CN110704960A - Method and device for calculating natural frequency of three-layer cylindrical shell, storage medium and computer equipment - Google Patents

Abstract

The application discloses a method and a device for calculating the natural frequency of a three-layer cylindrical shell, a storage medium and computer equipment, and relates to the technical field of mechanical dynamics. The method comprises the following steps: establishing a three-layer cylindrical shell with a functional gradient layer as an intermediate layer; calculating potential energy and kinetic energy of the three layers of cylindrical shells under the elastic boundary and spring potential energy generated by the boundary spring according to resultant force of force and moment of the three layers of cylindrical shells; and calculating the natural frequency of the three layers of cylindrical shells by utilizing a Rayleigh Ritz method according to the calculated potential energy and kinetic energy of the three layers of cylindrical shells under the elastic boundary and the calculated spring potential energy generated by the boundary spring. The method and the device are suitable for aerospace, machinery and civil engineering to realize analysis of dynamic response.

Description

Method and device for calculating natural frequency of three-layer cylindrical shell, storage medium and computer equipment

technical field

The present application relates to the technical field of mechanical dynamics, and in particular to a method and device for calculating the natural frequency of a three-layer cylindrical shell, a storage medium, and a computer device.

Background technique

At present, the analysis of the natural frequencies of laminated shells containing functionally graded materials mainly focuses on the classical boundaries, but in practical engineering applications, most of the boundaries are very complex, and there are few ideal classical boundaries such as fixed support and simple support. Appear. Therefore, based on the prior art to analyze the natural frequency of the laminated shell containing the functionally graded material, the accuracy of the obtained analysis result is not high.

In addition, the desired properties of functionally graded materials within the laminated shell in any spatial direction can be obtained by changing the composition of the constituent materials, whereas the properties of the currently studied functionally graded materials are mostly confined to the radial direction (i.e., the thickness direction). The changes that occur have greater limitations on the requirements for the desired properties of functionally graded materials in other spatial directions.

SUMMARY OF THE INVENTION

In view of this, the present application provides a method and device, storage medium, and computer equipment for the natural frequency of a three-layer cylindrical shell, the main purpose of which is to solve the calculation accuracy of the natural frequency of the existing laminated shell containing functionally graded materials based on classical boundary conditions. The technical problems are relatively low, and the requirements for the desired properties of functionally graded materials in other spatial directions are relatively limited.

According to one aspect of the present application, a method for calculating the natural frequency of a three-layer cylindrical shell is provided, the method comprising:

Build a three-layer cylindrical shell with the middle layer as the functional gradient layer;

According to the resultant force of the force and moment of the three-layer cylindrical shell, the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring are obtained by calculation;

Using Rayleigh-Ritz method, according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring, the natural frequency of the three-layer cylindrical shell is calculated;

Wherein, the functionally graded layer is a two-dimensional functionally graded material.

According to another aspect of the present application, a device for calculating the natural frequency of a three-layer cylindrical shell is provided, the device comprising:

A building module is used to build a three-layer cylindrical shell whose middle layer is a functional gradient layer;

The boundary simulation module is used to calculate the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring according to the resultant force of the force and the moment of the three-layer cylindrical shell;

The natural frequency calculation module is used to use the Rayleigh-Ritz method to calculate the natural frequency of the three-layer cylindrical shell according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring;

Wherein, the functionally graded layer is a two-dimensional functionally graded material.

According to yet another aspect of the present application, a storage medium is provided on which a computer program is stored, and when the program is executed by a processor, the above-mentioned method for calculating the natural frequency of a three-layer cylindrical shell is provided.

According to yet another aspect of the present application, a computer device is provided, comprising a storage medium, a processor, and a computer program stored on the storage medium and running on the processor, wherein the processor implements the above three layers when executing the program. Calculation of the natural frequencies of cylindrical shells.

With the above technical solutions, the method and device, storage medium, and computer equipment for calculating the natural frequency of a three-layer cylindrical shell provided by the present application are compared with the existing technical solutions based on classical boundary-based natural frequency analysis. The layer is a three-layer cylindrical shell with a functional gradient layer. According to the resultant force of the force and moment of the three-layer cylindrical shell, the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring are calculated and used. The Rayleigh-Ritz method calculates the natural frequency of the three-layer cylindrical shell based on the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring. It can be seen that, different from the classical boundary conditions in the prior art, under elastic boundary conditions, a method for calculating the natural frequency of a three-layer cylindrical shell with a functionally graded material in the middle layer is provided, which can better simulate practical engineering applications. It can effectively improve the calculation accuracy of the natural frequency of the laminated shell containing the functionally graded material, and at the same time, the properties of the functionally graded material can be changed in both the axial and radial directions, thereby further enhancing the feasibility of the functionally graded material. Customization.

The above description is only an overview of the technical solution of the present application. In order to be able to understand the technical means of the present application more clearly, it can be implemented according to the content of the description, and in order to make the above-mentioned and other purposes, features and advantages of the present application more obvious and easy to understand , and the specific embodiments of the present application are listed below.

Description of drawings

The drawings described herein are used to provide further understanding of the present application and constitute a part of the present application. The schematic embodiments and descriptions of the present application are used to explain the present application and do not constitute an improper limitation of the present application. In the attached image:

1 shows a schematic flowchart of a method for calculating the natural frequency of a three-layer cylindrical shell provided by an embodiment of the present application;

2 shows a schematic flowchart of another method for calculating the natural frequency of a three-layer cylindrical shell provided by an embodiment of the present application;

FIG. 3 shows a schematic diagram of a three-layer cylindrical shell model in which the intermediate layer is a functionally graded material under elastic boundary conditions provided by an embodiment of the present application;

FIG. 4 shows a schematic structural diagram of a device for calculating the natural frequency of a three-layer cylindrical shell provided by an embodiment of the present application.

Detailed ways

Hereinafter, the present application will be described in detail with reference to the accompanying drawings and in conjunction with the embodiments. It should be noted that the embodiments in the present application and the features of the embodiments may be combined with each other in the case of no conflict.

For the existing classical boundary conditions based on the low accuracy of calculation of the natural frequency of the laminated shell containing functionally graded materials, and the technical problems that the requirements for the required properties of the functionally graded materials in other spatial directions are relatively limited. This embodiment provides a natural frequency method for a three-layer cylindrical shell, which can effectively avoid the above-mentioned technical problems existing in the prior art, better simulate the boundary conditions in practical engineering applications, and effectively improve the layers containing functionally graded materials. The calculation accuracy of the natural frequency of the closed shell also enables the properties of the functionally graded material to be changed in both the axial and radial directions, thereby further enhancing the customizability of the functionally graded material. As shown in Figure 1, the method includes:

101. Establish a three-layer cylindrical shell in which the intermediate layer is a functionally graded layer; wherein, the functionally graded layer is a two-dimensional functionally graded material.

In this embodiment, material parameters and two-dimensional volume fractions are applied to the functionally graded material of the intermediate layer, so that the properties of the functionally graded material of the intermediate layer can be changed in radial and axial directions at the same time. Among them, the two-dimensional volume fraction is used to describe the volume percentage of a material composing the functionally graded layer, which can enhance the customizability of the functionally graded material, so as to better simulate three layers in practical engineering applications based on elastic boundary conditions Cylindrical shell.

102. Calculate the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring according to the resultant force of the force and the moment of the three-layer cylindrical shell.

In this embodiment, multiple groups of springs are arranged at both ends of the three-layer cylindrical shell to form an elastic boundary, that is, different elastic boundary conditions are simulated based on artificial springs, so that the established three-layer cylindrical shell can be used in practical engineering applications. of various situations. Among them, according to the needs of the actual application scenario, the springs are set to 4 groups, and the number of springs is not specifically limited here.

103. Using the Rayleigh-Ritz method, according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring, calculate the natural frequency of the three-layer cylindrical shell.

For this embodiment, according to the above scheme, a three-layer cylindrical shell with a functional gradient layer as the middle layer can be established, and the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary can be calculated according to the resultant force of the force and moment of the three-layer cylindrical shell established. , and the spring potential energy generated by the boundary spring, and using the Rayleigh-Ritz method, according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring, the natural frequency of the three-layer cylindrical shell is calculated. . Compared with the existing technical solutions based on classical boundary natural frequency analysis, the boundary conditions in practical engineering applications can be better simulated, and the calculation accuracy of the natural frequency of laminated shells containing functionally graded materials can be effectively improved. At the same time, the properties of functionally graded materials can be changed in both the axial and radial directions, thereby further enhancing the customizability of functionally graded materials.

Further, as a refinement and extension of the specific implementation of the above-mentioned embodiment, in order to completely describe the specific implementation process of this embodiment, another calculation method for the natural frequency of a three-layer cylindrical shell is provided, as shown in FIG. include:

201. Establish a three-layer cylindrical shell whose intermediate layer is a functionally graded layer; wherein, the functionally graded layer is a two-dimensional functionally graded material.

In the specific implementation, under the elastic boundary conditions, a three-layer cylindrical shell model and coordinate system with the intermediate layer as the functional gradient layer are established. As shown in Figure 3, the total shell thickness h=0.002m, and each layer of the three-layer cylindrical shell The thicknesses are all the same, the length L=20m, and the radius R=1m. The inner and outer layers of the three-layer cylindrical shell are made of the same common metal material, Young's modulus E=68.95×10 ¹¹ , Poisson’s ratio μ=0.315, density ρ=2714.53; the middle layer is a functionally graded material, made of zirconia It is composed of nickel, the inner material is nickel, and the outer material is zirconia. Among them, the parameters of nickel are Young's modulus E=2.05098×10 ¹¹ , Poisson’s ratio μ=0.31, density ρ=8900; the parameters of zirconia are Young’s modulus E=1.68063×10 ¹¹ , Poisson’s ratio μ= 0.28, density ρ=5700.

202. Use Chebyshev polynomial to calculate the axial, circumferential and radial displacements of the three-layer cylindrical shell. In order to illustrate the specific implementation of step 202, as a preferred embodiment, step 202 may specifically include:

2021. Use the Chebyshev polynomial Chebyshev to fit the axial displacement of the three-layer cylindrical shell to obtain the axial displacement of the three-layer cylindrical shell after fitting.

According to the actual engineering application requirements, the number of terms of the Chebyshev polynomial is set to 5, and the spring stiffness is uniformly taken as 1×10 ²⁰ . The number of terms of the Chebyshev polynomial is not specifically limited here. The expression of the axial displacement of the cylindrical shell after fitting is as follows:

T ₀ (n)=1, T ₁ (n)=n, T _m+1 (n)=2nT _m (n)-T _m-1 (n), (m≥2);

T(η)为第一类切比雪夫多项式，其定义域为[-1，1]，若η的范围为[0，1]，则将T(η)转换成T^*(η)，具体计算公式为，

即由η到2η-1的坐标变换。Among them, η is the dimensionless axial coordinate of the three-layer cylindrical shell. In order to simplify the calculation of the mathematical formula, the axial coordinate is dimensionless and defined as η=x/L, that is

T(η) is a Chebyshev polynomial of the first kind, and its definition domain is [-1, 1]. If the range of η is [0, 1], then T(η) is converted into T ^* (η), specifically The calculation formula is,

That is, the coordinate transformation from n to 2n-1.

Using Chebyshev polynomial Chebyshev to simulate the axial mode shape of the three-layer cylindrical shell can make the established three-layer cylindrical shell model have good convergence and accuracy in the process of natural frequency, and by increasing the number of truncation terms Further improve the accuracy of the calculation results.

2022. According to the axial displacement of the fitted three-layer cylindrical shell, calculate the axial, circumferential and radial displacements of the three-layer cylindrical shell.

In the specific implementation, due to the geometric axis symmetry and the periodicity of deformation, the displacement of the three-layer cylindrical shell in the circumferential direction can be simply represented by a harmonic function. Therefore, the displacement calculation formula of a three-layer cylindrical shell containing a functionally graded layer under free vibration under arbitrary boundary conditions is as follows:

Among them, u, v, w are the displacements of the three-layer cylindrical shell in the axial, hoop, and radial directions, respectively. t is the time variable, n is the circumferential wave number, ω is the natural frequency, and U(η), V(η) and W(η) are the longitudinal mode functions, which represent the vibration modes of the cylindrical shell in the longitudinal direction. For example, the calculation formula for calculating the longitudinal modal function using the Chebyshev polynomial is as follows:

和

为预设系数。in,

and

is the default coefficient.

Further, the expansion coefficient vector q ⁱ (i=u, v, w) and the expansion function vector p ⁱ (i=u, v, w) are defined, and the calculation formula for calculating the longitudinal modal function using the Chebyshev polynomial is specifically:

U(η)=q ^u · ^pu (η),

V(η)=q ^v ·p ^v (η),

W(η)=q ^w ·p ^w (η),

Among them, qi and p ⁱ are ¹ ×(M+1) and (M+1)×1 vectors respectively, and the calculation formula is as follows:

p ⁱ (η)=[T ₀ ^* (η),T ₀ ^* (η),...T ₀ ^* (η),...,T ₀ ^* (η)] ^T ;

In summary, the calculation formula for the displacement of the three-layer cylindrical shell is as follows:

u(η,θ,t)=q ^u ·p ^u (η)cos(nθ)cos(ωt),

v(η,θ,t)=q ^v ·p ^v sin(nθ)cos(ωt),

w(η,θ,t)=q ^w ·p ^w cos(nθ)cos(ωt),

203. Calculate the resultant force of the force and moment of the three-layer cylindrical shell according to the axial, circumferential and radial displacements of the three-layer cylindrical shell. In order to illustrate the specific implementation of step 203, as a preferred embodiment, step 203 may specifically include:

2031. Calculate the stress, strain, and curvature vectors of the three-layer cylindrical shell according to the axial, hoop, and radial displacements of the three-layer cylindrical shell.

In the specific implementation, according to the Sanders thin shell theory, the strain calculation formula of any point on the surface of the three-layer cylindrical shell is specifically:

为中曲面应变，κ_x,κ_θ,κ_xθ为三层圆柱壳表面曲率，计算公式具体为：in,

is the mid-surface strain, κ _x , κ _θ , κ _xθ is the surface curvature of the three-layer cylindrical shell, and the calculation formula is as follows:

According to the generalized Hooke's law, the calculation formula of the stress-strain relationship of a three-layer cylindrical shell and a single-layer thin shell is as follows:

Among them, σ _x , σ _θ are the stress in the x and θ directions, σ _xθ is the shear stress on the xθ plane; ε _x , ε _θ , are the x and θ directions The strain, ε _xθ is the shear stress on the xθ plane shear strain.

For isotropic materials, the formula for calculating Q _ij is:

For functionally graded materials, the formula for calculating Q _ij is:

Among them, E is the Young's modulus of the material, _{E fgm} is the Young's modulus of the functionally graded material, and μ is the Poisson's ratio. For the cylindrical shell of the gradient material, the value of B _ij is determined according to the distribution of the functionally graded material, and the value of Q _ij is determined according to the performance of the functionally graded material.

Assuming that the functionally graded material in the middle layer of the three-layer cylindrical shell is composed of materials M1 and M2, the material properties of each point can be obtained by using the linear rule of the mixture. Specifically, the material property P of any point in the two-dimensional functionally graded cylindrical shell can be determined according to the linear combination of the volume fraction and the material properties of the basic material, and the calculation formula is as follows:

P _fgm =P _M1 V _f1 +P _M2 V _f2 ;

Among them, P is the physical parameter of the material. In order to make the material of the intermediate layer transition from M1 to M2 while being uniform in the radial and axial directions, the calculation formulas for the volume fractions of the materials M1 and M2 are as follows:

Among them, h is the thickness of the three-layer cylindrical shell, nz, nx are the volume fraction exponents. Taking different values can make the properties of functionally graded materials change at different speeds along the radial and axial directions.

Correspondingly, the calculation formula of the relevant parameters of the functionally graded material is as follows:

Since the parameters affecting the natural frequency are mainly Young's modulus E and density ρ, while the change of Poisson's ratio μ is small, the Poisson's ratio μ is set as a constant in practical engineering applications.

2032. Calculate the resultant force of the force and the moment of the three-layer cylindrical shell according to the stress, strain, and curvature vectors of the three-layer cylindrical shell.

In the specific implementation, the calculation formula of the resultant force of the force and the moment of the three-layer cylindrical shell is as follows:

Convert the resultant force calculation formula of the force and moment of the three-layer cylindrical shell into a matrix form, which is expressed as [N]=[S][ε], and expand to get:

Among them, A _ij , B _ij , and D _ij are the stretching matrix, the coupling matrix and the bending matrix, respectively. The calculation formula is as follows:

In practical engineering applications, the calculation formulas of A _ij , B _ij , and D _ij of the three-layer cylindrical shell are:

A _ij =A _ij (iso)+A _ij (FGM)+A _ij (iso)

B _ij =B _ij (iso)+B _ij (FGM)+B _ij (iso)

D _ij =D _ij (iso)+D _ij (FGM)+D _ij (iso);

Specifically,

Among them, iso is the isotropic material of the inner and outer layers of the three-layer cylindrical shell, and FGM is the functionally graded material of the middle layer of the three-layer cylindrical shell.

204. Calculate the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring according to the resultant force of the force and the moment of the three-layer cylindrical shell.

In the specific implementation, according to the resultant force of the force and the moment of the three-layer cylindrical shell, the potential energy and kinetic energy of the three-layer cylindrical shell whose middle layer is a functional gradient under the elastic boundary and the spring potential energy generated by the boundary spring are calculated. The calculation formula is specifically:

The formula for calculating the potential energy U _ε of the three-layer cylindrical shell is as follows:

After the simplification operation, expand to get:

The formula for calculating the kinetic energy T of the three-layer cylindrical shell is as follows:

Among them, ρ _t is the mass density per unit length of the three-layer cylindrical shell, and the calculation formula is as follows:

The calculation formula of the spring potential energy U _s generated by the springs arranged at both ends of the three-layer cylindrical shell is as follows:

为在三层圆柱壳壳体边界η＝0处不同方向的弹簧刚度的值；

为在三层圆柱壳壳体边界η＝1处不同方向的弹簧刚度的值。Among them, k _u , k _v , k _w , k _θ are the four groups of springs at both ends of the three-layer cylindrical shell, representing the linear springs in the u, v, and w directions, and the torsion springs with the v direction as the central axis.

is the value of the spring stiffness in different directions at the boundary η=0 of the three-layer cylindrical shell;

is the value of the spring stiffness in different directions at the boundary η=1 of the three-layer cylindrical shell.

205. Using the Rayleigh-Ritz method, obtain the frequency equation of the three-layer cylindrical shell according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring. In order to illustrate the specific implementation of step 205, as a preferred embodiment, step 205 may specifically include:

2051. According to the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring, obtain the maximum potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the maximum spring potential energy generated by the boundary spring.

In the specific implementation, using the Rayleigh-Ritz method, the displacement calculation formulas of the three-layer cylindrical shell in the axial, circumferential and radial directions are brought into the kinetic energy and potential energy of the three-layer cylindrical shell, and the three-layer cylindrical shell is calculated. Maximum values of kinetic and potential energies for layered cylindrical shells.

The calculation formula of the maximum value U _εmax of the potential energy U _ε is as follows:

The calculation formula of the kinetic energy T maximum value T _max is as follows:

The calculation formula of the spring elastic potential energy U _s maximum value U _smax is as follows:

2052. According to the obtained maximum potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, and the maximum potential energy of the spring generated by the boundary spring, obtain the frequency equation of the three-layer cylindrical shell.

In order to illustrate the specific implementation of step 2052, as a preferred embodiment, step 2052 may specifically include: according to the obtained maximum potential energy and maximum kinetic energy of the three-layer cylindrical shell under the elastic boundary, and the maximum potential energy of the spring generated by the boundary spring value, the Lagrangian energy equation of the three-layer cylindrical shell is obtained; the Lagrangian energy equation of the obtained three-layer cylindrical shell is minimized and separated, and the frequency equation of the three-layer cylindrical shell is obtained.

In the specific implementation, the potential energy and kinetic energy of the three-layer cylindrical shell whose intermediate layer is a functional gradient under the elastic boundary and the spring potential energy generated by the boundary spring are brought into the Lagrangian equation to obtain the motion equation of the three-layer cylindrical shell (that is, frequency equation).

The calculation formula of the Lagrangian energy equation L of the three-layer cylindrical shell is as follows:

In order to determine the natural frequency of the three-layer cylindrical shell, the Lagrangian energy equation L is minimized by taking partial derivatives for each unknown coefficient in q. The calculation formula is as follows:

Through the minimization and separation processing, the motion equation of the three-layer cylindrical shell is obtained, and the calculation formula is as follows:

(K+K _s -ω ² M)q ^T =0;

Among them, K is the stiffness matrix related to the strain energy of the three-layer cylindrical shell, and the calculation formula is as follows:

K _s is the stiffness matrix related to the potential energy of the three-layer cylindrical shell boundary spring, and the calculation formula is as follows:

M is the mass matrix of the three-layer cylindrical shell, and the calculation formula is as follows:

in,

206. Calculate the natural frequency in the frequency equation of the three-layer cylindrical shell according to the established size and material parameters of the three-layer cylindrical shell.

In the specific implementation, the cylindrical shell is a key element widely used in aerospace, mechanical and civil engineering. Because of its good heat resistance, functionally graded materials are also gradually applied in the structure of cylindrical shells. For the cylindrical shell of the gradient layer, the motion equation of the three-layer cylindrical shell is solved based on the free vibration characteristics of the cylindrical shell, and the natural frequency of the three-layer cylindrical shell is obtained, so as to further analyze the dynamic response.

By applying the technical solution of this embodiment, a three-layer cylindrical shell with the intermediate layer as a functional gradient layer is established, and the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary are calculated according to the resultant force of the established three-layer cylindrical shell and the moment. , and the spring potential energy generated by the boundary spring, and using the Rayleigh-Ritz method, according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring, the natural frequency of the three-layer cylindrical shell is calculated. . Compared with the existing technical solutions based on classical boundary natural frequency analysis, the boundary conditions in practical engineering applications can be better simulated, and the calculation accuracy of the natural frequency of laminated shells containing functionally graded materials can be effectively improved. At the same time, the properties of functionally graded materials can be changed in both the axial and radial directions, thereby further enhancing the customizability of functionally graded materials.

Further, as a specific implementation of the method in FIG. 1 , an embodiment of the present application provides a device for calculating the natural frequency of a three-layer cylindrical shell. As shown in FIG. 4 , the device includes: a building module 41 , a boundary simulation module 44 , a natural frequency Calculation module 45 .

The establishment module 41 is used to establish a three-layer cylindrical shell whose intermediate layer is a functionally graded layer; wherein, the functionally graded layer is a two-dimensional functionally graded material.

The boundary simulation module 44 is used for calculating the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring according to the resultant force of the three-layer cylindrical shell and the moment.

The natural frequency calculation module 45 is used for calculating the natural frequency of the three-layer cylindrical shell according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring by using the Rayleigh-Ritz method.

In a specific application scenario, the displacement module 42 and the resultant force module 43 are also included.

In a specific application scenario, the displacement module 42 is configured to use Chebyshev polynomials to calculate the axial, circumferential and radial displacements of the three-layer cylindrical shell.

The resultant force module 43 is used for calculating the resultant force of the force and the moment of the three-layer cylindrical shell according to the axial, circumferential and radial displacements of the three-layer cylindrical shell.

In a specific application scenario, the displacement module 42 is specifically used for: using Chebyshev polynomial to fit the axial displacement of the three-layer cylindrical shell to obtain the fitted axial displacement of the three-layer cylindrical shell; and , according to the axial displacement of the fitted three-layer cylindrical shell, the axial, circumferential and radial displacements of the three-layer cylindrical shell are calculated.

The resultant force module 43 is specifically used for: calculating the stress, strain and curvature vectors of the three-layer cylindrical shell according to the displacements of the three-layer cylindrical shell in the axial, circumferential and radial directions; and, according to the three-layer cylindrical shell The stress, strain, and curvature vectors of the three-layer cylindrical shell are calculated to obtain the resultant force of the force and moment of the three-layer cylindrical shell.

In a specific application scenario, the natural frequency calculation module 45 is specifically used to: using the Rayleigh-Ritz method, obtain three The frequency equation of the three-layer cylindrical shell; and, according to the established three-layer cylindrical shell size and material parameters, the natural frequency in the frequency equation of the three-layer cylindrical shell is obtained by calculating.

In a specific application scenario, the Rayleigh-Ritz method is used to obtain the frequency equation of the three-layer cylindrical shell according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring. Including: according to the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring, obtain the maximum potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, and the maximum value of the spring potential energy generated by the boundary spring; And, according to the obtained maximum potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the maximum spring potential energy generated by the boundary spring, the frequency equation of the three-layer cylindrical shell is obtained.

In a specific application scenario, the frequency equation of the three-layer cylindrical shell is obtained according to the obtained maximum potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, and the maximum spring potential energy generated by the boundary spring, Specifically, it includes: obtaining the Lagrangian energy equation of the three-layer cylindrical shell according to the obtained maximum potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the maximum spring potential energy generated by the boundary spring; and, for the obtained The Lagrangian energy equation of the three-layer cylindrical shell is minimized and separated to obtain the frequency equation of the three-layer cylindrical shell.

It should be noted that, for other corresponding descriptions of the functional units involved in the device for calculating the natural frequency of a three-layer cylindrical shell provided by the embodiments of the present application, reference may be made to the corresponding descriptions in FIG. 1 and FIG. 2 , and details are not repeated here.

Based on the above methods shown in FIGS. 1 and 2 , correspondingly, an embodiment of the present application further provides a storage medium on which a computer program is stored. The calculation method of the natural frequency of the three-layer cylindrical shell shown.

Based on this understanding, the technical solution of the present application can be embodied in the form of a software product, and the software product can be stored in a non-volatile storage medium (which may be CD-ROM, U disk, mobile hard disk, etc.), including several The instructions are used to cause a computer device (which may be a personal computer, a server, or a network device, etc.) to execute the methods described in various implementation scenarios of this application.

Based on the methods shown in FIG. 1 and FIG. 2 and the virtual device embodiment shown in FIG. 3 , in order to achieve the above purpose, the embodiment of the present application further provides a computer device, which may specifically be a personal computer, a server, a network Equipment, etc., the physical equipment includes a storage medium and a processor; a storage medium is used to store a computer program; a processor is used to execute the computer program to realize the above-mentioned calculation of the natural frequency of the three-layer cylindrical shell as shown in FIG. 1 and FIG. 2 method.

Optionally, the computer device may further include a user interface, a network interface, a camera, a radio frequency (Radio Frequency, RF) circuit, a sensor, an audio circuit, a WI-FI module, and the like. The user interface may include a display screen (Display), an input unit such as a keyboard (Keyboard), etc., and the optional user interface may also include a USB interface, a card reader interface, and the like. Optional network interfaces may include standard wired interfaces, wireless interfaces (such as Bluetooth interfaces, WI-FI interfaces), and the like.

Those skilled in the art can understand that the structure of a computer device provided in this embodiment does not constitute a limitation on the physical device, and may include more or less components, or combine some components, or arrange different components.

The storage medium may also include an operating system and a network communication module. An operating system is a program that manages the hardware and software resources of a computer device and supports the operation of information processing programs and other software and/or programs. The network communication module is used to realize the communication between various components inside the storage medium, as well as the communication with other hardware and software in the physical device.

From the description of the above embodiments, those skilled in the art can clearly understand that the present application can be implemented by means of software plus a necessary general hardware platform, and can also be implemented by hardware. By applying the technical solution of the present application, compared with the existing technical solution based on the classical boundary natural frequency analysis, the present embodiment can establish a three-layer cylindrical shell whose intermediate layer is a functionally gradient layer, according to the established three-layer cylindrical shell. The resultant force of the force and moment of the shell is calculated to obtain the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring, and the Rayleigh-Ritz method is used to calculate the three-layer cylindrical shell under the elastic boundary. The potential and kinetic energies, as well as the spring potential energy generated by the boundary spring, are calculated to obtain the natural frequencies of the three-layer cylindrical shell. It can be seen that, different from the classical boundary conditions in the prior art, under elastic boundary conditions, a method for calculating the natural frequency of a three-layer cylindrical shell with a functionally graded material in the middle layer is provided, which can better simulate practical engineering applications. It can effectively improve the calculation accuracy of the natural frequency of the laminated shell containing the functionally graded material, and at the same time, the properties of the functionally graded material can be changed in both the axial and radial directions, thereby further enhancing the feasibility of the functionally graded material. Customization.

Those skilled in the art can understand that the accompanying drawing is only a schematic diagram of a preferred implementation scenario, and the modules or processes in the accompanying drawing are not necessarily necessary to implement the present application. Those skilled in the art can understand that the modules in the device in the implementation scenario may be distributed in the device in the implementation scenario according to the description of the implementation scenario, or may be located in one or more devices different from the implementation scenario with corresponding changes. The modules of the above implementation scenarios may be combined into one module, or may be further split into multiple sub-modules.

The above serial numbers in the present application are only for description, and do not represent the pros and cons of the implementation scenarios. The above disclosures are only a few specific implementation scenarios of the present application, however, the present application is not limited thereto, and any changes that can be conceived by those skilled in the art should fall within the protection scope of the present application.

Claims (10)

1. a method for calculating the natural frequency of a three-layer cylindrical shell, is characterized in that, comprising: Build a three-layer cylindrical shell with the middle layer as the functional gradient layer; According to the resultant force of the force and moment of the three-layer cylindrical shell, the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring are obtained by calculation; Using Rayleigh-Ritz method, according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring, the natural frequency of the three-layer cylindrical shell is calculated; Wherein, the functionally graded layer is a two-dimensional functionally graded material. 2. The method according to claim 1, wherein, according to the resultant force of the three-layer cylindrical shell and the moment, the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary are calculated and obtained, and the boundary spring produces the resultant force. Before the spring potential energy, it also includes: Using Chebyshev polynomial to calculate the axial, circumferential and radial displacements of the three-layer cylindrical shell; According to the displacement of the three-layer cylindrical shell in the axial, circumferential and radial directions, the resultant force of the force and the moment of the three-layer cylindrical shell is calculated. 3. The method according to claim 2, wherein the calculating the displacement of the three-layer cylindrical shell in the axial direction, the circumferential direction and the radial direction by using the Chebyshev polynomial, specifically comprises: Using Chebyshev polynomial to fit the axial displacement of the three-layer cylindrical shell to obtain the axial displacement of the three-layer cylindrical shell after fitting; According to the axial displacement of the three-layer cylindrical shell after fitting, the axial, circumferential and radial displacements of the three-layer cylindrical shell are calculated. 4. The method according to claim 2, wherein, according to the displacement of the three-layer cylindrical shell in the axial direction, the circumferential direction and the radial direction, the resultant force of the force and the moment of the three-layer cylindrical shell is calculated, and the specific include: According to the axial, circumferential and radial displacements of the three-layer cylindrical shell, the stress, strain and curvature vectors of the three-layer cylindrical shell are obtained by calculation; According to the stress, strain and curvature vectors of the three-layer cylindrical shell, the resultant force of the force and the moment of the three-layer cylindrical shell is calculated. 5. method according to claim 1, is characterized in that, described utilizing Rayleigh Ritz method, according to the potential energy and kinetic energy of three-layer cylindrical shell under the elastic boundary that obtains, and the spring potential energy that boundary spring produces, calculates to obtain The natural frequencies of the three-layer cylindrical shell, including: Using the Rayleigh-Ritz method, according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring, the frequency equation of the three-layer cylindrical shell is obtained; According to the established size and material parameters of the three-layer cylindrical shell, the natural frequency in the frequency equation of the three-layer cylindrical shell is calculated. 6. method according to claim 5, is characterized in that, described utilizing Rayleigh Ritz method, according to the potential energy and kinetic energy of three-layer cylindrical shell under elastic boundary that obtains, and the spring potential energy that boundary spring produces, obtain three. The frequency equation of the layered cylindrical shell, including: According to the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring, obtain the maximum potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the maximum spring potential energy generated by the boundary spring; According to the obtained maximum potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the maximum spring potential energy generated by the boundary spring, the frequency equation of the three-layer cylindrical shell is obtained. 7. method according to claim 6 is characterized in that, described according to the potential energy maximum value and kinetic energy maximum value of the lower three-layer cylindrical shell obtained according to the elastic boundary, and the spring potential energy maximum value that the boundary spring produces, obtains three layers The frequency equation for a cylindrical shell, including: According to the obtained maximum potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the maximum spring potential energy generated by the boundary spring, the Lagrangian energy equation of the three-layer cylindrical shell is obtained; The Lagrangian energy equation of the obtained three-layer cylindrical shell is minimized and separated, and the frequency equation of the three-layer cylindrical shell is obtained. 8. A computing device for the natural frequency of a three-layer cylindrical shell, comprising: A building module is used to build a three-layer cylindrical shell whose middle layer is a functional gradient layer; The boundary simulation module is used to calculate the potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary and the spring potential energy generated by the boundary spring according to the resultant force of the force and the moment of the three-layer cylindrical shell; The natural frequency calculation module is used to use the Rayleigh-Ritz method to calculate the natural frequency of the three-layer cylindrical shell according to the calculated potential energy and kinetic energy of the three-layer cylindrical shell under the elastic boundary, as well as the spring potential energy generated by the boundary spring; Wherein, the functionally graded layer is a two-dimensional functionally graded material. 9 . A storage medium on which a computer program is stored, characterized in that, when the program is executed by a processor, the method for calculating the natural frequency of a three-layer cylindrical shell according to any one of claims 1 to 7 is implemented. 10 . 10. A computer device, comprising a storage medium, a processor and a computer program stored on the storage medium and running on the processor, wherein the processor implements the programs in claims 1 to 7 when executing the program The method for calculating the natural frequency of any one of the three-layer cylindrical shells.

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