A method for dynamic analysis of composite plates of arbitrary shape in a humid heat environment

Abstract

The present application belongs to the technical field of aerospace structure and structural dynamics modeling, and particularly relates to a kind of dynamics analysis method of arbitrary shape composite material plate under hygrothermal environment, the geometric profile of arbitrary shape composite material plate is obtained by image calibration, geometric feature recognition and boundary extraction;Plate domain is divided into multiple substructures, and Gaussian integration points are generated in the substructure;A discrete orthogonal admissible displacement function family suitable for each substructure is constructed;The kinetic energy, potential energy, hygrothermal potential energy and boundary potential energy of each substructure are assembled, and the natural frequency and modal shape of the structure are obtained by solving;The dynamics response of the structure under external excitation is solved by matrix condensation and modal superposition.The present application overcomes the deficiencies of the prior art, such as complex modeling of complex geometric composite material plate, poor boundary adaptability, and difficulty in considering hygrothermal effects, and can improve the efficiency, accuracy and engineering applicability of modeling of arbitrary shape composite material plate under complex hygrothermal environment.

Description

Technical Field

[0001] This invention belongs to the field of aerospace structure and structural dynamics modeling technology, specifically relating to a dynamic analysis method for composite material plates of arbitrary shape under humid and hot conditions. Background Technology

[0002] Composite material plate structures are widely used in aerospace and intelligent equipment fields due to their advantages such as high specific strength, high specific stiffness, and strong designability. In practical applications, due to constraints such as functional integration, installation space, and working conditions, composite material plate components often contain irregular openings, complex outer contours, and multiple local boundary constraints. The geometric complexity of such irregular plate structures can significantly change the stiffness distribution and dynamic response of the structure, while traditional analytical methods developed for regular plate structures are difficult to apply directly.

[0003] Furthermore, humid and hot environments can lead to the degradation of the material properties of composite materials and induce internal stresses, thereby affecting the dynamic behavior of the structure and significantly increasing the risk of structural damage. Accurately solving the dynamic characteristics of such irregular composite material plates is of great significance for assessing structural safety margins, ensuring functional reliability, and carrying out vibration reduction, noise reduction, and engineering design.

[0004] Currently, the dynamic analysis of complex geometric plate structures mainly employs numerical techniques such as the finite element method, isogeometric analysis methods, and meshless methods. While these methods are applicable to regular objects or simple working conditions, they still have significant limitations when dealing with composite material plates of arbitrary shapes. On one hand, the finite element method requires extensive mesh refinement when analyzing structures such as cracked plates, and the solution accuracy is highly dependent on mesh quality. On the other hand, isogeometric analysis methods have high modeling complexity and time-consuming preprocessing. Furthermore, meshless methods generate highly dense system matrices, resulting in large memory consumption.

[0005] More importantly, the above methods all require cumbersome preprocessing, which makes it difficult to meet the needs of rapid dynamic modeling of irregular plate structures in engineering applications. Furthermore, the general modeling capability for composite material plates of arbitrary shapes in complex environments is still limited, and there are common problems such as cumbersome modeling process, poor boundary adaptability, difficulty in taking into account damp heat effects, low computational efficiency, and difficulty in balancing accuracy and efficiency.

[0006] In summary, existing dynamic analysis methods suffer from bottlenecks when dealing with composite material plates of arbitrary shapes and complex geometries, including difficulties in capturing geometric features, insufficient adaptability to arbitrary boundary constraints, and the inability to incorporate humid and thermal environments into a unified modeling framework. Engineering practice urgently needs to develop an efficient dynamic analysis method that can quickly capture complex geometric features, adapt to arbitrary boundary constraints, and take into account humid and thermal effects. Summary of the Invention

[0007] To address the technical problems of cumbersome modeling of composite material plates of arbitrary shapes, poor adaptability to arbitrary boundaries, difficulty in incorporating hygrothermal effects into a unified modeling framework, and low computational efficiency in existing technologies, this invention provides a dynamic analysis method for composite material plates of arbitrary shapes under hygrothermal conditions.

[0008] This invention is implemented by providing a dynamic analysis method for composite material plates of arbitrary shape under humid and hot conditions, comprising the following steps: Step 1: Acquire image data of the composite material plate structure of arbitrary shape, as well as material parameters, layup configuration, boundary constraints, and ambient temperature and humidity parameters. Obtain a two-dimensional geometric model of the composite material plate of arbitrary shape through image calibration, geometric feature recognition, and boundary extraction. Step 2: Based on the two-dimensional geometric model obtained in Step 1, the plate domain of the composite material plate of arbitrary shape is divided into multiple substructures with convex polygons, and each substructure and its boundary are numbered to determine the common edge coupling relationship between the substructures. Step 3: Arrange discrete Gaussian integration points within each substructure using mapping techniques to establish a normalized mapping domain for the substructure; Step 4: Based on the normalized mapping domain of each substructure, construct a family of discrete orthogonal allowable displacement functions applicable to each substructure using an improved Schmidt orthogonalization method; Step 5: Substitute the material parameters, layup configuration, boundary constraints, ambient temperature and humidity parameters obtained in Step 1, and the discrete orthogonal allowable displacement function family obtained in Step 4 into the mechanical control equations of the arbitrary-shaped composite material plate. Calculate the kinetic energy, potential energy, hygrothermal potential energy, and boundary potential energy of each substructure. Apply displacement coordination constraints and arbitrary boundary constraints between substructures using virtual spring technology to assemble the total energy functional of the arbitrary-shaped composite material plate. Solve the total energy functional based on the Rayleigh-Ritz method to obtain the natural frequencies and mode shapes of the arbitrary-shaped composite material plate. Step 6: Based on the natural frequencies and mode shapes obtained in Step 5, establish and solve the dynamic differential equations of the arbitrary-shaped composite material plate through matrix condensation and modal superposition to obtain the dynamic response of the arbitrary-shaped composite material plate under external excitation, thus completing the dynamic analysis.

[0009] Preferably, step 1 specifically includes: Step 1.1: Lay the composite material plate of arbitrary shape flat on a horizontal surface, place a scale with a calibrated direction on the plate surface, align one side of the scale with the fiber direction of the composite material, use an imaging device to acquire images perpendicular to the plate surface, and transmit the acquired images to the image correction model. At the same time, obtain the material parameters, layup configuration, boundary constraints, ambient temperature and humidity parameters of the composite material plate of arbitrary shape. Step 1.2: Using the scale bar as a reference, perform perspective correction, scale restoration, and irrelevant background removal on the acquired image to obtain a front view image of the composite material plate of arbitrary shape according to its original size; Step 1.3: Binarize the front view image, and mark the inner and outer regions of the composite material plate of arbitrary shape as black and white respectively to obtain a binary mask; Step 1.4: Identify and extract the boundary contour of the arbitrary-shaped composite material plate using a binary mask, approximate the boundary contour using polyline segments, convert the geometric information into a two-dimensional geometric file, and obtain the two-dimensional geometric model of the arbitrary-shaped composite material plate.

[0010] Preferably, step 2 specifically includes: Step 2.1: Based on the two-dimensional geometric model, the plate domain of the composite material plate of arbitrary shape is initially divided into multiple substructures; Step 2.2: Perform concavity / convexity test on each substructure. When the substructure obtained by division is a concave polygon, perform a second subdivision at the maximum concave angle of the concave polygon, and repeat the process until all substructures are converted into convex polygons. Step 2.3: Number all substructures and substructure boundaries of the convex polygons, record the coordinate data of the corresponding boundary endpoints, and determine the common edge coupling relationship between substructures based on the recorded numbers and coordinate data.

[0011] Preferably, step 3 specifically includes: Step 3.1: Use a combination of rectangles or triangles and quadrilaterals to completely cover the integration domain of each substructure with as few filling units as possible; Step 3.2: Using a high-order numerical integration algorithm that combines the least squares Newton scheme and the node elimination method, discrete Gaussian integration points and their corresponding integration weights are generated within the filled cells. Step 3.3: Establish a local coordinate system at the geometric centroid of each substructure, determine the longest side of the substructure in each direction along the coordinate axis, use the longest side as the normalization scale, and map the substructure to the normalized square domain to obtain the normalized mapping domain of the substructure.

[0012] Further preferably, step 4 specifically includes: Step 4.1: Within the normalized mapping domain, construct one-dimensional discrete displacement polynomials along two directions respectively, and construct discrete displacement basis functions through tensor product, where the initial displacement function along the free boundary direction is taken as a constant; Step 4.2: Using the discrete displacement basis function as the original basis function, define the discrete inner product by combining the integral weights of the discrete Gaussian integral points, use the improved Schmidt orthogonalization method to eliminate non-orthogonal components term by term, and perform normalization to obtain the discrete orthogonal displacement function. Step 4.3: Map the discrete orthogonal displacement functions back to the original physical coordinate system of the substructure to obtain a family of discrete orthogonal allowable displacement functions applicable to each substructure.

[0013] Preferably, step 5 specifically includes: Step 5.1: Based on the classical Kirchhoff thin plate theory, substitute the material parameters, layup configuration, boundary constraints, ambient temperature and humidity parameters obtained in Step 1.1 and the discrete orthogonal allowable displacement function family applicable to each substructure obtained in Step 4.3 into the mechanical control equation of the composite material plate of arbitrary shape, and calculate the potential energy and kinetic energy of each substructure. Step 5.2: Calculate the resultant force in the hygrothermal surface of the composite material plate of arbitrary shape based on the ambient temperature and humidity parameters, and establish the hygrothermal potential energy of each substructure; Step 5.3: Using virtual spring technology, apply displacement coordination constraints on the common edge of the substructures and apply arbitrary stiffness constraints on the external boundary to establish the coupling potential energy and boundary constraint potential energy between the substructures; Step 5.4: Assemble the potential and kinetic energies of each substructure obtained in Step 5.1, the wet heat potential energy obtained in Step 5.2, and the coupling potential energy and boundary constraint potential energy obtained in Step 5.3 into a total energy functional of the composite material plate of arbitrary shape. Based on the Rayleigh-Ritz method, take the partial derivatives of each Ritz parameter of the total energy functional and set the partial derivatives of each Ritz parameter to zero to obtain the mass matrix and stiffness matrix of the structure and derive the eigenvalue equation. Solve the eigenvalue equation through matrix decomposition algorithm to obtain the natural frequencies and mode shapes of the composite material plate of arbitrary shape.

[0014] Further preferably, step 6 specifically includes: Step 6.1: Based on the natural frequencies and mode shapes, the mass matrix and stiffness matrix are reduced in order and normalized by matrix condensation and modal superposition. The dynamic differential equations of the composite material plate of arbitrary shape are established by combining the Lagrange equation. Step 6.2: By combining coordinate transformation with modal orthogonality, the dynamic differential equations are decoupled into the modal coordinate system, so that the mass matrix is ​​converted into an identity matrix and the stiffness matrix is ​​converted into a diagonal matrix form; Step 6.3: Determine the excitation force vector projected onto each modal component based on the external excitation, and solve the decoupled dynamic differential equation using the transfer function method or Runge-Kutta method to obtain the amplitude-frequency response curve or time-domain dynamic response of the composite material plate of arbitrary shape; Step 6.4: Output the dynamic response results of the composite material plate of arbitrary shape under external excitation to complete the dynamic analysis.

[0015] Compared with the prior art, the advantages of the present invention are as follows: 1. By introducing image calibration, geometric feature recognition and boundary extraction technologies, this invention can quickly capture the complex geometric features of composite material plates of arbitrary shapes and automatically generate two-dimensional geometric models. This overcomes the shortcomings of existing technologies, such as cumbersome modeling of complex geometric structures and large preprocessing workload, and is conducive to improving modeling efficiency. 2. By dividing the plate domain into convex polygonal substructures and arranging discrete Gaussian integration points using mapping technology, this invention can achieve high-precision solution of the energy integral of each substructure with fewer integration nodes, which is beneficial to reducing the system's degrees of freedom and improving computational convergence and solution efficiency. 3. The present invention uses an improved Schmidt orthogonalization method to construct a family of discrete orthogonal allowable displacement functions applicable to each substructure. This can eliminate the linear dependence between higher-order basis functions, improve the matrix condition number, accelerate the convergence of eigenvalue solutions, and reduce the risk of numerical divergence caused by non-orthogonality when higher-order truncation is performed. 4. This invention achieves displacement coordination and arbitrary boundary constraints between substructures through virtual spring technology. It can flexibly simulate various complex boundary constraint conditions by adjusting the stiffness of virtual springs of different positions and types, thereby improving the adaptability of the method to arbitrary boundary conditions. 5. This invention incorporates humid heat potential energy into the Rayleigh-Ritz modeling framework and solves the dynamic response through matrix condensation and modal superposition, forming a complete technical process from image input to response output, which can provide data support for the dynamic evaluation and engineering design of composite material plates of arbitrary shapes under complex humid heat environments. Attached Figure Description

[0016] Figure 1 This is a schematic diagram illustrating the application of dynamic analysis of composite material plates with arbitrary shapes in engineering structures provided by the present invention. Figure 2 This is a schematic diagram of the dynamic analysis method for composite material plates of arbitrary shape under humid and hot conditions provided by the present invention. Figure 3 This is a schematic diagram of the image recognition and geometric model generation process provided by the present invention; Figure 4 This is a schematic diagram of the substructure division and numbering provided by the present invention; Figure 5 This is a schematic diagram of the substructure integration point mapping provided by the present invention; Figure 6 This is a schematic diagram illustrating the construction and normalization mapping process of the orthogonal allowable displacement function provided by the present invention. Figure 7 This is a schematic diagram of the virtual spring technology provided by the present invention; Figure 8 This is a schematic diagram illustrating an example of modal shape and natural frequency calculation provided by the present invention; Figure 9This is a schematic diagram of the results of the modal shape and natural frequency calculation provided by the present invention; Figure 10 This is a schematic diagram of the dynamic response results under humid and hot conditions provided by the present invention. Detailed Implementation

[0017] The technical solutions of this invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are merely some embodiments of this invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without inventive effort are within the scope of protection of this invention.

[0018] like Figure 1 , Figure 2 As shown, this invention provides a dynamic analysis method for composite material plates of arbitrary shapes under humid and hot conditions, using a laminated composite material plate structure with an irregular internal cavity as an example. This composite material plate consists of several layers of fiber-reinforced composite material layup, with complex geometric features in both its outer contour and internal cavity. It operates in a humid and hot environment, requiring the analysis of its natural frequencies, mode shapes, and dynamic response using the method of this invention.

[0019] Step 1: Image Recognition and Geometric Model Generation like Figure 3 As shown, in step 1.1, the composite material plate of arbitrary shape is laid as flat as possible on a horizontal surface. A scale ruler is placed on the plate surface for subsequent image scaling and automatic calibration, and one side of the scale ruler is aligned with the fiber direction of the composite material as much as possible to provide a reference for determining the layup angle. Images are acquired approximately perpendicular to the plate surface using a smartphone or other imaging device, and the acquired images are wirelessly transmitted to the image correction model. At the same time, the material parameters, layup configuration, boundary constraints, and ambient temperature and humidity parameters of the composite material plate of arbitrary shape are obtained.

[0020] In step 1.2, using the scale position as a reference, perspective correction, scale restoration, and irrelevant background removal are performed on the acquired image to obtain a front view image of the composite material plate at its original size.

[0021] In step 1.3, the front view image is binarized, and the inner and outer regions of the plate are marked as black and white, respectively, to obtain a binary mask.

[0022] In step 1.4, the boundary contour of the structure is identified and extracted using a binary mask, the boundary is approximated using polyline segments, and the geometric information is converted into a two-dimensional geometric file (such as a DXF or STP file) to obtain the two-dimensional geometric model of the composite material plate for subsequent modeling and verification analysis.

[0023] Step 2: Substructure Partitioning like Figure 4 As shown, in step 2.1, based on the two-dimensional geometric model, the plate domain of the arbitrary-shaped composite material plate is initially divided into multiple substructures. When the arbitrary-shaped composite material plate is divided into convex polygonal substructures of similar size under possible conditions, more efficient convergence acceleration and higher computational efficiency can be obtained.

[0024] In step 2.2, the concavity / convexity of each substructure is checked. If the substructure obtained from the initial subdivision is a concave polygon, it needs to be further subdivided. The secondary subdivision is usually performed at the largest concave angle of the concave polygon to improve the subdivision efficiency, and this process is repeated until all substructures are transformed into convex polygons.

[0025] In step 2.3, all substructures and their boundaries are numbered, and the coordinate data of the corresponding boundary endpoints are recorded. The recorded numbers and coordinate data provide input for the location of substructures, the identification of common edge coupling relationships, and the subsequent automatic assembly of the overall matrix.

[0026] Step 3: Mapping the Integral Points and Constructing the Normalized Mapping Domain like Figure 5 As shown, due to the irregular geometry of the substructure, analytical integration of the corresponding integration domain is complex and inefficient. To overcome this difficulty, in step 3.1, a combination of rectangles or triangles and quadrilaterals is used to completely cover the substructure integration domain with as few filling units as possible, thereby reducing the number of integration points and improving integration efficiency while maintaining integration accuracy.

[0027] In step 3.2, a high-order numerical integration algorithm combining the least squares Newton scheme and node elimination is used to generate discrete Gaussian integration points and their corresponding integration weights within the filled cells. This algorithm is applicable to one-dimensional line segments and two-dimensional triangular and quadrilateral domains, and can generate high-precision integration sequences with fewer integration nodes. For polynomial displacement functions, it can maintain complete integration accuracy up to order 50.

[0028] like Figure 6 As shown, in step 3.3, a local coordinate system is established at the geometric centroid of the substructure. In this local coordinate system, along... y The short and long sides of the directional coordinate axes after division are denoted as follows: and ,along x The short and long sides of the directional coordinate axes after division are denoted as follows: and Using the longer sides in both directions and As a normalization scale, the substructure is mapped to a size of The normalized square domain. After mapping, the substructure domain is completely covered by the normalized basis function domain.

[0029] Step 4: Construct a family of dedicated discrete orthogonal admissible displacement functions In step 4.1, the first i The displacement field of each substructure is represented as follows: ; In the formula, t Indicates time, e It is the base of the natural logarithm. The imaginary unit, The natural angular frequency of a composite material plate of arbitrary shape. NTN The number of truncated terms. For the generalized coordinates to be determined, For the first i The first substructure n An orthogonal allowable displacement function.

[0030] Orthogonal allowable displacement function With a set of discrete displacement functions The original basis functions are constructed using an improved Schmidt orthogonalization method. Within the normalized domain, the discrete displacement basis functions are constructed using tensor products: ; in: ; In the formula, and respectively along x direction and y The dimensionless parameter of direction, and ; and respectively along x direction and y One-dimensional discrete displacement polynomial in direction; and These represent the solution orders for these two directions, respectively. Initial displacement function. (in Let the initial displacement function be a function that satisfies the specified boundary conditions. For free boundaries, the initial displacement function satisfies... .

[0031] Discretized higher-order displacement functions Its first and second derivatives are determined by the following recurrence relation, where the apostrophe indicates that the derivatives are related to the first and second derivatives. Differentiate: ; in: , ; In the formula, and These are the intermediate coefficients for the recursion. Substituting the above basis functions into the tensor product relation, we obtain the original basis function set expressed in discrete basis vectors: ; In the formula, For the first k 1 discrete basis vector.

[0032] In step 4.2, using the aforementioned discrete basis vectors as initial basis functions, a new set of orthogonal basis functions is constructed using the improved Schmidt orthogonalization method. For any two basis functions related to the numerical integration weight vectors... Associated discrete basis vectors Define the discrete inner product as: ; ; In the formula, This represents the total number of discrete integration points. , and These are the components of the corresponding vector; It is a diagonal matrix composed of integral weights.

[0033] For the k The original basis functions are set to the initial residuals. The projection coefficient is then calculated as follows: ; In the formula, For the first j A discrete orthogonal function; These are the projection coefficients. Using... After eliminating all projected components in sequence, the first k Discrete orthogonal functions Represented as: ; In the formula, For discrete norm, it is defined as: ; The discrete orthogonal displacement functions constructed by the above recursive process satisfy the following orthogonality relation: ; In step 4.3, the orthogonal functions obtained after discretization and orthogonalization are mapped back to the original physical coordinate system of the substructure, resulting in a family of discrete orthogonal allowable displacement functions applicable to each substructure. At this point, the substructure displacement field is further expressed as: ; In the formula, For the first i Discrete displacement magnitude vectors of each substructure. The constructed discrete orthogonal displacement function can eliminate non-orthogonal components between higher-order basis functions, thereby improving the matrix condition number, accelerating the convergence of eigenvalue solutions, and reducing the risk of numerical divergence during higher-order truncation.

[0034] Step 5: Assembly of energy equations and modal solution In step 5.1, based on the classical Kirchhoff thin plate small deflection theory, the displacement function and the material parameters, layup configuration, boundary constraints, ambient temperature and humidity parameters obtained in step 1 are substituted into the mechanical control equation. The potential energy of the arbitrary-shaped composite material plate structure can be written as: ; In the formula, N The total number of substructures; For the first i The integral field of a substructure; Let be the bending stiffness component of the laminate. The bending stiffness of the laminate is calculated by the following formula: ; In the formula, This represents the total number of composite material layups. For the first The distance from the layer interface to the neutral surface; For the first The transformed stiffness components of the layer. The transformation stiffness matrix of the layer is expressed as: ; In the formula, The transformation matrix; This represents the stiffness matrix of a single-layer composite material. For the ply angle... The two are single-layer composite materials: ; ; in: ; In the formula, and These are the Young's moduli in the principal directions of the material; It is the in-plane shear modulus; and It is Poisson's ratio and satisfies .

[0035] The kinetic energy of the arbitrary-shaped composite material plate is expressed as: ; In the formula, The density of the material; h The thickness of the plate.

[0036] like Figure 7 As shown, in step 5.3, virtual spring technology is used to achieve coupling between substructures. The elastic potential energy of the virtual spring used to force displacement coordination at the connection boundary of adjacent substructures is: ; In the formula, and They are respectively with x , y Rotational stiffness related to directional angle; The lateral displacement stiffness at the boundary of the substructure intersection; The total number of common edges; and d s The first The length of the common edge and the differential arc length element along that edge; and Adjacent substructures g and f Displacement function along the common edge. To fully enforce displacement compatibility at the internal boundary, the translational spring stiffness is taken as... The stiffness of the rotating spring is taken as .

[0037] The energy of an arbitrary elastically constrained boundary is expressed as: ; In the formula, and These are rotational stiffness parameters related to the boundary rotation angle; The lateral displacement stiffness parameter at the constraint boundary; To constrain the total number of boundaries; and The first c The length and displacement function of the constraint boundary. When the virtual spring stiffness is taken as... When the constraint is fixed, it is considered a fixed support; by adjusting the stiffness values ​​of virtual springs at different positions and of different types, a variety of complex boundary constraint conditions can be flexibly simulated.

[0038] In step 5.2, the influence of the humid and thermal environment on the composite material is incorporated into the modeling process, and the following assumptions are made: First, the layers of the composite material are completely bonded together, and no relative slippage occurs between the layers; second, the temperature and humidity fields are uniformly distributed; third, the fibers in each single layer are uniformly distributed within the resin matrix, and the material is considered homogeneous and continuous. The humid and thermal potential energy of the structure is expressed as: ; In the formula, , They are respectively x , y The resultant force in the humidified surface in the direction of the directional heat; This represents the resultant shear force within the humid surface. The resultant force induced by humid heat strain within the laminate is: ; In the formula, , and For the first The hydrothermal strain components of the layer in the global coordinate system are obtained by the following formula: ; In the formula, For the first The transformation matrix of the layer from the material principal coordinate system to the global coordinate system; and These represent the hydrothermal strain along the principal direction of the material and the transverse direction, respectively. The transformation matrix and the hydrothermal strain are as follows: ; ; In the formula, and The coefficient of thermal expansion; and The coefficient of wet expansion; The temperature difference relative to the reference temperature; C The moisture absorption rate of a material is defined as: ; In the formula, and The values ​​are the mass of the laminate under completely dry and humid conditions, respectively.

[0039] In step 5.4, the displacement functions described above are substituted into the expressions for potential energy, kinetic energy, coupled potential energy, boundary potential energy, and hydrothermal potential energy, respectively, to obtain the various energies expressed in discrete integral form: ; ; ; ; ; In the formula, superscript x , y , xy , xx and yy These represent the first and second spatial derivatives of the displacement function vector, respectively; For the firsti The integral weight vector of each substructure.

[0040] According to the dynamic equilibrium condition that the maximum kinetic energy equals the maximum potential energy during free vibration. The Rayleigh-Ritz quotient is used to solve this problem, and it is written as: ; In the formula, This is the Rayleigh-Ritz quotient; the superscript * indicates the magnitude of the corresponding energy term. Taking stationary values ​​for each Ritz parameter, we get: ; This leads to the eigenvalue equation: ; In the formula, , , and These are the elastic potential energy matrix, the coupled potential energy matrix, the boundary potential energy matrix, and the hydrothermal potential energy matrix, respectively. This is the quality matrix; This is the modal coefficient matrix; all of the above matrices are of dimension 1. A square matrix. Construct the matrix. The eigenvalues ​​and eigenvectors are solved using QR decomposition. eigenvalues ​​corresponding to eigenvectors constitute Therefore, the natural angular frequency vector is determined. With the modal coefficient matrix : ; ; ; ; In the formula, For the first g First natural angular frequency; For the first g First-order mode vector; For the first g The first mode i Substructure n The modal coefficients of the term. Substituting the above modal coefficients into the displacement field, we obtain the modal coefficients of the term. i Substructure g First-order mode shape function: ; like Figure 8 , Figure 9 As shown, by assembling the mode shapes of each substructure, the complete structure's mode shape can be determined. gThe mode shape corresponding to the first natural frequency.

[0041] Step 6: Solving for the dynamic response like Figure 10 As shown, in step 6.1, the aforementioned high-dimensional stiffness matrix is ​​directly used. With the mass matrix Calculating the dynamic response is computationally expensive and resource-intensive. Therefore, we reduce the order of the mass and stiffness matrices through matrix condensation and modal superposition, and normalize the mass matrix. Then, we establish the following dynamic ordinary differential equations using the Lagrange equations: ; In the formula, , and These are the reduced-order mass matrix, damping matrix, and stiffness matrix, respectively. For modal generalized coordinate vectors; This is the excitation force vector projected onto each modal component of the structure.

[0042] In step 6.2, the dynamic equations are decoupled to the modal coordinate system through coordinate transformation combined with modal orthogonality. At this point, the mass matrix... The stiffness matrix changes to the identity matrix. The damping matrix is ​​in diagonal form. Represented as ( ),in Let the damping ratio vector be... For the first Modal stiffness.

[0043] In step 6.3, the excitation force vector Represented as: ; In the formula, In order to be with the first r Generalized forces related to mode shapes; For the upper edge of the neutral surface z External forces acting in a specific direction; For the first r Normalized modal displacement function; Let be the load-bearing area. Based on the above dynamic differential equations, the amplitude-frequency response curve of the structure under harmonic excitation can be obtained by solving the transfer function method; for transient excitation conditions, the time-domain dynamic response of the structure can be obtained by numerically solving the Runge-Kutta method.

[0044] In step 6.4, the dynamic response of the arbitrary-shaped composite material plate under external excitation is output. In this embodiment, a given ambient temperature difference... With moisture absorption rateC By applying harmonic uniform excitation, the natural frequencies, mode shapes, and amplitude-frequency response curves of the structure under different humid and hot conditions are calculated through the above steps, thus providing data support for the dynamic evaluation, vibration reduction, noise reduction, and engineering design of composite material plates of arbitrary shapes under complex humid and hot conditions.

[0045] 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. For those skilled in the art, the present invention can have various modifications and variations. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A dynamic analysis method for composite material plates of arbitrary shape under humid and hot conditions, characterized in that, Includes the following steps: Step 1: Acquire image data of the composite material plate structure of arbitrary shape, as well as material parameters, layup configuration, boundary constraints, and ambient temperature and humidity parameters. Obtain a two-dimensional geometric model of the composite material plate of arbitrary shape through image calibration, geometric feature recognition, and boundary extraction. Step 2: Based on the two-dimensional geometric model obtained in Step 1, the plate domain of the composite material plate of arbitrary shape is divided into multiple substructures with convex polygons, and each substructure and its boundary are numbered to determine the common edge coupling relationship between the substructures. Step 3: Arrange discrete Gaussian integration points within each substructure using mapping techniques to establish a normalized mapping domain for the substructure; Step 4: Based on the normalized mapping domain of each substructure, construct a family of discrete orthogonal allowable displacement functions applicable to each substructure using an improved Schmidt orthogonalization method; Step 5: Substitute the material parameters, layup configuration, boundary constraints, ambient temperature and humidity parameters obtained in Step 1, and the discrete orthogonal allowable displacement function family obtained in Step 4 into the mechanical control equations of the arbitrary-shaped composite material plate. Calculate the kinetic energy, potential energy, hygrothermal potential energy, and boundary potential energy of each substructure. Apply displacement coordination constraints and arbitrary boundary constraints between substructures using virtual spring technology to assemble the total energy functional of the arbitrary-shaped composite material plate. Solve the total energy functional based on the Rayleigh-Ritz method to obtain the natural frequencies and mode shapes of the arbitrary-shaped composite material plate. Step 6: Based on the natural frequencies and mode shapes obtained in Step 5, establish and solve the dynamic differential equations of the arbitrary-shaped composite material plate through matrix condensation and modal superposition to obtain the dynamic response of the arbitrary-shaped composite material plate under external excitation, thus completing the dynamic analysis.

2. The dynamic analysis method for a composite material plate of arbitrary shape under humid and hot conditions according to claim 1, characterized in that, Step 1 specifically includes: Step 1.1: Lay the composite material plate of arbitrary shape flat on a horizontal surface, place a scale with a calibrated direction on the plate surface, align one side of the scale with the fiber direction of the composite material, use an imaging device to acquire images perpendicular to the plate surface, and transmit the acquired images to the image correction model. At the same time, obtain the material parameters, layup configuration, boundary constraints, ambient temperature and humidity parameters of the composite material plate of arbitrary shape. Step 1.2: Using the scale bar as a reference, perform perspective correction, scale restoration, and irrelevant background removal on the acquired image to obtain a front view image of the composite material plate of arbitrary shape according to its original size; Step 1.3: Binarize the front view image, and mark the inner and outer regions of the composite material plate of arbitrary shape as black and white respectively to obtain a binary mask; Step 1.4: Identify and extract the boundary contour of the arbitrary-shaped composite material plate using a binary mask, approximate the boundary contour using polyline segments, convert the geometric information into a two-dimensional geometric file, and obtain the two-dimensional geometric model of the arbitrary-shaped composite material plate.

3. The dynamic analysis method for a composite material plate of arbitrary shape under humid and hot conditions according to claim 1, characterized in that, Step 2 specifically includes: Step 2.1: Based on the two-dimensional geometric model, the plate domain of the composite material plate of arbitrary shape is initially divided into multiple substructures; Step 2.2: Perform a concavity / convexity test on each substructure. When the substructure obtained by division is a concave polygon, perform a second subdivision at the maximum concave angle of the concave polygon, and repeat the process until all substructures are converted into convex polygons. Step 2.3: Number all substructures and substructure boundaries of the convex polygons, record the coordinate data of the corresponding boundary endpoints, and determine the common edge coupling relationship between substructures based on the recorded numbers and coordinate data.

4. The dynamic analysis method for a composite material plate of arbitrary shape under humid and hot conditions according to claim 1, characterized in that, Step 3 specifically includes: Step 3.1: Use a combination of rectangles or triangles and quadrilaterals to completely cover the integration domain of each substructure with as few filling units as possible; Step 3.2: Using a high-order numerical integration algorithm that combines the least squares Newton scheme and the node elimination method, discrete Gaussian integration points and their corresponding integration weights are generated within the filled cells. Step 3.3: Establish a local coordinate system at the geometric centroid of each substructure, determine the longest side of the substructure in each direction along the coordinate axis, use the longest side as the normalization scale, and map the substructure to the normalized square domain to obtain the normalized mapping domain of the substructure.

5. The dynamic analysis method for a composite material plate of arbitrary shape under humid and hot conditions according to claim 4, characterized in that, Step 4 specifically includes: Step 4.1: Within the normalized mapping domain, construct one-dimensional discrete displacement polynomials along two directions respectively, and construct discrete displacement basis functions through tensor product, where the initial displacement function along the free boundary direction is taken as a constant; Step 4.2: Using the discrete displacement basis function as the original basis function, define the discrete inner product by combining the integral weights of the discrete Gaussian integral points, use the improved Schmidt orthogonalization method to eliminate non-orthogonal components term by term, and perform normalization to obtain the discrete orthogonal displacement function. Step 4.3: Map the discrete orthogonal displacement functions back to the original physical coordinate system of the substructure to obtain a family of discrete orthogonal allowable displacement functions applicable to each substructure.

6. The dynamic analysis method for a composite material plate of arbitrary shape under humid and hot conditions according to claim 1, characterized in that, Step 5 specifically includes: Step 5.1: Based on the classical Kirchhoff thin plate theory, substitute the material parameters, layup configuration, boundary constraints, ambient temperature and humidity parameters obtained in Step 1.1 and the discrete orthogonal allowable displacement function family applicable to each substructure obtained in Step 4.3 into the mechanical control equation of the composite material plate of arbitrary shape, and calculate the potential energy and kinetic energy of each substructure. Step 5.2: Calculate the resultant force in the hygrothermal surface of the composite material plate of arbitrary shape based on the ambient temperature and humidity parameters, and establish the hygrothermal potential energy of each substructure; Step 5.3: Using virtual spring technology, apply displacement coordination constraints on the common edge of the substructures and apply arbitrary stiffness constraints on the external boundary to establish the coupling potential energy and boundary constraint potential energy between the substructures; Step 5.4: Assemble the potential and kinetic energies of each substructure obtained in Step 5.1, the wet heat potential energy obtained in Step 5.2, and the coupling potential energy and boundary constraint potential energy obtained in Step 5.3 into a total energy functional of the composite material plate of arbitrary shape. Based on the Rayleigh-Ritz method, take the partial derivatives of each Ritz parameter of the total energy functional and set the partial derivatives of each Ritz parameter to zero to obtain the mass matrix and stiffness matrix of the structure and derive the eigenvalue equation. Solve the eigenvalue equation through matrix decomposition algorithm to obtain the natural frequencies and mode shapes of the composite material plate of arbitrary shape.

7. The dynamic analysis method for a composite material plate of arbitrary shape under humid and hot conditions according to claim 6, characterized in that, Step 6 specifically includes: Step 6.1: Based on the natural frequencies and mode shapes, the mass matrix and stiffness matrix are reduced in order by matrix condensation and modal superposition, and the mass matrix is ​​normalized. The dynamic differential equations of the composite material plate of arbitrary shape are established by combining the Lagrange equation. Step 6.2: By combining coordinate transformation with modal orthogonality, the dynamic differential equations are decoupled into the modal coordinate system, so that the mass matrix is ​​converted into an identity matrix and the stiffness matrix is ​​converted into a diagonal matrix form; Step 6.3: Determine the excitation force vector projected onto each modal component based on the external excitation, and solve the decoupled dynamic differential equation using the transfer function method or Runge-Kutta method to obtain the amplitude-frequency response curve or time-domain dynamic response of the composite material plate of arbitrary shape; Step 6.4: Output the dynamic response results of the composite material plate of arbitrary shape under external excitation to complete the dynamic analysis.

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