A method for analyzing vibration response of a plate assembly structure under a moving load

Abstract

The application discloses a kind of plate combination structure vibration response analysis methods under mobile load, comprising the following steps: S1, obtains structure geometric parameter, structure material parameter and mobile load parameter;S2, based on the structure geometric parameter and structure material parameter, dynamic stiffness method is used to construct plate combination structure dynamic stiffness matrix;S3, based on the mobile load parameter, establishes mobile load semi-analytical model, utilizes trigonometric function integral transformation and fourier transformation and moves load from time domain-space domain to frequency domain-wave number domain, obtains frequency domain-wave number domain mobile load;S4, the plate combination structure dynamic stiffness matrix and frequency domain-wave number domain mobile load are solved, and the vibration response of plate combination structure frequency domain-wave number domain is obtained;S5, based on the vibration response of plate combination structure frequency domain-wave number domain, utilizes trigonometric function integral inverse transformation and fourier inverse transformation and obtains plate combination structure time domain-space domain vibration response.

Description

Technical Field

[0001] This invention belongs to the field of structural dynamics technology, specifically relating to a method for analyzing the vibration response of a plate composite structure under moving loads. Background Technology

[0002] Currently, in the rail transit field, due to the high speed and heavy load of trains, the moving loads of vehicles easily cause severe vibrations in the load-bearing structure (which can be simplified as a plate composite structure). Therefore, more efficient and accurate analysis of the vibration response of plate composite structures under moving loads is crucial for structural condition assessment and full life cycle monitoring. Traditional methods for monitoring the overall vibration response of structures under moving loads mainly rely on placing sensors on the structural surface. On the one hand, the fixation effect and lifespan of the sensors make full life cycle monitoring of the structure impossible; on the other hand, the diversity of structures makes traditional monitoring schemes lack universality. Furthermore, compared to static loads, the changes in moving loads have temporal and spatial dimensions, making it difficult to mathematically analyze the structural vibration response under moving loads in the time-space domain. Traditional numerical analysis methods, represented by the finite element method, after discretizing a large number of structural elements, not only increase the complexity of setting the moving load application points but also increase the computational load on the computing equipment, leading to reduced computational efficiency.

[0003] Therefore, a new method for analyzing the vibration response of plate composite structures under moving loads is proposed. Summary of the Invention

[0004] The purpose of this invention is to provide a method for analyzing the vibration response of a plate composite structure under moving loads. This addresses the shortcomings of traditional sensor-based monitoring of the overall structural vibration response under moving loads, as proposed in the background art. On the one hand, the fixed effect and lifespan of sensors make full life-cycle monitoring of the structure impossible. On the other hand, the diversity of structures makes traditional monitoring schemes lack universality. Furthermore, the changes in moving loads have temporal and spatial dimensions, making it difficult to mathematically analyze the structural vibration response under moving loads in the time-space domain. Traditional numerical analysis methods, such as the finite element method, not only increase the complexity of setting the moving load loading points after a large number of discrete structural elements, but also increase the computational load of computing equipment, leading to reduced computational efficiency.

[0005] To achieve the above objectives, this invention provides a method for analyzing the vibration response of a plate composite structure under moving loads, comprising the following steps:

[0006] S1. Obtain structural geometric parameters, structural material parameters, and moving load parameters;

[0007] S2. Based on the aforementioned structural geometric parameters and structural material parameters, the dynamic stiffness matrix of the plate composite structure is constructed using the dynamic stiffness method.

[0008] S3. Based on the moving load parameters, establish a semi-analytical model of the moving load, and use trigonometric function integral transform and Fourier transform to project the moving load from the time domain to the space domain to the frequency domain to the wavenumber domain to obtain the frequency domain to wavenumber domain moving load.

[0009] S4. Combine the dynamic stiffness matrix of the plate composite structure with the frequency domain-wavenumber domain moving load to obtain the frequency domain-wavenumber domain vibration response of the plate composite structure.

[0010] S5. Based on the frequency-wavenumber domain vibration response of the plate assembly structure, the time-space domain vibration response of the plate assembly structure is obtained by using the inverse trigonometric function integral transform and the inverse Fourier transform.

[0011] In one specific implementation, the governing differential equation of the dynamic stiffness method considers structural Rayleigh damping, and its specific expression is:

[0012]

[0013] Where u and v represent the in-plane displacement variables of the plate composite structure in the x and y directions, respectively, and w represents the out-of-plane displacement variable of the plate composite structure in the z direction. All displacement variables are ternary functions of x, y, and t, where x and y represent displacement changes related to spatial position, and t represents displacement changes related to time. This represents the second-order partial derivative of the displacement variable u with respect to the x-direction. This represents the second-order partial derivative of the displacement variable u with respect to the y-direction. Let v represent the second-order partial derivatives of the displacement variable v with respect to the x and y directions, respectively. This represents the second-order partial derivative of the displacement variable u with respect to the time variable t. This represents the first-order partial derivative of the displacement variable u with respect to the time variable t. This represents the second-order partial derivative of the displacement variable v with respect to the x-direction. This represents the second-order partial derivative of the displacement variable v with respect to the y-direction. Let represent the second-order partial derivatives of the displacement variable u with respect to the x and y directions, respectively. This represents the second-order partial derivative of the displacement variable v with respect to the time variable t. This represents the first-order partial derivative of the displacement variable v with respect to the time variable t. This represents the fourth-order partial derivative of the displacement variable w with respect to the x-direction. Let w represent the fourth-order partial derivatives of the displacement variable w with respect to the x and y directions, respectively. This represents the fourth-order partial derivative of the displacement variable w with respect to the y-direction. This represents the second-order partial derivative of the displacement variable w with respect to the time variable t. E represents the first-order partial derivative of the displacement variable w with respect to the time variable t, μ is the Poisson's ratio of the material, ρ is the material density, h is the element section thickness, and α0 is the external viscous damping coefficient of the structure. * It is the dynamic elastic modulus, and its expression is:

[0014] E * =E(1+iβ0ω) (2)

[0015] Where E is Young's modulus, i is the imaginary unit, β0 is the internal viscous damping coefficient of the structure, ω represents the angular frequency of vibration, and D is the damping force after considering the damping force. * Bending stiffness is expressed as:

[0016] D * =E * h 3 / 12(1-μ 2 (3).

[0017] In one specific implementation, step S3 involves resolving the time-space domain moving load in the frequency-wavenumber domain, specifically including the following expressions:

[0018]

[0019]

[0020]

[0021] Equation (4) is the mathematical expression for the moving load, Equation (5) is the trigonometric function integral transform expression for the moving load, and Equation (6) is the Fourier transform expression for the moving load, where p(x,y,t) r Let be the mathematical expression of the moving load in the time-space domain, where x and y represent the changes in the spatial state of the moving load, and t is the value of t. r This indicates the correlation between the moving load and the time dimension, where the subscript r represents the number of time-domain sampling points for the moving load, and t... d This represents the duration of the moving load applied to the plate assembly structure, that is, the length *l* of the plate assembly structure in the direction of the moving load and the velocity *v* of the moving load. p The ratio of P z (t r ) represents the magnitude of the moving load as a function of time t r The expression for the change, δ(xg) x (x,t r )) and δ(yg y (y,t r )) represents the spatial state (x, y) of a moving load as a function of the Dirichlet function δ over time t. r The change expression of p m (αm ,y,t r α is the mathematical expression of the moving load in the time-wavenumber domain. m p represents the wave number along the length of the plate composite structure. mn (α m ,y,ω n ω is the mathematical expression of the moving load in the frequency-wavenumber domain. n This represents the discrete circular frequency value of the nth harmonic in the frequency domain, where N represents the number of frequency samples within the specified time range.

[0022] In one specific implementation, step S4 involves simultaneously establishing the dynamic stiffness matrix of the plate composite structure and the frequency-wavenumber domain moving load to obtain the frequency-wavenumber domain vibration response of the plate composite structure, including:

[0023] By utilizing the dynamic stiffness matrix of the plate composite structure and the frequency-wavenumber domain moving load, and after inverting the dynamic stiffness matrix of the plate composite structure and substituting it into the frequency-wavenumber domain moving load, the frequency-wavenumber domain vibration response of the plate composite structure is obtained.

[0024] In one specific implementation, in step S5, the frequency-wavenumber domain vibration response of the plate assembly structure is obtained by using the inverse trigonometric function integral transform and the inverse Fourier transform to obtain the time-space domain vibration response of the plate assembly structure, specifically expressed as follows:

[0025]

[0026]

[0027] Equation (7) is the inverse Fourier transform expression of the plate assembly structure, and Equation (8) is the inverse trigonometric function integral transform expression of the plate assembly structure, where W mn (α m ,y,ω n W represents the mathematical expression of the vibration response of a plate composite structure in the frequency-wavenumber domain. m (α m ,y,t r Let w(x,y,t) be the mathematical expression for the vibration response of a plate composite structure in the time-wavenumber domain. r ) is the mathematical expression of the vibration response of a plate composite structure in the time-space domain.

[0028] Compared with the prior art, the present invention has the following beneficial effects:

[0029] This invention derives exact shape functions from exact solutions of element differential equations, independent of the number of elements used in the analysis. Therefore, this invention requires very few elements to solve the dynamic characteristics of large structures, exhibiting high efficiency.

[0030] This invention is the first to employ a small-degree-of-freedom analytical dynamic stiffness method for analytical modeling of plate composite structures. The number of degrees of freedom is directly related to the size of the solution matrix; if the model has N degrees of freedom, then the size of the solution matrix is ​​N. 2 The more degrees of freedom there are, the more complex the model matrix becomes, leading to higher computational time complexity and longer computation time. Finite element methods or other analytical methods limited to simplifying plate composite structures require decomposing the plate composite structure into tens of thousands of regular elements, each with three or more degrees of freedom, to ensure modeling accuracy. The number of degrees of freedom required by this invention is approximately one-thousandth of that required by methods such as the finite element method, enabling analytical modeling of plate composite structures. This reduces the complexity of constructing and solving the model matrix of the plate composite structure using computing equipment, saving computational time costs.

[0031] This invention analyzes the moving load from the time-space domain to the frequency-wavenumber domain, enabling the moving load signal, which is difficult to express mathematically under real-world conditions, to be represented by a simple superposition of a sine wave and a wavenumber, thus simplifying the mathematical expression of the moving load.

[0032] This invention is applicable to modeling the dynamic response of complex plate composite structures caused by complex moving traffic loads, including but not limited to factors such as moving speed, moving acceleration, time-varying amplitude, and superposition of multiple loads. The combination of different factors can enrich the application scenarios in which this invention can play its role.

[0033] Compared with the finite element software ANSYS, under the same computing conditions, this invention achieves similar accuracy and solution results with only 0.24% of the number of elements and 1.2% of the time required by the ANSYS finite element model.

[0034] This invention uses the dynamic stiffness method to determine the stiffness matrix of the structure, which eliminates the need for fine meshing as required by the finite element method. This makes it applicable to complex structural models and speeds up computation time. At the same time, it performs semi-analytical modeling of moving loads, reducing the complexity of setting loading points due to the correlation between the time and spatial dimensions of moving loads. This makes it more suitable for real-time vibration monitoring and analysis of actual engineering structures.

[0035] In addition to the objectives, features, and advantages described above, the present invention has other objectives, features, and advantages. The invention will now be described in further detail with reference to the figures. Attached Figure Description

[0036] The accompanying drawings, which form part of this application, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0037] Figure 1This is a flowchart illustrating a method for analyzing the vibration response of a plate composite structure under moving load according to an embodiment of the present invention.

[0038] Figure 2 This is a schematic diagram of a rectangular plate and its boundary conditions. x, y, and z represent the three directions of the Cartesian coordinate system, l represents the length of the plate in the x-direction, b represents the width of the plate in the y-direction, and h represents the thickness of the plate in the z-direction.

[0039] Figure 3 This is a schematic diagram of the boundary conditions for the displacement of nodes on a non-simply supported rectangular plate. This represents the displacement magnitude of the boundary node in the z-direction at y=0 of the plate element. This represents the displacement magnitude of the boundary corner node at y=0 of the plate element. This represents the magnitude of the nodal displacement in the z-direction at the boundary of the plate element at y=b. This represents the displacement magnitude of the boundary corner node at y=b of the plate element. This represents the nodal force amplitude in the z-direction at the boundary of the plate element at y=0. This represents the force amplitude at the boundary corner node of the plate element at y=0. This represents the nodal force amplitude in the z-direction at the boundary of the plate element at y=b. This represents the force amplitude at the boundary corner node of the plate element at y=b;

[0040] Figure 4 This is a schematic diagram of the boundary conditions for the forces at the non-simply supported edges of a rectangular plate. This represents the magnitude of the nodal displacement in the x-direction at the boundary of the plate element at y=0. This represents the displacement magnitude of the boundary node in the y-direction at y=0 of the plate element. This represents the magnitude of the nodal displacement in the x-direction at the boundary of the plate element at y=b. This represents the magnitude of the nodal displacement in the y-direction at the boundary of the plate element at y=b. This represents the nodal force amplitude in the x-direction at the boundary of the plate element at y=0. This represents the nodal force amplitude in the y-direction at the boundary of the plate element at y=0. This represents the nodal force amplitude in the x-direction at the boundary of the plate element at y=b. This represents the nodal force amplitude in the y-direction at the boundary of the plate element at y=b.

[0041] Figure 5This is a schematic diagram of the assembly of a rectangular plate composite structure; f1 represents the nodal force vector of edge Line 1 of the plate composite structure, d1 represents the nodal displacement vector of edge Line 1 of the plate composite structure, f2 represents the nodal force vector of edge Line 2 of the plate composite structure, d2 represents the nodal displacement vector of edge Line 2 of the plate composite structure, f3 represents the nodal force vector of edge Line 3 of the plate composite structure, d3 represents the nodal displacement vector of edge Line 3 of the plate composite structure, K represents the element of the dynamic stiffness matrix, the superscript of K indicates the plate corresponding to the matrix, and the subscript of K indicates the position of different elements. Indicates the included angle between plate units E1 and E2;

[0042] Figure 6 This is a schematic diagram of a moving load acting on the central axis of the top plate of a plate composite structure; p(x,y,t) represents the moving load function related to the spatial position (x,y) and time t, and the dashed line represents the moving path of the moving load in this embodiment;

[0043] Figure 7 This is a schematic diagram of the cross-section and dimensional symbols of a plate combination structure according to an application embodiment of the present invention; b represents the width of the wing plate, h1 represents the thickness of the top plate and the wing plate, b1 represents the width of the bottom plate, h2 represents the thickness of the bottom plate, h3 represents the thickness of the web plate, θ represents the angle between the top plate and the web plate, and h represents the height of the lower surface of the top plate to the upper surface of the bottom plate.

[0044] Figure 8 This is a schematic diagram illustrating the amplitude variation of a moving load projected into the wavenumber domain in one application embodiment of the present invention; m represents the variation of the moving load in the wavenumber domain described by different wavenumbers;

[0045] Figure 9 This is a schematic diagram illustrating the amplitude variation of a moving load projected into the frequency-wavenumber domain in one application embodiment of the present invention;

[0046] Figure 10 This is a schematic diagram of the vibration response of the plate assembly structure in the frequency domain and wavenumber domain under the action of a moving load in one application embodiment of the present invention;

[0047] Figure 11 This is a schematic diagram of the vibration response of the top plate center of the plate composite structure in the time and space domains under the action of a moving load, according to an application embodiment of the present invention.

[0048] Figure 12 This is a schematic diagram of the vibration response of the center of the bottom plate of the plate composite structure in the time-space domain under the action of a moving load, according to an application embodiment of the present invention. Detailed Implementation

[0049] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings. The specific embodiments described herein are merely illustrative of the present invention and are not intended to limit the present invention.

[0050] Traditional numerical simulation methods for structural dynamics often employ the finite element method (FEM). The FEM essentially uses a weighted residual method or the principle of functional extrema to perform numerical discretization of structures, which means the discretization density of the structure directly affects the computational accuracy.

[0051] Example 1

[0052] This invention provides a method for analyzing the vibration response of a plate composite structure under moving loads. (See [link to relevant documentation]). Figure 1 The method includes the following steps:

[0053] S1. Obtain structural geometric parameters, structural material parameters, and moving load parameters;

[0054] S2. Based on the aforementioned structural geometric parameters and structural material parameters, the dynamic stiffness matrix of the plate composite structure is constructed using the dynamic stiffness method.

[0055] S3. Based on the moving load parameters, establish a semi-analytical model of the moving load, and use trigonometric function integral transform and Fourier transform to project the moving load from the time domain to the space domain to the frequency domain to the wavenumber domain to obtain the frequency domain to wavenumber domain moving load.

[0056] S4. Combine the dynamic stiffness matrix of the plate composite structure with the frequency domain-wavenumber domain moving load to obtain the frequency domain-wavenumber domain vibration response of the plate composite structure.

[0057] S5. Based on the frequency-wavenumber domain vibration response of the plate assembly structure, the time-space domain vibration response of the plate assembly structure is obtained by using the inverse trigonometric function integral transform and the inverse Fourier transform.

[0058] Suppose there exists a rectangular plate in a Cartesian coordinate system with length l in the x-direction, width b in the y-direction, and thickness h. (See [reference]) Figure 2 The plate is supported on two sides parallel to the y-axis (y = 0 and y = b), while the two sides parallel to the x-axis (x = 0 and x = l) can have arbitrary boundary conditions. In-plane vibration of the plate element considers the degrees of freedom (u and v) in the x and y directions. Out-of-plane vibration considers the degrees of freedom (w and Φ) in the z-direction and rotation about the plate boundary. This means that any node at y = 0 and y = b has four degrees of freedom.

[0059] The simply supported boundary conditions for in-plane vibration of a plate element are the normal strain u and the tangential stress N. xy The values ​​are zero, while the tangential strain v and normal stress N are zero. xx Non-zero. The boundary conditions for the simply supported edge of the out-of-plane vibration of the plate element are that the vertical strain w and bending moment M are zero, while the angular displacement θ and transverse strain T are not zero.

[0060] Based on Hamilton's principle and viscous damping theory, the governing differential equations for the in-plane and out-of-plane free vibrations of a flat plate element can be expressed as:

[0061]

[0062] Where u and v represent the in-plane displacement variables of the plate composite structure in the x and y directions, respectively, and w represents the out-of-plane displacement variable of the plate composite structure in the z direction. All displacement variables are ternary functions of x, y, and t, where x and y represent displacement changes related to spatial position, and t represents displacement changes related to time. This represents the second-order partial derivative of the displacement variable u with respect to the x-direction. This represents the second-order partial derivative of the displacement variable u with respect to the y-direction. Let v represent the second-order partial derivatives of the displacement variable v with respect to the x and y directions, respectively. This represents the second-order partial derivative of the displacement variable u with respect to the time variable t. This represents the first-order partial derivative of the displacement variable u with respect to the time variable t. This represents the second-order partial derivative of the displacement variable v with respect to the x-direction. This represents the second-order partial derivative of the displacement variable v with respect to the y-direction. Let represent the second-order partial derivatives of the displacement variable u with respect to the x and y directions, respectively. This represents the second-order partial derivative of the displacement variable v with respect to the time variable t. This represents the first-order partial derivative of the displacement variable v with respect to the time variable t. This represents the fourth-order partial derivative of the displacement variable w with respect to the x-direction. Let w represent the fourth-order partial derivatives of the displacement variable w with respect to the x and y directions, respectively. This represents the fourth-order partial derivative of the displacement variable w with respect to the y-direction. This represents the second-order partial derivative of the displacement variable w with respect to the time variable t. E represents the first-order partial derivative of the displacement variable w with respect to the time variable t, μ is the Poisson's ratio of the material, ρ is the material density, h is the element section thickness, and α0 is the external viscous damping coefficient of the structure. * It is the dynamic elastic modulus, and its expression is:

[0063] E * =E(1+iβ0ω) (2)

[0064] Where E is Young's modulus, i is the imaginary unit, β0 is the internal viscous damping coefficient of the structure, ω represents the angular frequency of vibration, and D is the damping force after considering the damping force. * Bending stiffness is expressed as:

[0065] D * =E * h 3 / 12(1-μ 2 (3).

[0066] The conversion formula between the damping parameters (α0 damping and β0 damping) of the dynamic stiffness method and Rayleigh damping is as follows:

[0067]

[0068] See Figure 2 Considering that the plate element has simply supported boundary conditions in the x and z directions at y = 0 and y = l, the vibration displacement of the plate element can be written as:

[0069]

[0070] Where, α m = mπ / l, where m represents the wave number of the plate element in the x-direction, and is a positive integer. U m V m and W m This represents the displacement component in the m-th wavenumber domain. Since there is no rigid body displacement in the y-direction of the spatial bridge, m > 0.

[0071] Related research indicates that the structure undergoes simple harmonic motion during free vibration. Therefore, it is assumed that the variation of the m-th wavenumber domain displacement with time t is a harmonic function of the angular frequency ω, i.e., the governing differential equation is transformed from the time domain to the frequency domain. According to the inverse discrete Fourier transform, the expression for the m-th wavenumber domain displacement is:

[0072]

[0073] and These represent the discrete circumfrequency ω of the nth harmonic in the frequency domain. n The displacement amplitude corresponding to (rad / s). ω n Defined as ω n = nΔω (n = 0, 1, ..., N-1), where Δω represents the frequency interval (fundamental frequency), and its expression is Δω = 2π / T. T represents the desired time window, and N represents the number of frequency samples within the specified time range.

[0074] Under simply supported boundary conditions, the relationship between nodal forces and nodal displacements on the non-simply supported edges of a plate element is expressed as follows:

[0075]

[0076] Where, N yx N represents the in-plane tangential stress. yy T represents the normal stress in the plane. y M represents the out-of-plane vertical stress. yy Indicates bending moment, This represents the first-order partial derivative of the displacement variable v with respect to the x-direction. This represents the first-order partial derivative of the displacement variable u with respect to the y-direction. This represents the first-order partial derivative of the displacement variable v with respect to the y-direction. This represents the first-order partial derivative of the displacement variable u with respect to the x-direction. This represents the third partial derivative of the displacement variable w with respect to the y-direction. This represents the second partial derivative of the displacement variable w with respect to the first partial derivative in the y-direction with respect to the x-direction. This represents the second-order partial derivative of the displacement variable w with respect to the y-direction. G represents the second-order partial derivative of the displacement variable w with respect to the x-direction. * G represents the shear modulus. * =E * / (2(1+μ)).

[0077] See Figure 3 The nodal displacements and nodal forces of the plate element's non-simply supported edges are defined with the following symbols:

[0078]

[0079] The sign setting in formula (13) is only for the convenience of subsequent matrix-vector representation. and These represent the discrete circumfrequency ω of the nth harmonic in the frequency domain. n (rad / s) corresponds to the displacement amplitude of the x, y, z directions and the corner node; 0 indicates the boundary of the plate element at y=0, and the subscript of its corresponding symbol is set to 1; b indicates the boundary of the plate element at y=b, and the subscript of its corresponding symbol is set to 2.

[0080] Based on the aforementioned process and the formulas above, the dynamic stiffness matrix of the plate element is derived as follows:

[0081]

[0082] Wherein, the superscript i represents the in-plane degree of freedom of the plate element, the superscript o represents the out-of-plane degree of freedom of the plate element, and j represents the degree of freedom number of the dynamic stiffness matrix of the plate element, j = 1, 2, ..., 6.

[0083] See Figure 5 In the global coordinate system, assuming two plate elements form a plate composite structure, the overall dynamic stiffness matrix of this plate composite structure can be represented as follows by sharing node edges:

[0084]

[0085] in, This represents the nodal force amplitude in the x-direction at edge Line 1 of plate E1. This represents the nodal force amplitude in the y-direction at edge Line 1 of plate E1. This represents the nodal force amplitude in the z-direction at edge Line 1 of plate E1. This represents the force amplitude at the corner node of edge Line 1 of plate E1, as shown in the figure. This represents the nodal force amplitude in the x-direction at Line 2, the intersection edge of plates E1 and E2. This represents the nodal force amplitude in the y-direction at Line 2, the intersection edge of plates E1 and E2. This represents the nodal force amplitude in the z-direction at Line 2, the intersection edge of plates E1 and E2. This represents the force amplitude at the corner node at Line 2, the intersection edge of plates E1 and E2, as shown in the diagram. This represents the nodal force amplitude in the x-direction at edge Line 3 of plate E2. This represents the nodal force amplitude in the y-direction at edge Line 3 of plate E2. This represents the nodal force amplitude in the z-direction at edge Line 3 of plate E2. This represents the force amplitude at the corner node of edge Line 3 of plate E2, as shown in the diagram. This represents the magnitude of the nodal displacement in the x-direction at edge Line 1 of plate E1. This represents the magnitude of the nodal displacement in the y-direction at edge Line 1 of plate E1. This represents the magnitude of the nodal displacement in the z-direction at edge Line 1 of plate E1. This represents the displacement magnitude at the corner node at edge Line 1 of plate E1, as shown in the diagram. This represents the magnitude of the nodal displacement in the x-direction at Line 2, the intersection edge of plates E1 and E2. This represents the magnitude of the nodal displacement in the y-direction at Line 2, the intersection edge of plates E1 and E2. This represents the magnitude of the nodal displacement in the z-direction at Line 2, the intersection edge of plates E1 and E2. This represents the displacement magnitude at the corner node Line 2, the intersection edge of plates E1 and E2, as shown in the figure. This represents the magnitude of the nodal displacement in the x-direction at edge Line 3 of plate E2. This represents the magnitude of the nodal displacement in the y-direction at edge Line 3 of plate E2. This represents the magnitude of the nodal displacement in the z-direction at edge Line 3 of plate E2. This represents the displacement magnitude at the corner node at edge Line 3 of plate E2, as shown in the diagram. △ represents the dynamic stiffness matrix of plate element E1, Ο represents the dynamic stiffness matrix of plate element E2, and Δ+Ο represents the matrix combination of the two plate elements at Line 2, where the corresponding degrees of freedom of the matrices are added together.

[0086] The unique characteristic of moving loads lies in the change of their spatial location over time, and it is precisely this close spatiotemporal correlation that makes their mathematical expression more complex than that of common static loads. To accurately describe this spatiotemporal correlation, this paper introduces the Dirichlet function δ to simulate moving loads, and its mathematical expression is:

[0087]

[0088] Where g(x,t) represents the time-space domain variation process of the continuous impact of the moving load on the structure.

[0089] If the plate composite structure is subjected to a moving load, the expression for the moving load in the time-space domain is:

[0090]

[0091] Where p(x,y,t) r Let be the mathematical expression of the moving load in the time-space domain, where x and y represent the changes in the spatial state of the moving load, and t is the value of t. r This indicates the correlation between the moving load and the time dimension, where the subscript r represents the number of time-domain sampling points for the moving load, and t... d The duration of the moving load applied to the plate assembly structure is typically expressed as the length *l* of the plate assembly structure in the direction of the moving load and the velocity *v* of the moving load. p The ratio of P z (t r ) represents the magnitude of the moving load as a function of time t r The expression for the change.

[0092] Trigonometric integral transform can convert a moving load from the spatial domain to the wavenumber domain. The moving load in the wavenumber domain can be expressed as:

[0093]

[0094] Where, p m (α m ,y,t r α is the mathematical expression of the moving load in the time-wavenumber domain. m The wave number in the length direction of the plate assembly structure.

[0095] The Discrete Fourier Transform can be used to transform a moving load from the time domain to the frequency domain. The moving load can be represented in the frequency-wavenumber domain as follows:

[0096]

[0097] Where, p mn (α m ,y,ω n ω is the mathematical expression of the moving load in the frequency-wavenumber domain. n denoted by , where represents the discrete circular frequency value of the nth harmonic in the frequency domain, and r represents the number of time-domain sampling points of the moving load.

[0098] In the global coordinate system, the dynamic stiffness model of the plate composite structure under moving load is expressed as:

[0099] f g =K g (α m ,ω n )d g (17)

[0100] Here, the superscript g denotes the global coordinate system. Since the non-simply supported edges of the plate element have four degrees of freedom, for the moving load applied to the plate composite structure, the nodal force vector f g The expression is:

[0101]

[0102] Where n1 = 1, 2, ..., (I-1), n2 = 1, 2, ..., (I-1), and n1 + n2 = I-1, where I represents the total number of plate units in the spatial composite bridge structure.

[0103] Therefore, the frequency-wavenumber domain vibration response of a spatial composite bridge structure can be expressed as:

[0104] d g =K g (α m ,ω n ) -1 f g (19)

[0105] Wherein, the nodal displacement vector d g Represented as:

[0106]

[0107] The time-space vibration response of the plate composite structure is obtained using the inverse trigonometric integral transform and the inverse Fourier transform. The specific expression is as follows:

[0108]

[0109]

[0110] Equation (7) is the inverse Fourier transform expression of the plate assembly structure, and Equation (8) is the inverse trigonometric function integral transform expression of the plate assembly structure, where W mn (α m ,y,ω n W represents the mathematical expression of the vibration response of a plate composite structure in the frequency-wavenumber domain. m (α m ,y,t r Let w(x,y,t) be the mathematical expression for the vibration response of a plate composite structure in the time-wavenumber domain. r ) is the mathematical expression of the vibration response of a plate composite structure in the time-space domain.

[0111] The vibration response of the plate assembly structure under moving load was obtained.

[0112] The solution of this invention uses the dynamic stiffness method to determine the stiffness matrix of the structure, which does not require fine meshing like the finite element method. It can be applied to complex structural models and speeds up the calculation time. At the same time, it performs semi-analytical modeling of moving loads, which reduces the complexity of setting loading points due to the correlation between the time and space dimensions of moving loads, making it more suitable for real-time vibration monitoring and analysis of actual engineering structures.

[0113] To more clearly illustrate the specific implementation process of the present invention, the present invention will be further described in detail below with reference to application embodiments.

[0114] Step 1: Obtain the structural geometric parameters, structural material parameters, and moving load parameters of the plate assembly.

[0115] See Figure 7 The structural geometric parameters of the plate combination structure in the application embodiment of the present invention are as follows: wing plate width b = 3m, top plate and wing plate thickness h1 = 0.35m, bottom plate width b1 = 7.3m, bottom plate thickness h2 = 0.3m, web plate thickness h3 = 0.5m, the angle between the top plate and the web plate θ = 67°, the height from the lower surface of the top plate to the upper surface of the bottom plate h = 2.28m, and the overall length of the plate combination structure l = 30m.

[0116] The structural material parameters of the plate composite structure applied in this invention are as follows: Young's modulus E = 34.5 GPa, Poisson's ratio μ = 0.2, and density ρ = 2600 kg / m³. 3 The external viscous damping coefficient α0 = 0.1, and the internal viscous damping coefficient β0 = 0.01.

[0117] Referring to the standard speed of my country's Fuxing bullet train (350 km / h and maximum speed of 400 km / h), the application example of this invention sets the speed of the moving load to v. p =360km / h, constant amplitude P z=24kN, see motion path Figure 6 .

[0118] Step 2: Based on the structural geometric parameters and material parameters, construct the dynamic stiffness matrix of the plate composite structure using the dynamic stiffness method. Select a range for the wave number m. Calculate the duration t of the moving load based on the length of the plate composite structure and the moving load velocity. d Set the time step, calculate the number of sampling points r and the discrete value of the harmonic angular frequency n. Based on the values ​​of m and n, substitute the parameters to obtain several corresponding dynamic stiffness matrices of the plate combination structure.

[0119] Step 3: Based on the aforementioned moving load parameters, see... Figure 8 First, the moving load is projected from the spatial domain to the wavenumber domain using trigonometric integral transform. (See [link to relevant documentation]). Figure 9 Then, the moving load is projected from the time domain to the frequency domain using Fourier transform.

[0120] Step 4: Combine the dynamic stiffness matrix of the plate composite structure with the frequency-wavenumber domain moving load to obtain the frequency-wavenumber domain vibration response of the plate composite structure. See [link to relevant documentation]. Figure 10 .

[0121] Step 5: Based on the frequency-wavenumber domain vibration response of the plate assembly structure, obtain the time-space domain vibration response of the plate assembly structure using inverse trigonometric function integral transform and inverse Fourier transform. See [link to relevant documentation]. Figure 11 and Figure 12 The overall length of the slab composite structure is l = 30m. The vibration pickup point is 15m from any end of the top slab of the slab composite structure along the length direction, and the vibration pickup point is 25m from any end of the bottom slab of the slab composite structure along the length direction.

[0122] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions and substitutions can be made without departing from the inventive concept, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. A method for analyzing the vibration response of a plate composite structure under moving load, characterized in that, Includes the following steps: S1. Obtain structural geometric parameters, structural material parameters, and moving load parameters; S2. Based on the aforementioned structural geometric parameters and structural material parameters, the dynamic stiffness matrix of the plate composite structure is constructed using the dynamic stiffness method. S3. Based on the moving load parameters, establish a semi-analytical model of the moving load, and use trigonometric function integral transform and Fourier transform to project the moving load from the time domain to the space domain to the frequency domain to the wavenumber domain to obtain the frequency domain to wavenumber domain moving load. In step S3, the time-space domain moving load is analyzed in the frequency-wavenumber domain, and the specific expression includes: （4） （5） （6） Equation (4) is the mathematical expression for the moving load, Equation (5) is the trigonometric function integral transform expression for the moving load, and Equation (6) is the Fourier transform expression for the moving load, where, This is the mathematical expression of the moving load in the time-space domain, where x and y represent the changes in the spatial state of the moving load, and t... r This indicates the correlation between the moving load and the time dimension, where the subscript r represents the number of time-domain sampling points for the moving load, and t... d This represents the duration of the moving load applied to the plate assembly structure, that is, the length *l* of the plate assembly structure in the direction of the moving load and the velocity *v* of the moving load. p The ratio, This indicates that the magnitude of the moving load varies with the time dimension t. r The change expression, and Indicated by Dirichlet function Describe the spatial state (x, y) of the moving load as a function of time dimension t r The change expression, For the mathematical expression of the moving load in the time-wavenumber domain, α m This indicates the wave number along the length of the plate composite structure. ω represents the mathematical expression of the moving load in the frequency-wavenumber domain. n This represents the discrete circular frequency value of the nth harmonic in the frequency domain, where N represents the number of frequency samples within the specified time range. S4. Combine the dynamic stiffness matrix of the plate composite structure with the frequency domain-wavenumber domain moving load to obtain the frequency domain-wavenumber domain vibration response of the plate composite structure. S5. Based on the frequency-wavenumber domain vibration response of the plate assembly structure, the time-space domain vibration response of the plate assembly structure is obtained by using the inverse trigonometric function integral transform and the inverse Fourier transform.

2. The method for analyzing the vibration response of a plate composite structure under moving load according to claim 1, characterized in that, The governing differential equation of the dynamic stiffness method considers structural Rayleigh damping, and its specific expression is as follows: （1） Where u and v represent the in-plane displacement variables of the plate composite structure in the x and y directions, respectively, and w represents the out-of-plane displacement variable of the plate composite structure in the z direction. All displacement variables are ternary functions of x, y, and t, where x and y represent displacement changes related to spatial position, and t represents displacement changes related to time. This represents the second-order partial derivative of the displacement variable u with respect to the x-direction. This represents the second-order partial derivative of the displacement variable u with respect to the y-direction. Let v represent the second-order partial derivatives of the displacement variable v with respect to the x and y directions, respectively. This represents the second-order partial derivative of the displacement variable u with respect to the time variable t. This represents the first-order partial derivative of the displacement variable u with respect to the time variable t. This represents the second-order partial derivative of the displacement variable v with respect to the x-direction. This represents the second-order partial derivative of the displacement variable v with respect to the y-direction. Let represent the second-order partial derivatives of the displacement variable u with respect to the x and y directions, respectively. This represents the second-order partial derivative of the displacement variable v with respect to the time variable t. This represents the first-order partial derivative of the displacement variable v with respect to the time variable t. This represents the fourth-order partial derivative of the displacement variable w with respect to the x-direction. Let w represent the fourth-order partial derivatives of the displacement variable w with respect to the x and y directions, respectively. This represents the fourth-order partial derivative of the displacement variable w with respect to the y-direction. This represents the second-order partial derivative of the displacement variable w with respect to the time variable t. denoted by w, which represents the first-order partial derivative of the displacement variable w with respect to the time variable t; μ is the Poisson's ratio of the material; ρ is the material density; h is the thickness of the unit section; and α0 is the external viscous damping coefficient of the structure. It is the dynamic elastic modulus, and its expression is: （2） Where E is Young's modulus, i is the imaginary unit, β0 is the internal viscous damping coefficient of the structure, and ω represents the angular frequency of vibration, after considering the damping force. Bending stiffness is expressed as: （3）。 3. The method for analyzing the vibration response of a plate composite structure under moving load as described in claim 1, characterized in that, In step S4, the dynamic stiffness matrix of the plate composite structure and the frequency-wavenumber domain moving load are combined to obtain the frequency-wavenumber domain vibration response of the plate composite structure, including: By utilizing the dynamic stiffness matrix of the plate composite structure and the frequency-wavenumber domain moving load, and after inverting the dynamic stiffness matrix of the plate composite structure and substituting it into the frequency-wavenumber domain moving load, the frequency-wavenumber domain vibration response of the plate composite structure is obtained.

4. The method for analyzing the vibration response of a plate composite structure under moving load as described in claim 1, characterized in that, In step S5, the frequency-wavenumber domain vibration response of the plate assembly structure is obtained by using inverse trigonometric function integral transform and inverse Fourier transform to obtain the time-space domain vibration response of the plate assembly structure. The specific expression is as follows: （7） （8） Equation (7) is the inverse Fourier transform expression of the plate composite structure, and Equation (8) is the inverse trigonometric function integral transform expression of the plate composite structure, wherein, This is the mathematical expression for the vibration response of a plate composite structure in the frequency domain and wavenumber domain. This is the mathematical expression for the vibration response of a plate composite structure in the time-wavenumber domain. This is the mathematical expression of the vibration response of a plate composite structure in the time-space domain.

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