A method for geometrically nonlinear equivalent plate dynamics modeling and response analysis of a truss structure

Abstract

The application discloses a kind of geometric nonlinear equivalent plate dynamics modeling and response analysis method of truss structure, its implementation steps are: for the truss structure containing geometric nonlinearity, first, construct linear equivalent Mindlin plate dynamics model;Then describe the geometric constraint relationship of Mindlin plate, carry out displacement and angle approximation and simulate general elastic boundary condition;Finally, according to the extended Hamilton principle, establish the equivalent plate dynamics model containing geometric nonlinearity, and carry out nonlinear dynamic response analysis.The application overcomes the defect that the installation position characteristics of distribution hinge in the width direction of cross section is not considered in the traditional equivalent beam model, and is suitable for geometric nonlinear large deformation analysis;It is suitable for truss structure with general boundary conditions, and has strong engineering applicability;The equivalent nonlinear model established by semi-analytical method is more efficient than traditional finite element numerical algorithm, and is more suitable for dynamic analysis and control law design.

Description

Technical Field

[0001] This invention relates to the field of structural dynamics modeling and analysis research technology, specifically to a method for geometric nonlinear equivalent plate dynamics modeling and response analysis of truss structures. Background Technology

[0002] Deployable space structures, due to their numerous advantages such as light weight, high specific stiffness, high storage ratio, and strong deployability, have been widely used in the aerospace field. With the increasing size and complexity of spacecraft payloads, large flexible space truss antenna structures, such as… Figure 1 As shown, this has gradually become a major development trend and a key research focus both domestically and internationally. Its truss antenna unit consists of a supporting truss and an antenna array, as shown... Figure 2 As shown, the supporting truss is interconnected with each antenna array via hinges distributed along the width of the truss cross-section. Therefore, the truss structure exhibits nonlinear connection characteristics at the hinge joints. Because the truss structure has a large dimension along its length, it is highly susceptible to large geometrical nonlinear deformation under external loads, exhibiting strong geometrical nonlinear characteristics. This poses significant challenges to the dynamic modeling and vibration control of spacecraft. Therefore, research on the large geometrical nonlinear deformation of such truss structures has important practical significance.

[0003] Currently, the finite element method (FEM) is commonly used in the study of the geometric nonlinear dynamic characteristics of truss structures. However, the FEM typically suffers from high degrees of freedom and large computational costs, making it unsuitable for the design of vibration control systems for truss structures. The equivalent continuum modeling method, as a semi-analytical approach, effectively avoids the shortcomings of the FEM and shows great promise for solving the problem of efficient and accurate modeling of the dynamics of truss structures with geometric nonlinear characteristics. Currently, there is limited research on equivalent modeling of truss structures with geometric nonlinearity; the existing research focuses on equating truss structures to continuous beam models. However, continuous beam models use the point section assumption, which cannot characterize the installation position characteristics of hinges distributed along the width of the cross-section, leading to difficulties in representing the connection nonlinearity caused by hinges. Furthermore, existing research on equivalent dynamic modeling of truss structures with geometric nonlinearity is limited to satisfying the boundary conditions of simply supported sides, suitable for the dynamic modeling of long-span bridges but not for spacecraft truss structures with cantilever boundary conditions. Existing methods have limited research on general boundary conditions with greater engineering significance. Therefore, for truss structures with geometric nonlinear characteristics under general boundary conditions and considering the characteristics of hinge installation positions distributed in the cross-sectional width direction, current equivalent dynamic modeling and response analysis techniques cannot solve this problem well. Summary of the Invention

[0004] To address or partially resolve problems in related technologies, this invention provides a method for modeling and analyzing the dynamics of a geometrically nonlinear equivalent plate for truss structures. To more accurately describe geometrically nonlinear characteristics and consider the installation position characteristics of distributed hinges along the width of the cross-section, a method for modeling and analyzing the dynamics of a geometrically nonlinear equivalent plate for truss structures is established. A linear equivalent Mindlin plate dynamic model of the truss structure is constructed to describe the geometric constraints of the Mindlin plate. Uniformly distributed springs are used to simulate general elastic boundary conditions, and Legendre polynomials are used to approximate the displacement and rotation of the equivalent plate. Based on the extended Hamiltonian principle, a dynamic model of the equivalent plate containing geometrically nonlinear characteristics is established, and nonlinear dynamic response analysis is performed. This lays the technical foundation for the dynamic modeling, analysis, and control of truss structures with geometrically nonlinear characteristics.

[0005] This invention provides a method for geometric nonlinear equivalent plate dynamic modeling and response analysis of truss structures, characterized by the following steps:

[0006] S1, Construct a linear equivalent Mindlin plate dynamic model for the truss structure;

[0007] S2 describes the geometric constraints of the Mindlin plate;

[0008] S3, perform Mindlin plate displacement and rotation approximation and artificial spring boundary setting;

[0009] S4. Construct a geometrically nonlinear equivalent plate dynamic model and perform response analysis.

[0010] Optionally, in step S1, an orthogonal anisotropic linear equivalent plate dynamic model of the truss structure is established. Since the cross-section of the truss structure is an equilateral triangle, the lateral vibration and side vibration of the truss structure are symmetrical. Therefore, for the geometric nonlinear analysis of this patent, only the lateral vibration needs to be considered.

[0011] For truss structures, based on Mindlin plate theory, the strain of periodic elements is expanded using a higher-order Taylor series to describe the strain of the equivalent plate's lateral vibration.

[0012] Based on the strain expression for transverse vibration, the strain energy and kinetic energy of the periodic truss element are calculated. Then, using the principle of energy equivalence, the equivalent elastic matrix and equivalent inertial matrix of the equivalent plate model for transverse vibration are solved.

[0013] The equivalent model is reduced in order using the static condensation method to obtain the equivalent elastic matrix D and equivalent inertia matrix G with the same dimension as the strain variables of the Mindlin plate, as follows:

[0014]

[0015] Where D1 and D2 are the out-of-plane bending stiffness, D3 is the torsional stiffness, and G... xz and G yz These represent the transverse shear stiffness; the off-diagonal element η ij (i,j=1,2,3,4,5,i≠j) represents the coupling stiffness. Since the cross-section of the truss structure is an equilateral triangle, the truss structure is symmetrical. Therefore, the influence of the off-diagonal coupling stiffness in the D matrix can be ignored.

[0016]

[0017] in, This represents the mass per unit area of ​​the equivalent plate. and Represent the moments of inertia per unit area along the x-axis and y-axis of the equivalent plate model, respectively; off-diagonal element m ij (i,j=1,2,3,4,5,i≠j) are the coupling mass parameters. Since the truss structure is symmetrical, the influence of off-diagonal coupling mass in the G matrix can be ignored.

[0018] In step S2, based on Mindlin plate theory, when considering the large geometric nonlinear deformation of the truss structure, the geometric constraint equations of the equivalent Mindlin plate are constructed as follows:

[0019]

[0020] Where u, v, and w are the displacements in the x, y, and z directions, respectively. Let be the angle of rotation along the x-axis. Let be the rotation angle along the y-axis.

[0021] In step S3, the five displacement and rotation variables u, v, w of the Mindlin plate are approximated using Legendre polynomials. As shown in the following formula:

[0022]

[0023] Where ξ = 2x / a - 1, η = 2y / b - 1; -1 < ξ, η < 1; a and b are the length and width of the plate, respectively; P i (ξ) and P j (η) is a Legendre polynomial; U ij (t),V ij (t),W ij (t),Φ ij (t),Ψ ij (t) represents the generalized coordinates of the variables; i and j represent the order of the Legendre polynomial, and I and J represent the maximum order of the truncated Legendre polynomial.

[0024] Displacement springs and rotational constraint springs are uniformly arranged along the four boundaries of a Mindlin plate. The boundary conditions of the Mindlin plate are simulated using an artificial spring boundary method, and arbitrary elastic boundary conditions are simulated by setting different spring stiffness values. The strain energy stored on the boundary springs is then:

[0025]

[0026] in, Let x = 0, x = a, y = 0, and y = b represent the stiffness values ​​of the springs with normal linear displacement along the edge, respectively. Let x = 0, x = a, y = 0, and y = b represent the stiffness values ​​of the springs with tangential linear displacement along the sides, respectively. Let x = 0, x = a, y = 0, and y = b represent the spring stiffness values ​​for the lateral displacement along the sides, respectively. Let represent the stiffness values ​​of the constraint springs rotating along the x-axis on the sides at x = 0, x = a, y = 0, and y = b, respectively. Let represent the stiffness values ​​of the constraint springs rotating along the y-axis on the sides x=0, x=a, y=0, and y=b, respectively. Changing the stiffness value to infinity or zero can simulate arbitrary general boundary conditions. For example, if we let... If the value is infinite and all other stiffness values ​​are 0, then the cantilever boundary condition is simulated with the side at x=0 fixed and the other three sides free.

[0027] In step S4, the total strain energy of the Mindin plate is:

[0028] V = V p +V b

[0029]

[0030] Where V b The strain energy stored on the boundary spring.

[0031] The total kinetic energy of the Mindin plate is:

[0032]

[0033] The displacement and rotation variables u, v, w after polynomial approximation Substituting the geometric constraint equations, we obtain the following equation:

[0034]

[0035] Multiply the left side of the above equation by P. k (ξ) and P lIntegrating (η) over [-1, 1], the geometric constraint equations can be transformed into:

[0036]

[0037]

[0038]

[0039]

[0040] in, These represent the four corresponding constraint equations; P k (ξ) and P l (η) represents the Legendre polynomial, and k and l also represent the order of the Legendre polynomial.

[0041] Using the extended Hamiltonian principle, the following dynamic equilibrium equations can be obtained:

[0042]

[0043]

[0044]

[0045]

[0046]

[0047] Where F is the coordinate point (x) acting on the Mindlin board. F ,y F External load along the deflection direction at the location, It is a Lagrange multiplier.

[0048] Strain energy V, kinetic energy T, and constraint equations Substituting the expression into the above equation yields the equivalent plate dynamics equation containing geometric nonlinearity.

[0049] The obtained equivalent plate dynamic equations are a system of differential-algebraic equations with Lagrange multipliers in index 3. To facilitate the solution, the constraint equations are differentiated twice, transforming the dynamic equations into a system of differential-algebraic equations in index 1, which is then transformed into a system of ordinary differential equations using an augmentation method. To control constraint violation, artificial damping is introduced using the Baumgart method to correct the constraints, achieving better stability.

[0050] The obtained dynamic ordinary differential equations are solved iteratively using the fourth-order Runge-Kutta method, and the generalized coordinate variable q = [U] can be obtained. i,j V i,j Wi,j Φ i,j Ψ i,j ] T (i = 0, ..., I, j = 0, ..., J). Substituting q into the expressions for displacement and rotation, we can obtain the dynamic response of the equivalent plate under external loads, which contains geometric nonlinearity.

[0051] Under the same load conditions, comparing the dynamic response calculated by the method of this patent with the geometric nonlinear dynamic response obtained by using commercial finite element software can verify the correctness of the method of this patent.

[0052] The technical solution provided by this invention may include the following beneficial effects:

[0053] (1) Compared with the equivalent beam model, the geometric nonlinear equivalent plate model proposed in this invention can calculate and analyze the dynamic response at different points on the cross-sectional width of the truss structure, thus overcoming the defects of the equivalent beam model.

[0054] (2) The method of the present invention can be applied to any boundary condition, not only the simply supported boundary condition, but also the cantilever and the boundary condition with both ends free, thus broadening the scope of application of the method;

[0055] (3) Compared with numerical calculation methods such as finite element method, this patent adopts the method of solving dynamic response by equivalent nonlinear model, which has higher calculation efficiency while ensuring accuracy;

[0056] (4) The present invention can efficiently and accurately describe the nonlinear dynamic characteristics of truss structures with geometric nonlinearity and considering the characteristics of hinge installation positions distributed in the cross-sectional width direction, and is more convenient for designing vibration control laws.

[0057] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and are not intended to limit the invention. Attached Figure Description

[0058] To more clearly illustrate the technical solutions of the embodiments of this invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0059] Figure 1 This is a schematic diagram of the space truss antenna structure in an embodiment of the present invention;

[0060] Figure 2 This is a schematic diagram of the truss antenna unit in an embodiment of the present invention;

[0061] Figure 3 This is a schematic diagram of the truss structure in an embodiment of the present invention;

[0062] Figure 4 This is a schematic diagram of the geometric model of the truss unit and equivalent plate in an embodiment of the present invention;

[0063] Figure 5 This is a flowchart of the geometric nonlinear equivalent plate dynamics modeling and response analysis of the truss structure in an embodiment of the present invention;

[0064] Figure 6 This is a schematic diagram of the geometric relationship of the plate micro-element before and after deformation in the x-direction in an embodiment of the present invention;

[0065] Figure 7 This is a schematic diagram of the geometric relationship of the plate micro-element before and after deformation in the y-direction in an embodiment of the present invention;

[0066] Figure 8 This is a schematic diagram of the uniform spring arrangement of the plate structure under arbitrary lateral boundary conditions in an embodiment of the present invention.

[0067] Figure 9 This is a schematic diagram of the uniform spring arrangement of a plate structure under arbitrary in-plane boundary conditions in an embodiment of the present invention. Detailed Implementation

[0068] Embodiments of the invention will now be described in more detail with reference to the accompanying drawings. While embodiments of the invention are shown in the drawings, it should be understood that the invention can be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that the invention will be more thorough and complete, and will fully convey the scope of the invention to those skilled in the art.

[0069] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The singular forms “a,” “the,” and “the” used in this invention and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term “and / or” as used herein refers to and includes any or all possible combinations of one or more of the associated listed items.

[0070] This invention provides a method for geometric nonlinear equivalent plate dynamic modeling and response analysis of truss structures, which can calculate and analyze the dynamic response at different points on the cross-sectional width of the truss structure, overcoming the defects of the equivalent beam model.

[0071] The technical solutions of the embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0072] Please refer to the figure. This embodiment provides a method for geometric nonlinear equivalent plate dynamic modeling and response analysis of truss structures, including:

[0073] S1. Construct a linear equivalent Mindlin plate dynamic model for the truss structure;

[0074] S2. Describe the geometric constraints of the Mindlin plate;

[0075] S3. Perform Mindlin plate displacement and rotation approximation and set artificial spring boundaries;

[0076] S4. Construct a geometrically nonlinear equivalent plate dynamic model and perform response analysis.

[0077] Figure 3 The diagram shows a typical truss structure without an antenna array, composed of periodic elements extending one-dimensionally along its length. A periodic truss element is isolated from this truss structure, as shown in the attached diagram. Figure 4 As shown, the truss unit consists of three types of components: longitudinal beams, transverse beams, and stay cables, with an equilateral triangular cross-section. The truss unit comprises 15 components, of which components (1)-(3) are longitudinal beams, components (4)-(9) are stay cables, and components (10)-(15) are transverse beams. The inertial coordinate system oxyz is located on the mid-surface ABCD of the equivalent plate, with the origin o at the centroid of the truss cross-section. The x-axis runs along the length of the truss, and the y-axis and z-axis run along the two transverse directions of the cross-section.

[0078] like Figure 5 The following is a method for geometric nonlinear equivalent plate dynamic modeling and response analysis of a truss structure according to the present invention. The main implementation steps are as follows:

[0079] In step S1, a linear equivalent Mindlin plate dynamic model of the truss structure is constructed.

[0080] Since the cross-section of the truss structure is an equilateral triangle, the lateral vibration and the side vibration of the truss structure are symmetrical. Therefore, for the geometric nonlinear analysis of this patent, only the lateral vibration needs to be considered.

[0081] For truss structures, based on Mindlin plate theory, the strain of periodic elements is expanded using a higher-order Taylor series to describe the strain of the equivalent plate under lateral vibration. According to the strain expression for lateral vibration, the strain energy and kinetic energy of the periodic truss elements are calculated. Then, using the principle of energy equivalence, the equivalent elastic matrix and equivalent inertia matrix of the equivalent plate model under lateral vibration are solved.

[0082] The equivalent model is reduced in order using the static condensation method to obtain the equivalent elastic matrix D and equivalent inertia matrix G with the same dimension as the strain variables of the Mindlin plate, as follows:

[0083]

[0084] Where D1 and D2 are the out-of-plane bending stiffness, D3 is the torsional stiffness, and G... xz and G yz These represent the transverse shear stiffness; the off-diagonal element η ij (i,j=1,2,3,4,5,i≠j) represents the coupling stiffness. Since the cross-section of the truss structure is an equilateral triangle, the truss structure is symmetrical. Therefore, the influence of the off-diagonal coupling stiffness in the D matrix can be ignored.

[0085]

[0086] in, This represents the mass per unit area of ​​the equivalent plate. and Represent the moments of inertia per unit area along the x-axis and y-axis of the equivalent plate model, respectively; off-diagonal element m ij (i,j=1,2,3,4,5,i≠j) are the coupling mass parameters. Since the truss structure is symmetrical, the influence of off-diagonal coupling mass in the G matrix can be ignored.

[0087] Step S2: Describe the geometric constraints of the Mindlin plate.

[0088] When considering the large geometrical nonlinear deformation of a truss structure, based on Mindlin plate theory, a micro-element is taken from the plate, with length dx and width dy, respectively. (Appendix) Figure 6 Let d represent the state of the infinitesimal element before and after deformation in the x-direction, and dw be the displacement in the deflection direction. Based on trigonometric relationships, we can obtain:

[0089]

[0090] Appendix Figure 7 This represents the state of the infinitesimal element before and after deformation in the y-direction. Based on trigonometric relationships, we can obtain:

[0091]

[0092] Where u, v, and w are the displacements in the x, y, and z directions, respectively. Let be the angle of rotation along the x-axis. Let be the rotation angle along the y-axis.

[0093] Dividing both sides of equations (3) and (4) by dx and dy respectively, we can obtain the geometric constraint equations of the Mindlin plate, as follows:

[0094]

[0095] Step S3: Perform Mindlin plate displacement and rotation approximation and set artificial spring boundaries.

[0096] The five displacement and rotation variables u, v, w of the Mindlin plate are approximated using Legendre polynomials. As shown in the following formula:

[0097]

[0098] Where ξ = 2x / a - 1, η = 2y / b - 1; -1 < ξ, η < 1; a and b are the length and width of the plate, respectively; P i (ξ) and P j (η) is a Legendre polynomial; U ij (t),V ij (t),W ij (t),Φ ij (t),Ψ ij (t) represents the generalized coordinates of the variables; i and j represent the order of the Legendre polynomial, and I and J represent the maximum order of the truncated Legendre polynomial.

[0099] Displacement springs and rotational constraint springs are evenly arranged on the four boundaries of the Mindlin plate, as follows: Figure 8 and Figure 9 As shown. Among them, Figure 8 The middle section shows a uniform spring arrangement for simulating a plate structure under arbitrary lateral boundary conditions. Figure 9 The diagram shows a uniform spring arrangement for simulating plate structures under arbitrary in-plane boundary conditions. The artificial spring boundary method is used to simulate the boundary conditions of the Mindlin plate, achieving arbitrary elastic boundary conditions by setting different spring stiffness values. The strain energy stored on the boundary springs is then:

[0100]

[0101] in, Let x = 0, x = a, y = 0, and y = b represent the stiffness values ​​of the springs with normal linear displacement along the edge, respectively. Let x = 0, x = a, y = 0, and y = b represent the stiffness values ​​of the springs with tangential linear displacement along the sides, respectively. Let x = 0, x = a, y = 0, and y = b represent the spring stiffness values ​​for the lateral displacement along the sides, respectively. Let represent the stiffness values ​​of the constraint springs rotating along the x-axis on the sides at x = 0, x = a, y = 0, and y = b, respectively. Let represent the stiffness values ​​of the constraint springs rotating along the y-axis on the sides x=0, x=a, y=0, and y=b, respectively. Changing the stiffness value to infinity or zero can simulate arbitrary general boundary conditions. For example, if we let... If the value is infinite and all other stiffness values ​​are 0, then the cantilever boundary condition is simulated with the side at x=0 fixed and the other three sides free.

[0102] Step S4: Construct a geometrically nonlinear equivalent plate dynamic model and perform response analysis.

[0103] The total strain energy of the Mindin plate can be written as:

[0104] V = V p +V b (8)

[0105]

[0106] Among them, V b The strain energy stored on the boundary spring as shown in formula (7) is .

[0107] The total kinetic energy of the Mindin plate is:

[0108]

[0109] Substituting equation (6) into the geometric constraint equation in equation (5), we obtain the following equation:

[0110]

[0111] Multiply the left side of equation (11) by P respectively k (ξ) and P l Integrating (η) over [-1, 1], the geometric constraint equations can be transformed into:

[0112]

[0113]

[0114]

[0115]

[0116] in, These represent the four corresponding constraint equations; P k (ξ) and P l (η) represents the Legendre polynomial, and k and l also represent the order of the Legendre polynomial.

[0117] Using the extended Hamilton's principle, the following dynamic equilibrium equations can be obtained:

[0118]

[0119]

[0120]

[0121]

[0122]

[0123] Where F is the coordinate point (x) acting on the Mindlin board. F ,y F External load along the deflection direction at the location, It is a Lagrange multiplier.

[0124] Substituting equations (8), (10), (12)-(15) into equations (16)-(20), we can obtain the equivalent plate dynamic equations containing geometric nonlinearity.

[0125] The obtained equivalent plate dynamic equations are a system of differential-algebraic equations with Lagrange multipliers in index 3. To facilitate the solution, the constraint equations are differentiated twice, transforming the dynamic equations into a system of differential-algebraic equations in index 1, which is then transformed into a system of ordinary differential equations using an augmentation method. To control constraint violation, artificial damping is introduced using the Baumgart method to correct the constraints, achieving better stability.

[0126] The obtained dynamic ordinary differential equations are solved iteratively using the fourth-order Runge-Kutta method, and the generalized coordinate variable q = [U] can be obtained. i,j V i,j W i,j Φ i,j Ψ i,j ] T ,(i=0,...,I,j=0,...,J). Substituting q into the displacement and rotation expressions in formula (6), we can obtain the dynamic response of the equivalent plate with geometric nonlinearity under external loads.

[0127] Under the same load conditions, comparing the dynamic response calculated by the method of this patent with the geometric nonlinear dynamic response obtained by using commercial finite element software can verify the correctness of the method of this patent.

[0128] The above description is merely an embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and scope of the present invention are included within the scope of protection of the present invention.

Claims

1. A method for geometric nonlinear equivalent plate dynamic modeling and response analysis of truss structures, characterized in that, include: S1. Construct a linear equivalent Mindlin plate dynamic model for the truss structure; S2. Describe the geometric constraints of the Mindlin plate; S3. Perform Mindlin plate displacement and rotation approximation and set artificial spring boundaries; S4. Construct a geometrically nonlinear equivalent plate dynamics model and perform response analysis; Step S1 includes: For truss structures, based on Mindlin plate theory, the strain of periodic elements is expanded using a higher-order Taylor series to describe the strain of the equivalent plate lateral vibration. Based on the strain expression for transverse vibration, the strain energy and kinetic energy of the periodic truss element are calculated, and then the equivalent elastic matrix and equivalent inertial matrix of the equivalent plate model for transverse vibration are solved using the energy equivalence principle. The equivalent model was reduced in order using a static condensation method to obtain an equivalent elastic matrix with the same dimension as the strain variables of the Mindlin plate. D and equivalent inertia matrix G ,as follows: in, and These are out-of-plane bending stiffness, To increase torsional stiffness, and These represent transverse shear stiffness; off-diagonal elements. For coupling stiffness; in, This represents the mass per unit area of ​​the equivalent plate. and These represent the upper edge per unit area of ​​the equivalent plate model, respectively. x shaft and y Moment of inertia of the axis; off-diagonal elements These are the coupling quality parameters; In step S3, Legendre polynomials are used to approximate the five displacement and rotation variables of the Mindlin plate. The details are as follows: in, , ; ; a and b These are the length and width of the board, respectively. and For Legendre polynomials; These represent the generalized coordinates of the variables; i and j Denotes the order of the Legendre polynomial. I and J This represents the maximum order of the truncated Legendre polynomial; Displacement springs and rotational constraint springs are uniformly arranged on the four boundaries of the Mindlin plate. The boundary conditions of the Mindlin plate are simulated using the artificial spring boundary method. By setting different spring stiffness values, arbitrary elastic boundary conditions can be simulated. The strain energy stored on the boundary springs is: in, , , , They represent x =0, x = a , y =0, y = b The stiffness value of the spring with normal linear displacement on the side. , , , They represent x =0, x = a , y =0, y = b The stiffness value of the spring with tangential linear displacement on the side. , , , They represent x =0, x = a , y =0, y = b The stiffness value of the lateral displacement spring on the side. , , , They represent x =0, x = a , y =0, y = b edge x The stiffness value of the constraint spring for shaft rotation. , , , They represent x =0, x = a , y =0, y = b edge y The stiffness value of the constraint spring for shaft rotation.

2. The method for geometric nonlinear equivalent plate dynamic modeling and response analysis of truss structures as described in claim 1, characterized in that, In step S1, an orthogonal anisotropic linear equivalent plate dynamic model of the truss structure is established.

3. The method for geometric nonlinear equivalent plate dynamic modeling and response analysis of truss structures as described in claim 1, characterized in that, In step S2, based on Mindlin plate theory, when considering the large geometric nonlinear deformation of the truss structure, the geometric constraint equations of the equivalent Mindlin plate are constructed as follows: in, u , v and w coordinates x , y and z Displacement in direction, For along x The angle of rotation of the axis, For along y The rotation angle of the axis.

4. The method for geometric nonlinear equivalent plate dynamic modeling and response analysis of truss structures as described in claim 1, characterized in that, In step S4, the total strain energy of the Mindin plate is: in, The strain energy stored on the boundary spring; The total kinetic energy of the Mindin plate is: The displacement and rotation variables after polynomial approximation Substituting the geometric constraint equations, we obtain the following equation: Multiply the left side of the above equation by... and Integrating between [-1, 1], the geometric constraint equations can be transformed into: in, , , , These represent the four corresponding constraint equations; and For Legendre polynomials, k and l It also represents the order of the Legendre polynomial; Using the extended Hamiltonian principle, the following dynamic equilibrium equations can be obtained: in, F To act on the Mindlin board coordinate points External load along the deflection direction at the location, , , , For Lagrange multipliers; strain energy V ,kinetic energy T Constraint equations , , , Substituting the expression into the above equation, we can obtain the equivalent plate dynamic equation containing geometric nonlinearity. The obtained equivalent plate dynamic equation is a system of differential-algebraic equations with Lagrange multipliers in index 3. To facilitate the solution, we differentiate the constraint equation twice and transform the dynamic equation into a system of differential-algebraic equations with index 1. We then transform it into a system of ordinary differential equations through the augmentation method. The obtained dynamic ordinary differential equations are solved iteratively using the fourth-order Runge-Kutta method to obtain the generalized coordinate variables. ;Will q By substituting the displacement and rotation expressions, the dynamic response of the equivalent plate with geometric nonlinearity under external loads can be obtained.

Images