Method and system for predicting natural frequency of three-phase functionally graded composite conical plate

Abstract

The present application relates to the technical field of composite structure dynamics analysis, and provides a method and system for predicting the natural frequency of a three-phase functionally graded composite conical plate, which comprises the following steps: calculating the equivalent parameters of a two-phase composite substrate containing GPL by using an improved Halpin-Tsai model, solving the equivalent parameters of a three-phase composite layer, constructing the nonlinear geometric relationship between the displacement field and the strain field, calculating the stiffness coefficient of each layer, constructing the kinetic energy of the three-phase composite conical plate, calculating the energy coefficient according to the stiffness coefficient of each layer, constructing the potential energy of the conical plate according to the energy coefficient, constructing the modal displacement function by using Chebyshev polynomials and boundary functions, establishing the characteristic equation based on the principle of minimum potential energy and solving it, and obtaining the natural frequency of the three-phase composite conical plate. The present application realizes efficient calculation of the natural frequency of the three-phase composite conical plate, and significantly improves the reliability and engineering application value of structural dynamics analysis.

Description

Technical Field

[0001] This invention relates to the field of composite material structural dynamics analysis technology, and in particular to a method and system for predicting the natural frequencies of a three-phase functionally graded composite conical plate. Background Technology

[0002] Traditional composite materials suffer from high sensitivity to localized damage, interfacial failure, and insufficient resistance to low-velocity impacts under multi-field coupling conditions. Graphene (GPL), with its ultra-high Young's modulus, excellent mechanical stability, and extremely high specific surface area, is widely used in polymer-based composites to significantly improve the stiffness and strength of structures. Graphene's high specific surface area and excellent interfacial compatibility facilitate the formation of strong interfacial bonds between fibers and the matrix, effectively improving flexural strength, impact toughness, and hardness, while also increasing specific strength and specific stiffness. When GPL is uniformly dispersed in the matrix, it can form a continuous reinforcing network structure at the micro- and nano-scale, significantly enhancing the overall mechanical properties and buckling resistance of the material. Further synergistic reinforcement with carbon fibers can simultaneously achieve mechanical strengthening at both the macro and micro scales, resulting in excellent comprehensive properties in terms of load-bearing capacity, structural stability, and damage tolerance. To further enhance the functionality of materials, the introduction of functionally graded distributions has become an effective way to optimize the mechanical properties of composite materials, especially in static, buckling, vibration, and dynamic responses. The three-phase functionally graded composite system of carbon fiber, graphene, and matrix, through the rational design of the distribution of reinforcing components, enables the material properties to change continuously in the thickness direction, thereby effectively alleviating interfacial stress concentration, improving interlayer bonding, and enhancing the overall structure's damage resistance.

[0003] However, existing dynamic analyses of composite structures mostly focus on traditional two-phase systems, making it difficult to accurately characterize the multi-scale mechanical properties under the synergistic effect of macroscopic fiber reinforcement and nanoscale reinforcement, resulting in insufficient accuracy in predicting the structure's natural frequencies. Furthermore, the geometric nonlinearity and thickness gradient effect of conical components complicate the calculation of natural frequencies, and existing methods often suffer from low computational efficiency, poor model applicability, and large prediction bias when dealing with multi-layer layups, functionally graded composite material distributions, and coupled boundary conditions. Therefore, there is an urgent need for a method that can accurately and efficiently calculate the natural frequencies of three-phase functionally graded composite conical plates to address the insufficient accuracy and limited applicability of existing technologies in the dynamic prediction of multi-scale reinforced structures.

[0004] Current research mainly focuses on the vibrational properties of two-phase composite or nano-reinforced functionally graded structures, while studies on three-phase functionally graded laminated shell structures with synergistic macro- and micro-scale effects remain relatively limited. To further reveal the dynamic laws of shell structures under multi-scale reinforcement mechanisms, it is necessary to establish a more universal theoretical model and systematically study the influence of graphene content, distribution, and lamination parameters on the natural frequencies and modal characteristics of the structure. Summary of the Invention

[0005] This invention aims to at least solve one of the technical problems existing in related technologies. To this end, this invention provides a method and system for predicting the natural frequencies of a three-phase functionally graded composite conical plate, achieving efficient calculation of the natural frequencies of the three-phase composite conical plate, thereby significantly improving the reliability of structural dynamics analysis and its engineering application value.

[0006] This invention provides a method for predicting the natural frequencies of a three-phase functionally graded composite conical plate, comprising:

[0007] S1: Construct a three-phase composite material composition model and use the improved Halpin-Tsai model to calculate the equivalent Young's modulus, equivalent Poisson's ratio and equivalent density of the GPL two-phase composite matrix in the three-phase composite material composition model;

[0008] S2: Add carbon fiber reinforcements distributed at a preset layup angle to a GPL-containing two-phase composite matrix to form an anisotropic three-phase composite layer. Based on the equivalent Young's modulus, equivalent Poisson's ratio, and equivalent density of the GPL-containing two-phase composite matrix, use the Mori-Tanaka micromechanical model to solve for the equivalent Young's modulus, equivalent in-plane shear modulus, equivalent Poisson's ratio, and equivalent density of the three-phase composite layer.

[0009] S3: Construct the nonlinear geometric relationship between the displacement field and the strain field, and construct the constitutive equation based on the nonlinear geometric relationship between the displacement field and the strain field;

[0010] S4: Calculate the stiffness coefficient of each layer based on the constitutive equation, the equivalent Young's modulus, the equivalent in-plane shear modulus, and the equivalent Poisson's ratio of the three-phase composite layer;

[0011] S5: Construct the kinetic energy of the three-phase composite conical plate based on the equivalent density of the three-phase composite layer, calculate the energy coefficient based on the stiffness coefficient of each layer, and construct the potential energy of the conical plate based on the energy coefficient.

[0012] S6: Modal displacement functions are constructed using Chebyshev polynomials and boundary functions. Based on the modal displacement functions, the kinetic energy and potential energy of the cone plate, characteristic equations are established using the Rayleigh-Ritz method and the principle of minimum potential energy. The natural frequencies of the three-phase composite material cone plate are obtained by solving the characteristic equations.

[0013] Furthermore, by incorporating the length, width, and thickness of graphene nanosheets, we established longitudinal and transverse enhancement efficiency models to improve the Halpin-Tsai model.

[0014] Furthermore, the volume fraction of graphene nanosheets is symmetrically distributed along the thickness direction of the conical plate, decreasing linearly from the top layer to the middle surface to a minimum value, and then symmetrically increasing to the bottom layer.

[0015] Furthermore, step S1 includes:

[0016] S11: The three-phase composite material composition model includes a three-phase system of carbon fiber, epoxy resin matrix and graphene nanosheets;

[0017] S12: The equivalent Young's modulus of the two-phase composite matrix containing GPL in the three-phase composite material composition model was calculated using the improved Halpin-Tsai model.

[0018] S13: Calculate the equivalent Poisson's ratio and equivalent density of the two-phase composite matrix containing GPL in the three-phase composite material composition model by using the mixing rule.

[0019] Furthermore, step S12 includes:

[0020] S121: Calculate the longitudinal reinforcement efficiency parameter, the transverse reinforcement efficiency parameter, the longitudinal reinforcement correction factor, and the transverse reinforcement correction factor of graphene nanosheets based on their geometric parameters.

[0021] S122: Calculate the equivalent Young's modulus of the GPL-containing two-phase composite matrix based on the longitudinal reinforcement efficiency parameters of graphene nanosheets, the transverse reinforcement efficiency parameters of graphene nanosheets, the longitudinal reinforcement correction factor, and the transverse reinforcement correction factor.

[0022] Furthermore, the energy coefficient is determined based on the stiffness coefficient and geometric parameters.

[0023] Furthermore, the kinetic energy of the three-phase composite conical plate, constructed based on the equivalent density of the three-phase composite layer, includes:

[0024] S511: Define the inertia matrix coefficients based on the layer density of three-phase functionally graded composite materials;

[0025] S512: Construct the kinetic energy of a three-phase composite conical plate based on the inertia matrix coefficients and the equivalent density of the three-phase composite layer.

[0026] Furthermore, the potential energy for constructing a conical plate based on its energy coefficient includes:

[0027] S521: Based on the stress-strain relationship, the strain energy of the conical plate is constructed by integrating the stress and strain within the volume domain of the conical plate.

[0028] S522: The potential energy of a conical plate is obtained by expressing the strain components of the strain energy in terms of displacement components, rotational components, and energy coefficients using the Green strain tensor.

[0029] Furthermore, in step S6,

[0030] S61: The displacement and rotation components are described by exponential functions and spatial modal functions, and the spatial modal functions are expanded by Chebyshev polynomials and boundary functions;

[0031] S62: Substitute the spatial mode function and the components describing displacement and rotation into the potential energy and kinetic energy, ignore the nonlinear terms, and take time t equal to 0 to obtain the maximum kinetic energy and maximum potential energy;

[0032] S63: Using the Ritz minimization principle, take the partial derivative of the maximum potential energy minus the maximum kinetic energy with respect to the undetermined coefficients, and set the partial derivative result to 0 to obtain the characteristic equation;

[0033] S64: Solve for the eigenvalues ​​of the characteristic equation to obtain the natural frequencies of the three-phase composite conical plate.

[0034] This invention also provides a system for predicting the natural frequencies of a three-phase functionally graded composite conical plate, for performing the aforementioned method for predicting the natural frequencies of a three-phase functionally graded composite conical plate, comprising:

[0035] The first calculation module constructs a three-phase composite material composition model and uses the improved Halpin-Tsai model to calculate the equivalent Young's modulus, the equivalent Poisson's ratio of the two phases and the equivalent density of the GPL two-phase composite matrix in the three-phase composite material composition model.

[0036] The second calculation module adds carbon fiber reinforcements distributed at a preset layup angle to the GPL-containing two-phase composite matrix to form an anisotropic three-phase composite layer. Based on the equivalent Young's modulus, the equivalent Poisson's ratio of the two phases, and the equivalent density of the GPL-containing two-phase composite matrix, the Mori-Tanaka micromechanical model is used to solve for the equivalent Young's modulus, equivalent in-plane shear modulus, equivalent Poisson's ratio, and equivalent density of the three-phase composite layer.

[0037] The constitutive equation construction module constructs the nonlinear geometric relationship between the displacement field and the strain field, and constructs constitutive equations based on the nonlinear geometric relationship between the displacement field and the strain field.

[0038] The third calculation module calculates the stiffness coefficient of each layer based on the constitutive equation, the equivalent Young's modulus of the three-phase composite layer, the equivalent in-plane shear modulus, and the equivalent Poisson's ratio.

[0039] The kinetic and potential energy construction module constructs the kinetic energy of the three-phase composite material cone plate based on the equivalent density of the three-phase composite layer, calculates the energy coefficient based on the stiffness coefficient of each layer, and constructs the potential energy of the cone plate based on the energy coefficient.

[0040] The natural frequency solving module uses Chebyshev polynomials and boundary functions to construct modal displacement functions. Based on the modal displacement functions, the kinetic energy and potential energy of the cone plate, the characteristic equation is established using the Rayleigh-Ritz method and the principle of minimum potential energy. The natural frequency of the three-phase composite material cone plate is obtained by solving the characteristic equation.

[0041] The above-described one or more technical solutions in the embodiments of the present invention have at least one of the following technical effects:

[0042] This invention accurately quantifies the two-dimensional nano-reinforcement efficiency of GPL through an improved Halpin-Tsai method, combines the Mori-Tanaka model to calculate the stiffness of three-phase materials in layers, and integrates first-order shear deformation theory and Von-Karman large deformation theory, fully considering transverse shear deformation and geometric nonlinearity, to achieve high-precision prediction of the natural frequencies of three-phase functionally graded composite conical plates. This invention employs a modal function construction method based on Chebyshev polynomials and boundary functions, further shortening the computation cycle. Frequency calculations of complex structures can be performed on ordinary engineering computers, significantly reducing the computing power cost for industrial applications. This invention represents a significant breakthrough over existing technologies in terms of prediction accuracy, computational efficiency, scenario adaptability, and engineering value, and can be widely applied to the dynamic design and performance optimization of composite conical plates in aerospace, energy equipment, and other fields.

[0043] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0044] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0045] Figure 1 This is a flowchart illustrating a method for predicting the natural frequency of a three-phase functionally graded composite conical plate provided by the present invention.

[0046] Figure 2 This is a schematic diagram of the structure of the prediction system for the natural frequency of a three-phase functionally graded composite conical plate provided by the present invention.

[0047] Figure label:

[0048] 101. First calculation module; 102. Second calculation module; 103. Constitutive equation construction module; 104. Third calculation module; 105. Kinetic and potential energy construction module; 106. Natural frequency solution module. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention. The following embodiments are used to illustrate this invention but cannot be used to limit the scope of this invention.

[0050] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., refer to specific features, structures, materials, or characteristics described in connection with that embodiment or example, which are included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0051] The following is combined Figures 1 to 2 This invention describes a method and system for predicting the natural frequencies of a three-phase functionally graded composite conical plate.

[0052] like Figure 1 As shown, a method for predicting the natural frequencies of a three-phase functionally graded composite conical plate includes:

[0053] S1: Construct a three-phase composite material composition model and use the improved Halpin-Tsai model to calculate the equivalent Young's modulus, equivalent Poisson's ratio of the two phases and equivalent density of the GPL-containing two-phase composite matrix in the three-phase composite material composition model.

[0054] S11: The three-phase composite material composition model includes a three-phase system of carbon fiber, epoxy resin matrix and graphene nanosheets;

[0055] The three-phase composite material composition model includes a three-phase system of carbon fiber (CF), epoxy resin matrix and graphene nanosheets (GPL), defining its volume fraction, distribution pattern and functional gradient law.

[0056] The volume fraction of graphene nanosheets is symmetrically distributed along the thickness direction of the conical plate, decreasing linearly from the top layer to the middle surface to the minimum value, and then increasing symmetrically to the bottom layer.

[0057] The traditional Halpin-Tsai model primarily targets short fiber or particle-reinforced composites, using a single shape parameter to describe the reinforcement effect, failing to adequately consider the geometric size effects and anisotropic characteristics of two-dimensional nanosheet reinforcements in different directions. Therefore, it is unsuitable for this three-phase composite system. This invention employs an improved Halpin-Tsai model, introducing geometric parameters such as the length, width, and thickness of graphene nanosheets to establish longitudinal and transverse reinforcement efficiency models respectively. This allows for a more accurate description of the anisotropic nanoreinforcement effect of graphene nanosheets in the polymer matrix, improving the accuracy of equivalent mechanical parameter predictions.

[0058] S12: The equivalent Young's modulus of the two-phase composite matrix containing GPL in the three-phase composite material composition model was calculated using the improved Halpin-Tsai model.

[0059] S121: Calculate the longitudinal reinforcement efficiency parameter, the transverse reinforcement efficiency parameter, the longitudinal reinforcement correction factor, and the transverse reinforcement correction factor of graphene nanosheets based on their geometric parameters.

[0060] The geometric characteristics of GPL determine its enhanced shape parameters, which are calculated as follows:

[0061]

[0062]

[0063] in, For longitudinal enhancement correction factor, The Young's modulus of graphene nanosheet materials. This represents the Young's modulus of the epoxy resin matrix material. The parameter represents the longitudinal reinforcement efficiency of graphene nanosheets. As a horizontal enhancement correction factor, The parameter represents the reinforcement efficiency of graphene nanosheets in the lateral direction. The length of the graphene nanosheet. The width of the graphene nanosheet is [value missing]. The thickness of the graphene nanosheet;

[0064] S122: The equivalent Young's modulus of the GPL-containing two-phase composite matrix is ​​calculated based on the longitudinal reinforcement efficiency parameters of graphene nanosheets, the transverse reinforcement efficiency parameters of graphene nanosheets, the longitudinal reinforcement correction factor, and the transverse reinforcement correction factor. The calculation expression is as follows:

[0065]

[0066] in, The equivalent Young's modulus of a two-phase composite matrix (HM) containing graphene nanosheets (GPL) is given. GPL volume fraction;

[0067] S12: Calculate the equivalent Poisson's ratio and equivalent density of the two-phase composite matrix using the mixing rule;

[0068] The Poisson's ratio and density of each HM layer can be calculated using the mixing rule, and the calculation expression is:

[0069]

[0070]

[0071] in, The equivalent Poisson's ratio of a two-phase composite matrix containing GPL. The Poisson's ratio of graphene nanosheets. This represents the volume fraction of the epoxy group. Poisson's ratio of the epoxy resin matrix. The equivalent density of the two-phase composite matrix containing GPL. The density of the resin matrix, The density is the graphene nanosheet.

[0072] The Mori–Tanaka model was used to solve for the equivalent mechanical parameters of the three-phase composite material, including the equivalent Young's modulus, equivalent shear modulus, equivalent Poisson's ratio, and equivalent density. Further addition of carbon fiber reinforcement to the aforementioned two-phase composite matrix creates an anisotropic three-phase composite layer. This reinforcement structure maintains a uniform interlayer distribution in the thickness direction, with the carbon fibers distributed at a predetermined layup angle. Considering that the carbon fibers are transversely isotropic, the longitudinal Young's modulus of the carbon fibers is... , transverse Young's modulus of carbon fiber Transverse Young's modulus of carbon fiber in the thickness direction In-plane shear modulus of carbon fiber and the longitudinal out-of-plane shear modulus of carbon fiber In-plane Poisson's ratio of carbon fiber Longitudinal out-of-plane Poisson's ratio of carbon fiber They need to be processed separately. Carbon fiber meets the requirements. and The characteristic relationship.

[0073] S2: Add carbon fiber reinforcements distributed at a preset layup angle to a GPL-containing two-phase composite matrix to form an anisotropic three-phase composite layer. Based on the equivalent Young's modulus, equivalent Poisson's ratio of the two phases, and equivalent density of the GPL-containing two-phase composite matrix, the Mori-Tanaka micromechanical model is used to solve for the equivalent Young's modulus, equivalent in-plane shear modulus, equivalent Poisson's ratio, and equivalent density of the three-phase composite layer.

[0074] The expression for calculating the effective longitudinal Young's modulus is:

[0075]

[0076] in, For effective longitudinal Young's modulus, This represents the volume fraction of carbon fiber. This represents the longitudinal Young's modulus of carbon fiber. The in-plane Poisson's ratio of the carbon fiber is its longitudinal direction. The shear deformation related combination coefficient is jointly determined by the Young's modulus and Poisson's ratio of the fiber phase and the matrix phase;

[0077] The expression for calculating the effective transverse Young's modulus is:

[0078]

[0079] in, For effective transverse Young's modulus, The transverse Poisson's ratio of carbon fiber. The shear stiffness weighting coefficient is jointly determined by the Young's modulus, volume fraction of the fiber phase, and equivalent Young's modulus of the matrix. This refers to the transverse Young's modulus of the carbon fiber in the radial direction.

[0080] The formula for calculating the equivalent Poisson's ratio is:

[0081]

[0082]

[0083] in, This represents the longitudinal equivalent Poisson's ratio of carbon fiber. This represents the transverse equivalent Poisson's ratio of carbon fiber.

[0084] The equivalent in-plane shear modulus and the equivalent out-of-plane shear modulus are:

[0085]

[0086]

[0087]

[0088]

[0089]

[0090] in, This is the equivalent in-plane shear modulus. This is the equivalent out-of-plane shear modulus. It is the transverse shear modulus. This represents the in-plane shear modulus of the carbon fiber. For the equivalent longitudinal out-of-plane shear modulus,

[0091] The equivalent density is calculated by linearly superimposing volume fractions, and the expression is as follows:

[0092]

[0093] in, For equivalent density, The density of carbon fiber, This represents the volume fraction of carbon fiber. The volume fraction of the two-phase composite matrix containing GPL;

[0094] S3: Construct the nonlinear geometric relationship between the displacement field and the strain field, and construct the constitutive equation based on the nonlinear geometric relationship between the displacement field and the strain field;

[0095] A displacement field is established, and the displacement-strain relationship of the conical plate geometric model is constructed using the first-order shear deformation theory; a cylindrical coordinate system is established with the mid-surface of the conical plate as the reference surface. , representing the generatrix direction, circumferential direction, and thickness direction, respectively. According to the first-order shear deformation theory (FSDT), considering the relationship between mid-surface displacement and rotation, the displacement components of any particle in three-dimensional space can be expressed as:

[0096]

[0097]

[0098]

[0099] in, for The displacement component of any particle in the direction of the generatrix at any given moment. The coordinates of the generatrix of the conical plate are as follows: Let be the coordinates along the circumference of the cone plate. The coordinates are along the thickness direction of the conical plate. for The displacement component of any particle in the circumferential direction at any given moment. for The displacement component of any particle in the thickness direction at any given time. for The displacement of the mid-surface in the direction of the generatrix at time t. for The displacement of the mid-surface in the circumferential direction at time t. for The displacement of the mid-surface in the thickness direction at time t. for The angle of rotation of the horizontal normal around the circumference at any given moment. for The angle of rotation of the horizontal normal around the generatrix at any given moment;

[0100] Based on the Von-Karman large deformation theory, the nonlinear geometric relationship between the displacement field and the strain field can be expressed as follows:

[0101]

[0102] in, For axial normal strain, For circumferential positive strain, For in-plane shear strain, For axial-thickness shear strain For circumferential-thickness shear strain, For the axial strain of the mid-surface, For the circumferential strain of the mid-surface, For the in-plane shear strain of the mid-plane, The transverse shear strain (axial) of the mid-surface. The transverse shear strain (circumferential) of the mid-surface. For axial curvature-dependent strain. For circumferential curvature-related strain, For in-plane shear curvature related components, This is the transverse shear gradient component (axial). This represents the transverse shear gradient component (circumferential).

[0103] Neglecting thermal stress, the constitutive equation for a rotary pre-torsion GPL-reinforced functionally graded composite conical plate is as follows:

[0104]

[0105] in, For the first layer Stress in the direction, For the first layer Stress in the direction, for Shear stress in a plane for Shear stress in a plane for Shear stress in a plane This refers to the layer number of a functionally graded three-phase composite material. This is the axial equivalent tensile stiffness matrix. This is the axial-circumferential tensile coupling stiffness matrix. This is the circumferential equivalent tensile stiffness matrix. The in-plane equivalent shear stiffness matrix is... This is the transverse equivalent shear stiffness matrix (radial-circumferential). This is the transverse equivalent shear stiffness matrix (radial–axial).

[0106] S4: Calculate the stiffness coefficient of each layer based on the constitutive equation, the equivalent Young's modulus, the equivalent in-plane shear modulus, and the equivalent Poisson's ratio of the three-phase composite layer;

[0107] In stiffness matrix calculation, k is defined as the number of layers in the three-phase composite material. Compared with the traditional single-layer stiffness matrix calculation method, the stiffness matrix calculation used in this invention requires separate calculation for each layer. The stiffness matrix expression in the constitutive equation is as follows:

[0108]

[0109]

[0110]

[0111]

[0112]

[0113]

[0114]

[0115]

[0116] in, The layup angle of each layer of carbon fiber in a functionally graded three-phase composite conical plate.

[0117] The definition is as follows:

[0118] , ,

[0119] , ,

[0120] in, For the first Directional stress and the first Directional strain The equivalent stiffness matrix of the layer, , .

[0121] Strain can be expressed as:

[0122] ,

[0123] in, Let be the displacement of the mid-plane in the direction of the generatrix. Let be the displacement of the mid-surface in the circumferential direction. This represents the displacement of the mid-surface in the thickness direction. The angle of rotation of the horizontal normal around the circumference. for The angle of rotation of the horizontal normal around the generatrix at any given moment. The angle between the generatrix of the cone and the axis. Let be the circumferential radius of curvature of the conical plate.

[0124] S5: Construct the kinetic energy of the three-phase composite conical plate based on the equivalent density of the three-phase composite layer, calculate the energy coefficient based on the stiffness coefficient of each layer, and construct the potential energy of the conical plate based on the energy coefficient.

[0125] The kinetic energy of a three-phase composite conical plate constructed based on the equivalent density of the three-phase composite layer includes:

[0126] S511: Define the inertia matrix coefficients based on the layer density of three-phase functionally graded composite materials;

[0127] For laminated three-phase composite materials, the moment of inertia coefficient is defined as follows:

[0128] ,

[0129] in, Here is the moment of inertia coefficient. This represents the number of layers in the three-phase composite material divided along the thickness direction of the conical plate. For the first The boundary coordinates of the layer in the thickness direction. For the first The boundary coordinates of the layer in the thickness direction. For the first Layer density, for of Power of;

[0130] S512: The kinetic energy of the three-phase composite conical plate is constructed based on the inertia matrix coefficients and the equivalent density of the three-phase composite layer. The calculation expression for the kinetic energy of the three-phase composite conical plate is as follows:

[0131]

[0132] in, The total kinetic energy of the system. The length in the direction of the busbar. The in-plane moment of inertia coefficient is... The moment of inertia coefficient for tension-bending coupling. The bending moment of inertia coefficient, Let be the displacement variable of the mid-surface in the direction of the generatrix. Let be the displacement variable of the mid-surface in the circumferential direction. Let be the displacement variable of the mid-surface in the thickness direction;

[0133] The energy coefficient is determined based on the stiffness coefficient and geometric parameters. The formula for calculating the energy coefficient is:

[0134] , , ,

[0135] , , ,

[0136] , , , , ,

[0137] , , , , ,

[0138] , , , , ,

[0139] , , , , , , ,

[0140] , , , , , , ,

[0141] , , ,

[0142] , , , ,

[0143] , , , , ,

[0144] , , ,

[0145] , , ,

[0146] in, The generalized strain energy coefficient is related to the coupling of axial displacement and circumferential rotation of the mid-surface. The generalized strain energy coefficient is related to the coupling between the circumferential displacement gradient and the circumferential rotation angle. The generalized strain energy coefficient related to the axial tension-bending gradient coupling. The membrane energy coefficient corresponding to the square term of the mid-surface axial displacement. The generalized strain energy coefficient is related to the coupling between axial rotation gradient and axial displacement. The generalized strain energy coefficient is related to the coupling of axial displacement and its gradient. The generalized strain energy coefficient is related to the coupling of mid-surface circumferential displacement and circumferential rotation. The membrane energy coefficient is related to the circumferential displacement and its axial gradient. The generalized strain energy coefficient is related to the coupling of circumferential displacement and rotation gradient. The energy coefficient is related to the coupling of axial rotation gradient and circumferential displacement gradient. The membrane strain energy coefficient is related to the axial-circumferential tensile coupling. The energy coefficient corresponding to the square term of the mid-surface lateral displacement. The bending energy coefficient is related to the coupling of axial rotation gradient and lateral displacement. The energy coefficient related to the coupling of axial tension and deflection. The bending energy coefficient is related to the interaction with the rotation gradient. The energy coefficient related to the coupling of axial tension and circumferential bending. The energy coefficient related to circumferential tension-bending coupling, The bending energy coefficient is related to the second-order axial curvature of the lateral displacement. The bending energy coefficient is related to the axial rotation angle and its gradient coupling. The energy coefficient related to axial tension-rotation coupling, The bending energy coefficient is related to the circumferential curvature of the lateral displacement. The energy coefficient related to the circumferential rotation-axial tensile coupling, The energy coefficient related to circumferential tension-bending coupling, The energy coefficient related to the stretch-rotation gradient coupling, The energy coefficient is related to the square of the circumferential rotation angle. The energy coefficient related to corner crossing coupling, The energy coefficient related to the rotation-circumferential displacement coupling, The energy coefficient related to circumferential tension-displacement coupling. The energy coefficient related to the circumferential rotation-deflection coupling, The energy coefficient related to circumferential tension-deflection coupling, The energy coefficient related to corner crossing coupling, The energy coefficient related to the circumferential tension-axial rotation coupling, The energy coefficient related to the deflection slope-axial rotation coupling, The energy coefficient related to the axial displacement–axial rotation coupling, The energy coefficient related to the coupling of circumferential rotation and axial displacement. The energy coefficient related to the coupling of circumferential tension and axial displacement. The energy coefficient is related to the circumferential curvature of the axial rotation angle. The energy coefficient is related to the axial curvature of the circumferential rotation angle. The energy coefficient related to the nonlinear coupling of tension and rotation. The energy coefficient related to the cross-coupling of the corner gradient. The energy coefficient is related to the circumferential curvature of the axial displacement. The energy coefficient is related to the axial curvature of the circumferential displacement. The energy coefficient related to the axial-circumferential tensile nonlinear coupling. The energy coefficient related to the rotation-tension cross-coupling, The energy coefficient corresponding to the square of the circumferential rotation angle. The energy coefficient is related to the circumferential rotation angle and circumferential curvature. The energy coefficient corresponding to the square of the axial rotation angle. The energy coefficient is related to the circumferential curvature of the axial rotation angle. The energy coefficient corresponding to the square term of the mid-surface circumferential displacement. The energy coefficient is related to the axial rotation angle and axial curvature. The energy coefficient is related to the axial displacement and axial curvature. The energy coefficient related to deflection-circumferential displacement coupling, The energy coefficient related to the lateral displacement–axial rotation coupling, The energy coefficient related to axial displacement-deflection coupling. For the laminated conical plate, the first Directional stress and the first Tensile stiffness coefficient of directional strain For the laminated conical plate, the first Directional stress and the first The tensile-bending coupling stiffness coefficient of directional strain, For the laminated conical plate, the first Directional stress and the first Bending stiffness coefficient of directional strain;

[0147] , and The calculation expression is:

[0148] , ,

[0149] in, This represents the total thickness of the conical plate.

[0150] The potential energy for constructing a conical plate based on its energy coefficient includes:

[0151] S521: Based on the stress-strain relationship, the strain energy of the conical plate is constructed by integrating the stress and strain within the volume domain of the conical plate.

[0152] S522: The strain components of the strain energy are expressed in terms of displacement components, rotational components, and energy coefficients using the Green strain tensor to obtain the potential energy of the conical plate;

[0153] The expression for calculating the potential energy of a conical plate is:

[0154]

[0155]

[0156]

[0157]

[0158]

[0159]

[0160]

[0161]

[0162]

[0163]

[0164]

[0165] in, Let be the potential energy of the conical plate. Let be the radius of the small end of the conical plate.

[0166] S6: Modal displacement functions are constructed using Chebyshev polynomials and boundary functions. Based on the modal displacement functions, the kinetic energy of the cone plate, and the potential energy of the cone plate, eigenvalue equations are established using the Rayleigh-Ritz method and the principle of minimum potential energy. The natural frequencies of the three-phase composite material cone plate are obtained by solving the eigenvalue equations.

[0167] By applying Chebyshev polynomials to construct modal displacement functions and using the Rayleigh-Ritz method to establish energy functionals, the natural frequencies of different modes are obtained. The energy functional is then established using the Ritz method. Substituting the displacement field assumptions into the energy functional and utilizing the principle of minimum potential energy, the generalized eigenvalue problem controlling the natural frequencies can be derived.

[0168] S61: The displacement and rotation components are described by exponential functions and spatial modal functions, and the spatial modal functions are expanded by Chebyshev polynomials and boundary functions;

[0169] The displacement and rotation components are described using exponential functions and spatial modal functions, and the calculation expression is as follows:

[0170]

[0171] in, Let be the spatial mode function corresponding to the displacement in the direction of the generatrix. Let be the spatial mode function corresponding to the circumferential displacement. Let be the spatial mode function corresponding to the displacement in the thickness direction. Let θ be the spatial mode function corresponding to the rotation angle of the transverse normal around the direction θ. Let be the spatial mode function corresponding to the rotation angle of the transverse normal around the x-direction. Let be the natural frequency of the cone plate to be solved. The imaginary unit;

[0172] In the Ritz method, the spatial mode function is expanded using the product of the Chebyshev polynomial and the boundary function, and the calculation expression is as follows:

[0173]

[0174]

[0175]

[0176]

[0177]

[0178] in, Let be the Chebyshev polynomial in the direction of the busbar. Let be the Chebyshev polynomial in the circumferential direction. The coordinates are dimensionless coordinates along the generatrix direction. Dimensionless coordinates in the circumferential direction The boundary function is the one for the direction of the busbar. For the circumferential direction boundary function, For the thickness direction boundary function, Let θ be the boundary function of the transverse normal about the direction θ. Let x be the boundary function of the transverse normal around the x-direction. The axial displacement expansion coefficient is... The circumferential displacement expansion factor is... This is the lateral displacement expansion coefficient. The axial rotation expansion factor is . The circumferential rotation coefficient is the expansion factor. Let be the truncation order of the axial Chebyshev polynomial. Let be the truncation order of the circumdirectional Chebyshev polynomial. The boundary functions under different boundary conditions are shown in Table 1.

[0179] Table 1 Boundary functions under different boundary conditions

[0180]

[0181] Where F is a free boundary, S is a simply supported boundary, and C is a fixed boundary;

[0182] S62: Substitute the spatial mode function and the components describing displacement and rotation into the potential energy and kinetic energy, ignore the nonlinear terms, and take time t equal to 0 to obtain the maximum kinetic energy and maximum potential energy;

[0183] S63: Using the Ritz minimization principle, take the partial derivative of the maximum potential energy minus the maximum kinetic energy with respect to the undetermined coefficients, and set the partial derivative result to 0 to obtain the characteristic equation;

[0184] That is, for each unknown coefficient Find the partial derivative and set it to zero:

[0185]

[0186] in, For undetermined coefficients, including , , , , , For maximum potential energy, For the maximum kinetic energy, the characteristic equation is:

[0187]

[0188] in, The overall stiffness matrix consists of stiffness matrices A, B, and D, and the boundary springs. For the quality matrix, Includes all undetermined coefficients ( represents a single generalized coordinate coefficient, while q is composed of all (The vector form of the composition).

[0189] S64: Solve for the eigenvalues ​​of the characteristic equation to obtain the natural frequencies of the three-phase composite conical plate.

[0190] like Figure 2 As shown, a prediction system for the natural frequencies of a three-phase functionally graded composite conical plate is used to execute the aforementioned prediction method for the natural frequencies of a three-phase functionally graded composite conical plate, comprising:

[0191] The first calculation module 101 constructs a three-phase composite material composition model and uses the improved Halpin-Tsai model to calculate the equivalent Young's modulus, the equivalent Poisson's ratio of the two phases and the equivalent density of the GPL two-phase composite matrix in the three-phase composite material composition model.

[0192] The second calculation module 102 adds carbon fiber reinforcements distributed at a preset layup angle to the GPL-containing two-phase composite matrix to form an anisotropic three-phase composite layer. Based on the equivalent Young's modulus, the equivalent Poisson's ratio of the two phases, and the equivalent density of the GPL-containing two-phase composite matrix, the Mori-Tanaka micromechanical model is used to solve for the equivalent Young's modulus, the equivalent in-plane shear modulus, the equivalent Poisson's ratio, and the equivalent density of the three-phase composite layer.

[0193] Constituent equation construction module 103 constructs the nonlinear geometric relationship between the displacement field and the strain field, and constructs constituent equations based on the nonlinear geometric relationship between the displacement field and the strain field;

[0194] The third calculation module 104 calculates the stiffness coefficient of each layer based on the constitutive equation, the equivalent Young's modulus, the equivalent in-plane shear modulus, and the equivalent Poisson's ratio of the three-phase composite layer.

[0195] The kinetic and potential energy construction module 105 constructs the kinetic energy of the three-phase composite material cone plate based on the equivalent density of the three-phase composite layer, calculates the energy coefficient based on the stiffness coefficient of each layer, and constructs the potential energy of the cone plate based on the energy coefficient.

[0196] The natural frequency solving module 106 uses Chebyshev polynomials and boundary functions to construct modal displacement functions. Based on the modal displacement functions, the kinetic energy and potential energy of the cone plate, the characteristic equation is established based on the Rayleigh-Ritz method and the principle of minimum potential energy. The natural frequency of the three-phase composite material cone plate is obtained by solving the characteristic equation.

[0197] Through the collaborative work of the above modules, the improved Halpin-Tsai method accurately quantifies the two-dimensional nano-reinforcement efficiency of GPL, combines the Mori-Tanaka model to calculate the stiffness of three-phase materials in layers, and integrates the first-order shear deformation theory and the Von-Karman large deformation theory. By fully considering transverse shear deformation and geometric nonlinearity, high-precision prediction of the natural frequencies of three-phase functionally graded composite conical plates is achieved. This invention adopts the modal function construction method of Chebyshev polynomials and boundary functions to further shorten the calculation cycle. The frequency calculation of complex structures can be completed on ordinary engineering computers, which greatly reduces the computing power cost of industrial applications. This invention has achieved significant breakthroughs over existing technologies in terms of prediction accuracy, calculation efficiency, scenario adaptability, and engineering value. It can be widely used in the dynamic design and performance optimization of composite conical plates in aerospace, energy equipment and other fields.

[0198] In a specific embodiment of the present invention, the laminate structure predicted by the prediction method of the present invention is composed of a 16-layer three-phase composite material, with graphene (GPL) and carbon fiber distributed as multi-scale reinforcing phases in an epoxy resin matrix. The material properties of each layer exhibit a functional gradient along the thickness direction, which stems from the specific design of the graphene volume fraction. Its content is symmetrically distributed along the thickness, linearly decreasing to a minimum from the top layer to the middle layer, and then symmetrically increasing to the bottom layer. The distribution pattern is as follows:

[0199]

[0200]

[0201] in, For GPL The volume fraction of the layer, This is the baseline volume fraction for GPL;

[0202]

[0203] in, This represents the weight fraction of graphene.

[0204] Given parameters R=2m and thickness h=0.06m, the Young's modulus of the epoxy resin matrix material is... Density of epoxy resin matrix material Poisson's ratio of epoxy resin matrix material Young's modulus of graphene materials Graphene material density Poisson's ratio of graphene materials Graphene weight fraction The length of graphene nanosheets The width of graphene nanosheets The thickness of graphene nanosheets Longitudinal Young's modulus of carbon fiber Transverse Young's modulus of carbon fiber In-plane shear modulus of carbon fiber Transverse shear modulus of carbon fiber The density of carbon fiber The in-plane and out-of-plane Poisson's ratios of carbon fiber The transverse Poisson's ratio of carbon fiber carbon fiber volume fraction The boundary conditions are fixed at one end and free at the other three ends. The theoretical predictions and experimental values ​​of the first five frequencies of the three-phase material conical plate are compared in Table 2.

[0205] Table 2 Comparison of theoretical and experimental values ​​of the first five frequencies of a three-phase material conical plate.

[0206]

[0207] As shown in Table 2, the theoretical values ​​calculated using the prediction method of this invention have a maximum error of no more than 4% compared with the simulation experimental values, and the errors of the 2nd, 3rd, and 4th order modal frequency values ​​are all below 1%, indicating that the prediction effect of this invention is quite good.

[0208] To verify the technical effects of the present invention, the following were selected: as well as Two cross-layout configurations were compared and calculated. The comparison between the predicted values ​​and simulation results of the two cross-layout configurations is shown in Table 3. Based on the first-order shear deformation theory, the Chebyshev and Ritz methods were used to solve the problem. The first five dimensionless natural frequencies were calculated in a systematic manner, taking into full account the effects of transverse shear and rotational inertia.

[0209] Table 3 Comparison of predicted values ​​and simulation results for two cross-layout configurations

[0210]

[0211] As shown in Table 3, the maximum relative deviation between the frequencies obtained by this invention and the simulation experiments does not exceed 5.41%, which is within the accepted error range for high-precision vibration analysis. This result fully demonstrates that this invention can accurately characterize the influence of layup sequence and fiber orientation on the vibration characteristics of composite conical plates, further solidifying the applicability and reliability of the model.

[0212] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for predicting the natural frequencies of a three-phase functionally graded composite conical plate, characterized in that, include: S1: Construct a three-phase composite material composition model and use the improved Halpin-Tsai model to calculate the equivalent Young's modulus, equivalent Poisson's ratio and equivalent density of the GPL two-phase composite matrix in the three-phase composite material composition model; By introducing the length, width, and thickness of graphene nanosheets, the Halpin-Tsai model is improved by establishing longitudinal and transverse enhancement efficiency models. S2: Add carbon fiber reinforcements distributed at a preset layup angle to a GPL-containing two-phase composite matrix to form an anisotropic three-phase composite layer. Based on the equivalent Young's modulus, equivalent Poisson's ratio, and equivalent density of the GPL-containing two-phase composite matrix, use the Mori-Tanaka micromechanical model to solve for the equivalent Young's modulus, equivalent in-plane shear modulus, equivalent Poisson's ratio, and equivalent density of the three-phase composite layer. S3: Construct the nonlinear geometric relationship between the displacement field and the strain field, and construct the constitutive equation based on the nonlinear geometric relationship between the displacement field and the strain field; S4: Calculate the stiffness coefficient of each layer based on the constitutive equation, the equivalent Young's modulus, the equivalent in-plane shear modulus, and the equivalent Poisson's ratio of the three-phase composite layer; S5: Construct the kinetic energy of the three-phase composite conical plate based on the equivalent density of the three-phase composite layer, calculate the energy coefficient based on the stiffness coefficient of each layer, and construct the potential energy of the conical plate based on the energy coefficient. S6: Modal displacement functions are constructed using Chebyshev polynomials and boundary functions. Based on the modal displacement functions, the kinetic energy and potential energy of the cone plate, characteristic equations are established using the Rayleigh-Ritz method and the principle of minimum potential energy. The natural frequencies of the three-phase composite material cone plate are obtained by solving the characteristic equations.

2. The method for predicting the natural frequencies of a three-phase functionally graded composite conical plate according to claim 1, characterized in that, The volume fraction of graphene nanosheets is symmetrically distributed along the thickness direction of the conical plate, decreasing linearly from the top layer to the middle surface to the minimum value, and then increasing symmetrically to the bottom layer.

3. The method for predicting the natural frequencies of a three-phase functionally graded composite conical plate according to claim 1, characterized in that, Step S1 includes: S11: The three-phase composite material composition model includes a three-phase system of carbon fiber, epoxy resin matrix and graphene nanosheets; S12: The equivalent Young's modulus of the two-phase composite matrix containing GPL in the three-phase composite material composition model was calculated using the improved Halpin-Tsai model. S13: Calculate the equivalent Poisson's ratio and equivalent density of the two-phase composite matrix containing GPL in the three-phase composite material composition model by using the mixing rule.

4. The method for predicting the natural frequencies of a three-phase functionally graded composite conical plate according to claim 3, characterized in that, Step S12 includes: S121: Calculate the longitudinal reinforcement efficiency parameter, the transverse reinforcement efficiency parameter, the longitudinal reinforcement correction factor, and the transverse reinforcement correction factor of graphene nanosheets based on their geometric parameters. S122: Calculate the equivalent Young's modulus of the GPL-containing two-phase composite matrix based on the longitudinal reinforcement efficiency parameters of graphene nanosheets, the transverse reinforcement efficiency parameters of graphene nanosheets, the longitudinal reinforcement correction factor, and the transverse reinforcement correction factor.

5. The method for predicting the natural frequencies of a three-phase functionally graded composite conical plate according to claim 1, characterized in that, The energy coefficient is determined based on the stiffness coefficient and geometric parameters.

6. The method for predicting the natural frequencies of a three-phase functionally graded composite conical plate according to claim 1, characterized in that, The kinetic energy of a three-phase composite conical plate constructed based on the equivalent density of the three-phase composite layer includes: S511: Define the inertia matrix coefficients based on the layer density of three-phase functionally graded composite materials; S512: Construct the kinetic energy of a three-phase composite conical plate based on the inertia matrix coefficients and the equivalent density of the three-phase composite layer.

7. The method for predicting the natural frequencies of a three-phase functionally graded composite conical plate according to claim 1, characterized in that, The potential energy for constructing a conical plate based on its energy coefficient includes: S521: Based on the stress-strain relationship, the strain energy of the conical plate is constructed by integrating the stress and strain within the volume domain of the conical plate. S522: The potential energy of a conical plate is obtained by expressing the strain components of the strain energy in terms of displacement components, rotational components, and energy coefficients using the Green strain tensor.

8. The method for predicting the natural frequencies of a three-phase functionally graded composite conical plate according to claim 1, characterized in that, In step S6 S61: The displacement and rotation components are described by exponential functions and spatial modal functions, and the spatial modal functions are expanded by Chebyshev polynomials and boundary functions; S62: Substitute the spatial mode function and the components describing displacement and rotation into the potential energy and kinetic energy, ignore the nonlinear terms, and take time t equal to 0 to obtain the maximum kinetic energy and maximum potential energy; S63: Using the Ritz minimization principle, take the partial derivative of the maximum potential energy minus the maximum kinetic energy with respect to the undetermined coefficients, and set the partial derivative result to 0 to obtain the characteristic equation; S64: Solve for the eigenvalues ​​of the characteristic equation to obtain the natural frequencies of the three-phase composite conical plate.

9. A prediction system for the natural frequencies of a three-phase functionally graded composite conical plate, characterized in that, A method for predicting the natural frequencies of a three-phase functionally graded composite conical plate as described in any one of claims 1 to 8, comprising: The first calculation module constructs a three-phase composite material composition model and uses the improved Halpin-Tsai model to calculate the equivalent Young's modulus, equivalent Poisson's ratio, and equivalent density of the GPL two-phase composite matrix in the three-phase composite material composition model. The second calculation module adds carbon fiber reinforcements distributed at a preset layup angle to the GPL-containing two-phase composite matrix to form an anisotropic three-phase composite layer. Based on the equivalent Young's modulus, equivalent Poisson's ratio, and equivalent density of the GPL-containing two-phase composite matrix, the Mori-Tanaka micromechanical model is used to solve for the equivalent Young's modulus, equivalent in-plane shear modulus, equivalent Poisson's ratio, and equivalent density of the three-phase composite layer. The constitutive equation construction module constructs the nonlinear geometric relationship between the displacement field and the strain field, and constructs constitutive equations based on the nonlinear geometric relationship between the displacement field and the strain field. The third calculation module calculates the stiffness coefficient of each layer based on the constitutive equation, the equivalent Young's modulus of the three-phase composite layer, the equivalent in-plane shear modulus, and the equivalent Poisson's ratio. The kinetic and potential energy construction module constructs the kinetic energy of the three-phase composite material cone plate based on the equivalent density of the three-phase composite layer, calculates the energy coefficient based on the stiffness coefficient of each layer, and constructs the potential energy of the cone plate based on the energy coefficient. The natural frequency solving module uses Chebyshev polynomials and boundary functions to construct modal displacement functions. Based on the modal displacement functions, the kinetic energy and potential energy of the cone plate, the characteristic equation is established using the Rayleigh-Ritz method and the principle of minimum potential energy. The natural frequency of the three-phase composite material cone plate is obtained by solving the characteristic equation.

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