； in: (19)； (20)； (21)； (22)； (23)； (24)； in Thermal capacity array related to the temperature of thin plate unit and neutral layer For thin-plate elements and temperature gradient-related thermal capacity arrays; Thermal conduction arrays related to the temperature of thin plate units and neutral layer For thin-plate units and temperature gradient-dependent heat conduction arrays; Heat source array related to the temperature of thin plate unit and neutral layer For thin plate units and temperature gradient-related heat source arrays; Similarly, the heat capacity array related to the neutral layer temperature in the thin plate unit is... Temperature gradient related heat capacity array Neutral layer temperature-dependent thermal conductivity array Temperature gradient-dependent thermal conductivity array Neutral layer temperature-related heat source array Temperature gradient related heat source array By assembling the mesh according to the traditional finite element method, the thermal capacity matrix related to the neutral layer temperature in the thin plate system is obtained. Temperature gradient related heat capacity array Neutral layer temperature-dependent thermal conductivity array Temperature gradient-dependent thermal conductivity array Neutral layer temperature-related heat source array Temperature gradient related heat source array Finally, the heat conduction equation based on the thin-plate system is obtained: (25)； in For nodal temperature coordinate arrays; For the heat capacity array of the thin plate system; For heat conduction arrays in thin-plate systems; It is a heat source array for a thin plate system.

4. The simulation method for the dynamic behavior of a rigid-flexible-thermal coupled thin-plate system according to claim 1, characterized in that, Step 4: Apply the generalized α method to perform finite difference on the dynamic equations of the rigid-flexible-thermal coupled thin plate system, and use the Newton-Raphson method to iteratively solve the finite difference scheme to obtain the position and temperature coordinate arrays of the thin plate system nodes, specifically: Combining the nodal position coordinate array and the nodal temperature coordinate array, the thin-plate element with coupled temperature is represented as follows: The nodal coordinate array of coupled temperature , represented as ； Step 3 yields the rigid-flexible-thermal coupled dynamic equations of the thin-plate system. A difference scheme is constructed using the generalized α method, and the Newton-Raphson method is used for iterative processing. The specific steps are as follows: (1) Difference scheme for constructing dynamic equations of rigid-flexible-thermal coupled thin plate system based on generalized-α method Nodal position coordinate array of thin plate system under initial conditions Node temperature coordinate array The nodal coordinate array of the coupling temperature under initial conditions Introducing parameters: generalized acceleration *a* and Lagrange multipliers. ; (34)； Time has passed h later, the nodal coordinate array obtained from the kinetic equations and heat conduction equations of the previous time step is used to estimate the next time step. , , : (35)； in h is the time step. For spectral radius The relevant parameters are in the following format: (36)； spectral radius This determines the distribution range of the frequency of energy dissipation in the algorithm. Equation (34) is a difference scheme constructed based on the generalized-α method for the rigid-flexible-thermal coupling dynamic equation of the thin plate system; (2) Solving the position coordinate matrix and temperature coordinate matrix of the nodes of the thin plate system based on the Newton-Raphson method The difference scheme based on the generalized-α method constructed in step 4(1) is used to construct... , Substituting into equation (33), we get: (37)； in It is the difference matrix of the rigid-flexible-thermal coupling dynamic equations of the thin plate system after the (n+1)th time step, without considering the thermodynamic constraint equations. It is the difference matrix of the rigid-flexible coupling dynamic equation of the thin plate system after the (n+1)th time step; This is the difference matrix of the heat conduction equation of the thin plate system after the (n+1)th time step; Using the iterative method proposed by Arnold, the solution is obtained iteratively: (38)； in, This is the correction result of the kth iteration after the (n+1)th time step; This is the difference matrix of the rigid-flexible-thermal coupling dynamic equations of the thin plate system in the kth iteration after the (n+1)th time step; Indicates the kth time During iteration Jacobian matrix of partial derivatives; Introducing correction factors , The results of the (k+1)th iteration after correction are as follows: (39)； in The node coordinate array is the coupling temperature of the (k+1)th iteration after the (n+1)th time step; The Lagrange multiplier for the (k+1)th iteration after the (n+1)th time step; If after k+1 iterations, the following conditions are met: This indicates that the accuracy of the iteration result meets the requirements, where tol is the allowable error. Use this as the initial condition for the next time step and update the parameters. Start the loop for the next time step, until the time expires; (40)； By iteratively solving equation (39) and satisfying the allowable error, the final nodal temperature coordinate array in the thin plate system is obtained. and node position coordinate array , and The position coordinate array and temperature coordinate array of the nodes of the thin plate system are obtained by solving the rigid-flexible-thermal coupling dynamic equation of the thin plate system based on the Newton-Raphson method.

5. The simulation method for the dynamic behavior of a rigid-flexible-thermal coupled thin-plate system according to claim 1, characterized in that, Step 5: Perform coordinate transformation on the position coordinates of the thin plate system, and plot the transformed results to obtain the lateral deformation-time curve of the end point of the thin plate system and the deformation diagram of the thin plate system, specifically: Perform coordinate transformation on the obtained node position coordinate matrix: (41)； Where u, v, w represent the deformation of the thin plate system in the x, y, z directions of the inertial coordinate system; and the initial coordinates of point P are... Generalized coordinates of point P ; The angle through which the thin-plate system has rotated; and Node position coordinates In and ; Plot the graph with time t and lateral deformation w as coordinate regions to obtain the lateral deformation of the post-processed thin plate system. Using the generalized coordinates of point P Using the coordinate region as the coordinate area, the graph is drawn at set step intervals to obtain the deformation history of the thin plate system in post-processing.

6. A simulation method system for the dynamic behavior of a rigid-flexible-thermal coupled thin plate system, characterized in that, Based on the simulation method for the dynamic behavior of the rigid-flexible-thermal coupled thin plate system according to any one of claims 1-5, the dynamic behavior of the thin plate system under the action of a thermal field is simulated.

7. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it realizes the dynamic response simulation of the rigid-flexible-thermal coupled thin plate system based on the simulation method for the dynamic behavior of the rigid-flexible-thermal coupled thin plate system according to any one of claims 1-5.

8. A computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, it realizes the dynamic response simulation of the rigid-flexible-thermal coupled thin plate system based on the simulation method for the dynamic behavior of the rigid-flexible-thermal coupled thin plate system according to any one of claims 1-5.

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