Method for analyzing dynamic characteristics of rigid-flexible coupling box type structure of composite material

Abstract

The invention belongs to the technical field of hydrogen energy aircraft box type structure dynamic characteristic research, and particularly relates to a composite material rigid-flexible coupling box type structure dynamic characteristic analysis method which comprises the steps that a mathematical model and a coordinate system of composite material rigid-flexible coupling box type structure dynamics are established, and size parameters and material parameters are input; calculating a displacement equation of the rigid-flexible coupling box type structure of the composite material; calculating the stress strain of the rigid-flexible coupling box type structure of the composite material; the kinetic energy and potential energy of the composite rigid-flexible coupling box type structure are solved based on a displacement equation and stress-strain; the inherent characteristics of the composite material rigid-flexible coupling box type structure are solved by applying Rayleigh-Ritz, and the dynamic characteristics of the composite material rigid-flexible coupling box type structure are analyzed. The analysis method can solve the problem that analysis and evaluation research on the dynamic characteristics of the rigid-flexible coupling box type structure of the hydrogen energy aircraft composite material is insufficient.

Description

Technical Field

[0001] The present invention belongs to the technical field of research on the dynamic characteristics of the box structure of hydrogen energy aircraft, and particularly relates to a method for analyzing the dynamic characteristics of a composite rigid-flexible coupling box structure. Background Art

[0002] With the increasing emphasis on reducing carbon emissions and transitioning to clean energy, hydrogen propulsion systems have become a promising solution for achieving environmentally friendly air transportation. The overall structural configuration and materials of hydrogen energy aircraft are completely different. They adopt a fully composite laminated fuselage and wing box structure, accounting for about 80% of the aircraft weight, aiming to increase the flight range. However, when the aircraft is subjected to unpredictable external loads, the high proportion of composite materials will become a new dynamic problem. Especially for hydrogen energy aircraft, hydrogen fuel cells are installed in the box structure of hydrogen energy aircraft, which may lead to non-linear changes in the stiffness of lightweight composite structures, thereby affecting the vibration performance of hydrogen energy aircraft. In addition, during the operation of the aircraft, hydrogen fuel cells will generate a large amount of water at high temperatures. In this complex environment, hydrogen energy aircraft may exhibit some unpredictable vibration behaviors. Therefore, studying the rigid-flexible coupling box structure is the key to promoting the technological development of hydrogen-powered aircraft.

[0003] To effectively analyze the influence of rigid bodies on the vibration of composite structures, many researchers have studied the vibration of composite structures through theoretical calculations and finite element simulations. In addition to rectangular single-plate structures, the unique characteristics of composite structures have been widely applied in the aerospace field. Many excellent researchers have shown great scientific interest in the dynamic evolution and characteristic analysis under complex environments. For example, Zhong (Mechanical Systems and Signal Processing, 2023, 192) et al. used the meshless method to study the stochastic thermal vibration characteristics of laminated composite plate structures, but did not consider the influence of hygrothermal environments on composite structures. Patent CN202110536485.6 proposed a method for optimizing the layout of temporary fasteners during the pre-connection stage of composite wing boxes, but could not establish a computational model for the rigid-flexible coupled box-type composite structure considering hygrothermal environments and elastic boundaries. Patent CN201510523133.1 provided a three-dimensional vibration analysis method for composite laminated structures applied in the fields of engineering mechanics and vibration engineering, but did not consider the complex working environment, boundary conditions of hydrogen energy aircraft, and the influence of hydrogen fuel cells on the vibration of composite box-type structures. Moreover, for more advanced and complex box-type structures, this method has obvious deficiencies. Liu et al. (Aerospace Science and Technology, 2016, 52: 102-114) used a rigid-flexible-thermal coupled dynamic model to characterize the flutter phenomenon under thermal environments and studied the rigid-flexible coupled composite laminated plate system considering warping effects and transverse shear stresses, but could not analyze the influence of rigid bodies and hygrothermal environments on composite box-type structures. Through the numerical method of the TVD difference scheme, Patent CN202011234648.7 proposed a method for analyzing the rigid-flexible coupled vibration of turnout, but the limitation of this method is that it only calculates through a large number of sample data and does not obtain an analytical model of rigid-flexible coupling.

[0004] As described above, although the above-mentioned literature and patents have studied the vibration of composite assemblies and rigid-flexible coupling problems to varying degrees, there is little research on the rigid-flexible coupling mechanism of composite plate structures and its evolution law under hygrothermal environments, which seriously restricts the wide application of lightweight composite structures in the aviation field. Therefore, it is necessary to provide an effective modeling method for complex rigid-flexible coupled thin-walled composite plate systems, starting from the reliability and safety of hydrogen energy aircraft, to predict the dynamic mechanism of composite laminated box-type structures considering hygrothermal environments during the design process. Summary of the Invention

[0005] To solve the above technical problems, the present invention provides a method for analyzing the dynamic characteristics of a composite rigid-flexible coupled box structure, so as to solve the problem of insufficient research on the dynamic mechanism analysis and evaluation of composite laminated box structures considering the hygrothermal environment.

[0006] The present invention is implemented as follows. A method for analyzing the dynamic characteristics of a composite rigid-flexible coupled box structure is provided, including the following steps:

[0007] Step 1: Establish a mathematical model and coordinate system for the dynamics of the composite rigid-flexible coupled box structure, and input dimensional parameters and material parameters;

[0008] Step 2: Calculate the displacement equation of the composite rigid-flexible coupled box structure;

[0009] Step 3: Calculate the stress and strain of the composite rigid-flexible coupled box structure;

[0010] Step 4: Solve the kinetic energy and potential energy of the composite rigid-flexible coupled box structure based on the displacement equation and stress and strain;

[0011] Step 5: Apply Rayleigh-Ritz to solve the natural characteristics of the composite rigid-flexible coupled box structure and perform dynamic characteristic analysis on the composite rigid-flexible coupled box structure.

[0012] Preferably, the specific content of Step 1 is as follows:

[0013] Step 1.1: According to the fact that the composite rigid-flexible coupled box structure is composed of 23 composite laminated plates with a certain number of layers and 1 rigid square plate, establish a dynamic model of the composite rigid-flexible coupled box structure; when establishing the coordinate system, first establish the local coordinate system O g -x g -y g -z g , where g = 1 - 23, and the local coordinate system O r -x r -y r -z r of the rigid square plate. Secondly, select an arbitrary point in the composite rigid-flexible coupled box structure as the origin of the global coordinate system O-x-y-z, and the direction of the global coordinate system is consistent with that of the local coordinate system. Let the included angle between the positive direction of each composite laminated plate and the x-axis direction of its local coordinate be β;

[0014] Step 1.2: Input dimensional parameters into the dynamic model of the composite rigid-flexible coupled box structure, including the length a g , width b g , and thickness h g of each carbon fiber composite laminated plate, and the length a r of the rigid square plate., width b r and thickness h r ;

[0015] Step 1.3: Input the material parameters of the composite laminate into the dynamic model of the composite rigid-flexible coupled box structure, including density ρ, Poisson's ratio ν 12 and ν 21 , shear modulus G 12 and the elastic moduli E g in the x g direction and E 11 in the y 22 direction of the composite laminate.

[0016] Preferably, the specific content of Step 2 is as follows:

[0017] Step 2.1: Based on the thin plate theory, the displacement functions of each composite laminate in the dynamic model of the composite rigid-flexible coupled box structure and the displacement function of the rigid square plate are respectively:

[0018]

[0019]

[0020] where u g and v g are the displacements of any point on the composite laminate in the x g , y g directions, w g is the transverse displacement, u g0 and v g0 are the in-plane displacements of the neutral plane of the composite laminate. Similarly, u r and v r are the displacements of any point on the rigid square plate in the x r and y r directions, w r is the transverse displacement, u r0 and v r0 are the in-plane displacements of the neutral plane of the rigid square plate, t is the time variable, z p-1 and z p represent the vertical distances from the upper and lower surfaces of the p-th layer of the composite laminate to the reference plane, and z represents the distance of any point on the composite laminate from the neutral plane;

[0021] Step 2.2: u g , v g , w g , u r , v r and w r are assumed to be in the following forms:

[0022]

[0023] In the formula, ω is the frequency of the rigid-flexible coupling box structure of the composite material, and ξ B , η B are dimensionless variables, and U B (ξ B , η B ), V B (ξ B , η B ), W B (ξ B , η B ) are the assumed displacement equations in each local coordinate system, expressed as:

[0024]

[0025] In the formula, a hk , b hk , c hk are undetermined polynomial coefficients, H and K represent the orders of the truncated characteristic polynomials, and are Schmidt functions that satisfy the orthogonality of geometric and mechanical boundary conditions, and h and k are variables representing the orders of the undetermined characteristic polynomials respectively.

[0026] Preferably, step 3 is specifically as follows:

[0027] Step 3.1: Calculate the displacement-strain relationship of the composite laminate of the rigid-flexible coupling box structure of the composite material according to the following formula:

[0028]

[0029] In the formula, χ gx and χ gy respectively represent the linear strains along the x g and y g directions, δ gxy represents the shear strain, φ gx , φ gy , φ gxy are the normal strains and shear strains in the local coordinate system O g -x g -y g -z g respectively, are the bending strain and the torsional strain respectively, and are defined by the following formulas:

[0030]

[0031]

[0032]

[0033] Where ξ g = x g / a g , η g = y g / b g ;

[0034] Step 3.2: According to Hooke's law, the stress of the composite laminate is expressed as:

[0035]

[0036] Where σ gx and σ gy are the stresses in the x g , y g directions respectively, τ gxy is the shear stress in the z g direction, and the material coefficient matrix is expressed as:

[0037]

[0038] Where the material coefficient is expressed as:

[0039]

[0040] Where Θ is the transformation matrix, defined as:

[0041]

[0042] Step 3.3: By integrating the stress on the composite laminate, the forces and moments can be obtained, and their expressions are as follows:

[0043]

[0044] Where N gx , N gy and N gxy are the internal forces per unit width in the same plane, M gx , M gy and M gxy represent the internal moments per unit width in the same plane, and A ij , B ij and D ij represent the tensile stiffness coefficient, the tensile-bending coupling coefficient, and the bending stiffness coefficient respectively, and are expressed as:

[0045]

[0046] Step 3.4: The composite laminate is sensitive to temperature and humidity. When the temperature and humidity in the environment change uniformly, the thermal expansion coefficient and hygrothermal expansion coefficient can be expressed as:

[0047]

[0048]

[0049] where α1 and α2 are the thermal expansion coefficients in the main directions of the composite laminate, β1 and β2 are the moisture expansion coefficients in the main directions of the composite laminate, α x 、α y and α xy are the thermal expansion coefficients in the directions other than the main directions of the composite laminate, β x 、β y and β xy are the moisture expansion coefficients in the directions other than the main directions of the composite laminate, and [Ω] is the coordinate transformation matrix of the composite laminate in the main directions and hygrothermal environment, expressed as:

[0050]

[0051] Step 3.5: When the temperature and humidity change uniformly, the hygrothermal strain of the composite laminate is defined as:

[0052]

[0053] where ΔT is the difference between the environmental temperature and the reference temperature, and C is the moisture absorption rate of the composite material, given by the following formula:

[0054]

[0055] where Mx is the mass of the composite laminate in the composite rigid-flexible coupling structure after absorbing moisture, Mc represents the initial mass of the composite laminate in the composite rigid-flexible coupling structure, and ΔM D is the mass of moisture absorbed by the composite laminate in the composite rigid-flexible coupling structure;

[0056] Step 3.6: The stress-strain relationship of the composite laminate under humidity environment and temperature change is defined as:

[0057]

[0058]

[0059] where represents the equivalent in-plane force vector required to produce a hygrothermal strain equivalent to that in the direction other than the main material direction of the composite monolayer in the composite laminate, and Denote the equivalent moment vector required to generate bending and torsional strains in the composite laminate that are equivalent to the hygrothermal strains of the composite monolayer in the non-material principal direction, t k is the thickness of a single layer of any composite laminate, z k = 0.5(z p + z p-1 ) is the z-axis coordinate at the center of the k-th layer of any composite laminate; g axis coordinate;

[0060] Step 3.7: Calculate the displacement-strain relationship of the rigid square plate of the composite rigid-flexible coupling box structure according to the following formula:

[0061] [ε r = [ε x ε y γ xy T = [ε0] + h r [κ]

[0062] where ε x and ε y are linear strains, γ xy is the shear strain, [ε0] and [κ] are the strain vector and the bending curvature vector, respectively, expressed as:

[0063]

[0064]

[0065] where ξ r = x r / a r , η r = y r / b r ;

[0066] Step 3.8: According to Hooke's law, the constitutive equation of the rigid square plate is:

[0067]

[0068] where Q 11 = Q 22 = E r / (1 - ν 2 ), Q 21 = Q 12 = νE r / (1 - ν 2 ) and Q 66 = E r / [2(1 + ν)] are the stiffness coefficients, E r ​is the elastic modulus of the rigid square plate, and ν is the Poisson's ratio of the rigid square plate. The Poisson's ratio ν can be ignored during the modeling of the rigid square plate. Therefore, the material coefficient matrix of the rigid square plate is expressed as:

[0069]

[0070] Preferably, step 4 is specifically as follows:

[0071] Step 4.1: The strain energy of the composite laminate of the composite rigid-flexible coupled box structure is given by the following formula:

[0072]

[0073] Step 4.2: The strain energy of the rigid square plate is expressed as:

[0074]

[0075] Step 4.3: The kinetic energy of the composite laminate of the composite rigid-flexible coupled box structure is defined as:

[0076]

[0077] In the formula, ρ is the density of the composite laminate of the composite rigid-flexible coupled box structure;

[0078] Step 4.4: The kinetic energy of the rigid square plate of the composite rigid-flexible coupled box structure is expressed as:

[0079]

[0080] In the formula, M R is the mass of the rigid square plate, and ρ r is the density of the rigid square plate;

[0081] Step 4.5: U H is the hygrothermal potential energy, expressed as:

[0082]

[0083] Step 4.6: On the two support boundaries of the composite rigid-flexible coupled box structure, four artificial springs are added. By selecting the stiffness values, different boundary conditions are simulated. The elastic potential energy at the boundary of the composite rigid-flexible coupled box structure is expressed as:

[0084]

[0085] In the formula, k bu 、k bv and k bw respectively represent the stiffness values of the boundary artificial springs per unit length in the directions parallel to the local coordinate axes, and k bθThe stiffness value of the boundary artificial spring per unit length in the rotational direction;

[0086] Step 4.7: Artificial springs are also used at the interfaces between adjacent plates to connect the 24 plates into a unified structure. Four other types of connecting artificial springs are added to the neutral planes of the four sides of the composite material plate. By adjusting the stiffness values of the four springs, various situations at the corners of the simulated composite rigid-flexible coupled box structure are simulated. Therefore, the potential energy U of the interconnected boundaries in the x and y directions CKX , U CKY is expressed as:

[0087]

[0088]

[0089] where θ X and θ Y are the coupling angles in the x and y directions respectively, k cu , k cv and k cw represent the stiffness values of the connecting artificial springs per unit length parallel to the local coordinate axis directions respectively, and k cθ represents the stiffness value of the boundary artificial spring per unit length in the rotational direction.

[0090] Preferably, step 5 is specifically as follows:

[0091] Step 5.1: Substitute the assumed displacement function into all potential energy and kinetic energy expressions. According to the Rayleigh-Ritz method, the Lagrangian function of the composite rigid-flexible coupled box structure is written as:

[0092]

[0093] Step 5.2: Then find the extreme value of the Lagrangian function. By taking the derivatives of these coefficients, we get:

[0094]

[0095] Step 5.3: The eigenvalue equation of the composite rigid-flexible coupled box structure is expressed as:

[0096] [K Ο +K R +K H +K BS +K CHX +K CHY -ω 2 (M O +M R )]Φ = 0

[0097] where Φ is the Ritz vector, KO , K R , K H , K BS , K CHX , K CHY , M O , M R are respectively the stiffness matrix of the composite laminate, the stiffness matrix of the rigid square plate, the hygrothermal stiffness matrix, the boundary spring stiffness matrix, the connection spring stiffness matrix, the mass matrix of the composite laminate and the mass matrix of the rigid square plate.

[0098] Compared with the prior art, the advantages of the present invention are as follows:

[0099] In the analysis method of the present invention, when ignoring the Poisson's ratio of the rigid square plate, a suitable elastic modulus of the rigid square plate is selected to establish the dynamic model of the rigid square plate. By using the Schmidt orthogonalization method in combination with the thin plate theory and the Rayleigh-Ritz method, the dynamic equation of the composite rigid-flexible coupling box structure is derived, and the inherent characteristics of the composite rigid-flexible coupling box structure are solved. Description of the Drawings

[0100] Figure 1 is the flowchart of the analysis method for the dynamic characteristics of a composite rigid-flexible coupling box structure of the present invention;

[0101] Figure 2 is the schematic diagram of the geometric model of the composite rigid-flexible coupling box structure;

[0102] Figure 3a is the frequency curve of the composite rigid-flexible coupling box structure with an artificial spring installed on one side and fixed on the other side and the ply angle of [0 / 30] 4S ;

[0103] Figure 3b is the frequency curve of the composite rigid-flexible coupling box structure with an artificial spring installed on one side and fixed on the other side and the ply angle of [0 / 90] 4S ;

[0104] Figure 4a The frequency curve of the composite rigid-flexible coupling box structure with the size of the rigid square plate being 0.05m × 0.05m;

[0105] Figure 4b The frequency curve of the composite rigid-flexible coupling box structure with the size of the rigid square plate being 0.10m × 0.10m;

[0106] Figure 4c The frequency curve of the composite rigid-flexible coupling box structure with the size of the rigid square plate being 0.15m × 0.15m;

[0107] Figure 4dFrequency curve of the rigid square plate with dimensions of 0.20m×0.20m in the composite rigid-flexible coupled box structure;

[0108] Figure 5 Curves of different temperatures of the composite rigid-flexible coupled box structure under moisture absorption rates of 0% and 1.2%; Specific implementation mode

[0109] The following will further elaborate on the method for analyzing the dynamic characteristics of the composite rigid-flexible coupled box structure through specific implementation examples.

[0110] As Figure 1 shown, the present invention provides a method for analyzing the dynamic characteristics of a composite rigid-flexible coupled box structure, including the following steps:

[0111] Step 1: Establish a mathematical model and coordinate system for the dynamics of the composite rigid-flexible coupled box structure, and input dimensional parameters and material parameters; specifically:

[0112] Step 1.1: As Figure 2 shown, according to the composite rigid-flexible coupled box structure being composed of 23 composite laminates with a certain number of layers and 1 rigid square plate, establish a dynamic model of the composite rigid-flexible coupled box structure; when establishing the coordinate system, first establish the local coordinate system O g -x g -y g -z g of any one composite laminate, where g = 1 - 23, and the local coordinate system O r -x r -y r -z r of the rigid square plate. Secondly, select an arbitrary point in the composite rigid-flexible coupled box structure as the origin of the global coordinate system O-x-y-z, and the direction of the global coordinate system is the same as that of the local coordinate system. Let the angle between the positive direction of each composite laminate and the x-axis direction of its local coordinate be β;

[0113] Step 1.2: Input dimensional parameters into the dynamic model of the composite rigid-flexible coupled box structure, including the length a g , width b g , thickness h g of each carbon fiber composite laminate, and the length a r , width b r and thickness h r of the rigid square plate;

[0114] Step 1.3: Input the material parameters of the composite laminate into the dynamic model of the composite rigid-flexible coupled box structure, including density ρ, Poisson's ratio ν 12 and ν21 、Shear modulus G 12 and the elastic moduli E g in the x g direction and E 11 in the y 22 direction of the composite laminate, respectively.

[0115] During the specific implementation, experiments with different moisture absorption rates and temperatures were conducted on the same test piece, and the inherent characteristics were solved using the analysis method for the inherent characteristics of the rigid-flexible coupled box structure of the composite material. The moisture absorption rate in the first experiment was 0%, and the moisture absorption rate in the second experiment was 1.2%. The dimensions and material parameters of the test piece are shown in Table 1 below:

[0116] Table 1

[0117]

[0118]

[0119] In the self-written Matlab program, input the length, width, and thickness of the composite laminate in the composite box structure in Table 1, the elastic modulus, shear modulus, Poisson's ratio, density, hygrothermal expansion coefficient of each layer of the composite material, and the length, width, thickness, and density of the rigid square plate, etc. parameters to provide the necessary data for the subsequent solution.

[0120] Step 2: Calculate the displacement equation of the rigid-flexible coupled box structure of the composite material; specifically: According to Schmidt orthogonalization and thin plate theory, respectively represent the assumed displacement functions of the composite laminate and the rigid square plate in the rigid-flexible coupled box structure of the composite material:

[0121] Step 2.1: Based on thin plate theory, the displacement functions of each composite laminate and the displacement function of the rigid square plate in the dynamic model of the rigid-flexible coupled box structure of the composite material are respectively:

[0122]

[0123]

[0124] In the formula, u g and v g are the displacements of any point on the composite laminate in the x g , y g directions, w g is the transverse displacement, u g0 and v g0 are the in-plane displacements of the neutral plane of the composite laminate. Similarly, u r and v r are the displacements of any point on the rigid square plate in the x r and y rDisplacement in the w direction r is the lateral displacement, u r0 and v r0 are the in-plane displacements of the neutral plane of the rigid square plate, t is the time variable, z p-1 and z p The values represent the vertical distances from the upper and lower surfaces of the p-th layer of the composite laminate to the reference plane, and z represents the distance of any point on the composite laminate from the neutral plane;

[0125] Step 2.2: u g 、v g 、w g 、u r 、v r and w r are assumed to be in the following forms:

[0126]

[0127] In the formula, ω is the frequency of the composite rigid-flexible coupled box structure, ξ B 、η B are dimensionless variables, U B (ξ B ,η B ), V B (ξ B ,η B ), W B (ξ B ,η B ) are the assumed displacement equations in each local coordinate system, expressed as:

[0128]

[0129] In the formula, a hk , b hk , c hk are undetermined polynomial coefficients, H and K represent the orders of truncation of the characteristic polynomial, and are Schmidt functions that satisfy the orthogonality of geometric boundary conditions and mechanical boundary conditions, and h and k are variables representing the orders of the undetermined characteristic polynomials respectively.

[0130] Step 3: Calculate the stress and strain of the composite rigid-flexible coupled box structure; specifically:

[0131] Step 3.1: Calculate the displacement-strain relationship of the composite laminate of the composite rigid-flexible coupled box structure according to the following formula:

[0132]

[0133] In the formula, χ gx and χ gyrespectively represent the linear strains along the x g and y g directions, δ gxy represents the shear strain, φ gx , φ gy , φ gxy are respectively the normal strains and shear strains in the local coordinate system O g -x g -y g -z g , are respectively the bending strain and the torsional strain, φ gx , φ gy , φ gxy , and are defined by the following equations:

[0134]

[0135]

[0136]

[0137] where ξ g = x g / a g , η g = y g / b g ;

[0138] Step 3.2: According to Hooke's law, the stresses of the composite laminate are expressed as:

[0139]

[0140] where σ gx and σ gy are respectively the stresses in the x g and y g directions, τ gxy is the shear stress in the z g direction, and the material coefficient matrix is expressed as:

[0141]

[0142] where the material coefficient is expressed as:

[0143]

[0144] where Θ is the transformation matrix, defined as:

[0145]

[0146] Step 3.3: By integrating the stress on the composite laminate, the forces and moments can be obtained, and their expressions are as follows:

[0147]

[0148] where N gx , N gy and N gxy are the internal forces per unit width in the same plane, M gx , M gy and M gxy represent the internal moments per unit width in the same plane, A ij , B ij and D ij represent the tensile stiffness coefficient, the tensile-bending coupling coefficient, and the bending stiffness coefficient respectively, and are expressed as:

[0149]

[0150] Step 3.4: The composite laminate is sensitive to temperature and humidity. When the temperature and humidity in the environment change uniformly, the thermal expansion coefficient and the hygrothermal expansion coefficient can be expressed as:

[0151]

[0152]

[0153] where α1 and α2 are the thermal expansion coefficients in the main directions of the composite laminate, β1 and β2 are the hygroscopic expansion coefficients in the main directions of the composite laminate, α x , α y and α xy are the thermal expansion coefficients in the non-main directions of the composite laminate, β x , β y and β xy are the hygroscopic expansion coefficients in the non-main directions of the composite laminate, and [Ω] is the coordinate transformation matrix of the composite laminate in the main directions and the hygrothermal environment, and is expressed as:

[0154]

[0155] Step 3.5: When the temperature and humidity change uniformly, the hygrothermal strain of the composite laminate is defined as:

[0156]

[0157] where ΔT is the difference between the environmental temperature and the reference temperature, and C is the moisture absorption rate of the composite material, which is given by the following formula:

[0158]

[0159] Wherein, Mx is the mass of the composite laminate in the rigid-flexible coupling structure of the composite material after absorbing moisture, Mc represents the initial mass of the composite laminate in the rigid-flexible coupling structure of the composite material, and ΔM D is the mass of moisture absorbed by the composite laminate in the rigid-flexible coupling structure of the composite material;

[0160] Step 3.6: The stress-strain relationship of the composite laminate under humidity environment and temperature change is defined as:

[0161]

[0162]

[0163] Wherein, and represent the equivalent in-plane force vectors required to generate hygrothermal strains in the composite laminate equivalent to those in the non-material principal direction of the single layer of the composite material. and represent the equivalent moment vectors required to generate bending and torsional strains in the composite laminate equivalent to those in the non-material principal direction of the single layer of the composite material. t k is the single layer thickness of any composite laminate, and z k = 0.5(z p + z p-1 ) is the z g axis coordinate at the center of the kth layer of any composite laminate;

[0164] Step 3.7: Calculate the displacement-strain relationship of the rigid square plate of the rigid-flexible coupling box structure of the composite material according to the following formula:

[0165] [ε r = [ε x ε y γ xy T = [ε0] + h r [κ]

[0166] Wherein, ε x and ε y are linear strains, γ xy is the shear strain, and [ε0] and [κ] are the strain vector and the bending curvature vector, respectively, which are expressed as:

[0167]

[0168]

[0169] Wherein, ξ r = x​r / a r ,η r =y r / b r ;

[0170] Step 3.8: According to Hooke's law, the constitutive equation of the rigid square plate is:

[0171]

[0172] where Q 11 =Q 22 =E r / (1 - ν 2 ), Q 21 =Q 12 =νE r / (1 - ν 2 ), and Q 66 =E r / [2(1 + ν)] are stiffness coefficients, E r is the elastic modulus of the rigid square plate, and ν is the Poisson's ratio of the rigid square plate. The Poisson's ratio ν can be ignored during the modeling process of the rigid square plate. Therefore, the material coefficient matrix of the rigid square plate is expressed as:

[0173]

[0174] Step 4: Solve the kinetic and potential energies of the composite rigid-flexible coupled box structure based on the displacement equation and stress-strain; specifically:

[0175] Step 4.1: The strain energy of the composite laminate of the composite rigid-flexible coupled box structure is given by the following equation:

[0176]

[0177] Step 4.2: The strain energy of the rigid square plate is expressed as:

[0178]

[0179] Step 4.3: The kinetic energy of the composite laminate of the composite rigid-flexible coupled box structure is defined as:

[0180]

[0181] where ρ is the density of the composite laminate of the composite rigid-flexible coupled box structure;

[0182] Step 4.4: The kinetic energy of the rigid square plate of the composite rigid-flexible coupled box structure is expressed as:

[0183]

[0184] where MR is the mass of the rigid square plate, ρ r is the density of the rigid square plate;

[0185] Step 4.5: U H is the hygrothermal potential energy, expressed as:

[0186]

[0187] Step 4.6: On the two support boundaries of the composite rigid-flexible coupled box structure, add four types of artificial springs. By selecting the stiffness values, different boundary conditions are simulated. The elastic potential energy at the boundary of the composite rigid-flexible coupled box structure is expressed as:

[0188]

[0189] In the formula, k bu , k bv and k bw respectively represent the stiffness values of the boundary artificial springs per unit length in the directions parallel to the local coordinate axes, and k bθ represents the stiffness value of the boundary artificial spring per unit length in the rotational direction;

[0190] Step 4.7: Artificial springs are also used at the interfaces between adjacent plates to connect the 24 plates into a unified structure. Another four types of connecting artificial springs are added on the neutral planes of the four sides of the composite material plates. By adjusting the stiffness values of the four springs, various situations at the corners of the simulated composite rigid-flexible coupled box structure are considered. Therefore, the potential energies U CKX , U CKY in the x and y directions of the interconnected boundaries are expressed as:

[0191]

[0192]

[0193] In the formula, θ X and θ Y are the coupling angles in the x and y directions respectively, and k cu , k cv and k cw respectively represent the stiffness values of the connecting artificial springs per unit length in the directions parallel to the local coordinate axes, and k cθ represents the stiffness value of the boundary artificial spring per unit length in the rotational direction.

[0194] Step 5: Apply Rayleigh-Ritz to solve the natural characteristics of the composite rigid-flexible coupled box structure and conduct a dynamic characteristic analysis of the composite rigid-flexible coupled box structure, specifically:

[0195] Step 5.1: Substitute the assumed displacement function into all potential and kinetic energy expressions. According to the Rayleigh-Ritz method, the Lagrangian function of the composite rigid-flexible coupled box structure is written as:

[0196]

[0197] Step 5.2: Then, find the extremum of the Lagrangian function. By taking the derivatives with respect to these coefficients, we get:

[0198]

[0199] Step 5.3: The eigenvalue equation of the composite rigid-flexible coupled box structure is expressed as:

[0200] [K Ο +K R +K H +K BS +K CHX +K CHY -ω 2 (M O +M R )]Φ=0

[0201] In the formula, Φ is the Ritz vector, and K O , K R , K H , K BS , K CHX , K CHY , M O , M R are the stiffness matrices of the composite laminate, the rigid square plate, the hygrothermal stiffness, the boundary spring stiffness, the connection spring stiffness, the mass matrix of the composite laminate, and the mass matrix of the rigid square plate, respectively.

[0202] Figure 3a -3b compares the frequency change curves of the composite laminate with ply angles of [0 / 90] 4S and [0 / 30] 4S . The results show that when only the stiffness value of the artificial spring k w is changed, the frequencies of the composite rigid-flexible coupled box structures with different ply angles increase by 6.0 Hz and 1.5 Hz, respectively. When the four types of artificial springs are changed simultaneously, the increments of the frequency curves are 14.4 Hz and 3.1 Hz, respectively. Therefore, when increasing the stiffness value of the artificial spring, there is no significant difference in the frequencies between the two groups of ply schemes.

[0203] Figure 4a -4d compares the vibration conditions of rigid square plates with different sizes (from 0.05 m to 0.20 m). For the ply scheme [0 / 90] 4SFor example, as the size of the rigid square plate increases, the frequency of the composite rigid-flexible coupled box structure becomes less and less sensitive to the mass of the rigid square plate. Therefore, the size of the rigid square plate will affect the vibration characteristics of the entire structure. Reasonably designing the size of the hydrogen fuel cell can reduce the vibration transmission and reflection of the box structure, thereby improving the structural performance.

[0204] Figure 5 The curves of the experimental and theoretical calculations of the moisture absorption rates of 0% and 1.2% of the composite rigid-flexible coupled box structure at different temperatures are compared. The results show that the experimental results of the eighth-order natural frequency are consistent with the theoretical calculations in the range of environmental temperatures from 20°C to 125°C. It is worth noting that the theoretically calculated frequency is higher than the experimental results because many manufacturing errors are inevitable. For example, problems such as non-parallel carbon fiber directions, carbon fiber overlaps, carbon fiber fractures, and uneven matrix distributions. Therefore, the influence of the hygrothermal environment needs to be considered when designing the composite box structure.

[0205] The above are only the preferred embodiments of the present invention and are not intended to limit the idea of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for analyzing dynamic characteristics of a composite rigid-flexible coupling box structure, characterized in that: The following steps are involved: Step 1: Establish the mathematical model and coordinate system of the dynamics of the rigid-flexible coupled box-type structure of composite materials, and input the size parameters and material parameters; Step 2: Calculate the displacement equation of the composite rigid-flexible coupled box structure; Step 3: Calculate the stress and strain of the composite rigid-flexible coupled box structure; Step 4: Solve the kinetic potential energy of the composite rigid-flexible coupling box structure based on the displacement equation and stress-strain; Step 5: Apply Rayleigh-Ritz to solve the inherent characteristics of the composite rigid-flexible coupled box structure and analyze the dynamic characteristics of the composite rigid-flexible coupled box structure.

2. The method for analyzing dynamic characteristics of composite rigid-flexible coupled box-type structures according to claim 1 is characterized in that: The step 1 is specifically as follows: Step 1.1: According to the fact that the composite rigid-flexible coupling box structure is composed of 23 composite laminates with a certain number of layers and 1 rigid square plate, a dynamic model of the composite rigid-flexible coupling box structure is established; when establishing the coordinate system, first establish the local coordinate system O of any composite laminate and g -x g -y g -z g , g = 1-23, and the local coordinate system O of the rigid square plate r -x r -y r -z r , Secondly, select any point in the composite rigid-flexible coupling box structure as the coordinate origin to establish the global coordinate system Oxyz, and the direction of the global coordinate system is consistent with the direction of the local coordinate system. Let the angle between the positive direction of each composite laminate and the x-axis direction of its local coordinate be β; Step 1.2: Input the dimensional parameters into the composite rigid-flexible coupled box structure dynamics model, including the length a of each carbon fiber composite laminate. g , width b g , thickness h g , and the length a of the rigid square plate r Width b r and thickness h r ; Step 1.3: Input the material parameters of the composite laminate into the composite rigid-flexible coupled box structure dynamics model, including density ρ, Poisson's ratio ν 12 and ν 21 , shear modulus G 12 and composite laminates at x g Direction and y g Elastic modulus E in the direction 11 and E 22 .

3. The method for analyzing dynamic characteristics of composite rigid-flexible coupled box-type structures according to claim 2, characterized in that: The step 2 is specifically as follows: Step 2.1: Based on the thin plate theory, the displacement function of each composite laminate and the displacement function of the rigid square plate in the composite rigid-flexible coupled box structure dynamics model are: In the formula, u g and v g Any point on the composite laminate at x g ,y g Displacement in the direction, w g is the lateral displacement, u g0 and v g0 is the in-plane displacement of the neutral plane of the composite laminate. Similarly, u r and v r For any point on the rigid square plate at x r and r Displacement in the direction, w r is the lateral displacement, u r0 and v r0 is the in-plane displacement of the neutral surface of the rigid square plate, t is the time variable, z p-1 and z p The value of represents the vertical distance from the upper and lower surfaces of the pth layer of the composite laminate to the reference plane, and z represents the distance from any point of the composite laminate to the neutral plane; Step 2.2: g 、v g 、w g 、u r 、v r and w r Assume the following form: Where ω is the frequency of the composite rigid-flexible coupled box structure, ξ B , η B is a dimensionless variable, U B (ξ B ,η B ), V B (ξ B ,η B ), W B (ξ B ,η B ) is the assumed displacement equation in each local coordinate system, expressed as: In the formula, a hk , b hk , c hk are the coefficients of the unknown polynomial, H and K represent the order of the characteristic polynomial being truncated, and To satisfy the Schmidt function that satisfies the geometric boundary conditions and the mechanical boundary conditions orthogonal, h and k represent the variables of the order of the characteristic polynomial to be determined.

4. The method for analyzing dynamic characteristics of composite rigid-flexible coupled box-type structures according to claim 3 is characterized in that: The step 3 is specifically as follows: Step 3.1: Calculate the displacement-strain relationship of the composite rigid-flexible coupled box-type composite laminate according to the following formula: In the formula, χ gx and χ gy Respectively represent along x g and g Linear strain in the direction, δ gxy represents shear strain, φ gx ,φ gy ,φ gxy The local coordinate system O g -x g -y g -z g The normal strain and shear strain under are the bending strain and torsional strain, φ gx ,φ gy ,φ gxy , and It is defined by the following formula: where ξ g = x g / a g , η g = y g / b g ; Step 3.2: According to Hooke's law, the stress in the composite laminate is expressed as: In the formula, σ gx and σ gy x g ,y g Stress in the direction, τ gxy For z g Shear stress in the direction, material coefficient matrix It is expressed as: In the formula, the material coefficient It is expressed as: Where Θ is the transformation matrix, defined as: Step 3.3: The forces and moments can be obtained by integrating the stresses on the composite laminate as shown below: Where N gx , N gy and N gxy is the internal force per unit width in the same plane, M gx ,M gy and M gxy It represents the internal moment per unit width in the same plane, A ij , B ij and D ij They represent the tensile stiffness coefficient, the tensile-bending coupling coefficient and the bending stiffness coefficient respectively, and are expressed as: Step 3.4: Composite laminates are sensitive to temperature and humidity. When the temperature and humidity in the environment change uniformly, the thermal expansion coefficient and the hygrothermal expansion coefficient can be expressed as: Where α1 and α2 are the thermal expansion coefficients of the composite laminate in the main direction, β1 and β2 are the moisture expansion coefficients of the composite laminate in the main direction, and α x , α y and α xy is the thermal expansion coefficient of the non-composite laminate in the main direction, β x , β y and β xy is the moisture expansion coefficient of the non-composite laminate in the main direction, [Ω] is the coordinate transformation matrix of the composite laminate in the main direction and in the wet and hot environment, expressed as: Step 3.5: When the temperature and humidity change uniformly, the hygrothermal strain of the composite laminate is defined as: Where ΔT is the difference between the ambient temperature and the reference temperature, and C is the moisture absorption rate of the composite material, which is given by the following formula: Where Mx is the mass of the composite laminate in the composite rigid-flexible coupling structure after absorbing water, Mc represents the initial mass of the composite laminate in the composite rigid-flexible coupling structure, and ΔM D It is the mass of water absorbed by the composite laminate in the composite rigid-flexible coupled structure; Step 3.6: The stress-strain relationship of the composite laminate under humidity environment and temperature changes is defined as: In the formula, and It represents the equivalent in-plane force vector required to produce mechanical in-plane strain in the composite laminate equivalent to the hygrothermal strain of the composite single layer in the non-material main direction. and It represents the equivalent moment vector required to make the composite laminate produce bending and torsional strains equivalent to the hygrothermal strains of the composite single layer in the non-material main direction, t k is the thickness of a single layer of any composite laminate, z k =0.5(z p +z p-1 ) is the z center of the kth layer of any composite laminate g Axis coordinates; Step 3.7: Calculate the displacement-strain relationship of the rigid square plate of the composite rigid-flexible coupling box structure according to the following formula: [e r ]=[e x e y c xy ] T =[ε0]+h r [k] In the formula, ε x and ε y is the linear strain, γ xy is the shear strain, [ε0] and [κ] are the strain vector and the bending curvature vector, respectively expressed as: where ξ r = x r / a r , η r = y r / b r ; Step 3.8: According to Hooke's law, the constitutive equation of the rigid square plate is: In the formula, Q 11 =Q 22 =E r / (1-ν 2 ), Q 21 =Q 12 =νE r / (1-ν 2 ) and Q 66 =E r / [2(1+ν)] is the stiffness coefficient, E r is the elastic modulus of the rigid square plate, and ν is the Poisson's ratio of the rigid square plate. The Poisson's ratio ν can be ignored in the modeling process of the rigid square plate, so the material coefficient matrix of the rigid square plate is expressed as:

5. The method for analyzing dynamic characteristics of composite rigid-flexible coupled box-type structures according to claim 4, characterized in that: The step 4 is specifically as follows: Step 4.1: The strain energy of the composite laminate of the rigid-flexible coupled box structure is given by the following formula: Step 4.2: The strain energy of the rigid square plate is expressed as: Step 4.3: The kinetic energy of the composite laminate of the rigid-flexible coupled box structure is defined as: Where, ρ is the density of the composite laminate of the rigid-flexible coupled box structure; Step 4.4: The kinetic energy of the rigid square plate of the composite rigid-flexible coupling box structure is expressed as: Where M R is the mass of the rigid square plate, ρ r is the density of the rigid square plate; Step 4.5: U H is the moist heat potential energy, expressed as: Step 4.6: Add four artificial springs to the two supporting boundaries of the composite rigid-flexible coupling box structure. Simulate different boundary conditions by selecting stiffness values. The elastic potential energy at the boundary of the composite rigid-flexible coupling box structure is expressed as: In the formula, k bu , k bv and k bw They represent the stiffness of the boundary artificial spring per unit length parallel to the local coordinate axis, k bθ Represents the stiffness value of the boundary artificial spring per unit length in the rotation direction; Step 4.7: Artificial springs are also used at the interface between two adjacent plates to connect the 24 plates into a unified structure. Another four types of connecting artificial springs are added to the neutral planes of the four edges of the composite plate. By adjusting the stiffness values ​​of the four springs, various situations at the corners of the composite rigid-flexible coupled box structure are simulated. Therefore, the potential energy U of the interconnected boundaries in the x and y directions is CKX , U CKY It is expressed as: In the formula, θ X and θ Y are the coupling angles in the x and y directions, respectively, and k cu , k cv and k cw They represent the stiffness of the artificial spring per unit length parallel to the local coordinate axis, k cθ Represents the stiffness value of the boundary artificial spring per unit length in the rotation direction.

6. The method for analyzing dynamic characteristics of composite rigid-flexible coupled box-type structures according to claim 5, characterized in that: The step 5 is specifically as follows: Step 5.1: Substitute the assumed displacement function into all potential energy and kinetic energy expressions. According to the Rayleigh-Ritz method, the Lagrangian function of the composite rigid-flexible coupled box structure is written as: Step 5.2: Then find the extreme value of the Lagrangian function, and by differentiating these coefficients, we get: Step 5.3: The characteristic value equation of the composite rigid-flexible coupled box structure is expressed as: [K Ο +K R +K H +K BS +K CHX +K CHY -oh 2 (M O +M R )]Φ=0 Where Φ is the Ritz vector, K O , K R , K H , K BS , K CHX , K CHY 、M O 、M R They are the composite laminate stiffness matrix, rigid square plate stiffness matrix, moist thermal stiffness matrix, boundary spring stiffness matrix, connection spring stiffness matrix, composite laminate mass matrix and rigid square plate mass matrix.

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