Thin shell structure vibration analysis method based on energy method and finite element method

Abstract

The invention discloses a thin-shell structure vibration analysis method based on an energy method and a finite element method.The thin-shell structure vibration analysis method based on the energy method and the finite element method.The thin-shell structure vibration analysis method based on the energy method and the finite element methodcomprises the steps that firstly, input parameters of a thin-shell structure are determined according to the shape size, the material and the working condition of the thin-shell structure; setting an error limit; establishing a thin-shell structure finite element model through a finite element method, obtaining a stiffness matrix and a mass matrix of the thin-shell structure finite element model, and reducing the rotational degree of freedom in the thin-shell structure finite element model through a modal synthesis method to obtain a reduced-order model; substituting a displacement formula based on Fourier series expansion into the reduced-order model according to an energy equality principle to obtain a thin-shell structure hybrid model; and finally, thin shell vibration analysis is carried out according to a kinetic equation in the thin shell structure hybrid model.

Description

Technical Field

[0001] The invention belongs to the technical field of structural vibration analysis, and in particular, relates to a thin shell structure vibration analysis method based on an energy method and a finite element method. Background Art

[0002] Thin shell structures are important load-bearing components in core areas of strategic or livelihood development such as the power industry, aerospace, and underwater navigation structures. Their vibrations will affect the stability and safety of the overall mechanical operation.

[0003] Generally speaking, the experimental method is an important means to obtain the vibration characteristics of thin shell structures. However, considering the diverse size arrangements of thin shells, the complex operating conditions and the cumbersome experimental measurement procedures, the difficulty and cost of analyzing the vibration of thin shell structures by experimental methods are greatly increased. The development of computer technology has enabled the vibration analysis of thin shell structures to be carried out through theoretical derivation and numerical simulation methods, which can not only overcome the deficiencies in experiments, but also fully understand the vibration mechanism of thin shell structures. There are many simulation methods for the analysis of thin shell vibration problems, which can be mainly divided into three categories: analytical method, numerical method and semi-analytical method.Analytical methods are obtained through rigorous mathematical derivation, and the resulting solutions are accurate and have important significance in both theory and engineering practice. Analytical solutions not only accurately describe the essence of physical quantities, but also promote the development of various numerical solutions as standard solutions. However, analytical methods are only effective for certain specific loads, specific boundary conditions and specific physical problems. It is difficult to analyze complex engineering problems using analytical methods. Numerical methods reduce the restrictions on solutions that simultaneously satisfy the computational domain control equations and boundary conditions through appropriate mathematical methods. In the early days, there were mainly finite difference methods, Ritz methods and Galerkin methods. With the development of computer technology, numerical methods such as boundary element method, meshless method, differential integral method and finite element method have also been developed and applied in thin shell vibration analysis. Among them, finite element method is the most widely used method for analyzing thin shell problems. The most widely used numerical method, its equation can be derived from the variational principle, can also be derived by the weighted residual method, this method has the advantages of the difference method and the variational method, and fully estimates the contribution of the unit to the node parameters, making the calculation results more accurate. The finite element method also has some disadvantages when analyzing thin shell structures. For example, for thinner structures, the difficulty lies mainly in the construction of C1 type units, especially the plate and shell problems. For thicker structures, the difficulty lies mainly in the treatment of shear self-locking problems, which requires specialized construction units such as the reduced integration method. The semi-analytical method combines the precise calculation of the analytical method with the flexible boundary processing of the numerical method. Compared with the numerical method, it can reduce the amount of calculation and achieve the required calculation accuracy on some problems. Commonly used semi-analytical methods include the finite strip method (Finite Strip Method). Strip Method), Combinatory Strip-Element Method, Finite Element Method of Lines, Improved Fourier Series Method, etc. The idea of ​​the improved Fourier Series Method is to express the displacement as a linear combination of Fourier cosine series and an auxiliary polynomial, or a Fourier series product. The expansion method is related to the problem being analyzed, and combined with the Rayleigh-Ritz method, the plate and shell dynamics problems with different boundaries are analyzed. This method solves the continuity problem of the displacement function of the thin shell structure at the elastic boundary and has a very good convergence speed. The semi-analytical method mentioned above has the advantages of accurate calculation results and fast calculation speed, but the main problem of the semi-analytical method is that it is not adaptable to complex structures, and it is not easy to handle complex loads and boundaries, so it is difficult to apply it to actual engineering calculations. Summary of the invention

[0004] In view of this, the purpose of the present invention is to provide a vibration analysis method for a thin shell structure based on an energy method and a finite element method. The method constructs a hybrid model of a thin shell structure based on the energy method and the finite element method, and can mesh-free the finite element model of the thin shell structure constructed by the finite element method, thereby obtaining a displacement formula that satisfies the entire domain, and performing vibration analysis on the thin shell structure. The model overcomes the limitations of traditional methods, and while ensuring the accuracy of the calculation results, its calculation cost is lower than that of the finite element method.

[0005] In order to achieve the above-mentioned invention object, the present invention provides the following technical solution: a vibration analysis method of a thin shell structure based on an energy method and a finite element method, comprising:

[0006] Step S1: determining the geometric structure dimensions, material parameters and working conditions of the thin shell structure, and setting the error limit of the thin shell structure hybrid model as input parameters of the thin shell structure hybrid model;

[0007] Step S2: Establishing a thin shell structure hybrid model

[0008] ① Establish the displacement function of thin shell structure represented by improved Fourier series;

[0009] X E =Dα (4)

[0010] Among them, α is the Fourier expansion coefficient, D is the structural displacement constant;

[0011] ② Establish a finite element model of the thin shell structure through the finite element method, and obtain its stiffness matrix and mass matrix;

[0012] ③ Reduce the rotational degrees of freedom in the finite element model of the thin shell structure through the modal synthesis method to obtain a reduced-order model;

[0013] ④ Establish a hybrid model of thin shell structure based on energy method and finite element method;

[0014] ⑤ Error analysis; the error is composed of truncation error and discrete error caused by finite element method, and the truncation error is composed of modal truncation error and series truncation error;

[0015] Step S3: Performing vibration analysis of the thin shell structure according to the thin shell structure hybrid model in step S2.

[0016] Further, in the thin shell structure vibration analysis method based on the energy method and the finite element method, the thin shell structure finite element model established in step S2 (2) is:

[0017]

[0018] Among them, M F and K Fare the mass matrix and stiffness matrix required by the finite element method, X F is the node displacement, is the second-order derivative of the node displacement with respect to time, F ext is the force vector;

[0019] X F =[u,v,w,θ x ,θ y ,θ z ] T (6)

[0020] Among them, u, v and w are the displacements in the x-axis, y-axis and z-axis directions respectively, θ x ,θ y and θ z are the rotation angles around the x-axis, y-axis, and z-axis respectively.

[0021] Furthermore, in the thin shell structure vibration analysis method based on the energy method and the finite element method, the modal synthesis method in step S2 (3) adopts the Craig-Bampton method, and the process of reducing the rotational degrees of freedom in the finite element model of the thin shell structure by using the Craig-Bampton method is as follows:

[0022] Divide the matrix in equation (5) into the translational degrees of freedom and rotational degrees of freedom:

[0023]

[0024] The subscripts t and r represent the translational and rotational degrees of freedom, respectively, and u t =[u,v,w] T ,θ r =[θ x ,θ y ,θ z ] T , and u t and θ r The second derivative with respect to time;

[0025] The displacement vector in equation (7) is transformed into:

[0026]

[0027] In the formula, Ψ rt =-K rr -1 K rt , E tt and 0 tk are the identity matrix and the zero matrix respectively, η k is the modal coordinate, ΦCB is the Craig-Bampton transformation matrix, Φ rk The modal truncation is performed using formula (9), that is, the first k-order modes are taken to obtain;

[0028] (K rr -ω 2 M rr )Φ rk =0 (9)

[0029] Where ω is the angular frequency;

[0030] Substituting equation (8) into equation (7) yields the reduced-order model:

[0031]

[0032] Among them, the mass matrix in the finite element model of the thin shell structure is defined as Defining the stiffness matrix in a finite element model of a thin shell structure is η k The second derivative with respect to time.

[0033] Further, in the thin shell structure vibration analysis method based on the energy method and the finite element method, the process of establishing a thin shell structure hybrid model based on the energy method and the finite element method in step S2 (4) is:

[0034] Substituting equation (4) into equation (10), we can get u t Performing a secondary transformation yields:

[0035]

[0036] Among them, Φ D is the secondary transformation matrix;

[0037] Similarly, substituting equation (11) into equation (10), we get the hybrid model of thin shell structure:

[0038]

[0039] Among them, the mass matrix defining the hybrid model of the thin shell structure is Defining the stiffness matrix for a hybrid model of a thin shell structure

[0040] Further, in the thin shell structure vibration analysis method based on the energy method and the finite element method, in step S2 (5), for the truncation error generated by the modal synthesis method, the displacement response of all degrees of freedom under the pseudo-static load F is used as the modal index to estimate the modal truncation error;

[0041] Under the action of the quasi-static load F, the approximate solution of the flexibility matrix obtained from the first N modes is:

[0042]

[0043] Among them, ω i and Φ i is the circular frequency and vibration shape of the i-th mode, and F is the pseudo-static load;

[0044] Its exact solution is:

[0045] X=K -1 F (14)

[0046] The truncation error produced by the modal synthesis method is obtained as follows:

[0047]

[0048] Among them, ε CB is the truncation error produced by the modal synthesis method;

[0049] The error caused by truncating the Fourier series is;

[0050]

[0051] Among them, ε Fourier The error caused by truncation of the Fourier series.

[0052] Further, in the thin shell structure vibration analysis method based on the energy method and the finite element method, the thin shell structure vibration analysis in step S3 includes modal analysis and response analysis;

[0053] set up Substituting into equation (12) we get its characteristic equation:

[0054]

[0055] where Φ FE is the modal coordinate matrix, by solving Φ FE Modal analysis can be performed;

[0056] Frequency response function G between measurement point O and excitation point P OP for:

[0057]

[0058] in, is the Rth order mode at point O, is the Rth order mode at point P, m R is the Rth order modal mass, ω R is the Rth order natural frequency, ξ is the damping ratio;

[0059] According to formula (18), response analysis can be performed.

[0060] Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention constructs a hybrid model of thin shell structure based on the energy method and the finite element method, which can mesh-free the finite element model of thin shell structure constructed by the finite element method, and then obtain the Fourier series displacement formula that satisfies the entire domain. The model overcomes the limitations of traditional methods, and while ensuring the accuracy of the calculation results, its calculation cost is lower than that of the finite element method. It can be used for vibration analysis of thin shell structures, and provide important reference and basis for the design optimization of thin shell structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of the present application. The exemplary embodiments of the present invention and their descriptions are used to understand the present invention and do not constitute improper limitations of the present invention. In the drawings:

[0062] Figure 1 This is a flow chart of the vibration analysis method of thin shell structure based on energy method and finite element method;

[0063] Figure 2 The free mode vibration shape comparison diagram of the finite element model, energy method and hybrid model of the Kirchhoff plate;

[0064] Figure 3 This is a comparison diagram of the free mode vibration shapes of the finite element model, energy method and hybrid model of the cylindrical shell structure. DETAILED DESCRIPTION

[0065] In order to make the purpose, features and advantages of the present invention more obvious and easy to understand, the technical scheme in the present invention is clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the present invention is not limited to the following embodiments, and the specific implementation method can be determined according to the technical scheme of the present invention and the actual situation. In order to avoid confusing the essence of the present invention, the known methods, processes and flows are not described in detail.

[0066] like Figure 1 , Figure 2 and Figure 3 As shown, the present invention proposes a thin shell structure vibration analysis method based on energy method and finite element method, including:

[0067] Step S1: Determine the geometric structure dimensions, material parameters and working conditions of the thin shell structure, and set the error limit of the thin shell structure hybrid model as the input parameters of the thin shell structure hybrid model.

[0068] Step S2: Establishing a thin shell structure hybrid model

[0069] In order to establish a hybrid model of thin shell structure, firstly, the displacement function of the thin shell structure represented by the improved Fourier series and the finite element model of the thin shell structure are established according to the input parameters in step S1, and then the rotational degrees of freedom in the finite element model of the thin shell structure are reduced by the modal synthesis method to obtain a reduced-order model. Finally, according to the principle of equal node energy, a hybrid model of the thin shell structure based on the energy method and the finite element method is established.

[0070] ① Establish the displacement function of thin shell structures such as flat plates and cylindrical shells represented by improved Fourier series

[0071] Using the energy principle to solve the vibration characteristics of thin shells, selecting a suitable displacement function is the key to ensuring calculation accuracy and improving calculation efficiency. The displacement function must not only ensure the continuity of the function in the integral interval, but also take into account the convergence and accuracy of the function. When the traditional series is used as the displacement function in theoretical calculation, there will be breakpoints, complex calculations, and low efficiency, and the boundary constraints and structural coupling constraints will not be met. The present invention adopts an improved Fourier series, which is composed of a basic series and an auxiliary function, to ensure continuity during the integration process, while ensuring the convergence of the function and improving calculation efficiency.

[0072] The displacement function of the cylindrical shell is:

[0073]

[0074] Among them, u(x i ,θ i ,t),v(x i ,θ i ,t) and w(x i ,θ i , t) are axial displacement, circumferential displacement and bending vibration displacement respectively, x i is the axial coordinate, θ i is the circumferential coordinate, t represents time, j 2 =-1, ω is the angular frequency, A mni , B mni , C mni , and is the Fourier expansion coefficient of the structural displacement, λ m =mπ / l i , l i is the generatrix length of the thin shell structure along the axial direction, It is an auxiliary function, where m, n, and l are tool symbols used to represent the series, and they serve to distinguish the numbers of items in different series.

[0075] The displacement function of the rectangular plate is similar. For the convenience of expression, the displacement function of thin shell structures such as flat plates and cylindrical shells is written as:

[0076] X E =Dα (4)

[0077] Among them, α is the Fourier expansion coefficient and D is the structural displacement constant.

[0078] Building a finite element model of a thin shell structure

[0079]

[0080] Among them, M F and K F are the mass matrix and stiffness matrix required by the finite element method, X F is the node displacement, is the second-order derivative of the node displacement with respect to time, F ext is the force vector.

[0081] X F =[u,v,w,θ x ,θ y ,θ z ] T (6)

[0082] Among them, u, v and w are the displacements in the x-axis, y-axis and z-axis directions respectively, θ x ,θ y and θ z are the rotation angles around the x-axis, y-axis, and z-axis respectively.

[0083] ② Using modal synthesis methods, such as the Craig-Bampton method, to reduce the rotational degrees of freedom in the finite element model of the thin shell structure to obtain a reduced-order model;

[0084] Divide the matrix in equation (5) into the translational degrees of freedom and rotational degrees of freedom:

[0085]

[0086] The subscripts t and r represent the translational and rotational degrees of freedom, respectively, and u t =[u,v,w] T ,θ r =[θ x ,θ y ,θ z ] T , and u t and θ r The second derivative with respect to time.

[0087] The displacement vector in equation (7) is transformed into:

[0088]

[0089] In the formula, Ψ rt =-K rr -1 K rt , E tt and 0 tk are the identity matrix and the zero matrix respectively, η k is the modal coordinate, Φ CB is the Craig-Bampton transformation matrix, Φ rk The modal truncation is performed using formula (9), that is, the first k-order modes are taken to obtain;

[0090] (K rr -ω 2 M rr )Φ rk =0 (9) Substituting formula (8) into formula (7) we get the reduced-order model:

[0091]

[0092] Among them, the mass matrix in the finite element model of the thin shell structure is defined as Defining the stiffness matrix in a finite element model of a thin shell structure is η k The second derivative with respect to time.

[0093] ③Establish a hybrid model of thin shell structure based on energy method and finite element method

[0094] Substituting equation (4) into equation (10), we can get u t Performing a secondary transformation yields:

[0095]

[0096] Among them, Φ D is the quadratic transformation matrix.

[0097] Similarly, substituting equation (11) into equation (10), we get the hybrid model of thin shell structure:

[0098]

[0099] Among them, the mass matrix defining the hybrid model of the thin shell structure is Defining the stiffness matrix for a hybrid model of a thin shell structure

[0100] ④Error analysis

[0101] In order to estimate the modal truncation error (Equation (9)) generated by the modal synthesis method, the displacement response of all degrees of freedom under the pseudo-static load F is used as the modal index to estimate the modal truncation error.

[0102] Under the action of the quasi-static load F, the approximate solution of the flexibility matrix obtained from the first N modes is:

[0103]

[0104] Among them, ω i and Φ i is the circular frequency and vibration shape of the i-th mode, and F is the pseudo-static load.

[0105] Its exact solution is:

[0106] X=K -1 F (14)

[0107] The truncation error produced by the modal synthesis method is obtained as follows:

[0108]

[0109] The error caused by truncating the Fourier series (Equation (4)) is:

[0110]

[0111] In summary, the error of the hybrid model of the thin shell structure is mainly composed of the truncation error and the discrete error caused by the finite element method, and the truncation error is mainly composed of the modal truncation error and the series truncation error. The discrete error caused by the finite element method can be verified by mesh independence, and the truncation error can be determined by equations (15) and (16) and the error limit in the input parameters of step S1 to determine the truncation number.

[0112] Step S3: performing vibration analysis of the thin shell structure such as modal analysis and response analysis according to the thin shell structure hybrid model in step S2;

[0113] set up Substituting into equation (12) we get its characteristic equation:

[0114]

[0115] where Φ FE is the modal coordinate matrix, by solving Φ FE Modal analysis can be performed.

[0116] The frequency response function between the measurement point O and the excitation point P is:

[0117]

[0118] in, is the Rth order mode at point O, is the Rth order mode at point P, m R is the Rth order modal mass, ω R is the Rth order natural frequency, ξ is the damping ratio.

[0119] According to formula (18), response analysis can be performed.

[0120] Step S4: Example verification

[0121] The thin shell structure vibration analysis method based on energy method and finite element method proposed in this invention is implemented as follows: Figure 1 As shown. First, according to the shape, size, material and working conditions of the thin shell structure, its input parameters are clarified and the error limits are set; then, the finite element model of the thin shell structure is established by the finite element method, and its stiffness matrix and mass matrix are obtained. The rotational degrees of freedom in the finite element model of the thin shell structure are reduced by the Craig-Bampton method to obtain a reduced-order model. According to the energy equality principle, the displacement formula based on the Fourier series expansion is substituted into the reduced-order model to obtain a hybrid model of the thin shell structure; finally, the thin shell vibration analysis is performed according to the dynamic equations in the hybrid model of the thin shell structure.

[0122] ① Flat plate structure

[0123] Taking the Kirchhoff plate as an example, a finite element model and a hybrid numerical model are constructed for the Kirchhoff plate. The natural frequencies and vibration modes obtained by solving the two models are compared with the calculation results of the model in reference [1] as shown in Tables 1 and Figure 2 Show.

[0124] Table 1 Comparison of natural frequencies of three models of Kirchhoff plate

[0125]

[0126] Figure 2 The figure shows the comparison of free mode vibration shapes of three models of Kirchhoff plate (finite element model on the left, energy method in the middle, and hybrid model on the right).

[0127] ②Cylindrical shell structure

[0128] Finite element models and hybrid models are constructed for cylindrical shells with different geometric structures and material parameters. The natural frequencies and vibration modes obtained by solving the two models are compared with the calculation results of the model in reference [2] as shown in Table 2 and Figure 3 Show, Figure 3(a) shows the 15th-order vibration mode diagram; (b) shows the 17th-order vibration mode diagram; (c) shows the 15th-order circumferential vibration mode comparison diagram; (d) shows the 17th-order circumferential vibration mode comparison diagram;

[0129] The cylindrical shells with different geometric structures and material parameters are divided into four groups: 1, 2, 3 and 4, where r is the radius of the bottom surface of the cylindrical shell, h is the height of the cylinder, T is the shell thickness, Y is the elastic modulus, Pr is the Poisson's ratio, and ρ is the density:

[0130] Condition 1: r = 1m, h = 3m, T = 0.01m, Y = 210GPa, Pr = 0.3, ρ = 7850kg / m 3 ;

[0131] Condition 2: r = 2m, h = 3m, T = 0.01m, Y = 210GPa, Pr = 0.3, ρ = 7850kg / m 3 ;

[0132] Condition 3: r = 1m, h = 2m, T = 0.01m, Y = 216GPa, Pr = 0.28, ρ = 7800kg / m 3 ;

[0133] Condition 4: r = 1m, h = 3m, T = 0.05m, Y = 210GPa, Pr = 0.3, ρ = 7850kg / m 3 ;

[0134] Table 2 Comparison of natural frequencies of three models of cylindrical shell structure

[0135]

[0136] Among them, a is the number of annular half waves, and b is the number of axial waves. Figure 3 The free mode vibration shapes of three models, namely, the finite element model, the energy method and the hybrid model of the cylindrical shell structure are compared;

[0137] References [1] and [2]:

[0138] [1] Shi S, Guo T, Zhang M, et al. Analysis of the vibro-acoustic behaviors of the periodically stiffened double panel-cavity coupled system [J]. MechanicalSystems and Signal Processing, 2024, 208: 110993.

[0139] [2]Shi

[0140] The present invention constructs a finite element hybrid numerical model based on Fourier series expansion, which can de-mesh the finite element model of the thin shell structure constructed by the finite element method, and then obtain the displacement formula that satisfies the entire domain, and perform vibration analysis on the thin shell structure. The model overcomes the limitations of the traditional method, and its calculation cost is lower than that of the finite element method while ensuring the accuracy of the calculation results. Research and analysis of the vibration of thin shell structures can serve as an important reference for their design optimization. For the vibration analysis of thin shell structures, it is worth studying to construct an effective and accurate thin shell structure model. The present invention forms a hybrid model of thin shell structures based on the energy method and the finite element method, which can realize the vibration analysis of thin shell structures such as flat plates and cylindrical shells, thereby providing an important reference for the design optimization of thin shell structures.

[0141] The present invention constructs a hybrid model of thin shell structure based on energy method and finite element method for vibration analysis of thin shell structure. The validity and accuracy of the model are verified by numerical examples.

Claims

1. A vibration analysis method for thin shell structures based on energy method and finite element method, characterized in that: include: Step S1: determining the geometric structure dimensions, material parameters and working conditions of the thin shell structure, and setting the error limit of the thin shell structure hybrid model as input parameters of the thin shell structure hybrid model; Step S2: Establishing a thin shell structure hybrid model ① Establish the displacement function of thin shell structure represented by improved Fourier series; X E =Dα (4) Among them, α is the Fourier expansion coefficient, D is the structural displacement constant; ② Establish a finite element model of the thin shell structure through the finite element method, and obtain its stiffness matrix and mass matrix; ③ Reduce the rotational degrees of freedom in the finite element model of the thin shell structure through the modal synthesis method to obtain a reduced-order model; ④ Establish a hybrid model of thin shell structure based on energy method and finite element method; ⑤ Error analysis; the error is composed of truncation error and discrete error caused by finite element method, and the truncation error is composed of modal truncation error and series truncation error; Step S3: Performing vibration analysis of the thin shell structure according to the thin shell structure hybrid model in step S2.

2. The thin shell structure vibration analysis method based on energy method and finite element method according to claim 1 is characterized in that: The thin shell structure finite element model established in step S2 (2) is: Among them, M F and K F are the mass matrix and stiffness matrix required by the finite element method, X F is the node displacement, is the second-order derivative of the node displacement with respect to time, F ext is the force vector; X F =[u,v,w,θ x ,i y ,i z ] T (6) Among them, u, v and w are the displacements in the x-axis, y-axis and z-axis directions respectively, θ x ,θ y and θ z are the rotation angles around the x-axis, y-axis, and z-axis respectively.

3. The thin shell structure vibration analysis method based on energy method and finite element method according to claim 2 is characterized in that: The modal synthesis method in step S2 (3) adopts the Craig-Bampton method. The process of reducing the rotational degrees of freedom in the finite element model of the thin shell structure by using the Craig-Bampton method is as follows: Divide the matrix in equation (5) into the translational degrees of freedom and rotational degrees of freedom: The subscripts t and r represent the translational and rotational degrees of freedom, respectively, and u t =[u,v,w] T , θ r =[θ x ,θ y ,θ z ] T , and u t and θ r The second derivative with respect to time; The displacement vector in equation (7) is transformed into: In the formula, Ψ rt =-K rr -1 K rt , E tt and 0 tk are the identity matrix and the zero matrix respectively, η k is the modal coordinate, Φ CB is the Craig-Bampton transformation matrix, Φ rk The modal truncation is performed using formula (9), that is, the first k-order modes are taken to obtain; (K rr -ω 2 M rr )Φ rk =0(9)where ω is the angular frequency; Substituting equation (8) into equation (7) yields the reduced-order model: Among them, the mass matrix in the finite element model of the thin shell structure is defined as Defining the stiffness matrix in a finite element model of a thin shell structure is η k The second derivative with respect to time.

4. The thin shell structure vibration analysis method based on energy method and finite element method according to claim 3 is characterized in that: The process of establishing the hybrid model of the thin shell structure based on the energy method and the finite element method in step S2 (4) is as follows: Substituting equation (4) into equation (10), we can get u t Performing a secondary transformation yields: Among them, Φ D is the secondary transformation matrix; Similarly, substituting equation (11) into equation (10), we get the hybrid model of thin shell structure: Among them, the mass matrix defining the hybrid model of the thin shell structure is Defining the stiffness matrix for a hybrid model of a thin shell structure 5. The thin shell structure vibration analysis method based on energy method and finite element method according to claim 4 is characterized in that: In step S2, in step ⑤, for the truncation error generated by the modal synthesis method, in order to estimate the modal truncation error, the displacement response of all degrees of freedom under the pseudo-static load F is used as the modal index to estimate the modal truncation error; Under the action of the quasi-static load F, the approximate solution of the flexibility matrix obtained from the first N modes is: Among them, ω i and Φ i is the circular frequency and vibration shape of the i-th mode, and F is the pseudo-static load; Its exact solution is: X=K -1 F (14) The truncation error produced by the modal synthesis method is obtained as follows: Among them, ε CB is the truncation error produced by the modal synthesis method; The error caused by truncating the Fourier series is; Among them, ε Fourier The error caused by truncation of the Fourier series.

6. The thin shell structure vibration analysis method based on energy method and finite element method according to claim 5 is characterized in that: The thin shell structure vibration analysis in step S3 includes modal analysis and response analysis; set up Substituting into equation (12) we get its characteristic equation: where Φ FE is the modal coordinate matrix, by solving Φ FE Modal analysis can be performed; Frequency response function G between measurement point O and excitation point P OP for: in, is the Rth order mode at point O, is the Rth order mode at point P, m R is the Rth order modal mass, ω R is the Rth order natural frequency, ξ is the damping ratio; According to formula (18), response analysis can be performed.

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