A method for analyzing the vibration response of a carbon nanotube reinforced composite plate

Abstract

The application discloses a vibration response analysis method of a carbon nanotube reinforced composite material plate, and proposes the analysis method based on the nonlinear vibration response problem of a carbon nanotube reinforced functionally graded surface-layer-lattice sandwich hyperbolic plate in a thermal environment, establishes a mechanical model of the carbon nanotube reinforced functionally graded surface-layer-lattice sandwich hyperbolic plate, constructs a nonlinear vibration control equation of the functionally graded sandwich hyperbolic plate, and solves the nonlinear vibration control equation by using a direct integration method and a Runge-Kuta method. The vibration response analysis method of the carbon nanotube reinforced composite material plate can find the influence law of temperature, CNT distribution and component parameters, lattice sandwich layer structure parameters and sandwich hyperbolic plate geometric parameters on the nonlinear vibration behavior of the structure.

Description

Technical Field

[0001] This invention relates to the field of composite material structural dynamics technology, specifically to a method for analyzing the vibration response of carbon nanotube-reinforced composite plates. Background Technology

[0002] Functionally graded materials (FGRs) were initially developed to address the thermal stress mitigation problem in rocket propulsion systems. They are a new type of material developed to meet the specific material requirements of high-tech fields such as aerospace, and currently have broad application prospects in engineering. A basic FGR consists of a high-heat-resistant ceramic matrix and a high-strength metallic matrix. The two matrices serve as the surfaces of each other, gradually blending and transitioning into one another in a gradient manner. This smooth transition of material components achieves a gradient change in the structural physical properties, particularly the smooth matching of thermal expansion parameters, effectively improving and enhancing the thermal stress concentration problem caused by rapid cooling and drastic temperature rises. It is this material design philosophy that has facilitated the emergence and rapid technological development and application of various specific component FGR composites.

[0003] Hyperbolic plates are a common engineering structure, which can be transformed into various engineering structures such as flat plates, cylindrical shells, and variable curvature shells by adjusting the curvature. Due to its bidirectional curvature element, this structure exhibits geometric nonlinearity in mechanical principles. Especially in nonlinear dynamics problems, considering the influence of local structural reinforcement, nonlinear elastic base compensation, material parameter nonlinearity, and load nonlinearity, it easily forms a complex nonlinear dynamic problem. Due to the needs of practical engineering applications, the influence of elastic bases on plate and shell structures has also attracted attention.

[0004] Sandwich-type structures exhibit differentiated anisotropic mechanical properties, demonstrating superior stiffness and energy absorption characteristics for the same mass, among other excellent mechanical properties. They can support more demanding lateral and longitudinal combined load requirements, attracting significant attention from researchers and resulting in a series of representative research findings. Benefiting from their unique structural characteristics, sandwich structures, when combined with functionally graded materials (FGMs), can be further mixed to create diverse high-performance composite material structures. These are widely used in packaging, civil engineering, shipbuilding, aerospace, and construction industries. Performance evaluation and structural design of FGMs are currently promising research topics, particularly the urgent need to obtain descriptive and quantitative evaluation methods for their mechanical performance in extreme environments.

[0005] A comprehensive examination of its dynamic mechanical properties is therefore particularly urgent and important. As a lightweight structure, sandwich panels hold the promise of partially replacing the outer shells and large components of current ship, satellite, spacecraft, and automobile structures, achieving structural weight reduction and multifunctionality. Research on the mechanical properties of structures in terms of impact resistance and energy absorption is essential, as actual structures frequently endure various dynamic loads during service, such as explosions and impacts. For example, the hull design of ships and submarines not only requires the ability to withstand wind and wave loads but also to prevent catastrophic disasters in the event of collisions, underwater explosions, or other sudden events. This necessitates excellent energy absorption and impact resistance. Various spacecraft operating in space require a certain degree of resistance to high-speed impacts from birds and space debris. Automobile structures, in the event of an impact, require the ability to withstand significant plastic deformation and possess good cushioning performance.

[0006] Based on composite material mechanics, elasticity, heat and mass transfer, nonlinear dynamics, and nonlinear plate and shell theory, a multi-field response mechanical model is established for FGM sandwich hyperbolic plate structures with various geometric structures and composite material reinforcement. Addressing nonlinear dynamics problems, the model considers the influence of thermal environment, sandwich structure design, and the distribution of reinforcing materials. It clarifies the influence of environmental factors, component content and distribution, and structural geometric elements on multi-field dynamic behavior, fully leveraging the superior performance of functionally graded materials. This will promote the development of research on nonlinear mechanics of composite material structures with performance enhancement effects and provide a theoretical foundation and support for the design of functionally graded laminated hyperbolic structures, possessing significant scientific research and engineering application value. Summary of the Invention

[0007] The technical problem to be solved by this invention is: This invention provides a vibration response analysis method for carbon nanotube reinforced composite material plates, which can discover the influence of temperature, CNT distribution and component parameters, lattice core layer structural parameters, and sandwich hyperboloid plate geometric parameters on the nonlinear vibration behavior of the structure.

[0008] To solve the above-mentioned technical problems, the technical solution proposed by this invention is as follows:

[0009] A vibration response analysis method for a carbon nanotube reinforced composite material plate, the vibration response analysis method comprising the following steps:

[0010] Step S1: Construct the mechanical model of the CNT-enhanced functional gradient lattice sandwich hyperboloid plate;

[0011] The constitutive and geometric relations of the CNT-enhanced functional gradient lattice sandwich hyperboloid plate were obtained, and the corresponding mechanical model relations were provided.

[0012] Step S2: Construct a mathematical expression for the distribution of heterogeneous carbon nanotubes;

[0013] Step S3: Construct the equivalent mechanical relationship expression of the lattice sandwich structure;

[0014] The geometric parameters of the sandwich structure were numerically optimized, and the elastic constitutive relationship of the lattice material of the sandwich structure was analyzed. The sandwich structure is composed of multiple individual cells spliced ​​together, and the individual cells are taken as the analysis object.

[0015] Step S4: Construct the dynamic equilibrium equations for the impact dynamics problem;

[0016] Based on Hamilton's variational principle, the dynamic equilibrium equations of the sandwich hyperbolic plate are established.

[0017] Step S5: Use the Galerkin method, Gauss-Legendre integration technique and Runge-Kutta method to obtain the vibration response of the hyperbolic plate; analyze the influence of geometric parameters, material parameters, CNT distribution and boundary conditions on the frequency and amplitude of the composite plate vibration.

[0018] A further improvement to the technical solution is as follows:

[0019] Preferably, the carbon nanotube reinforced composite material plate has a core layer and isotropic metal surface layers located on both sides of the core layer. In step S1, the mechanical model relationship is as follows:

[0020] (1) Material properties of the sandwich layer It can be expressed by the following formula:

[0021]

[0022] and These represent the material properties of the outer and inner layers of the FGM intermediate layer, respectively. The thickness of the sandwich layer;

[0023] (2) The constitutive relations of each layer of material are expressed as follows:

[0024] = (5)

[0025] The stiffness coefficient is expressed as follows:

[0026] = = ,

[0027] In the formula, For membrane stress, For transverse shear stress, Indicates the stiffness coefficient;

[0028] (3) The displacement of any point in the sandwich hyperbolic plate can be expressed as:

[0029]

[0030]

[0031]

[0032] In the above formula, , and For accompanying time Associated functional gradient sandwich hyperbolic plate mid-surface edge , and Displacement in the axial direction, and The normals of the mid-surface are respectively around and Rotation of the shaft.

[0033] Preferably, in step S2, the mass-to-gravity relationship of CNTs is as follows:

[0034]

[0035] In the formula, This refers to the specific gravity parameter of CNTs. and These are the density parameters of CNTs and the matrix material FGM, respectively.

[0036] Preferably, in step S3, the core layer containing a large number of periodic unit cells is equivalent to a solid structure, and at least three types of core lattice hyperbolic plates are constructed under ideal impact loads. The dynamic response of the composite structure is solved respectively, and the elastic constitutive relationship of the lattice material is analyzed.

[0037] Preferably, in step S4, the dynamic equilibrium equation of the composite structure is:

[0038]

[0039] in, The strain energy represents the structure. Represents thermal potential energy. The work done by external forces Represents kinetic energy. This represents the first-order variation;

[0040] The strain energy is expressed as follows:

[0041]

[0042] The work done by the external force is:

[0043]

[0044] in, For the lateral load on the upper surface of the functionally graded sandwich hyperbolic plate;

[0045] The kinetic energy is described as follows:

[0046] .

[0047] Preferably, in step S5, by combining the boundary conditions and using the Galerkin method, the following set of nonlinear differential governing equations is obtained:

[0048]

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055] In the formula, For each coefficient, , , , as well as For an unknown dimensionless amplitude, and These are the half-wave numbers in the x and y directions, respectively.

[0056] Preferably, in step S5, the Gauss-Legendre integration technique is used, and the dimensionless nonlinear natural frequency of the composite structure is:

[0057]

[0058]

[0059] The ratio of the dimensionless nonlinear natural frequency to the dimensionless linear natural frequency of a functionally graded sandwich hyperbolic plate is:

[0060]

[0061] The nonlinear differential control equations can be solved using the Runge-Kutta method based on the following initial conditions:

[0062] .

[0063] The vibration response analysis method for carbon nanotube reinforced composite material plates provided by this invention has the following advantages compared with existing technologies:

[0064] (1) The vibration response analysis method of the carbon nanotube reinforced composite material plate of the present invention establishes a mechanical model of a CNT-reinforced FGM surface-lattice sandwich hyperbolic plate, constructs the nonlinear vibration control equation of the FGM sandwich plate, and solves the nonlinear vibration control equation using the direct integration method and the Runge-Kuta method. For comparative studies, the influence of temperature, CNT distribution and component parameters, lattice sandwich layer structural parameters, and sandwich hyperbolic plate geometric parameters on the nonlinear vibration behavior of the structure can be found.

[0065] (2) The vibration response analysis method of carbon nanotube reinforced composite material plate of the present invention establishes a vibration response mechanical model of reinforced FGM sandwich hyperbolic plate, and uses distribution and composition to differentiate the material properties of the reinforcing material layer, thus accurately and meticulously characterizing the structural material properties; and constructs a quantitative relationship between reinforcing material parameters, reinforcing structural parameters and material properties.

[0066] (3) The vibration response analysis method of carbon nanotube reinforced composite material plate of the present invention provides a material mechanical model of CNT hybrid functional gradient lattice sandwich structure considering temperature influence coefficient and distribution form, and constructs a precise quantitative relationship between material parameters and multiple physical quantities of lattice sandwich hyperbolic plate structure. Through numerical solution, it reveals the influence relationship between design parameters such as temperature parameters, material parameters, and geometric parameters and vibration displacement response and natural frequency of CNT reinforced lattice sandwich functional gradient sandwich hyperbolic plate. It can efficiently and accurately guide the design of corresponding composite material structure and provide the optimal matching combination solution of complex technical parameters. Attached Figure Description

[0067] Figure 1 This is a schematic diagram of the sandwich composite structure of the present invention subjected to an explosive impact.

[0068] Figure 2(a) shows the distribution of carbon nanotubes in the sandwich layer. A schematic diagram of type characteristics.

[0069] Figure 2(b) shows the distribution of carbon nanotubes in the sandwich layer. A schematic diagram of type characteristics.

[0070] Figure 2(c) shows the distribution of carbon nanotubes in the sandwich layer. A schematic diagram of type characteristics.

[0071] Figure 3(a) is a schematic diagram of the cone-shaped core unit cell structure model in an embodiment of the present invention.

[0072] Figure 3(b) is a schematic diagram of the tetrahedral core unit cell structure model in an embodiment of the present invention.

[0073] Figure 3(c) is a schematic diagram of the 3D-Kagome type core unit cell structure model in an embodiment of the present invention.

[0074] Figure 4(a) shows the effect of the structural radius of the 3D-Kagome core structure rod on the dimensionless nonlinear amplitude of the sandwich hyperboloid plate in the embodiment of the present invention.

[0075] Figure 4(b) shows the effect of the structural radius of the pentahedral core structure member on the dimensionless nonlinear amplitude of the sandwich hyperboloid plate in the embodiment of the present invention.

[0076] Figure 4(c) shows the effect of the structural radius of the tetrahedral core structure member on the dimensionless nonlinear amplitude of the sandwich hyperboloid plate in the embodiment of the present invention. Detailed Implementation

[0077] The following provides a detailed description of specific embodiments of the present invention. It should be understood that the specific embodiments described herein are for illustrative and explanatory purposes only and are not intended to limit the scope of the invention.

[0078] This invention discloses a vibration response analysis method for carbon nanotube-reinforced composite material plates. First, an analytical model of a sandwich hyperboloid plate with carbon nanotube-reinforced functionally graded material is constructed. The sandwich hyperboloid plate has three layers: a core layer containing composite material and isotropic metallic surface layers located on both sides of the core layer. The core layer is constructed with isotropic metallic materials in a lattice structure design, gradually transitioning from the isotropic metallic materials of the core layer to isotropic ceramic materials on the top and bottom surfaces.

[0079] The vibration response analysis method in this embodiment includes the following steps:

[0080] Step S1: Construct a mechanical model of a sandwich hyperbolic plate with a gradient lattice core structure reinforced by carbon nanotubes (CNTs).

[0081] Step S1-1: Establish the geometric parameters of the sandwich hyperboloid plate.

[0082] like Figure 1 As shown, the Cartesian orthogonal curvilinear coordinate system Built on the middle layer of the hyperboloid plate, The positive direction points to the outer side of the hyperbola, and in the coordinate system... , and Displacement in the axial direction is respectively represented by , and Let represent it. The principal radius of curvature of the sandwich hyperboloid is . and The side length is and Thickness is .

[0083] General material properties of sandwich layers mass density is The elastic modulus is Material properties Poisson's ratio changes along the thickness direction of the hyperboloid plate. This is a fixed value. In this embodiment, the outer surface layer of the hyperbolic plate is set. Rich in Material Type-1, Inner Surface of Hyperbola Rich in material types - 2.

[0084] Step S1-2: Construct the mechanical model of the sandwich hyperbolic plate.

[0085] The material properties of each component, expressed by its volume composition, can be represented by the following functions:

[0086] (1)

[0087] in, and These represent the material properties of the outer and inner layers of the FGM intermediate layer, respectively. and These refer to the volume components of material type-1 and material type-2, respectively, and their relationship can be defined by the following formula:

[0088] (2)

[0089] Volume composition It varies along the normal direction of the plate thickness in the form of an exponential function, and the specific functional relationship can be described as follows:

[0090] (3)

[0091] in, It is the volume change index and is limited to positive values. This is used to describe the distribution relationship of anisotropic materials in the sandwich layer. When The core layer material will degenerate into a metallic material when... Approaching At that time, the sandwich layer material will become a non-metallic material.

[0092] Material properties of the sandwich layer It can be expressed by the following formula:

[0093] (4)

[0094] FGM material properties are based on volume parameters Smoothly from the thickness direction of the hyperboloid plate Transition to .

[0095] The constitutive relations of each layer of material are expressed as follows:

[0096] = (5)

[0097] The stiffness coefficient is expressed as follows:

[0098] = = ,

[0099] In the formula, For membrane stress, For transverse shear stress, This represents the stiffness coefficient.

[0100] CNT (carbon nanotube) reinforced composites with functionally graded materials as the matrix are characterized by multiple independent, discrete small particles distributed within the matrix material (FGM). While CNTs are isotropic, the CNT-reinforcing core material is anisotropic due to their distribution. To obtain the equivalent mechanical properties of the functionally graded CNT material layer and more accurately describe the mechanical behavior of the hyperboloid plate, revealing its underlying mechanical principles, an improved hybrid constitutive model is used to express the effective Young's modulus and shear modulus of the CNT-reinforced composite plate, as shown in the following formula:

[0101] + (6a)

[0102] (6b)

[0103] (6c)

[0104] Where 'c' refers to the parameters related to the CNT enhancement center layer. , , These are the elastic modulus and shear modulus of CNT, respectively. and For the elastic modulus and shear modulus of the matrix material, The effectiveness parameters of CNT material in each layer of the hyperboloid plate are set; based on existing experimental data and published results, the following parameters are defined. = , = .

[0105] The effective Poisson's ratio of the CNT-reinforced hyperboloid core layer can be expressed as follows:

[0106] = + (7)

[0107] in, and These represent the Poisson's ratios of the CNT and the matrix material, respectively.

[0108] The coefficient of thermal expansion of CNT-reinforced composite materials can be defined by the following expression:

[0109] (8a)

[0110] (8b)

[0111] In the formula , and Let be the coefficients of thermal expansion of the CNT sandwich layer and the substrate material, respectively. For the substrate material, Poisson's ratio and density are set to fixed values, and the coefficients of thermal expansion and Young's modulus properties are related to temperature.

[0112] Temperature expression ,in Represents the temperature increment of the working environment, initial temperature CNT effective coefficient It depends on the parameter values ​​in the CNT volume fraction, and the corresponding parameters are shown in Table 1.

[0113] Table 1. Values ​​of different CNT volume ratios and CNT effective coefficients

[0114]

[0115] Young's modulus of CNT shear modulus and coefficient of thermal expansion The settings are associated with temperature, and the corresponding relationships are shown below:

[0116] =6.3998 4.33817 +7.43

[0117] 8.02155 5.420375 +9.725

[0118] 1.40755 3.476208

[0119] 1.12515 0.02291688

[0120] 0.984625

[0121] Based on the first-order shear deformation theory, the displacement of any point in the sandwich hyperboloid plate can be expressed as:

[0122] (9a)

[0123] (9b)

[0124] (9c)

[0125] In the above formula, , and For accompanying time Associated functional gradient sandwich hyperbolic plate mid-surface edge , and Displacement in the axial direction, and The normals of the mid-surface are respectively around and Rotation of the shaft.

[0126] The normal strain and shear strain components of a multi-directional functionally graded sandwich hyperboloid in an orthogonal curvilinear coordinate system can be expressed as:

[0127] (10)

[0128] in, Indicates membrane strain, This indicates lateral strain.

[0129] (11a)

[0130] (11b)

[0131] (12a)

[0132] (12b)

[0133] (12c)

[0134] The expressions for the force and torque components are as follows:

[0135] (13a)

[0136] (13b)

[0137] in, and This represents membrane stress and transverse shear force. , and These represent higher-order bending moments and shear forces, respectively.

[0138] Based on the thin-shell assumption, the internal forces, internal moments, and shear forces of a multi-directional functionally graded sandwich hyperbolic plate are defined as follows:

[0139] (14)

[0140] in, is the radial shear correction factor for the functionally graded sandwich hyperbolic plate.

[0141] tensile stiffness coefficient Flexural coupling stiffness coefficient Bending stiffness coefficient The definition is as follows:

[0142] (15a)

[0143] (15b)

[0144] Step S2: Construct a mathematical expression for the distribution of heterogeneous carbon nanotubes.

[0145] In this embodiment, four CNT-enhanced functionally graded material distribution functions are selected to describe the distribution of CNTs in the functionally graded layer: uniform distribution (as shown in Figure 2(a)), dense gradient distribution at the bottom (as shown in Figure 2(b)), and dense gradient distribution at the top and bottom (as shown in Figure 2(c)). The layer has dimensions a and b, and a thickness h, comprising isotropic top and bottom surface layers, and a core layer with a gradient distribution of CNTs. The thicknesses of the surface layer and the core layer are respectively... and The core layer is based on a tetrahedron / pentahedron / The three types of sandwich designs use different materials, and the CNTs are distributed in a gradient along the layer thickness direction.

[0146] The reinforced CNTs are distributed along the X-axis length of the hyperbola, and the total mass of the CNTs contained in the sandwich hyperbola is... The volume fraction of CNTs is For hyperbolic plates employing a dense gradient distribution at the bottom and a dense gradient distribution at both the top and bottom, the reinforcement CNTs follow an exponential distribution along the plate thickness.

[0147] Assume that the volumetric weight ratio of CNTs to the matrix material FGM follows a gradient distribution along the thickness direction of the plate according to a specific function. and The functions representing different volumetric specific gravities are expressed as follows:

[0148] (16)

[0149] (17)

[0150] The specific gravity relationship of CNTs is as follows:

[0151] (18)

[0152] In the formula, This refers to the specific gravity parameter of CNTs. and These are the density parameters of CNTs and the matrix material (FGM), respectively.

[0153] Step S3: Construct the equivalent mechanical relationship expression of the lattice sandwich structure.

[0154] The sandwich hyperboloid plate involved in this embodiment has a core layer surface composed of a lattice sandwich structure, while both the top and bottom surfaces of the hyperboloid plate are made of functionally graded polymer (FGM). The top surface of the sandwich hyperboloid plate gradually transitions from a ceramic (or metal) layer to a metal (or ceramic) core layer, and further transitions from a metal (or ceramic) core layer to a ceramic (or metal) bottom surface layer through the FGM bottom layer. The evolution trend of the functionally graded sandwich hyperboloid plate material composition can be expressed by the following volumetric component parameters:

[0155]

[0156] in, for , .

[0157] This embodiment treats the core layer containing a large number of periodic unit cells as an equivalent solid structure, as shown in the carbon nanotube distribution characteristics in Figures 3(a)-3(c). Analytical models of cone-shaped (Figure 3(a)), tetrahedral (Figure 3(b)), and 3D-kagome (Figure 3(c)) core lattice sandwich panels under ideal impact loads were established, and the dynamic responses of the three sandwich panel structures were analyzed and solved. The geometric parameters of the three sandwich panels were numerically optimized, including core and panel thicknesses, and the relative density of the core layer. The elastic constitutive relationship of the lattice material was analyzed for the three sandwich panels. Because the lattice sandwich structure is a typical periodic structure, it is considered as a combination of many individual cells, and the individual cell is taken as the object of analysis.

[0158] In this embodiment, the tetrahedral lattice is composed of three rods of equal length. The structural parameters are defined as follows: the spacing between each unit cell is... The cross-sectional area of ​​the rod is The length of the rod structure is The radius of the member structure is rods and The included angle between the planes is The elastic modulus of the rod material is Poisson's ratio is The relative density of the core , and Let be the shear stiffness coefficient of the lattice sandwich layer. Further, the specific expression is as follows:

[0159] Conical core:

[0160] (19a)

[0161] (19b)

[0162] Tetrahedral core:

[0163] (19c)

[0164] (19d)

[0165] 3D-kagome core:

[0166] (19e)

[0167] (19f)

[0168] Step S4: Construct the dynamic equilibrium equations for the impact dynamics problem.

[0169] Based on Hamilton's variational principle, the dynamic equilibrium equations of a sandwich hyperbolic plate with a buffer compensation system are as follows:

[0170] (20)

[0171] in, The strain energy represents the structure. Represents thermal potential energy. The work done by external forces Represents kinetic energy. This represents the first-order variation.

[0172] The strain energy of a functionally graded sandwich hyperbolic plate is expressed as follows:

[0173] (twenty one)

[0174] The work done by the impact force is:

[0175] (twenty two)

[0176] The lateral load is applied to the upper surface of the functionally graded sandwich hyperbolic plate.

[0177] The kinetic energy of the structure is expressed as follows:

[0178] (twenty three)

[0179] For the core layer in a sandwich hyperboloid slab, the stiffness coefficients for different structural types need to be treated differently. The material stiffness of the sandwich hyperboloid plate with multi-directional functional gradient characteristics of the core layer.

[0180] The nonlinear governing equations for a sandwich hyperboloid plate made of multi-directional functionally graded material are as follows:

[0181] (24a)

[0182] (24b)

[0183] (25c)

[0184] (25d)

[0185] (25e)

[0186] In the formula, For mass inertia.

[0187] (26)

[0188] Furthermore, the expressions for the mid-surface internal forces and moments of the functionally graded sandwich hyperboloid are as follows:

[0189] (27a)

[0190] (27b)

[0191] (27c)

[0192] (27d)

[0193] (27e)

[0194] (27f)

[0195] The nonlinear control equations in the form of the displacement function of the functionally graded sandwich hyperbolic plate are obtained as follows:

[0196] (28a)

[0197] (28b)

[0198] (28c)

[0199] (28d)

[0200] (28e)

[0201] By introducing the following dimensionless quantity, the above nonlinear governing equations can be transformed into dimensionless governing equations:

[0202]

[0203]

[0204]

[0205] The dimensionless form is as follows:

[0206] (29a)

[0207] (29b)

[0208] (29c)

[0209] (29d)

[0210] (29e)

[0211] Step S5: Use the Galerkin method, Gauss-Legendre integration technique and Runge-Kutta method to obtain the vibration response of the hyperbolic plate.

[0212] According to the dimensionless boundary conditions of formula (29), the dimensionless generalized displacement (including 5 variables) is defined as follows:

[0213] (30a)

[0214] (30b)

[0215] (30c)

[0216] (30d)

[0217] (30e)

[0218] in, , , , as well as For an unknown dimensionless amplitude, , and These are the half-wave numbers in the x and y directions, respectively.

[0219] By combining the boundary conditions and using the Galerkin method, the following set of nonlinear differential governing equations is obtained:

[0220] (31a)

[0221] (31b)

[0222]

[0223]

[0224] (31c)

[0225] (31d)

[0226] (31e)

[0227] In the formula, "and" "These are represented as the first and second derivatives of each displacement amplitude with respect to time, respectively." These are the coefficients for each item.

[0228] Due to the lateral inertia term In shell structures, inertia plays a dominant role in nonlinear vibration response. In this embodiment, to ensure accuracy, in-plane inertia and rotational inertia terms are ignored. The equation (31) is... , , and Represented as The function, then

[0229] (32a)

[0230] (32b)

[0231] Among them, the coefficients of each item It is expressed as follows:

[0232]

[0233]

[0234]

[0235]

[0236]

[0237] In the above formula, This represents finding the inverse of a matrix.

[0238] The nonlinear differential control equations can then be written as follows:

[0239] (33)

[0240] Solving for the dimensionless nonlinear natural frequency yields the homogeneous nonlinear differential governing equation:

[0241] (34)

[0242] in, , Let be the dimensionless linear natural frequency of the functionally graded sandwich hyperbolic plate.

[0243] The dimensionless nonlinear natural frequencies of the functionally graded sandwich hyperbolic plate are obtained using the direct integration method. The energy balance equation is as follows:

[0244] (35)

[0245] In the formula, It is a constant, and its value depends on the initial conditions. , and If we obtain it, then we have

[0246] (36)

[0247] The energy balance equation can be revised to:

[0248] (37)

[0249] Using the Gauss-Legendre integration technique, the dimensionless nonlinear natural frequency of the functionally graded sandwich hyperbolic plate is:

[0250] (38a)

[0251] (38b)

[0252] The ratio of the dimensionless nonlinear natural frequency to the dimensionless linear natural frequency of a functionally graded sandwich hyperbolic plate is:

[0253] (39)

[0254] Considering the influence of the external excitation force on the functionally graded sandwich hyperbolic plate, the nonlinear differential governing equations can be solved using the Runge-Kutta method based on the following initial conditions:

[0255] (40)

[0256] This implementation method provides a material mechanics model for CNT hybrid functionally graded lattice sandwich structures, considering the influence coefficient and distribution pattern of temperature. It constructs a precise quantitative relationship between material parameters and multiple physical quantities of the lattice sandwich hyperbolic plate structure. Through numerical solution, it reveals the influence relationship between design parameters such as temperature parameters, material parameters, and geometric parameters and the vibration displacement response and natural frequency of the CNT-reinforced lattice sandwich functionally graded hyperbolic plate. This method can efficiently and accurately guide the design of corresponding composite material structures and provide the optimal matching combination solution for complex technical parameters.

[0257] In this embodiment, Figures 4(a)-4(c) illustrate the influence of the structural radius of the member on the vibration response of the sandwich hyperbolic plate. (Transformation radius) They are respectively To adjust the structural stiffness. As shown in Figures 4(a)-4(c), an increase in vibration response frequency indicates an increase in stiffness; however, the dimensionless nonlinear amplitude of the sandwich hyperbolic plate is accompanied by… The increase in frequency exhibits a limited decrease, and the dimensionless nonlinear frequency ratio of the sandwich hyperboloid plate is basically unaffected. This is attributed to the limited improvement effect of dynamic response produced by the stable structure of the lattice core and the adjustment of the micro-structural parameters of the core.

[0258] The above embodiments are merely preferred examples of the present invention and are not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Therefore, any simple modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention should fall within the protection scope of the present invention.

Claims

1. A method for analyzing the vibration response of a carbon nanotube-reinforced composite material plate, characterized in that, The vibration response analysis method includes the following steps: Step S1: Construct the mechanical model of the CNT-enhanced functional gradient lattice sandwich hyperboloid plate; The constitutive and geometric relations of the CNT-enhanced functional gradient lattice sandwich hyperboloid plate were obtained, and the corresponding mechanical model relations were provided. Step S2: Construct a mathematical expression for the distribution of heterogeneous carbon nanotubes; Step S3: Construct the equivalent mechanical relationship expression of the lattice sandwich structure; Numerical optimization was performed on the geometric parameters of the sandwich structure. The elastic constitutive relationship of the lattice material of the sandwich structure was analyzed. The sandwich structure is composed of multiple individual cells, and the individual cell is taken as the object of analysis; specifically: The surface layer of the sandwich hyperboloid panel is composed of a lattice sandwich structure, while the top and bottom surfaces are made of FGM (fiber optic gluing). The evolution trend of the material composition of the sandwich hyperboloid panel is expressed as follows: in, for , The elastic constitutive relations of lattice materials for three types of sandwich panels were analyzed. The lattice sandwich structure is composed of many individual cells spliced ​​together. Structural parameters are defined as follows: the spacing between each cell is assumed to be... The cross-sectional area of ​​the rod is The length of the rod structure is The radius of the member structure is rods and The included angle between the planes is The elastic modulus of the rod material is Poisson's ratio is The relative density of the core , and The shear stiffness coefficient of the lattice sandwich layer is expressed as: Conical core: Tetrahedral core: 3D-kagome core: Step S4: Construct the dynamic equilibrium equations for the impact dynamics problem; Based on Hamilton's variational principle, the dynamic equilibrium equations of the sandwich hyperbolic plate are established. Step S5: Use the Galerkin method, Gauss-Legendre integration technique and Runge-Kutta method to obtain the vibration response of the hyperbolic plate; analyze the influence of geometric parameters, material parameters, CNT distribution and boundary conditions on the frequency and amplitude of the composite plate vibration.

2. The vibration response analysis method for carbon nanotube reinforced composite material plates according to claim 1, characterized in that, The carbon nanotube reinforced composite material plate has a core layer and isotropic metal surface layers located on both sides of the core layer. In step S1, the mechanical model relationship is: (1) Material properties of the sandwich layer It can be expressed by the following formula: and These represent the material properties of the outer and inner layers of the FGM intermediate layer, respectively. The thickness of the sandwich layer; (2) The constitutive relations of each layer of material are expressed as follows: = The stiffness coefficient is expressed as follows: In the formula, For membrane stress, For transverse shear stress, Indicates the stiffness coefficient; (3) The displacement of any point in the sandwich hyperbolic plate can be expressed as: In the above formula, , and For accompanying time Associated functional gradient sandwich hyperbolic plate mid-surface edge , and Displacement in the axial direction, and The normals of the mid-surface are respectively around and Rotation of the shaft.

3. The vibration response analysis method for carbon nanotube reinforced composite material plates according to claim 2, characterized in that, In step S2, the mass ratio of CNTs is as follows: In the formula, This refers to the specific gravity parameter of CNTs. and These are the density parameters of CNTs and the matrix material FGM, respectively.

4. The vibration response analysis method for carbon nanotube reinforced composite material plates according to claim 1, characterized in that, In step S3, the core layer containing a large number of periodic unit cells is equivalent to a solid structure, and at least three types of core lattice hyperbolic plates are constructed under ideal impact loads. The dynamic response of the composite structure is solved respectively, and the elastic constitutive relationship of the lattice material is analyzed.

5. The vibration response analysis method for carbon nanotube reinforced composite material plates according to claim 1, characterized in that, In step S4, the dynamic equilibrium equation of the composite structure is: in, The strain energy represents the structure. Represents thermal potential energy. The work done by external forces Represents kinetic energy. This represents the first-order variation; The strain energy is expressed as follows: The work done by the external force is: in, For the lateral load on the upper surface of the functionally graded sandwich hyperbolic plate; The kinetic energy is described as follows: 。 6. The vibration response analysis method for carbon nanotube reinforced composite material plates according to claim 1, characterized in that, In step S5, combining the boundary conditions and using the Galerkin method, the following set of nonlinear differential governing equations is obtained: In the formula, For each coefficient, , , , as well as For an unknown dimensionless amplitude, and These are the half-wave numbers in the x and y directions, respectively.

7. The vibration response analysis method for carbon nanotube reinforced composite material plates according to claim 6, characterized in that, In step S5, using the Gauss-Legendre integration technique, the dimensionless nonlinear natural frequency of the composite structure is: The ratio of the dimensionless nonlinear natural frequency to the dimensionless linear natural frequency of a functionally graded sandwich hyperbolic plate is: The nonlinear differential control equations can be solved using the Runge-Kutta method based on the following initial conditions: 。

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