Vibration Characteristics Prediction Method of Carbon-Glass Hybrid Laminated Plates Based on HSDT

Abstract

The present invention discloses a method for predicting the vibration characteristics of a carbon-glass hybrid laminate based on HSDT, comprising the following steps: S1. Establish a transverse shear function model of the high-order shear deformation theory according to the stress-free conditions of the upper and lower boundaries of the carbon-glass hybrid laminate; S2. Establish a high-order shear deformation displacement field, and combine the geometric equations, constitutive relations, and Hamilton's energy principle to obtain the governing differential equations; S3. Establish displacement variables that satisfy the boundary conditions, and combine the governing differential equations to obtain the eigenvalue equations, and finally determine the vibration frequencies and corresponding vibration modes of the carbon-glass hybrid laminate; The present invention takes the high-order shear effect into account in the prediction analysis of the vibration characteristics of the carbon-glass hybrid laminate, and can more accurately and effectively provide more accurate and reliable vibration characteristic analysis results for the carbon-glass hybrid laminate.

Description

Technical Field

[0001] The present invention belongs to the technical field of composite material structure dynamics, and particularly relates to a method for predicting the vibration characteristics of a carbon-glass hybrid laminate. Background Art

[0002] Due to their outstanding specific strength and specific stiffness, advanced composite laminates have been widely used in the aerospace, shipbuilding, automotive, and other industries. However, the high price of carbon fiber composites with excellent mechanical properties limits the large-scale use of carbon fiber in the above industries. For example, although carbon fiber has high strength and stiffness, it has a high price and poor toughness, while glass fiber has low cost and high toughness, but its stiffness and strength are far inferior to those of carbon fiber. In order to meet the strong demand for new lightweight materials with high strength and high toughness, in recent years, a composite hybrid concept mainly based on carbon-glass hybridization has emerged, that is, fibers of different materials are fused when manufacturing a single laminate. In recent years, with the development of composite manufacturing technology, the hybridization of advanced fibers (such as glass, carbon, and aramid) has become the focus of attention because it significantly improves the mechanical properties.

[0003] In recent years, many researchers have proposed theoretical, numerical, and experimental methods to study the bending, buckling, and vibration responses of carbon-glass hybrid composites. For example, carbon-wave hybrid laminates used in engineering applications are often subjected to various dynamic loads, and the structure will be damaged due to material fatigue under severe vibration, and in severe cases, the strength and stability of the structure will be reduced. Therefore, it is very important to use advanced theoretical methods (such as classical plate theory, first-order shear deformation theory, high-order shear deformation theory, etc.) to accurately predict its vibration characteristics.

[0004] However, the theoretical methods currently used to predict the vibration characteristics of carbon-glass hybrid laminates have the problem of low accuracy. How to establish an accurate analysis model for the vibration characteristics of carbon-glass hybrid laminates based on the high-order shear deformation theory (HSDT) is a major challenge currently. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for predicting the vibration characteristics of a carbon-glass hybrid laminate based on HSDT, which can more accurately and effectively provide the analysis results of the vibration characteristics of the carbon-glass hybrid laminate.

[0006] To achieve the above object, the following technical solutions are adopted:

[0007] A method for predicting the vibration characteristics of a carbon-glass hybrid laminate based on HSDT, comprising the following steps:

[0008] S1. Establish a high-order shear deformation theory transverse shear function model according to the stress-free conditions of the upper and lower boundaries of the carbon-glass hybrid laminate;

[0009] S2. Establish a high-order shear deformation displacement field, and combine the geometric equations, constitutive relations, and Hamilton's energy principle to obtain the governing differential equations;

[0010] S3. Establish displacement variables that satisfy the boundary conditions, combine the governing differential equations to obtain the eigenvalue equation, and finally determine the vibration frequencies and corresponding vibration modes of the carbon-glass hybrid laminated plate.

[0011] According to the above scheme, step S1 further includes:

[0012] S11. Construct an even function A(z / h) with z / h as the independent variable, and shift it along the ordinate. The shift distance is A(0.5);

[0013] S12. Scale the shifted curve in the horizontal direction to satisfy f'(0) = 1. The scaling ratio is Integrate the scaled f'(z) along the plate thickness direction to obtain the newly constructed shear function f(z).

[0014] According to the above scheme, step S2 further includes:

[0015] S21. Establish the high-order shear deformation displacement field of the carbon-glass hybrid laminated plate:

[0016]

[0017]

[0018] W(x, y, z, t) = w0(x, y, t)

[0019] where U, V, and W respectively represent the displacement states of each position point inside the carbon-glass hybrid laminated plate, and u0, v0, and w0 respectively represent the displacements of each position point on the physical mid-plane along the x, y, and z directions. respectively represent the rotations of each position point on the physical mid-plane along the y and x axes;

[0020] S22. Calculate the strains of each point of the carbon-glass hybrid laminated plate:

[0021]

[0022]

[0023]

[0024]

[0025]

[0026] S23. Establish the stress-strain relationship of the carbon-glass hybrid laminated plate:

[0027]

[0028] where σ = {σ x , σ y , τ yz , τ xz , τ xy} T and ε = {ε x , ε y , γ yz , γ xz , γ xy} T are stress components and strain components respectively, and the elastic constant Q ij is calculated by the following formula:

[0029]

[0030]

[0031] where E(z) is the elastic modulus and v is the Poisson's ratio;

[0032] S24. Calculate the variation of strain energy and the variation of kinetic energy, and obtain the governing differential equations:

[0033] Calculate the variation of strain energy δU of the carbon-glass hybrid laminate:

[0034]

[0035] where (N x , N y , N xy ) are the tensile terms, (M x , M y , M xy ) are the bending terms, (S x , S y , S xy ) are the higher-order terms, (N xz , N yz ) are the shear terms, and are expressed by the following formulas respectively:

[0036] ζ = x, y, xy

[0037]

[0038] Calculate the variation of kinetic energy δK of the carbon-glass hybrid laminate:

[0039]

[0040] where the dot superscript represents the differentiation with respect to time, i.e., ρ(z) is the mass density, and (I0, I1, I2, I3, I4, I5) are the inertia terms, which can be calculated by the following formula:

[0041]

[0042] where z represents the thickness coordinate, and f(z) is the transverse shear function; then substitute the variation of strain energy and the variation of kinetic energy into the energy equation as follows:

[0043]

[0044] Then the expression of the governing differential equation is:

[0045]

[0046]

[0047]

[0048]

[0049]

[0050] where (A ij , B ij , C ij , D ij , E ij , F ij , G ij ) are the stiffness coefficients, (I0, I1, I2, I3, I4, I5) are the inertia coefficients, d i , d ij , d ijm , d ijmn are variational operators, which are calculated by the following formulas respectively:

[0051]

[0052]

[0053]

[0054] ζ = x, (i, j, m, n = 1), ζ = y, (i, j, m, n = 2).

[0055] According to the above scheme, step S3 further includes:

[0056] S31. Establish displacement variables that satisfy the boundary conditions:

[0057] u0 = U mn cos(λx)sin(μy)e iwt

[0058] v0 = V mn sin(λx)cos(μy)e iwt

[0059] w0 = W mn sin(λx)sin(μy)e iwt

[0060]

[0061]

[0062] wherein, λ = mπ / a, μ = nπ / b, (U mn , V mn , W mn , X mn , Y mn ) are mode shape coefficients;

[0063] S32. Substitute the displacement variables into the governing differential equation to calculate the eigenvalue equation:

[0064]

[0065] wherein, a ij (i, j = 1, 2, 3, 4, 5) are stiffness matrix coefficients, m ij (i, j = 1, 2, 3, 4, 5) are mass matrix coefficients, w is the frequency, and the stiffness matrix coefficients and mass matrix coefficients are calculated by the following formulas respectively:

[0066] a 11 = -A 11 λ 2 - A 66 μ 2 , a 12 = -(A 12 + A 66 )λμ, a 13 = B 11 λ 3 + (B 12 + 2B 66 )λμ 2

[0067] a 14 = -C 11 λ 2 - C 66 μ 2 , a 15 = -(C 12 + C 66 )λμ, a 22 = -A 22 μ2 -A 66 λ 2

[0068] a 23 = B 22 μ 3 +(B 12 + 2B 66 )λ 2 μ, a 24 = -(C 12 + C 66 )λμ, a 25 = -C 22 μ 2 -C 66 λ 2

[0069] a 33 = -(D 11 λ 4 + 2D 12 λ 2 μ 2 + D 22 μ 4 ) - 4D 66 λ 2 μ 2 , a 34 = E 11 λ 3 + E 12 λμ 2 + 2E 66 λμ 2

[0070] a 35 = E 12 λ 2 μ + E 22 μ 3 + 2E 66 λ 2 , a 44 = -F 11 λ 2 -F 66 μ 2 -G 55 , a 55 = -F 22 μ 2 -F 66 λ 2 -G 44

[0071] m 11 = m 22 = -I0, m 13 = I1λ, m 14 = -I3, m 23 = I1μ, m25 =-I3,m 33 =-I0 - I2(λ 2 +μ 2 )

[0072] m 34 =I4λ,m 35 =I4μ,m 44 =-I5,m 55 =-I5;

[0073] S33. Calculate the vibration frequency w and the mode shape coefficients U mn ,V mn ,W mn ,X mn ,Y mn :

[0074] [w 2 ,D]=eig(K,M)

[0075] wherein, is the stiffness matrix, is the mass matrix, D = U mn ,V mn ,W mn ,X mn ,Y mn are the mode shape coefficients, then the vibration frequency and the response mode shape of the carbon - glass hybrid laminate can be obtained.

[0076] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0077] Aiming at the problem that it is difficult to establish an accurate prediction of the vibration characteristics of carbon - glass hybrid laminates based on the high - order shear deformation theory, the present invention completes the high - order modeling of the displacement field of carbon - glass hybrid laminates based on the plate - shell theory and the energy variational principle; constructs the governing differential equation based on the above - mentioned high - order model; and finally realizes the prediction of the vibration characteristics of carbon - glass hybrid laminates by transforming the governing differential equation into an eigenvalue equation and solving it.

[0078] In addition, the high - order plate - shell model in the present invention will gradually improve the prediction accuracy with the continuous optimization of the transverse shear function model, and can be extended to different types of plate - shell structures such as hybrid composite sandwich plates. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 is the flow chart of the present invention;

[0080] Figure 2 is the comparative analysis diagram of the vibration frequencies of each order in the embodiment of the present invention;

[0081] Figure 3 is the influence diagram of each ply - layup method on the vibration frequency in the embodiment of the present invention. Detailed implementation manners

[0082] The following embodiments further illustrate the technical solutions of the present invention, but do not limit the protection scope of the present invention.

[0083] The present invention will be further described in detail below in conjunction with the drawings and embodiments.

[0084] The present invention is a method for predicting the vibration characteristics of a carbon-glass hybrid laminate based on HSDT. As Figure 1 shown, the specific steps of the method are as follows:

[0085] S1. Establish a transverse shear function model of the high-order shear deformation theory according to the stress-free conditions of the upper and lower boundaries of the carbon-glass hybrid laminate;

[0086] S2. Establish a high-order shear deformation displacement field, and combine the geometric equations, constitutive relations, and Hamilton energy principle to obtain the governing differential equations;

[0087] S3. Establish displacement variables that satisfy the boundary conditions, combine the governing differential equations to obtain the eigenvalue equations, and finally determine the vibration frequencies and corresponding vibration modes of the carbon-glass hybrid laminate.

[0088] Further, the establishment of the transverse shear function model of the high-order shear deformation theory according to the stress-free conditions of the upper and lower boundaries of the carbon-glass hybrid laminate in step S1 further includes:

[0089] S11. Construct an even function with z / h as the independent variable and translate it vertically by a distance of A(0.5) to obtain A(z / h) - A(0.5).

[0090] S12. Scale the translated curve in the horizontal direction to satisfy f'(0) = 1, and the scaling ratio is After scaling, obtain Finally, integrate f'(z) along the thickness direction to obtain f(z).

[0091] In the embodiment of the present invention, the transverse shape function in step S12 is specifically shown in Table 1.

[0092] Table 1 Transverse shape functions f(z) and g(z), g(z) = f'(z)

[0093]

[0094] Further, the establishment of the high-order shear deformation displacement field in step S2, and the combination of the geometric equations, constitutive relations, and the governing differential equations obtained according to the Hamilton energy principle, the steps further include:

[0095] S21. Establish the high-order shear deformation displacement field of the carbon-glass hybrid laminate:

[0096]

[0097]

[0098] W(x, y, z, t) = w0(x, y, t)

[0099] where U, V, and W respectively represent the displacement states of each position point inside the carbon-glass hybrid laminate, and u0, v0, and w0 respectively represent the displacements of each position point on the physical mid-plane along the x, y, and z directions. respectively represent the rotations of each position point on the physical mid-plane along the y and x axes.

[0100] S22. Calculate the strains of each point of the carbon-glass hybrid laminate:

[0101]

[0102]

[0103]

[0104]

[0105]

[0106] S23. Establish the stress-strain relationship of the carbon-glass hybrid laminate:

[0107]

[0108] where σ = {σ x , σ y , τ yz , τ xz , τ xy} T and ε = {ε x , ε y , γ yz , γ xz , γ xy} T are respectively the stress components and strain components, and the elastic constants Q ij are calculated by the following formula:

[0109]

[0110]

[0111] where E(z) is the elastic modulus and v is the Poisson's ratio.

[0112] S24. Calculate the variation of strain energy and the variation of kinetic energy, and obtain the governing differential equations:

[0113] Calculate the variation of strain energy δU of the carbon-glass hybrid laminate:

[0114]

[0115] where (N x , N y , N xy ) are the stretching terms, (M x , M y , M xy ) are the bending terms, (S x , S y , S xy ) are the higher-order terms, and (N xz , N yz ) are the shear terms, which are expressed by the following equations respectively

[0116] ζ = x, y, xy

[0117]

[0118] Calculate the variation of kinetic energy δK of the carbon-glass hybrid laminate:

[0119]

[0120] where the dot superscript represents the differentiation with respect to time, i.e., ρ(z) is the mass density, and (I0, I1, I2, I3, I4, I5) are the inertia terms, which can be calculated by the following equations

[0121]

[0122] where z represents the thickness coordinate, and f(z) is the transverse shear function. Then substitute the variation of strain energy and the variation of kinetic energy into the energy equation as follows:

[0123]

[0124] The expression of the governing differential equations is:

[0125]

[0126]

[0127]

[0128]

[0129]

[0130] where (A ij , B ij , C ij , D ij , E ij , F ij , G ij ) are stiffness coefficients, (I0, I1, I2, I3, I4, I5) are inertia coefficients, d i , d ij , d ijm , d ijmn are variational operators, and are calculated by the following formulas respectively:

[0131]

[0132]

[0133]

[0134] ζ = x, (i, j, m, n = 1), ζ = y, (i, j, m, n = 2)

[0135] In the embodiment of the present invention, the stiffness coefficients of the carbon wave hybrid laminate described in step S24 are specifically shown in Table 2, and the inertia coefficients of the carbon wave hybrid laminate are shown in Table 3.

[0136] Table 2 Stiffness Coefficients of Carbon Wave Hybrid Laminate

[0137] A11 A12 A22 A66 B11 B12 B22 <![CDATA[1.18×10 10 > <![CDATA[2.99×10 9 > <![CDATA[1.18×10 10 > <![CDATA[6.00×10 8 > <![CDATA[-2.84×10 7 > <![CDATA[-6.63×10 6 > <![CDATA[-2.84×10 7 > B66 C11 C12 C22 C66 D11 D12 <![CDATA[-6.25×10 5 > <![CDATA[-2.72×10 7 > <![CDATA[-6.36×10 6 > <![CDATA[-2.72×10 7 > <![CDATA[-5.99×10 5 > <![CDATA[1.13×10 7 > <![CDATA[2.82×10 6 > D22 D66 E11 E12 E22 E66 F11 <![CDATA[1.13×10 7 > <![CDATA[5.31×10 5 > <![CDATA[8.94×10 6 > <![CDATA[2.24×10 6 > <![CDATA[8.94×10 6 > <![CDATA[4.23×10 5 > <![CDATA[7.17×10 6 > F12 F22 F66 G44 G55 <![CDATA[1.80×10 6 > <![CDATA[7.17×10 6 > <![CDATA[3.41×10 5 > <![CDATA[3.04×10 8 > <![CDATA[3.04×10 8 >

[0138] Table 3 Inertia Coefficients of Carbon Wave Hybrid Laminate

[0139] <![CDATA[I0]]> <![CDATA[I1]]> <![CDATA[I2]]> <![CDATA[I3]]> <![CDATA[I4]]> <![CDATA[I5]]> 167.75 0.13 0.13 0.13 0.11 0.09

[0140] Further, establish displacement variables that satisfy the boundary conditions according to step S3, and combine the governing differential equations to obtain the eigenvalue equation, and finally determine the vibration frequency and corresponding vibration mode of the carbon glass hybrid laminate, which further includes:

[0141] S31. Establish displacement variables that satisfy the boundary conditions:

[0142] u0 = U mn cos(λx)sin(μy)e iwt

[0143] v0 = V mn sin(λx)cos(μy)e iwt

[0144] w0 = W mn sin(λx)sin(μy)e iwt

[0145]

[0146]

[0147] wherein, λ = mπ / a, μ = nπ / b, (U mn , V mn , W mn , X mn , Y mn ) are mode shape coefficients.

[0148] S32. Substitute the displacement variables into the governing differential equation to calculate the eigenvalue equation:

[0149]

[0150] wherein, a ij (i, j = 1, 2, 3, 4, 5) are stiffness matrix coefficients, m ij (i, j = 1, 2, 3, 4, 5) are mass matrix coefficients, w is the frequency, and the stiffness matrix coefficients and mass matrix coefficients are calculated by the following equations respectively:

[0151] a 11 = -A 11 λ 2 -A 66 μ 2 , a 12 = -(A 12 + A 66 )λμ, a 13 = B 11 λ 3 +(B 12 + 2B 66 )λμ 2

[0152] a 14 = -C 11 λ 2 -C 66 μ 2 , a 15 = -(C 12 + C 66 )λμ, a 22 = -A 22 μ 2 -A 66 λ 2

[0153] a 23 = B 22 μ 3 +(B 12 + 2B 66 )λ 2μ, a 24 = -(C 12 + C 66 )λμ, a 25 = -C 22 μ 2 - C 66 λ 2

[0154] a 33 = -(D 11 λ 4 + 2D 12 λ 2 μ 2 + D 22 μ 4 ) - 4D 66 λ 2 μ 2 , a 34 = E 11 λ 3 + E 12 λμ 2 + 2E 66 λμ 2

[0155] a 35 = E 12 λ 2 μ + E 22 μ 3 + 2E 66 λ 2 μ, a 44 = -F 11 λ 2 - F 66 μ 2 - G 55 , a 55 = -F 22 μ 2 - F 66 λ 2 - G 44

[0156] m 11 = m 22 = -I0, m 13 = I1λ, m 14 = -I3, m 23 = I1μ, m 25 = -I3, m 33 = -I0 - I2(λ 2 + μ 2 )

[0157] m 34 = I4λ, m 35 = I4μ, m 44 = -I5, m55 = -I5

[0158] S33. Calculate the vibration frequency f = w / (2π) and the mode shape coefficients U mn , V mn , W mn , X mn , Y mn :

[0159] [w 2 , D] = eig(K, M)

[0160] where is the stiffness matrix, is the mass matrix, D = U mn , V mn , W mn , X mn , Y mn are the mode shape coefficients, then the vibration frequency and the response mode shape of the carbon - glass hybrid laminate can be obtained.

[0161] In the embodiment of the present invention, the length, width and thickness of the carbon - glass hybrid laminate are 1, 1, 0.1 respectively, and it is laid according to the CCGC four - layer layup, where C represents carbon fiber and G represents glass fiber. Through step S32, the calculation results of the stiffness matrix coefficients and the mass matrix coefficients are obtained, as shown in Table 4 and Table 5. Through step S33, the vibration frequency and the response mode shape of the carbon - glass hybrid laminate are obtained, as shown in Table 6. It can be seen from Figure 2 that the first - order and second - order curves are close, and the third - order and fourth - order curves are close. In addition, as a / b increases, the frequency growth rate increases. Figure 3 Showing the frequencies of hybrid laminates with different hybrid layups, a / h and elastic modulus ratios, it can be seen that the frequency increases with the increase of the hybrid ratio. In the range of a / h = 0.5 - 2, the frequency increases significantly. When a / h is greater than 2, the frequency almost remains unchanged.

[0162] Table 4 Stiffness matrix coefficients of the carbon - wave hybrid laminate

[0163] a11 a12 a13 a14 a15 <![CDATA[-3.02×10 11 > <![CDATA[-8.74×10 10 > <![CDATA[-8.84×10 8 > <![CDATA[6.77×10 8 > <![CDATA[1.69×10 8 > a21 a22 a23 a24 a25 <![CDATA[-8.74×10 10 > <![CDATA[-3.02×10 11 > <![CDATA[-8.84×10 8 > <![CDATA[1.69×10 8 > <![CDATA[6.77×10 8 <!-- 9 -->]]> a31 a32 a33 a34 a35 <![CDATA[-8.84×10 8 > <![CDATA[-8.84×10 8 > <![CDATA[-7.38×10 8 > <![CDATA[2.93×10 8 > <![CDATA[2.93×10 8 > a41 a42 a43 a44 a45 <![CDATA[6.77×10 8 > <![CDATA[6.77×10 8 > <![CDATA[2.93×10 8 > <![CDATA[-9.34×10 8 > <![CDATA[-5.21×10 7 > a51 a52 a53 a54 a55 <![CDATA[1.69×10 8 > <![CDATA[6.77×10 8 > <![CDATA[2.93×10 8 > <![CDATA[-5.21×10 8 > <![CDATA[-9.34×10 8 >

[0164] Table 5 Mass matrix coefficients of the carbon - wave hybrid laminate

[0165] m11 m12 m13 m14 m15 -414 0.00 0.332 -0.318 0.00 m21 m22 m23 m24 m25 0.00 -414 0.332 0.00 -0.318 m31 m32 m33 m34 m35 0.332 0.332 -42.6 0.264 0.264 m41 m42 m43 m44 m45 -0.318 0.00 0.264 -0.214 0.00 m51 m52 m53 m54 m55 0.00 -0.318 0.264 0.00 -0.214

[0166] Table 6 Frequencies and mode shapes of the carbon - wave hybrid laminate

[0167] Output result First order Second order Third order Fourth order Fifth order <![CDATA[w 2 > <![CDATA[1.33×10 7 > <![CDATA[6.62×10 7 > <![CDATA[6.61×10 7 > <![CDATA[1.25×10 8 > <![CDATA[1.88×10 8 > f 580 1300 1290 1780 2190 Umn -0.002 -0.001 0.001 0.001 0.000 Vmn -0.002 -0.001 0.001 0.001 -0.001 Wmn 1.000 1.000 -1.000 -1.000 1.000 Xmn 0.292 0.626 -0.388 -0.681 0.854 Ymn 0.293 0.390 -0.627 -0.682 0.461

[0168] Those of ordinary skill in the art will realize that the embodiments described herein are provided to assist the reader in understanding the principles of the present invention, and it should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various other specific deformations and combinations that do not depart from the essence of the present invention based on these technical revelations disclosed in the present invention, and these deformations and combinations are still within the scope of protection of the present invention.

Claims

1. A vibration characteristic prediction method for carbon-glass hybrid laminated plates based on HSDT, characterized in that It includes the following steps: S1. Establish a transverse shear function model of the high-order shear deformation theory according to the stress-free conditions of the upper and lower boundaries of the carbon-glass hybrid laminate; S2. Establish a high-order shear deformation displacement field, and combine the geometric equations, constitutive relations, and Hamilton's energy principle to obtain the governing differential equations; specifically including: S21. Establish the high-order shear deformation displacement field of the carbon-glass hybrid laminate: ； Among them, U , V , W respectively represent the displacement states of each position point inside the carbon-glass hybrid laminate, u 0, v 0, w 0 respectively represent the displacements of each position point on the physical mid-plane along x , y , z directions, φ x , φ y respectively represent the rotations of each position point on the physical mid-plane along y , x axes; S22. Calculate the strains of each point of the carbon-glass hybrid laminate: ； S23. Establish the stress-strain relationship of the carbon-glass hybrid laminate: ； Among them, and are stress components and strain components respectively, and the elastic constants Q ij are calculated by the following formula: ； Among them, E ( z ) is the elastic modulus, v is the Poisson's ratio; S24. Calculate the variation of strain energy and the variation of kinetic energy, and obtain the governing differential equations: Calculating the Strain Energy Variation of Carbon-Glass Hybrid Laminates δU : ； wherein ( , , ) are stretching terms, ( , , ) are bending terms, ( , , ) are high-order terms, ( , ) are shear terms, and are respectively represented by the following formulas: , ； Calculating the kinetic energy variation of carbon-glass hybrid laminates δK : ； where the superscript dot represents differentiation with respect to time, i.e., ( • ) = ∂() / ∂ t , ρ ( z ) is the mass density, and ( I 0, I 1, I 2, I 3, I 4, I 5) are the inertia terms, calculated by: ； Among them, z represents the thickness coordinate, f ( z ) is the transverse shear function; then substitute the variation of strain energy and the variation of kinetic energy into the energy equation: ； Then the expression of the governing differential equation is: ； wherein is the stiffness coefficient, is the inertia coefficient, is the variational operator, and they are calculated respectively by the following formulas: ； S3. Establish the displacement variables that satisfy the boundary conditions, and combine the governing differential equations to obtain the eigenvalue equation, and finally determine the vibration frequencies and corresponding vibration modes of the carbon-glass hybrid laminate.

2. The vibration characteristic prediction method of the carbon-glass hybrid laminate based on HSDT according to claim 1, wherein Step S1 further includes: S11. Construct any z / h is an even function of the independent variable , and translate it to the vertical coordinate by a distance of ; S12. Scale the translated curve in the horizontal coordinate direction to satisfy , with a scaling ratio of . Integrate the scaled in the plate thickness direction to obtain a newly constructed shear function .

3. The vibration characteristic prediction method of the carbon-glass hybrid laminate based on HSDT according to claim 1, characterized in that Step S3 further includes: S31. Establish the displacement variables that satisfy the boundary conditions: ； Among them, is the mode shape coefficient; S32. Substitute the displacement variables into the governing differential equations and calculate the eigenvalue equation: ； Among them, a ij ( i , j = 1, 2, 3, 4, 5) are the stiffness matrix coefficients, m ij ( i , j = 1, 2, 3, 4, 5) are the mass matrix coefficients, w is the frequency, and the stiffness matrix coefficients and the mass matrix coefficients are calculated by the following formulas respectively: ； S33. Calculate the vibration frequency according to the eigenvalue equation w and the mode shape coefficient : Among them, is the stiffness matrix, is the mass matrix, is the mode shape coefficient, and the vibration frequency and response mode shape of the carbon-glass hybrid laminate can be obtained.

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