Semi-analytical method suitable for analyzing vibration characteristics of cylindrical shell with non-circular section configuration

Abstract

A semi-analytical method suitable for analyzing vibration characteristics of a cylindrical shell with a non-circular section belongs to the technical field of algorithm calculation in computational mechanics, and comprises the following steps: firstly, giving a detailed process of vibration analysis modeling of the cylindrical shell with the non-circular section through an energy method; and then, through selection of a primary function and optimization of a matrix element integration strategy, a Gaussian wavelet primary function method is improved, and verification is carried out through comparison with a finite element simulation result. And finally, analyzing the influence of the section shape on the structure frequency and the annular vibration mode through a numerical result, and finding that the vibration characteristics of the complex-section cylindrical shell can be effectively solved by the improved semi-analytical method. According to the semi-analytical method suitable for processing the circumferential variable curvature cylindrical shell structure, on the basis of Gaussian expansion method, translation parameters of a primary function are directly associated with a physical solution domain, the bandwidth processing process is introduced by utilizing local properties of a wavelet function, the integral processing process of matrix elements is simplified, and the solution efficiency is effectively improved.

Description

Technical Field

[0001] The invention belongs to the technical field of algorithm calculation in computational mechanics, and in particular relates to a semi-analytical method suitable for analyzing vibration characteristics of a cylindrical shell with a non-circular cross-section configuration. Background Art

[0002] As an important component of aerospace equipment, the shell structure with light weight and high strength has always attracted much attention. At the same time, with the increasing consideration of multi-field coupling in the design and the continuous development of equipment manufacturing level, the application of complex configuration shells in the aerospace field has become more and more common. For example, in the lobe nozzle injection hybrid infrared suppression system used in the engine exhaust system, its related component lobe nozzle has a very complex geometric configuration. [1-3] For these structures with complex surfaces, their curvature parameters often no longer remain constant, and engineers can only solve them with the help of numerical methods such as finite element methods. Traditional analytical methods are usually unable to handle such structures.

[0003] Due to its light weight, high strength, excellent load-bearing performance and good space utilization, cylindrical shell structures have been widely used in aerospace and marine engineering. Cylindrical shell is the most common type of cylindrical shell structure. There are countless studies related to it, and even a large number of monographs have special chapters to introduce it. [5-7] For a long time, thanks to its wide application in practical engineering, the research on cylindrical shells has been kept hot. Early related research mainly focused on the analysis of cylindrical shell structures made of isotropic materials. [8-11] With the advancement of computational methods and materials science, more and more research has begun to study the optimization and improvement of cylindrical shell performance by different material compositions based on the structure and combined with the actual material application, such as typical functional gradient materials, composite materials and other advanced materials. [12-15] Among these studies, vibration-related issues are an important branch. Structural vibration analysis is a very important tool in the field of engineering. First, through vibration analysis, engineers can understand the natural vibration characteristics of cylindrical shell structures.

[16] , which can be used to optimize structural design, reduce resonance risks, and improve structural performance and stability [17,18] Secondly, through vibration analysis, the behavior of different materials and their distribution in the structure can also be evaluated, so as to make more appropriate material layout choices.

[18] In addition, when the column shell structure has vibration problems, by analyzing the response of the structure, possible defects and faults in the structure can be found.

[19] In summary, the research related to vibration characteristics is of great significance for designing, evaluating and improving the performance of cylindrical shell structures.

[0004] At present, most of the research on cylindrical shell structures is focused on circular cylindrical shell structures. On the one hand, it is because cylindrical shell structures are indeed widely used in engineering practice, and on the other hand, it is also due to the relatively simple geometric configuration of cylindrical shell structures. For cylindrical shell structures, their curvature-related parameters and Lamé coefficients are constants. Therefore, the circumferential vibration mode of cylindrical shells is relatively simple, and is usually directly described by trigonometric functions. However, for cylindrical shells with non-circular cross-sections, since the circumferential curvature varies with the angular coordinates, the corresponding circumferential vibration mode will become more complicated. In the literature

[20] The calculation results given by the transfer matrix method in the paper show that even for non-cylindrical shells with circumferential periodicity, most of their circumferential vibration modes no longer have circumferential periodicity. This result preliminarily demonstrates the complex characteristics of vibration modes of non-cylindrical shell structures. For complex configuration structures, the essence lies in how to accurately model complex structures with varying configuration parameters. As a non-cylindrical shell structure with a configuration very similar to that of a cylindrical shell, the elliptical shell has also attracted the attention of researchers. [21-23] Bochkarev et al.

[24] The free vibration and stability of an elliptical cylindrical shell containing a fluid were studied. Kumar and Patel

[25] By combining experiments with the finite element method, the nonlinear dynamic response of elliptical cylindrical shells under lateral harmonic excitation is studied. Within the excitation range considered in the experiment, the results show the softening nonlinearity of the elliptical cylindrical shell. In general, analytical research on complex cross-section cylindrical shells is still very rare. The main difficulty lies in the complexity of vibration modes caused by variable curvature configurations. For such variable parameter problems, the use of wavelet basis functions has very significant advantages. Because wavelet basis functions have local properties, they can effectively capture the response of the structure in areas where parameters change drastically. Some commonly used wavelet basis functions include Haar wavelets, Mexican hat wavelets, and Gaussian wavelets. [26-32] Among them, the Gaussian expansion method based on Gaussian wavelets has been used by Deng et al. [33-35] It has been successfully applied to the relevant research and analysis of variable thickness plate and shell structures, and has achieved a series of very ideal results. Summary of the invention

[0005] In order to solve the above problems, the present invention proposes: a semi-analytical method suitable for analyzing the vibration characteristics of a cylindrical shell with a non-circular cross-section configuration, comprising the following steps:

[0006] S1. Establish mathematical model: geometric description of non-circular cross section;

[0007] S2. Describe the energy of thin-walled non-circular shells;

[0008] S3. Determine the vibration modes and frequencies of thin-walled non-cylindrical shells.

[0009] The beneficial effects of the present invention are as follows: the present invention develops a semi-analytical method suitable for processing annular variable curvature cylindrical shell structures. Based on the Gaussian expansion method, the method directly associates the translation parameters of the basis function with the physical solution domain, and introduces the bandwidth processing process by using the local properties of the wavelet function, simplifies the integral processing process of the matrix elements, and effectively improves the solution efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 It is a schematic diagram of the non-circular cross-section shape of the present invention;

[0011] δ = 0.5, N = 3, p = 3, (a) cross-sectional function: C1, (b) cross-sectional function: C2;

[0012] Figure 2 This is a schematic diagram of the non-cylindrical shell shape of the present invention;

[0013] δ = 0.5, N = 3, p = 3, (a) cross-sectional function: C1, (b) cross-sectional function: C2;

[0014] Figure 3 Schematic diagram of different Gaussian basis functions at the center position of the present invention;

[0015] (q j : Function center position; c: Function scope radius parameter);

[0016] Figure 4 It is a comparative verification curve diagram of the modal frequency of the cylindrical shell with both ends fixed in the present invention (parameter combination 1 case);

[0017] (a) Frequency comparison, (b) relative error

[0018] Figure 5 is a curve diagram of the modal frequency and calculation error of the cantilever column shell of the present invention (parameter combination 2);

[0019] (a) Frequency characteristics (C1), (b) Relative error (C1), (c) Frequency characteristics (C2), (d) Relative error (C2);

[0020] Figure 6 is the axial vibration mode curve diagram of the cantilever cylindrical shell of the present invention;

[0021] Figure 7 is a circumferential vibration mode curve diagram of the cantilever cylindrical shell of the present invention;

[0022] (Red line: M-order mode; blue line: M+1-order mode; green dashed line: the result after the red line is rotated by π / (2n) or -π / (2n), n is the circumferential half-wave number);

[0023] Figure 8 It is the modal diagram of the vibration type displacement component in different directions of the 131st order mode of the present invention;

[0024] Fig. 9 is a circumferential vibration mode curve diagram of the cantilever non-cylindrical shell of the present invention;

[0025] C1, δ = 0.1 (black line: undeformed configuration; red line: modal vibration shape; green line: vibration shape after rotation of 2π / N, where N is the number of lobes)

[0026] Fig.10 is a circumferential vibration mode curve diagram of the cantilever non-cylindrical shell of the present invention;

[0027] C1, δ = 0.5 (black line: undeformed configuration; red line: modal vibration shape; green line: vibration shape after rotation of 2π / N, where N is the number of lobes)

[0028] Fig.11 is a circumferential vibration mode curve diagram of the cantilever non-cylindrical shell of the present invention;

[0029] C2, δ = 0.1 (black line: undeformed configuration; red line: modal vibration shape; green line: vibration shape after rotation of 2π / N, where N is the number of lobes; blue line: vibration shape after the displacement amplitude becomes negative)

[0030] Fig.12 It is a circumferential vibration mode curve diagram of the cantilever non-cylindrical shell 12 of the present invention;

[0031] C2, δ = 0.5 (black line: undeformed configuration; red line: modal vibration shape; green line: vibration shape after rotation of 2π / N, where N is the number of lobes; blue line: vibration shape after the displacement amplitude becomes negative)

[0032] Fig.13 It is the modal diagram of the vibration type displacement component in different directions of the 7th order mode of the present invention;

[0033] C1, δ = 0.1;

[0034] Fig.14 It is the modal diagram of the vibration type displacement component in different directions of the 10th order mode of the present invention;

[0035] C1, δ = 0.5;

[0036] Fig.15 It is the modal diagram of the radial vibration type displacement component of mode 15 of the present invention;

[0037] C1;

[0038] Fig.16 It is the modal diagram of the radial vibration type displacement component of the 16th mode of the present invention;

[0039] C2. DETAILED DESCRIPTION

[0040] In order to make the technical means and objectives of the present invention easy to understand, the present invention is further described below in combination with a specific embodiment. A semi-analytical method suitable for analyzing the vibration characteristics of a cylindrical shell with a non-circular cross-section configuration includes the following steps:

[0041] S1. Establish mathematical model: geometric description of non-circular cross section;

[0042] S2. Describe the energy of thin-walled non-circular shells;

[0043] S3. Determine the vibration modes and frequencies of thin-walled non-cylindrical shells.

[0044] 2.1 Geometric description of non-circular cross sections

[0045] Establish as Figure 1 The smooth non-circular cross section shown in the figure is given by the function expression: r = r (R0, δ, N, p, θ), where R0, δ, N, p and θ are the maximum radius, radius ratio (δ = R min / R0), number of lobes, lobe shape parameters and angular coordinates.

[0046] for Figure 1 A schematic diagram of a non-circular cross section is given, and its specific function expression is:

[0047]

[0048] exist Figure 1 Based on the cross section shown, it is easy to obtain the cylindrical shell with a length of L as Figure 2 shown.

[0049] for Figure 2 The non-circular cross-section cylindrical shell shown in the figure establishes a cylindrical coordinate system at the center of the end section and takes an infinitesimal surface element on the surface, which can be expressed as

[36] :

[0050]

[0051] Among them A i (i=1,2) represents the Lamé coefficient, that is:

[0052]

[0053] Where K i (i=1,2) represents the curvature. It should be noted that its value can be positive or negative [6].

[0054] 2.2 Energy description of thin-walled non-circular shells

[0055] Assuming that the mid-surface displacement of the thin-walled shell is (u0, v0, w0), the displacement field (u1, u2, u3) at any point of the shell can be expressed as follows [7] :

[0056] u1=u+zβ1,u2=v+zβ2,u3=w (5)

[0057]

[0058] For thin shells, considering the Love shell theory, the corresponding generalized linear strain-displacement relationship is

[37] :

[0059]

[0060] By analyzing equation (7), it can be found that for the cylindrical shell structure, its Lamé coefficient A1 is 1 and the curvature K1 is 0. Further, according to equations (3) to (6), equation (7) can be simplified as follows:

[0061]

[0062] The new vector notation is introduced as follows:

[0063] f u =(u,0,0),f v =(0,v,0),f w =(0,0,w) (9)

[0064] The strain can be further organized into the following form:

[0065]

[0066] In addition, the stress-strain relationship for thin-walled shells is given as follows:

[0067]

[0068] Where E and ν are Young's modulus and Poisson's ratio, respectively, and E t =E / (1-υ 2 ). According to the displacement field, strain and stress given above, the kinetic energy T and elastic potential energy U of the thin-walled shell can be obtained: s as follows:

[0069]

[0070]

[0071] Where ρ, L and h are the density, length and thickness of the shell respectively. The elastic potential energy can be further organized into a function expression related to the mid-surface displacement as follows:

[0072]

[0073] The specific expressions of f1 and f2 can be found in the appendix. In addition, for any boundary situation, artificial springs can be used for simulation. In this case, the potential energy of the boundary spring can be recorded as:

[0074]

[0075] in and is the boundary spring coefficient. According to equations (12), (14) and (15), we can get

[0076] The Lagrange equation of the system is as follows:

[0077] L=T-(U s +U b ) (16)

[0078] And the Euler-Lagrange equations are as follows:

[0079]

[0080] 2.3 Vibration shapes and frequencies of thin-walled non-cylindrical shells

[0081] In order to determine the vibration mode of the non-circular cross-section thin shell, it is first necessary to discretize the system displacement in the form of basis function series expansion, which is as follows:

[0082]

[0083] where a k (k=u,v,w) is the time-dependent weight coefficient to be determined, ψ k (k=u,v,w) is a spatial shape function, and q=[a u ; a v ; a w ], Furthermore, the spatial shape function can be separated into variables as follows:

[0084]

[0085] in represents Kronecker's product, function α k (x) and β k(θ)(k=u,v,w) are Gaussian wavelet functions in the axial and circumferential directions respectively, and their expressions are

[35] :

[0086]

[0087] where c λ and Represent the scaling and translation parameters respectively, and have:

[0088]

[0089] According to formula (21), it is not difficult to find that the physical solution domain is actually divided into (S x +4)×(S θ +4) grids. The number of basis functions here is directly based on the size parameters of the physical solution domain, similar to the meshing in the finite element method. Therefore, the number of grids is more flexible, that is, the difference in the number of grids under different but similar parameter conditions is small.

[35] The method of selecting basis functions will result in a greater difference in the number of grids under different but similar parameter conditions, which will result in a large number of redundant basis functions. A more specific comparison will be further explained in the subsequent comparison verification section. It should be noted that since the displacement of the cylindrical shell in the circumferential direction is continuous, the following continuity conditions need to be considered

[35] :

[0090] f (i) (x,-π)=f (i) (x,π),f=u,v,w (22)

[0091] The expression express After considering this continuity condition, the number of linearly independent cyclic basis functions will be reduced by 3.

[0092] Based on the displacement discretization given above, the corresponding Lagrange equation can be further organized as:

[0093]

[0094] The specific forms of the elements in the mass and stiffness matrices are given in detail in the appendix.

[0095] The corresponding mass matrix and stiffness matrix, and then combined with the Euler-Lagrange equations, we can get:

[0096]

[0097] According to equation (24), the corresponding eigenvalues ​​and eigenvectors are calculated and sorted to obtain the frequency and vibration mode of the non-circular cross-section cylindrical shell.

[0098] 2.4 Integration Optimization Strategy of Matrix Elements

[0099] It should be noted that when calculating the matrix integral, Kronecker's product is used to separate the radial and annular spatial variables, which greatly simplifies the integral operation. Therefore, when calculating each matrix block, the submatrix after the two variables are separated is obtained by Kronecker's product (i.e. ). Since each submatrix is ​​the integral of the product of Gaussian basis functions, and the Gaussian function image has a local property, that is, most of the area of ​​the function is concentrated within a certain width from the center (such as Figure 3 As shown), this inspires us to convert the submatrix (i.e. K x and K θ ) is processed into a band matrix, and the elements outside the set bandwidth are assigned to 0 in advance before integration, thereby improving the efficiency of integral calculation. The bandwidth setting can be adjusted appropriately, because the concentration of the area may change to a certain extent after the Gaussian function is multiplied by the relevant weight function, and the weight function may be related to the specific form of the material and structural parameters. Figure 3 As shown in the figure, the function graphs of Gaussian basis functions with different center positions are given when the weight function is a constant 1. The first thing that can be observed is that the radius of the function scope is about 4c, because the function value outside this radius basically tends to 0. In addition, the shaded parts shown in the figure are different basis functions and basis function q j =0. If there is a large intersection between the two functions, the integral value of the inner product of the two functions is also large. It can be found that when the center distance of the two functions is less than 8c, there is a visible intersection between the two functions, and the smaller the center distance, the larger the intersection. If the distance between the centers of the two functions is greater than 8c, it can be approximately considered that their inner product is approximately 0. The matrix elements correspond to the inner products of the two functions, and the row and column numbers of the matrix elements correspond to the distance between the function centers. For example, for the matrix element For , the distance between the two corresponding functions is |ij|c. Considering that the inner product is approximately 0 when the distance is greater than 8c, the matrix element You can assign 0 before integration, which can reduce the amount of subsequent integration operations and obtain a band matrix with limited bandwidth.

[0100] Although the efficiency of the eigenvalue calculation part is higher for a band matrix with a narrow bandwidth, the accuracy of the eigenvalue calculation can be improved by appropriately increasing the bandwidth. For the convenience of subsequent discussion, the definition of half bandwidth is introduced here, that is, the distance between the centers of two functions is greater than b.w ×c, the value b when the inner product of the two is approximately 0 w In addition, it should be noted that when the weight function is no longer a constant, if the half-bandwidth is 8, it may not be accurate enough, so it is necessary to further increase the half-bandwidth of the band matrix.

[0101] 3. Comparative Verification

[0102] In this section, the accuracy of the improved method in this paper is verified by comparing with the literature data and finite element simulation results; and it is also demonstrated that the improved method in this paper has the advantage of computational efficiency while ensuring the accuracy of the results. Considering the source of the literature data, the frequency characteristics of the cylindrical shell are first verified here. After this step of verification, the frequency characteristics of the complex non-circular cross-section cylindrical shell will be further compared and verified with the finite element simulation results. Table 1 gives two groups of cylindrical shell parameters of different sizes, of which the first group is the literature data and the second group is the structural parameters of this paper. In addition, when using the method in this paper for calculation, it is necessary to assign values ​​to the boundary spring parameters. The literature

[35] The parameter value given in is k z =1×10 10 , that is, for the fixed boundaries at both ends, After calculation and comparison, this paper selects the parameter value as the value of the elastic modulus, that is, k z =2.1×10 11 , and the spring stiffness in the axial displacement direction is increased by an additional 2 orders of magnitude. Therefore, for the cantilever boundary structure, the corresponding parameter values ​​are:

[0103] Table.1 Geometry and material parameters of cylindrical shell

[0104]

[0105] Figure 4 Given the literature

[35] Comparison of the modal frequencies of the cylindrical shell structure with the finite element values ​​under the parameter conditions, where the characteristic mesh unit size of the finite element is 0.02m. Figure 4 In the calculation example, The Gaussian basis has been constructed taking S x =20,S θ =36in Eq(21), and the stiffness matrix dimension calculation formula is: N = 3(S x +5)(S θ +2), the dimension of the matrix is ​​2850, which is much smaller than that in the literature.

[35] The dimension of the calculation is 14685 (also smaller than that of the literature

[35] The matrix dimension of the mid-frequency convergence is 8035). Moreover, by comparing the relative errors, it can be found that the relative error of the low-order modal frequency in the literature is about 1%, while the error is less than 0.5% when the solution is adopted in this paper. In addition, in the verification part and the subsequent analysis part, when calculating the integral value of the matrix element, the half bandwidth is taken as b w =12.

[0106] The good consistency of the frequency comparison results preliminarily illustrates the accuracy of the calculation of the improved method in this paper. At the same time, observing the relative error, it can be found that with the increase of the modal order, the error is also gradually increasing. The reason for this phenomenon is that for high-order modes, the circumferential displacement field corresponding to its vibration shape is often more complex, so it is necessary to compare the number of radial and circumferential Gaussian basis functions (S x =20,S θ =36) It can also be found that the complexity of the circumferential vibration mode is much higher than that of the axial vibration mode.

[0107] 4. Vibration characteristics analysis

[0108] In order to use the improved method in this paper to analyze the vibration characteristics of non-circular cross-section cylindrical shells, the influence of relevant geometric parameters on the structural frequency characteristics will be discussed in this section. In addition, unless otherwise stated, the material GH3044 (solid solution strengthened nickel-based oxidation-resistant alloy) and structural parameters in the subsequent analysis all adopt the values ​​of parameter case 2 (see Table 2).

[0109] 4.1 Frequency characteristics It is not difficult to find from formula (1) that when the radius ratio δ is taken as 0, the non-circular cross section will degenerate into a circular cross section. Therefore, the size of the radius ratio δ describes the degree to which the non-circular cross section deviates from the circular cross section. By analyzing the results under the condition of this parameter change, some more general conclusions about the response characteristics of the cylindrical shell structure can be drawn.

[0110] Figure 5 The frequencies of the first 500 modes of the non-circular cross-section under two different function forms (i.e., C1 and C2) are given, as well as the relative error analysis between the frequencies and the finite element results. At this time, the number of lobes is 3. In addition, Table 2 lists the values ​​of the number of axial and annular wavelet basis functions under the relevant parameter conditions. By analyzing the number of discrete wavelet basis functions selected, it can be found that the number of discrete annular wavelet basis functions will increase significantly with the increase of the radius ratio parameter. This can actually be understood by analogy with the number of mesh divisions in the finite element, that is, with the increase of the radius ratio parameter, the length of the mid-surface arc corresponding to the annular direction of the non-circular cylindrical shell increases; if the size of the mesh division in the finite element remains unchanged, then more meshes are needed to divide the longer arc, and the increase in the number of meshes corresponds to the increase in the number of discrete wavelet basis functions.

[0111] In addition, according to the case where the radius ratio is 0 in Table 2, under the condition that the calculation frequency converges, S x =16,S θ =70. Although the discrete number of the axial wavelet basis function is slightly lower than the value in the verification part, the discrete number of the annular wavelet basis function is much larger than the value S in the verification part. θ =36, which is caused by the size difference of the structure under the two calculation conditions. This can also be understood by comparing with the finite element method. For example, for structures with different size parameters, the number of grids required is usually different; and if the geometric shape of the structure is more complex, the number of corresponding grids is often larger.

[0112] Table.2 Relationship between discrete number and parameters of wavelet basis function (N=3, p=3)

[0113]

[0114] Through the data given in Table 2, we can compare the number of axial wavelet basis functions S of cylindrical shells with different cross-sectional shapes. x Obviously, the value of this parameter has not changed, which means that the change of the annular geometric parameters will not affect the vibration mode in the axial direction, that is, the annular and axial vibrations are separable.

[0115] By comparing the relative frequency errors of cylindrical shells with different cross-sectional shapes, it can be found that the cross-sectional shape function has a certain influence on the number of wavelet basis functions required for frequency calculation, especially when the radius ratio parameter is large. Figure 5 It is very obvious in (d) that the circumferential wavelet basis function is taken as S θ =180, the relative error still exceeds 1%, which is greater than Figure 5 (b) S θ =180.

[0116] Table.3 Frequency of non-cylindrical shell (C1, N = 3, p = 3)

[0117]

[0118]

[0119] Table 3 shows the frequencies of the first 10 modes when the cross-sectional shape is curve C1. By analyzing the data, it can be found that when the radius ratio parameter δ = 0 (the non-circular cross section degenerates into a circular cross section), the modal response of the structure has two identical frequencies (there are some calculation errors in the data in the table). The reason for this phenomenon is that for the annular vibration mode, the forward and backward waves in the annular direction are superimposed on each other to form two standing waves with a phase difference of π / (2n), where n is the number of half waves. Since the annular standing wave is formed by the superposition of the forward and backward waves, the cylindrical shell produces two natural frequencies with the same value. After the radius ratio parameter is greater than 0, since the cross section loses the special axial symmetry of the circular cross section, the nodes where the standing wave is formed after the interaction of the forward and backward waves of the annular direction change, so more unpaired natural frequencies will appear; and even if the natural frequencies are the same, the phase difference of the vibration mode is likely to no longer be maintained at π / (2n). In addition, for the convenience of the following discussion, the two modes with the same frequency are defined as the forward wave mode and the backward wave mode.

[0120] In general, the frequency comparison in this section not only further verifies the accuracy of the semi-analytical method proposed in this paper for calculating the frequency of non-circular cylindrical shell structures, but also reveals the relationship between the selection of the discrete number of wavelet basis functions and the structural parameters, that is, the more complex the geometric shape of the structure, the more discrete number of wavelet basis functions are required, which is similar to the meshing properties of the finite element method.

[0121] 4.2 Vibrational characteristics

[0122] In this section, the structural vibration modes calculated by the semi-analytical method in this paper will be given, and some unique properties of non-circular cylindrical shells will be further demonstrated by analyzing some characteristics of the vibration modes, in order to provide theoretical guidance for subsequent vibration analysis and structural design.

[0123] 4.2.1 Vibration modes of cylindrical shells

[0124] The vibration modes of the cylindrical shell can be divided into axial and circumferential directions, so the vibration modes of different modes in these two directions will be shown separately below.

[0125] Figure 6 The axial vibration modes of different order modes are given, and it is not difficult to observe that the vibration modes are almost consistent with the shape of the cantilever beam, which further indirectly explains that the vibration of the cylindrical shell structure can be separated in the axial and circumferential directions. In addition, Figure 6 The modal vibration modes with order difference of 1 are given respectively. It is not difficult to find that the axial vibration modes of the modes are exactly the same. This also corresponds to the analysis results of the circumferential vibration modes when analyzing the data in Table 3.

[0126] Figure 7The circumferential vibration modes of different modes of the cantilever cylindrical shell are given. The red and blue lines correspond to the forward-traveling wave mode mode and the backward-traveling wave mode mode defined above, respectively. The green dashed line is the result of rotating one of the modes by π / (2n). Obviously, the rotated vibration mode overlaps well with the other vibration mode, which further verifies the above conclusion that the phase difference between the forward-traveling wave mode mode and the backward-traveling wave mode mode is only π / (2n). It should be noted that, if Figure 7 As shown in (h), for the 150th mode vibration shape, the overlap between the blue line and the green line is slightly lower than that of the previous low-order mode, which means that under the premise of numerical convergence of frequency, the convergence of high-order mode displacement is relatively poor. In addition, since 150 is an even number and the other modal orders are odd numbers, there may be a special mode before the 150th mode. Therefore, we will find this mode in the data and analyze its characteristics.

[0127] By checking the calculated frequency data, it is found that the frequencies of the 129th to 133rd order modes are 1809.6498, 1809.6498, 1821.4342, 1822.2572 and 1822.2572 Hz, respectively. Obviously, the frequency of the 131st order mode is unpaired. Therefore, in Figure 8 The displacement components in different directions of the 131st mode are given separately. Figure 8 (a) and (c) show that the axial and radial displacements of this mode are almost zero, there is only annular torsional displacement, and the torsional displacement only changes with the axial coordinate. At this time, the cylindrical shell is a torsional mode.

[0128] This section mainly discusses the vibration mode characteristics of the cylindrical shell and verifies that there is a phase difference of π / (2n) between the front and rear traveling wave modes. The results further illustrate the accuracy of the proposed method in calculating the vibration mode of the structure.

[0129] 4.2.2 Vibration modes of non-cylindrical shells

[0130] In the previous section, the analysis of the vibration mode of the cylindrical shell preliminarily showed that the axial vibration mode is basically consistent with the vibration mode of the cantilever beam. Therefore, this section mainly analyzes the circumferential vibration mode characteristics of the non-cylindrical shell. In addition, when discussing the frequency characteristics given in Table 3, it has been found that the frequency of the non-circular cross-section cylindrical shell has more unpaired frequencies, and the phase difference change of the circumferential vibration mode is preliminarily analyzed. Therefore, this section will further discuss the general characteristics of the structural vibration mode in the case of non-circular cross-section.

[0131] Fig. 9 The circumferential vibration modes of the first 10 modes of the cantilever cylindrical shell are given when the radius ratio is δ = 0.1 and the non-circular cross-section curve function is C1. Fig. 9For modes with similar intermediate frequencies (such as a and b, d and e, and h and i), it can be found that although the number of wave peaks is the same, there are some differences in the shapes, so it is impossible to make the two circumferential vibration modes coincide after rotating with a fixed phase difference. In other words, the vibration modes of the forward wave mode and the backward wave mode are differentiated at this time, which is essentially different from the case of cylindrical shells. The reason for this difference may be that for circular curves, there are countless symmetry axes, but for non-circular cross-section curves, the number of symmetry axes is limited, so the standing wave shape formed by the superposition of traveling waves is different.

[0132] In addition, for Fig. 9 The frequencies of the modes shown in (c), (f) and (g) do not appear in pairs. It can be found that these modes have rotational symmetry, that is, they can coincide with themselves after rotating by an angle of 2π / N. Using this symmetry, when analyzing, the original structure can be divided into N equal parts, and 1 / N of the substructures can be analyzed. It should be noted that this method of simplifying the analysis of the structure by using symmetry is only applicable to these modes with rotational symmetry; however, for rotationally symmetric structures, not all of their vibration modes have rotational symmetry. If only 1 / N of the structure is analyzed according to rotational symmetry, some structural information may be lost. Figure 10-12 The circumferential vibration modes of cantilever non-cylindrical shells with different configurations under the other three sets of parameter conditions are given. It should be noted that the vibration mode represents the ratio of the displacements of each point. Therefore, when compared with the finite element results, the maximum and minimum values ​​may be exactly opposite. In this case, it is only necessary to negative the displacement amplitude (corresponding to Fig.11 and 12 In the figure, the vibration modes corresponding to the red and blue lines); in addition, since the non-cylindrical shell structure studied in this paper has rotational symmetry, when compared with the finite element method, the circumferential vibration modes may differ by a phase angle related to multiples of 2π / N.

[0133] In general, these images further indicate that: for unpaired frequencies, the corresponding modes have rotational symmetry; for paired frequencies, the modes corresponding to one of them may have certain symmetry, such as Fig. 9 (e), 10(d), 10(h), 11(b), 12(b) and 12(e). In addition, by observing these vibration modes, it can be found that the curves of some vibration modes are almost completely consistent with the undeformed configuration. Fig.12Take (i) as an example. For the left lobe, that is, in the range of about [2π / 3,4π / 3], the red solid line almost coincides with the black solid line. This shows that this part of the structure has almost no displacement response, that is, it is almost static, and only has local displacement response. The reason for this phenomenon is that for non-circular cylindrical shells, the curvature of the structure in the circumferential direction changes very dramatically, and the curvature has a significant effect on the vibration response of the structure. Generally speaking, the greater the local curvature, the greater the local stiffness, and therefore the smaller the displacement. Fig.12 For the corresponding undeformed configuration, the three inward concave points have a larger local curvature, so Fig.12 In the vibration mode shown, the displacements at these three special points are all small. Similar phenomena can also be seen in Fig.10 It is observed that Fig.10 The displacement at the tip of the "triangle" of the initial configuration is always smaller.

[0134] In the discussion and analysis of Section 4.2.1, it is also analyzed that the torsional modes of cylindrical shells have unpaired frequencies. Therefore, the modes corresponding to the unpaired frequencies of non-cylindrical shells will be categorized and discussed to verify whether the mode at this time is a torsional mode. At the same time, its displacement characteristics will also be analyzed.

[0135] Fig.13 and 14 The displacement components of some rotationally symmetric modes of non-cylindrical shells are given. Figure 8 It is not difficult to find from the results that the displacements of these modes are still dominated by radial displacement, while in the case of non-torsion mode, the displacement is dominated by torsional displacement, which means that the mode at this time is different from the torsional mode in the case of cylindrical shell. In order to further demonstrate the vibration mode characteristics of non-cylindrical shell, Fig.15 and 16 The distribution of some modal radial displacements in the entire physical domain is given. First, by comparing the cases with different cross-sectional functions, it can be found that some vibration modes corresponding to the cross-sectional function C2, such as Fig.16 (a), (c) and (f), even at the right side of the free end (i.e. x = L = 0.34), the corresponding displacement response is still close to 0 at some angle coordinates. This also corresponds to the previous Fig.12 The analysis conclusion of some vibration modes is that the mode at this time has a local displacement response, and this characteristic is caused by the curvature characteristics of the cross-sectional function.

[0136] In general, for cylindrical shell structures with non-circular cross-sections, their circumferential vibration modes have very unique shape characteristics, such as rotational symmetry and local vibration characteristics, under the action of circumferential curvature.

[0137] 5. Summary

[0138] This paper develops a semi-analytical method for solving the vibration characteristics of non-cylindrical shells using Gaussian wavelet basis functions. Since the wavelet basis function has a local nature, it has a similar effect to the grid in the finite element method; at the same time, unlike the finite element method in which the displacement field is only continuous but not smooth at the grid boundary, the displacement field constructed by the wavelet basis function is continuous and smooth in the entire definition domain. In addition, considering the local nature of the wavelet basis function, this paper further introduces a matrix bandwidth processing strategy to improve the computational efficiency, thereby improving and developing this semi-analytical method. Through the analysis of numerical calculation results, the following conclusions are drawn:

[0139] 1. By analogy with the idea of ​​finite element meshing, the improved Gaussian wavelet basis function method directly discretizes functions from the physical domain, effectively reducing the number of discrete basis functions and laying the foundation for improving computational efficiency; 2. By considering the local properties of the wavelet function, the corresponding sub-matrix elements of the mass and stiffness matrices are subjected to bandwidth restriction, thereby effectively reducing the amount of integral operations and effectively saving computational time;

[0140] 3. Through the analysis of the vibration response of non-circular cross-section cylindrical shells, the wavelet basis function method can effectively describe the characteristics of the local response of the structure, and the significant influence of curvature on the circumferential vibration mode is found. The numerical analysis results show that even for cylindrical shell structures with rotational symmetry, not all vibration modes have rotational symmetry; therefore, when analyzing such structures, it is necessary to carefully consider the operation of selecting only part of the structure instead of the whole structure for analysis.

[0141] Appendix A

[0142]

[0143]

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[0183] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical solutions and concepts of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A semi-analytical method for analyzing the vibration characteristics of cylindrical shells with non-circular cross-section configurations, characterized in that: The steps include: S1. Establish mathematical model: geometric description of non-circular cross section; S2. Describe the energy of thin-walled non-circular shells; S3. Determine the vibration modes and frequencies of thin-walled non-cylindrical shells.

2. The semi-analytical method for analyzing the vibration characteristics of a cylindrical shell with a non-circular cross-section configuration as claimed in claim 1, characterized in that: In step S1, a smooth non-circular cross section is established, and its function expression is given as: r = r (R0, δ, N, p, θ), where R0, δ, N, p and θ are the maximum radius, the radius ratio (δ = R min / R0), the number of lobes, lobe shape parameters and angle coordinates, the specific function expression for non-circular cross-section is: Based on the cross section, the cylindrical shell with length L is obtained. For a cylindrical shell with a non-circular cross-section, a cylindrical coordinate system is established at the center of the end section, and an infinitesimal surface element is taken on the surface to represent it as: dS=ds1ds2, ds1=A1dξ1=dx, Among them A i (i=1,2) represents the Lamé coefficient, that is: ξ1=x,ξ2=θ, A1＝1, K1(x,θ)=K z (x,θ)=0, Where K i (i=1,2) represents the curvature, which can be positive or negative.

3. The semi-analytical method for analyzing the vibration characteristics of a cylindrical shell with a non-circular cross-section configuration as claimed in claim 1, characterized in that: In step S2, the energy of the thin-walled non-circular shell is described as follows: Assuming that the mid-surface displacement of the thin-walled shell is (u0, v0, w0), the displacement field (u1, u2, u3) at any point of the shell is expressed as follows: u1=u+zβ1,u2=v+zβ2,u3=w (5) For thin shells, considering the Love shell theory, the corresponding generalized linear strain-displacement relationship is: The analysis formula shows that for the cylindrical shell structure, the Lamé coefficient A1 is 1 and the curvature K1 is 0. Then, according to equations (3) to (6), equation (7) is simplified as follows: ε 11 =u ,x -zw ,xx , ε 22 (A -1 v ,θ +K2w+z(K2A -1 v ,θ +A -1 K 2,θ vA -2 w ,θθ +A -3 AM ,θ w ,θ ), γ 12 =A -1 u ,θ +v ,x +z(K2v ,x -2A -1 w ,xθ ) (8) The new vector notation is introduced as follows: f u =(u,0,0),f v =(0,v,0),f w =(0,0,w) (9) The strain is further organized into the following form: In addition, the stress-strain relationship for thin-walled shells is given as follows: Where E and ν are Young's modulus and Poisson's ratio, respectively, and E t =E / (1-υ 2 ); According to the displacement field, strain and stress given above, the kinetic energy T and elastic potential energy U of the thin-walled shell are obtained s as follows: Where ρ, L and h are the density, length and thickness of the shell respectively; the elastic potential energy is further organized into a function expression related to the mid-surface displacement as follows: The specific expressions of f1 and f2 are as follows: In addition, for any boundary situation, artificial springs are used for simulation, and the potential energy of the boundary spring can be recorded as: in and is the boundary spring coefficient; according to equations (12), (14) and (15), the Lagrange function of the system is as follows: L=T-(U s +U b ) (16) And the Euler-Lagrange equation is as follows:

4. The semi-analytical method for analyzing the vibration characteristics of a cylindrical shell with a non-circular cross-section configuration as claimed in claim 1, characterized in that: In step S3, the vibration mode and frequency of the thin-walled non-cylindrical shell are determined as follows: In order to determine the vibration mode of the non-circular cross-section thin shell, the system displacement is first discretized in the form of basis function series expansion, which is as follows: where a k (k=u,v,w) is the time-dependent weight coefficient to be determined, ψ k (k=u,v,w) is a spatial shape function, and q=[a u ; a v ; a w ], Furthermore, the spatial shape function performs the following variable separation: in represents the Kronecker product, function α k (x) and β k (θ)(k=u,v,w) are Gaussian wavelet functions in the axial and circumferential directions, respectively, and their expressions are: where c λ and Represent the scaling and translation parameters respectively, and have: According to formula (21), it is not difficult to find that the physical solution domain is actually divided into (S x +4)×(S θ +4) grids. It should be noted that since the displacement of the cylindrical shell in the circumferential direction is continuous, the following continuity conditions need to be considered: f (i) (x,-π)=f (i) (x,π),f=u,v,w (22) The expression express After considering this continuity condition, the number of linearly independent cyclic basis functions will be reduced by 3; Based on the displacement discretization given above, the corresponding Lagrange equation is further organized as: The specific form of the elements in the mass and stiffness matrices is as follows: According to the above expressions, the corresponding mass matrix and stiffness matrix can be calculated, and then combined with Euler-Lagrange equations, we can get: According to equation (24), the corresponding eigenvalues ​​and eigenvectors are calculated and sorted to obtain the frequency and vibration mode of the non-circular cross-section cylindrical shell.

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