Research Article | Open Access

Volume 2019 |Article ID 4924306 | https://doi.org/10.1155/2019/4924306

Fuzhen Pang, Ruidong Huo, Haichao Li, Cong Gao, Xuhong Miao, Yi Ren, "Wave-Based Method for Free Vibration Analysis of Orthotropic Cylindrical Shells with Arbitrary Boundary Conditions", Mathematical Problems in Engineering, vol. 2019, Article ID 4924306, 15 pages, 2019. https://doi.org/10.1155/2019/4924306

# Wave-Based Method for Free Vibration Analysis of Orthotropic Cylindrical Shells with Arbitrary Boundary Conditions

Revised19 Aug 2019
Accepted04 Sep 2019
Published22 Sep 2019

#### Abstract

The wave-based method (WBM) is a feasible method which investigates the free vibration characteristics of orthotropic cylindrical shells under general boundary conditions. Based on Reissner–Naghid’s shell theory, the governing motion equation is established, and the displacement variables are transformed into wave functions formed to satisfy the governing equations. On the basis of the kinematic relationship between the force resultant and displacement vector, the overall matrix of the shell is established. Comparison studies of this paper with the solutions in the literatures were carried out to validate the accuracy of the present method. Furthermore, by analyzing some numerical examples, the free vibration characteristics of orthogonal anisotropic cylindrical shells under classical boundary conditions, elastic boundary conditions, and their combinations are studied. Also, the effects of the material parameter and geometric constant on the natural frequencies for the orthotropic circular cylindrical shell under general boundary conditions are discussed. The conclusions obtained can be used as data reference for future calculation methods.

#### 1. Introduction

Orthotropic materials have good material properties, and they are very popular in the engineering application field. As a common engineering geometry, cylindrical shell structure has certain applications in petroleum equipment, coal development, and marine equipment. Because of the development of research in recent years, the free vibration analysis of the orthotropic circular cylindrical shells under general boundary conditions has gradually deepened. With the development of cylindrical shell theory in recent decades, more and more theories are put forward and developed. At present, the main shell theory related to free vibration of cylindrical shell mainly includes three types, namely, classical shell theory , first-order shear deformation shell theory , and high-order shear deformation shell theory .

Over the decades, many researchers have put in a lot of time and effort on the vibration analysis of the orthotropic circular cylindrical shells and obtained many excellent research results. Chen et al.  analyzed the free vibration characteristics of the fluid-filled FG orthotropic cylindrical shells, and the boundary condition was set as simply supported. The 3D anisotropic elasticity fundamental equations were used to state equations and some numerical examples are presented. Ding et al.  extended the nonhomogeneous orthotropic elastic solution for the axisymmetric plane strain cylindrical shell dynamic problems, and the orthogonal expansion technique was adopted to derive the time variable and the solutions were obtained. Najafov et al.  conducted the Galerkin method to investigate the vibration and stability characteristics of the FG orthotropic cylindrical shells on elastic foundations. Sofiyev and Kuruoglu  proposed the vibration and buckling of FG orthotropic cylindrical shells in the same way. The Donnell shell theory and Galerkin method were adopted to derive the governing equation. Mallon et al.  proposed the coupled shaker-structure model to study the dynamic stable problem of the harmonically base-excited thin orthotropic cylindrical shell. The numerical observations were qualitatively confirmed by comparing the results with the experimental solutions. Prado et al.  presented the nonlinear vibrations and dynamic instability problem for the orthotropic circular cylindrical shell under simply supported boundary conditions. Donnell’s shell theory and Galerkin method were used to establish the differential equations of the shell. Lakis and Selmane  conducted the thin shell theory, fluid theory, and the hybrid FEM to study the influence of large amplitude vibration of orthotropic, circumferentially nonuniform cylindrical shells. Ahmed  analyzed the isotropic and orthotropic cylindrical shell with variable circumferentially thickness and complex curvature radius by Flugge’s shell theory. The transfer-matrix method and the Longberg integral method were proposed in the establishment of the motion control equation. Liu et al.  proposed the S-DQFEM to investigate the free vibration problem of the orthotropic circular cylindrical shells under classical boundary conditions, and the Donnell–Mushtari shell theory was adopted. Furthermore, the natural frequencies of the closed-form are obtained by the method of variable separation. Sofiyev and Aksogan  extended the Galerkin method to investigate the free vibration characteristics of the nonhomogeneous orthotropic thin cylindrical shells with geometric nonlinearity. Fang et al.  investigated the vibration characteristics of nanosized piezoelectric double-shell structures under simply supported boundary condition by the Goldenveizer–Novozhilov shell theory, thin plate theory, and electroelastic surface theory. Zhu et al.  studied the surface energy effect on the nonlinear free vibration behavior of orthotropic piezoelectric cylindrical nanoshells by the classical shell theory. Also, for the shell and composite structures, some reported literatures have been discussed. Ghassabi et al.  presented the solution strategy based on the state vector technique to study the vibroacoustic performance of carbon nanotube- (CNT-) reinforced composite doubly curved thick shells by three-dimensional theory. Talebitooti et al.  conducted the nondominated sorting genetic algorithm to optimize sound transmission loss of the laminated composite cylindrical shell by FSDST. Ghassabi et al.  investigated the vibroacoustic performance of the doubly curved thick shell by the three-dimensional sound propagation approach and state space solution. Talebitooti et al.  analyzed the acoustic characteristics of the doubly curved composite shell with full simply supported by the third-order shear deformation theory (TSDT). Talebitooti and Zarastvand  investigated the acoustic transmission of the infinitely long doubly curved shell based on HSDT. Talebitooti et al.  discussed the effect of compressing porous material on sound transmission loss of the multilayered cylindrical shell subjected to porous core and air-gap insulation in the presence of external flow based on the three-dimensional elasticity theory. Talebitooti and Zarastvand  investigated the wave propagation on infinite doubly curved laminated composite shell which is used in aerospace structures. Talebitooti et al.  analyzed the acoustic behavior of laminated composite infinitely long doubly curved shallow shells which is the acoustic behavior of laminated composite infinitely long doubly curved shallow shells. Talebitooti et al.  analyzed the acoustic behavior of the laminated composite cylindrical shell which is excited by an oblique plane sound wave by the third-order shear deformation theory.

For the wave-based method, it is an unfamiliar semianalytical method to investigate the dynamic characteristics of the engineering structure in some applications such as cylindrical shell structure , coupled structure , coupled vibroacoustic problem , composite structure , and so on.

#### 2. Theoretical Formulations

##### 2.1. Description of the Shell Model

The model of circular cylindrical shell is described in Figure 1. The circular cylindrical shell is composed of homogeneous and isotropic materials. h represents the cross section of the uniform circular cylindrical shell with thickness. L represents the length of the circular cylindrical shell. The model is described by curvilinear coordinate system (x, θ, and z), in which x and θ denote the axial and circumferential directions of the shell, respectively. The displacements in the direction of x, θ, and z of the middle surface are denoted as u, , and , respectively. For the elastic boundary conditions, there are three pair translational restrained springs (Ku, , and ) and one pair rotational restrained spring (Kθ) are set at two ends to simulate the arbitrary elastic boundary conditions.

##### 2.2. Kinematic Relations and Stress Resultants

According to Reissner–Naghid’s shell theory , the relationship between the strain resultant and curvature change resultant in the middle surface of the shell is shown as follows:

Furthermore, the strains of arbitrary point in the shell are given as follows:

Also, the stress-strain relationships of the orthotropic shell are obtained by Hooke’s law :where are the transform coefficients which are defined as follows:where E1 and E2 are the Young’s modulus, μ12 and μ21 are the Poisson’s ratios, and G12 is the shear modulus. The relationship between μ12 and μ21 is determined as μ21E1 = μ12E2. For the isotropic cylindrical shell, the relationship is defined as E1 = E2 and . To obtain the force vector and bending moment resultant of the orthotropic shell, the relationship between the force resultant, bending moment, and strain vector is given as follows:

So, the expression for the resultant forces and moments is as follows:where Aij and are the coefficients and are defined as follows:

##### 2.3. Governing Equations

Based on Reissner–Naghid’s theory, the governing equation of the orthotropic cylindrical shell in terms of the force resultant, moment vector, and displacement variables are given as follows :where ρ is the destiny of the orthotropic circular cylindrical shell. Submitting the expression of force and moment resultant into equation (8), the governing equation of the orthotropic cylindrical shell is given in the matrix form as follows:where are the transform coefficients, and the expression of them are given as follows:

##### 2.4. Implementation of the WBM

Using the WBM to analyze the free vibration characteristics of the orthotropic cylindrical shell under general boundary conditions, the displacement variables should be transformed into the wave functions form as follows :where U0, V0, and W0 are the displacement amplitude variables, kn is the axial wave number, t is the time variable, n is the circumferential number, and ω is the circular frequency. So, submitting equation (11) into equation (9), the governing equation of the orthotropic cylindrical shell in the wave function form is obtained as follows:where are the coefficients of equation (12) in the wave functions form, and the expressions are given as follows:

To make sure equation (13) has nontrivial solutions, the determinant of the coefficient matrix in equation (12) should be equal to zero. Also, the eighth-order equation of the axial wave number kn is obtained as follows:where are the coefficients of the eighth-order equation, and the solutions of equation (14) are calculated by Reference  as follows:where can be in real, imaginary, and combination form. Furthermore, the displacement variables in the wave function forms are converted as follows:where λn,i and κn,i are the wave contribution factors which are expressed as follows:

Particularly, the expressions for displacement variables are transformed as follows:where

On the basis of the relationship between the displacement variables and force and moment resultants, the detailed expression of the force resultant and moment vector is shown as follows:where

For the classical boundary conditions, four type of boundary conditions are considered which are widely used in some engineering applications, as follows :Free (F):Clamped (C):Simply supported (SS):Shear diaphragm (SD):

For the elastic boundary conditions, the relationship between the force, displacement, and elastic restrained stiffness is discussed. When the elastic restrained stiffness is in the axial direction, the boundary relationship is given as follows:where Ku is the stiffness constant, and the symbol ± means the elastic restrained spring at the boundary edge x = L and x = 0. Also, for the circumferential displacement, the normal displacement, and the transverse normal rotations of the θ and x axis, the symbol ± has the same meaning for the elastic boundary conditions. For the other elastic restrained states, the boundary relationships can refer to Reference . Related to the introduction of the elastic and classical boundary conditions, the final equation of the orthotropic cylindrical shell is given as follows:where K is the overall matrix, F is the external force resultant, and W = {W1; W2} is the wave contribution factor vector. W1 and W2 are the wave contribution factor vectors which are related to the boundary conditions at the two ends. The overall matrix K is shown as follows:where D1 and D2 are the boundary matrixes which depend on the boundary conditions and B1 and B2 are the segment matrixes which are given as follows:where and are the displacement and force matrix of the orthotropic shell and are expressed as follows:

When analyzing the free vibration characteristics of the orthotropic cylindrical shell under general boundary conditions, the external force vector F should be vanished. For each circumferential number n, searching the zero position of the overall matrix K, a series of values are calculated. When the sign occurs, the natural frequencies of the cylindrical shell are obtained.

The dichotomy method is a mathematical idea that uses flat partitions and infinite approximations. It is an effective algorithm for avoiding complex calculations and approximations of analytic one-dimensional functions. In this paper, the dichotomy method is adopted to get the natural frequencies of the cylindrical shell, and some numerical examples are established.

#### 3. Numerical Examples and Discussion

In this part, some numerical examples are presented to investigate the free vibration characteristics of the orthotropic cylindrical shell under general boundary conditions (i.e., classical, elastic, and combinations). Some numerical examples are extended to verify the correctness of the results by the presented method. Furthermore, the effects of the material parameter and geometric constants on the free vibration characteristics of the orthotropic circular cylindrical shell are studied.

##### 3.1. Orthotropic Cylindrical Shell with Classical and Elastic Boundary Conditions

First, the comparison of the natural frequencies for the homogeneous cylindrical shell under clamped-clamped boundary condition is presented. The results by the present method are compared with the solutions by Ref.  in Table 1. The geometric parameters and material constants are defined as follows: L = 511.2 mm, R = 216.2 mm, h = 1.5 mm, E = 1.83 × 1011 N/m2, μ = 0.3, and ρ = 7492 kg/m3. The range of the circumferential mode number n and longitudinal wave number m is set as 2–6 and 1–3. It can be seen that the results of the two methods are in good agreement, and the maximum error is 5.79% which appears with n = 3 and m = 1. Next, this paper discusses the free vibration characteristics of the orthotropic cylindrical shell under classical boundary conditions. To verify the correctness of the results by the presented method, the natural frequencies of the orthotropic cylindrical shell under several classical boundary conditions are compared with the solutions in the reported literature by Zhao et al. . There are three pair classical boundary conditions which are set as SS-SS, SD-SD, and C-C. The material parameters are given as follows: E1 = 120 GPa, E2 = 10 GPa, G12 = 5.5 GPa, μ12 = 0.27, and ρ = 1700 kg/m3. The geometric constants are set as follows: L = 5 m, h = 0.01 m, and R = 1 m. In Table 2, the natural frequencies of the first two longitudinal modes for the first six circumferential numbers are calculated. By comparing the MRRM in the reference literature and the results by the presented method, we can discover that the errors are small. It is obvious that the free vibration characteristics for the orthotropic circular cylindrical shell under classical boundary conditions calculated by the presented method are correct. From Table 1, we can find that for each circumferential number and longitudinal mode, the minimum natural frequency is associated with the boundary condition SD-SD and the maximum natural frequency is related to the boundary condition C-C. It can be concluded that the classical boundary conditions have a significant effect on the free vibration characteristics for the orthotropic circular cylindrical shell.

 n m = 1 m = 2 m = 3 Present Reference  Error (%) Present Reference  Error (%) Present Reference  Error (%) 2 1281.90 1299 1.32 2593.52 2660 2.50 4245.89 4364 2.71 3 2110.36 2240 5.79 2555.82 2574 0.71 3389.93 3483 2.67 4 3830.74 3872 1.07 4082.66 4043 −0.98 4473.30 4417 −1.27 5 6240.10 6262 0.35 6383.64 6491 1.65 6645.01 6728 1.23 6 9087.86 9125 0.41 9129.98 9276 1.57 9259.20 9540 2.94
 Boundary conditions Method n = 1 n = 2 n = 3 n = 4 n = 5 n = 1 n = 2 n = 3 n = 4 n = 5 m = 1 m = 2 C-C Present 122.7511 76.4642 52.8701 41.4866 39.4239 225.6351 146.3284 103.4431 79.3989 67.0333 Reference  122.75 76.464 52.869 41.486 39.423 225.63 146.33 103.44 79.398 67.033 Error (%) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 SS-SS Present 122.5910 76.1294 52.4352 41.0422 39.0609 225.2716 145.6195 102.4194 78.1700 65.7589 Reference  122.59 76.129 52.435 41.042 39.061 225.27 145.62 102.42 78.17 65.759 Error (%) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 SD-SD Present 117.9855 66.1809 40.8206 30.8837 32.3243 225.2414 144.7009 99.8179 74.0717 61.0129 Reference  117.99 66.181 40.821 30.884 32.324 225.24 144.7 99.818 74.072 61.01 Error (%) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Next, the free vibration characteristics of the orthotropic circular cylindrical shell under classical combination boundary conditions are concerned. The material parameters and geometric constants are equal to the numerical example in Table 2. For the classical combination boundary conditions, there are two type boundary conditions, C-F and SS-F, which are considered, and the results by the presented method are compared with the solutions by the MRRM method . From Table 2, the errors between the two numerical methods are small, and the maximum error is 0.95%. So, it can be concluded that the free vibration characteristics which are investigated by the presented method are right. From the two numerical examples in Tables 2 and 3, the free vibration characteristics for orthotropic circular cylindrical shell under classical boundary conditions and their combinations are discussed. Figure 2 shows some modes of orthotropic cylindrical shells under C-C boundary conditions. The purpose is to further study the free vibration characteristics of orthotropic cylindrical shells under classical boundary conditions.

 Boundary conditions Method n = 1 n = 2 n = 3 n = 4 n = 5 n = 1 n = 2 n = 3 n = 4 n = 5 m = 1 m = 2 C-F Present 57.2692 29.4248 18.4766 19.1837 27.1424 166.7386 96.3363 62.8738 46.8067 42.1131 Reference  57.272 29.445 18.566 19.365 27.376 166.74 96.347 62.922 46.952 42.414 Error (%) 0.00 −0.07 −0.48 −0.94 −0.86 0.00 −0.01 −0.08 −0.31 −0.71 SS-F Present 57.2303 29.3770 18.4436 19.1688 27.1366 166.6339 96.0876 62.5356 46.4639 41.8393 Reference  57.233 29.397 18.532 19.35 27.369 166.64 96.098 62.584 46.609 42.137 Error (%) 0.00 −0.07 −0.48 −0.95 −0.86 0.00 −0.01 −0.08 −0.31 −0.71

Next, the free vibration characteristics of the orthotropic circular cylindrical shell under elastic boundary conditions and their combinations are concerned. In this paper, there are three type elastic-restrained situations which are considered as follows:

First-elastic boundary condition (EBC_1): the axial displacement is under elastic restrained and others are fixed (u ≠ 0, ), and the elastic stiffness value is set as Ku = 109.

Second-elastic boundary condition (EBC_2): the circumferential displacement is under elastic restrained and others are fixed ( ≠ 0, u =  = ∂/∂x = 0), and the elastic stiffness value is set as  = 109.

Third-elastic boundary condition (EBC_3): the radial displacement is under elastic restrained and others are fixed ( ≠ 0, u =  = ∂/∂x = 0), and the elastic stiffness value is set as  = Kθ = 107.

In Table 4, there are six type elastic boundary conditions and their combinations (i.e., EBC1-EBC1, EBC2-EBC2, EBC3-EBC3, EBC1-EBC3, EBC2-EBC3, and EBC1-EBC2) are concerned. The first five circumferential numbers (i.e., n = 1–5) and the first four longitudinal modes (i.e., m = 1–4), natural frequencies for the orthotropic circular cylindrical shells under several elastic boundary conditions, are calculated. The material parameters and geometric constants are the same as the numerical example in previous discussion. For various elastic boundary conditions, the natural frequencies are relatively stable within a range and have a small variation range for various circumferential numbers and longitudinal modes. Also, some mode shapes of the orthotropic circular cylindrical shell with EBC1-EBC1 are shown in the Figure 3.

 n m Boundary conditions EBC1-EBC1 EBC2-EBC2 EBC3-EBC3 EBC1-EBC3 EBC2-EBC3 EBC1-EBC2 1 1 121.211 122.513 122.741 121.859 122.627 120.6801 2 225.634 225.269 225.618 225.626 225.444 223.8509 3 295.019 294.726 294.998 295.008 294.862 293.584 4 332.953 332.765 332.924 332.939 332.845 332.059 2 1 72.788 76.340 76.442 74.491 76.391 73.89489 2 146.075 146.074 146.289 146.194 146.182 145.0127 3 208.553 208.389 208.619 208.581 208.504 207.2762 4 255.979 255.727 255.920 255.950 255.823 254.7795 3 1 48.121 52.807 52.844 50.435 52.826 50.16391 2 102.408 103.281 103.387 102.929 103.334 102.2252 3 154.456 154.656 154.785 154.613 154.720 153.7046 4 199.167 199.040 199.161 199.166 199.100 198.2201 4 1 36.929 41.458 41.463 39.174 41.460 39.06892 2 77.503 79.304 79.336 78.464 79.320 78.08885 3 120.575 121.249 121.294 120.930 121.271 120.3684 4 160.463 160.585 160.623 160.546 160.604 159.9009 5 1 36.110 39.413 39.407 37.717 39.410 37.6851 2 64.670 66.982 66.973 65.868 66.978 65.69338 3 99.855 100.974 100.959 100.409 100.967 100.0973 4 134.854 135.270 135.241 135.051 135.256 134.6571
##### 3.2. Effect of the Material and Geometric Parameters on the Natural Frequencies

In addition, the effect of the material parameter and geometric constants on the natural frequencies for the orthotropic circular cylindrical shell under several boundary conditions (i.e., C-C, S-F, EBC_1-EBC_2, and EBC_3-EBC_3) is discussed.

First, the effect of the modulus ratio E1/E2 for the orthotropic circular cylindrical shell under various boundary conditions is discussed. The material parameter and geometric constants are similar to the numerical example in the previous study. The changing rule of the modulus ratio E1/E2 is from 1 to 12, the natural frequencies for the first two circumferential number (i.e., n = 1 and 2) and the first four longitudinal mode (i.e., m = 1–4) under various boundary conditions are calculated in Table 5. For analyzing the effect of the modulus ratio E1/E2, the changing rule of the natural frequencies under various boundary conditions is shown in the Figure 4 (n = 1). It is obvious that with the changing of the modulus ratio E1/E2, the natural frequencies are generally growing for various longitudinal modes under various boundary conditions. Also, for the boundary condition S-F, the change range of the natural frequencies is more obvious for various longitudinal modes. For other boundary conditions, the change range for the first two longitudinal modes (i.e., m = 1 and 2) are evident, and for the longitudinal mode m = 3, the change range stays in the basic stable range.

 E1/E2 Boundary conditions m C-C SS-F EBC1-EBC2 EBC3-EBC3 n = 1 n = 2 n = 1 n = 2 n = 1 n = 2 n = 1 n = 2 1 1 110.6992 55.2536 31.2626 11.6005 108.9950 54.0279 110.6879 55.2403 2 211.2475 120.7196 119.1539 58.5856 209.0331 118.8745 211.2434 120.7013 3 292.7848 188.8046 234.6133 132.9303 291.0075 186.7130 292.7825 188.7881 3 1 119.3153 68.6970 45.3992 18.7554 117.2422 66.4857 119.3085 68.6849 2 221.3334 136.5006 144.2771 78.3209 219.3547 134.6394 221.3285 136.4841 3 294.0729 201.8201 253.0665 157.9665 292.4997 200.0412 294.0666 201.8009 5 1 121.1147 72.5290 50.8336 22.7042 118.9951 70.0528 121.1077 72.5153 2 223.5331 140.9972 154.0669 85.6214 221.6405 139.3435 223.5256 140.9765 3 294.4017 205.0738 257.5586 165.3520 292.8816 203.4539 294.3914 205.0469 8 1 122.1522 74.9725 54.7245 26.3878 120.0403 72.3926 122.1439 74.9555 2 224.8418 144.1636 161.5715 91.4974 223.0119 142.7059 224.8302 144.1353 3 294.7082 207.2162 260.3054 170.3087 293.2317 205.7223 294.6917 207.1771 10 1 122.5081 75.8520 56.1950 28.0675 120.4149 73.2671 122.4986 75.8326 2 225.3073 145.4115 164.5223 94.0730 223.5032 144.0357 225.2928 145.3778 3 294.8730 208.0499 261.2583 172.1662 293.4169 206.6100 294.8524 208.0026 12 1 122.7511 76.4642 57.2303 29.3770 120.6801 73.8949 122.7406 76.4423 2 225.6351 146.3284 166.6339 96.0876 223.8509 145.0127 225.6179 146.2894 3 295.0230 208.6741 261.9059 173.4884 293.5837 207.2762 294.9984 208.6188

Next, the effect of the geometric constants on the natural frequencies for the orthotropic circular cylindrical shell under several boundary conditions is discussed. In this paper, the geometric constants are set as thickness to radius ratio h/R and length to radius ratio L/R. For the boundary conditions, geometric and material parameters are same as the numerical example in the previous study for the effect of the modulus ratio E1/E2. In Table 6, the natural frequencies for the orthotropic circular cylindrical shell with the changing of the thickness to radius ratios h/R are calculated. The changing range of the thickness to radius ratios h/R is set as 0.001 to 0.1. The natural frequencies are generally increased for various circumferential number and longitudinal modes for boundary condition C-C, SS-F, and ECB_3-ECB_3. Furthermore, the natural frequencies are first decreased and then increased with the growing of the thickness to radius ratios h/R under EBC_1-EBC_2. In order to reflect the law of natural frequency change more intuitively, the changing rule of the natural frequencies with the increase of the thickness to radius ratios h/R under several boundary conditions is shown in Figure 5. Especially, when the thickness to radius ratios h/R are set from 0.05 to 0.1 and related to the longitudinal wave number m = 3, the growth rates of the natural frequencies are more obvious than the longitudinal wave number m = 1 and 2. Also, for the boundary condition S-F and EBC3-EBC-3, when the longitudinal wave number m = 1 and 2, the natural frequencies have a small range of variation and is basically kept within a certain range. In particular, when the boundary condition is set to EBC1-EBC1 and longitudinal wave number m = 1, as the thickness to radius ratios h/R increases, the natural frequencies tend to decrease, but the variation is small.

 h Boundary conditions m C-C SS-F EBC1-EBC2 EBC3-EBC3 n = 1 n = 2 n = 1 n = 2 n = 1 n = 2 n = 1 n = 2 0.001 1 122.5913 76.0481 57.2287 29.2184 122.3553 75.6950 122.5913 76.0480 2 225.2229 145.4536 166.6168 95.9782 225.0297 145.2889 225.2229 145.4535 3 294.0861 207.0796 261.7840 173.2139 293.9218 206.8985 294.0861 207.0795 0.005 1 122.6467 76.1854 57.2295 29.2576 121.5370 74.6696 122.6451 76.1818 2 225.3556 145.7348 166.6211 96.0050 224.4220 144.9893 225.3533 145.7292 3 294.3756 207.5760 261.8143 173.2815 293.6008 206.7692 294.3724 207.5683 0.01 1 122.7511 76.4642 57.2303 29.3770 120.6801 73.8949 122.7406 76.4423 2 225.6351 146.3284 166.6339 96.0876 223.8509 145.0127 225.6179 146.2894 3 295.0230 208.6741 261.9059 173.4884 293.5837 207.2762 294.9984 208.6188 0.05 1 124.3256 81.4447 57.2392 32.9221 117.5104 76.1999 123.8317 80.5904 2 230.6216 157.0350 167.0094 98.6036 224.4088 154.2499 229.4403 154.9489 3 308.5174 230.8005 264.5660 179.5628 303.9226 227.8976 305.9156 226.6462 0.1 1 127.4686 92.6207 57.2553 42.0016 118.1959 87.4795 125.6479 89.6293 2 242.0943 181.4167 168.0745 105.8049 233.3506 178.7213 236.1977 171.4342 3 343.1045 283.8211 271.9447 196.1988 335.7079 280.6610 322.5592 255.6987

Furthermore, the influence of the length to radius ratios L/R on the natural frequencies for the orthotropic circular cylindrical shell under several boundary conditions is calculated in Table 7. The material constants and geometric properties are the same as the numerical study in previous part, and the changing range of the length to radius ratios L/R are set from 5 to 12. In Figure 6, the changing rule of the natural frequencies with respect to the length to radius ratios L/R under various boundary conditions is shown. It can be seen that with the changing of the length to radius ratios L/R, the natural frequencies are generally decreased for various boundary conditions related to three longitudinal wave number m = 1–3, at the same time, and as the length to radius ratios L/R changes, the attenuation effect of the frequency parameters is more obvious. When the length to radius ratios L/R changes from 5 to 8, the decreased attenuations are larger for four type boundary conditions.

 L Boundary conditions m C-C SS-F EBC1-EBC2 EBC3-EBC3 n = 1 n = 2 n = 1 n = 2 n = 1 n = 2 n = 1 n = 2 5 1 122.7511 76.4642 57.2303 29.3770 120.6801 73.8949 122.7406 76.4423 2 225.6351 146.3284 166.6339 96.0876 223.8509 145.0127 225.6179 146.2894 3 295.0230 208.6741 261.9059 173.4884 293.5837 207.2762 294.9984 208.6188 8 1 76.7055 45.4524 31.6944 14.1556 75.2519 43.4839 76.7009 45.4440 2 148.5210 90.2844 99.4478 53.6849 147.5756 89.3387 148.5141 90.2678 3 214.2263 137.1869 178.4795 107.3297 213.0963 136.2748 214.2159 137.1633 10 1 60.6792 34.7071 23.1038 9.8312 59.4140 33.0345 60.6761 34.7020 2 118.8628 70.3232 76.2998 39.8337 118.1806 69.3907 118.8582 70.3123 3 176.4915 109.1948 143.5250 82.7372 175.6285 108.4187 176.4847 109.1790 12 1 49.7780 27.4582 17.5069 7.3573 48.6440 26.0412 49.7758 27.4549 2 98.2672 56.7730 61.0112 30.8062 97.7181 55.8310 98.2638 56.7654 3 148.6260 89.5260 118.5957 65.9236 147.9499 88.8051 148.6211 89.5147

#### 4. Conclusions

In this paper, a semianalytical method is conducted to investigate the free vibration characteristics of the orthotropic cylindrical shell under general boundary conditions, including the classical boundary conditions, elastic boundary conditions, and their combinations. Reissner–Naghid’s shell theory is utilized to obtain the governing motion equations and the displacement variables are transformed into wave function forms to accurate the motion relationship. According to the motion relationship and boundary conditions, the final equation of the orthotropic circular cylindrical shell is established. Then the natural frequencies can be obtained by solving the zero position of the overall matrix determinant based on the dichotomy method. In the numerical examples and discussion parts, the effect of the material parameter and geometric constants on the free vibration characteristics of the orthotropic circular cylindrical shell is studied and some conclusions are obtained.

To verify the correctness of the calculation results by the presented method, some numerical examples are proposed by comparing with the solutions in the reported literatures. The effect of the material parameter modulus ratio E1/E2, geometric constants thickness to radius ratio L/R, and thickness to radius ratio h/R on the free vibration characteristics of the orthotropic cylindrical shell under several boundary conditions are discussed. For the effect of different parameters, various properties have its one influence impact on the free vibration characteristics of the orthotropic circular cylindrical shell. The changing ranges of the natural frequencies are different for various circumferential number and longitudinal modes under several boundary conditions.

This paper proposed a new numerical method to research the free vibration characteristics of the orthotropic circular cylindrical shell under general boundary conditions, and it provided a theoretical basis for the development of subsequent numerical studies.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This study was funded by the National Natural Science Foundation of China (nos. 51209052 and 51679053), National key Research and Development Program (2016YFC0303406), Fundamental Research Funds for the Central Universities (HEUCFD1515 and HEUCFM170113), Assembly Advanced Research Fund of China (6140210020105), Naval preresearch project, China Postdoctoral Science Foundation (2014M552661), and Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (HEUGIP201801).

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