«上一篇
 文章快速检索 高级检索

 哈尔滨工程大学学报  2019, Vol. 40 Issue (7): 1238-1244  DOI: 10.11990/jheu.201805099 0

### 引用本文

PANG Fuzhen, LI Haichao, PENG Dewei, et al. Transient vibration characteristics analysis of cylindrical shell structure with complex boundary conditions[J]. Journal of Harbin Engineering University, 2019, 40(7), 1238-1244. DOI: 10.11990/jheu.201805099.

### 文章历史

Transient vibration characteristics analysis of cylindrical shell structure with complex boundary conditions
PANG Fuzhen , LI Haichao , PENG Dewei , LI Yuhui , TIAN Hongye
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: Considering the deficiency in the research of the transient vibration characteristics of cylindrical shells under complex boundary conditions, the paper proposes a semi-analytical method to analyze the transient vibration characteristics of cylindrical shells. Based on the domain energy decomposition method and Reissner-Naghdi's linear thin shell theory, the cylindrical shell is decomposed into several shell sections along the generatrix direction, and the solution model for the general boundary condition of the shell is deduced. The displacement functions of the shell are represented by Chebyshev determinant and the Fourier series, and the least squares residual is introduced to eliminate the computational instability. The transient vibration characteristics of the cylindrical shell structure are analyzed and compared with the finite element simulation result, verifying the effectiveness of the method. The results show that in the same frequency range, the smaller the thickness of the structure, the greater the peak values of the displacement curve and the greater the transient response. The loss factor mainly affects the resonance peak value of the structure but has little effect on the natural frequency of the structure.
Keywords: cylindrical shell    domain decomposition method    displacement function    least squares residual    transient vibration    complex boundary conditions    thin shell theory    Chebyshev determinant

1 柱壳结构区域能量分析模型建立

 Download: 图 1 圆柱壳结构理论模型 Fig. 1 Theoretical model of cylindrical shell structure
1.1 柱壳结构的能量泛函建立

 ${\mathit{\Pi }_{{\rm{Tol}}}} = \int_{{t_0}}^{{t_1}} {\sum\limits_{i = 1}^{{N_L}} {\left( {{T_{L,i}} - {U_{L,i}} + {W_{L,i}}} \right){\rm{d}}t} } + \int_{{t_0}}^{{t_1}} {\sum\limits_{i,i + 1} {{\mathit{\Pi }_{K,L}}{\rm{d}}t} }$ (1)

 $T_{L, i}=\frac{1}{2} \iint_{S_{i}} \rho h_{i}\left[\left(\frac{\partial \boldsymbol{u}_{i}}{\partial t}\right)^{2}+\left(\frac{\partial \boldsymbol{v}_{i}}{\partial t}\right)^{2}+\left(\frac{\partial \boldsymbol{w}_{i}}{\partial t}\right)^{2}\right] R \mathrm{d} \boldsymbol{x} \mathrm{d} \boldsymbol{\theta}$ (2)

 $\begin{gathered} {U_{L,i}} = \frac{1}{2}\iiint {\left( {{\mathit{\boldsymbol{\sigma }}_1} \times {\mathit{\boldsymbol{\varepsilon }}_1} + {\mathit{\boldsymbol{\sigma }}_2} \times {\mathit{\boldsymbol{\varepsilon }}_2} + \mathit{\boldsymbol{\tau }} \times {\mathit{\boldsymbol{\varepsilon }}_3}} \right){\text{d}}V} = \hfill \\ \;\;\;\;\;\;\;\;\;\frac{1}{2}\iint_{{S_i}} {\left[ {\int_{ - \frac{h}{2}}^{\frac{h}{2}} {\left( {\frac{E}{{1 - {\mu ^2}}}\left[ {\left( {{\mathit{\boldsymbol{\varepsilon }}_1} + \mu {\mathit{\boldsymbol{\varepsilon }}_2}} \right) + } \right.} \right.} } \right.} \hfill \\ \;\;\;\;\;\;\;\;\;z\left( {{\mathit{\boldsymbol{\chi }}_1} + \mu {\mathit{\boldsymbol{\chi }}_2}} \right)] \times \left( {{\mathit{\boldsymbol{\varepsilon }}_1} + z{\mathit{\boldsymbol{\chi }}_1}} \right) + \hfill \\ \;\;\;\;\;\;\;\;\;\frac{E}{{1 - {\mu ^2}}}\left[ {\left( {{\mathit{\boldsymbol{\varepsilon }}_2} + \mu {\mathit{\boldsymbol{\varepsilon }}_1}} \right) + z\left( {{\mathit{\boldsymbol{\chi }}_2} + \mu {\mathit{\boldsymbol{\chi }}_1}} \right)} \right] \times \hfill \\ \;\;\;\;\;\;\;\;\;\left( {{\mathit{\boldsymbol{\varepsilon }}_2} + z{\mathit{\boldsymbol{\chi }}_2}} \right) + \frac{E}{{2(1 + \mu )}}\left[ {{\mathit{\boldsymbol{\varepsilon }}_3} + z\left( {{\mathit{\boldsymbol{\chi }}_3} + {\mathit{\boldsymbol{\chi }}_4}} \right)} \right] \times \hfill \\ \;\;\;\;\;\;\;\;\;\left[ {{\mathit{\boldsymbol{\varepsilon }}_3} + z\left( {{\mathit{\boldsymbol{\chi }}_3} + {\mathit{\boldsymbol{\chi }}_4}} \right)} \right]){\text{d}}z]R{\text{d}}\mathit{\boldsymbol{x}}{\text{d}}\mathit{\boldsymbol{\theta }} \hfill \\ \end{gathered}$ (3)

 $\begin{gathered} {U_{L,i}} = \frac{1}{2}\iint_{{S_i}} {{K_i}}\left[ {{{\left( {\frac{{\partial {\mathit{\boldsymbol{u}}_i}}}{{\partial \mathit{\boldsymbol{x}}}}} \right)}^2} + \frac{{2\mu }}{R}\frac{{\partial {\mathit{\boldsymbol{u}}_i}}}{{\partial \mathit{\boldsymbol{x}}}}\left( {\frac{{\partial {\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{\theta }}}} + {\mathit{\boldsymbol{w}}_i}} \right) + } \right. \hfill \\ \;\;\;\;\;\;\;\;\;\left. {\frac{1}{{{R^2}}}{{\left( {\frac{{\partial {\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{\theta }}}} + {\mathit{\boldsymbol{w}}_i}} \right)}^2} + \frac{{1 - \mu }}{2}{{\left( {\frac{{\partial {\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{x}}}} + \frac{1}{R}\frac{{\partial {\mathit{\boldsymbol{u}}_i}}}{{\partial \mathit{\boldsymbol{\theta }}}}} \right)}^2}} \right]R{\text{d}}\mathit{\boldsymbol{x}}{\text{d}}\mathit{\boldsymbol{\theta }} + \hfill \\ \;\;\;\;\;\;\;\;\;\frac{1}{2}\iint_{{S_i}} {{D_i}}\left[ {{{\left( {\frac{{{\partial ^2}{\mathit{\boldsymbol{w}}_i}}}{{\partial {\mathit{\boldsymbol{x}}^2}}}} \right)}^2} + \frac{{2\mu }}{{{R^2}}}\frac{{{\partial ^2}{\mathit{\boldsymbol{w}}_i}}}{{\partial {\mathit{\boldsymbol{x}}^2}}}\left( {\frac{{\partial {\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{\theta }}}} + \frac{{{\partial ^2}{\mathit{\boldsymbol{w}}_i}}}{{\partial {\mathit{\boldsymbol{\theta }}^2}}}} \right) + } \right. \hfill \\ \;\;\;\;\;\;\;\;\;\left. {\frac{1}{{{R^4}}}{{\left( {\frac{{\partial {\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{\theta }}}} - \frac{{{\partial ^2}{\mathit{\boldsymbol{w}}_i}}}{{\partial {\mathit{\boldsymbol{\theta }}^2}}}} \right)}^2} + \frac{{1 - \mu }}{{2{R^2}}}{{\left( {\frac{{\partial {\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{x}}}} - 2\frac{{{\partial ^2}{\mathit{\boldsymbol{w}}_i}}}{{\partial \mathit{\boldsymbol{x}}\partial \mathit{\boldsymbol{\theta }}}}} \right)}^2}} \right] \cdot \hfill \\ \;\;\;\;\;\;\;\;\;R{\text{d}}\mathit{\boldsymbol{x}}{\text{d}}\mathit{\boldsymbol{\theta }} \hfill \\ \end{gathered}$ (4)

 ${W_{L,i}} = \frac{1}{2}\iint_{{S_i}} {\left( {{\mathit{\boldsymbol{f}}_{u,i}}{\mathit{\boldsymbol{u}}_i} + {\mathit{\boldsymbol{f}}_{v,i}}{\mathit{\boldsymbol{v}}_i} + {\mathit{\boldsymbol{f}}_{w,i}}{\mathit{\boldsymbol{w}}_i}} \right)}R{\text{d}}\mathit{\boldsymbol{x}}{\text{d}}\mathit{\boldsymbol{\theta }}$ (5)

 ${\mathit{\Pi }_{\lambda ,L}} = \int_l {\left( {\lambda {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_u} + \beta {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_v} + \vartheta {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_w} + \psi {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_r}} \right)} R{\text{d}}\mathit{\boldsymbol{\theta }}$ (6)

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_u} = {\mathit{\boldsymbol{u}}_i} - {\mathit{\boldsymbol{u}}_{i + 1}},{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_v} = {\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_{i + 1}},\\ {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_w} = {\mathit{\boldsymbol{w}}_i} - {\mathit{\boldsymbol{w}}_{i + 1}},{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_r} = \frac{{\partial {\mathit{\boldsymbol{w}}_i}}}{{\partial \mathit{\boldsymbol{x}}}} - \frac{{\partial {\mathit{\boldsymbol{w}}_{i + 1}}}}{{\partial \mathit{\boldsymbol{x}}}} \end{array} \right.$ (7)

 $\lambda = {K_i}\left[ {\frac{{\partial {\mathit{\boldsymbol{u}}_i}}}{{\partial \mathit{\boldsymbol{x}}}} + \frac{\mu }{R}\left( {\frac{{\partial {\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{\theta }}}} + {\mathit{\boldsymbol{w}}_i}} \right)} \right]$ (8)
 $\begin{array}{l} \beta = \frac{{\left( {1 - \mu } \right){K_i}}}{2}\left( {\frac{1}{R}\frac{{\partial {\mathit{\boldsymbol{u}}_i}}}{{\partial \mathit{\boldsymbol{\theta }}}} + \frac{{\partial {\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{x}}}}} \right) + \\ \;\;\;\;\;\;\;\;\frac{{\left( {1 - \mu } \right){D_i}}}{2}\left( {\frac{{\partial {\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{x}}}} - 2\frac{{{\partial ^2}{\mathit{\boldsymbol{w}}_i}}}{{\partial \mathit{\boldsymbol{x}}\partial \mathit{\boldsymbol{\theta }}}}} \right) \end{array}$ (9)
 $\vartheta = {D_i}\left( {\frac{1}{{{R^2}}}\frac{{{\partial ^2}{\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{x}}\partial \mathit{\boldsymbol{\theta }}}} - \frac{{{\partial ^3}{\mathit{\boldsymbol{w}}_i}}}{{\partial {\mathit{\boldsymbol{x}}^3}}} - \frac{{2 - \mu }}{{{R^2}}}\frac{{{\partial ^3}{\mathit{\boldsymbol{w}}_i}}}{{\partial \mathit{\boldsymbol{x}}\partial {\mathit{\boldsymbol{\theta }}^2}}}} \right)$ (10)
 $\psi = {D_i}\left[ {\frac{{{\partial ^2}{\mathit{\boldsymbol{w}}_i}}}{{\partial {\mathit{\boldsymbol{x}}^2}}} - \frac{\mu }{{{R^2}}}\left( {\frac{{\partial {\mathit{\boldsymbol{v}}_i}}}{{\partial \mathit{\boldsymbol{\theta }}}} - \frac{{{\partial ^2}{\mathit{\boldsymbol{w}}_i}}}{{\partial {\mathit{\boldsymbol{\theta }}^2}}}} \right)} \right]$ (11)

 $\begin{array}{l} {{\mathit{\bar \Pi }}_{{\rm{Tol}}}} = \int_{{t_0}}^{{t_1}} {\sum\limits_{i = 1}^{{N_L}} {\left( {{T_{L,i}} - {U_{L,i}} + {W_{L,i}}} \right){\rm{d}}t} } + \\ \;\;\;\;\;\;\;\;\;\;\int_{{t_0}}^{{t_1}} {\sum\limits_{i,i + 1} {\int_l {\left( {{N_x}{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_u} + {N_\theta }{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_v} + {Q_x}{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_w} - {M_x}{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_r}} \right)R{\rm{d}}\mathit{\boldsymbol{\theta }}{\rm{d}}t} } } \end{array}$ (12)

 ${\mathit{\tilde \Pi }_{{\rm{Tol}}}} = {\mathit{\bar \Pi }_{{\rm{Tol}}}} + {\mathit{\Pi }_{\kappa ,L}}$ (13)
 $\begin{array}{*{20}{c}} {{\mathit{\Pi }_{\kappa ,L}} = - \frac{1}{2}\sum\limits_{i,i + 1} {\int_l {\left( {{\kappa_u}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_u^2 + {\kappa_v}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_v^2 + } \right.} } }\\ {\left. {{\kappa_w}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_w^2 + {\kappa_r}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_r^2} \right)R{\rm{d}}\mathit{\boldsymbol{\theta }}} \end{array}$ (14)

 $\begin{array}{l} {{\mathit{\tilde \Pi }}_{{\rm{Tol}}}} = \int_{{t_0}}^{{t_1}} {\sum\limits_{i = 1}^{{N_L}} {\left( {{T_{L,i}} - {U_{L,i}} + {W_{L,i}}} \right){\rm{d}}t} } + \\ \;\;\;\;\;\;\;\;\;\;\int_{{t_0}}^{{t_1}} {\sum\limits_{i,i + 1} {\int_l {\left( {{\zeta _u}{N_x}{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_u} + {\zeta _v}{N_\theta }{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_v} + {\zeta _w}{\theta _x}{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_w} - } \right.} } } \\ \;\;\;\;\;\;\;\;\;\;\left. {{\zeta _r}{M_x}{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_r}} \right)R{\rm{d}}\mathit{\boldsymbol{\theta }} - \frac{1}{2}\sum\limits_{i,i + 1} {\int_l {\left( {{\zeta _u}{\kappa _u}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_u^2 + {\zeta _v}{\kappa _v}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_v^2 + } \right.} } \\ \;\;\;\;\;\;\;\;\;\;\left. {{\zeta _w}{\kappa _w}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_w^2 + {\zeta _r}{\kappa _r}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_r^2} \right)R{\rm{d}}\mathit{\boldsymbol{\theta }} \end{array}$ (15)

2 柱壳结构振动求解 2.1 柱壳结构振动特性方程的求解

 $\mathit{\boldsymbol{M}}{\ddot \delta _{t + \Delta t}} + \mathit{\boldsymbol{C}}{\dot \delta _{t + \Delta t}} + \mathit{\boldsymbol{K}}{\delta _{t + \Delta t}} = {\mathit{\boldsymbol{F}}_{t + \Delta t}}$ (23)

 ${\dot \delta _{t + \Delta t}} = {\dot \delta _t} + \left[ {\left( {1 - \gamma } \right){{\ddot \delta }_t} + \gamma {{\ddot \delta }_{t + \Delta t}}} \right]\Delta t,\;\;\;\;\;0 \le \gamma \le 1$ (24)
 $\begin{array}{*{20}{c}} {{\delta _{t + \Delta t}} = {\delta _t} + {{\dot \delta }_t}\Delta t + \left[ {\left( {1/2 - \beta } \right){{\ddot \delta }_t} + \beta {{\ddot \delta }_{t + \Delta t}}} \right]\Delta {t^2},}\\ {0 \le 2\beta \le 1} \end{array}$ (25)

2.2 最小二乘参数与分段数收敛性

 $\mathit{\Omega } = \omega R{\left( {\rho \left( {1 - {\mu ^2}} \right)/E} \right)^{1/2}}$ (26)

3 柱壳结构瞬态振动响应研究 3.1 模型验证

 Download: 图 3 圆柱壳在C-C边界条件下受轴向阶跃载荷下圆柱壳瞬态位移响应曲线 Fig. 3 Transient displacement response of cylindrical shell under axis load with C-C boundary
 Download: 图 4 圆柱壳在不同边界条件下受轴向阶跃载荷下圆柱壳瞬态位移响应曲线 Fig. 4 Transient displacement response of cylindrical shell under axis load with various boundaries
3.2 边界条件对瞬态响应的影响

3.3 结构参数对瞬态响应的影响

 Download: 图 5 厚度改变时C-C边界受轴向阶跃载荷圆柱壳瞬态位移响应曲线 Fig. 5 Transient displacement response of cylindrical shell subjected to axial step load at the C-C boundary with various thickness

 Download: 图 6 结构损耗因子改变时C-C边界受轴向阶跃载荷圆柱壳瞬态位移响应曲线 Fig. 6 Transient displacement response of cylindrical shell subjected to axial step load at the C-C boundary with various loss factor

4 结论

1) 本文方法的柱壳分段数在NL=8时收敛，最小二乘残差加权系数取值达到κ=1×1014时结构的数值计算具有良好的稳定性；

2) 不同边界条件下，圆柱壳在相同的径向载荷作用时，结构的瞬态位移响应随边界条件的改变而发生变化，且发生变化的趋势保持一致；

3) 相同频率范围内，结构厚度越小，其位移响应曲线波峰的数量越多，并且其瞬态响应值越大；对于结构损耗因子的变化，只影响结构的共振峰值，对于结构固有频率影响较小。

 [1] RAYLEIGH J. The theory of sound[M]. New York: Dover Publications, 1945. (0) [2] LEISSA A W. Vibration of shells[M]. New York: Acoustic Society of America, 1993. (0) [3] 骆东平. 任意边界条件下圆柱壳体振动特性分析[J]. 固体力学学报, 1990, 11(1): 86-96. LUO Dongping. Vibration characteristics of cylindrical shells with arbitrary boundary conditions[J]. Acta mechnica solida sinica, 1990, 11(1): 86-96. (0) [4] YU Y Y. Free vibrations of thin cylindrical shells having finite lengths with freely supported and clamped edges[J]. Journal of applied mechanics, 1955, 22: 547-552. (0) [5] CHENG L, NICOLAS J. Free vibration analysis of a cylindrical shell-circular plate system with general coupling and various boundary conditions[J]. Journal of sound and vibration, 1992, 155(2): 231-247. (0) [6] QU Yegao, CHEN Yong, LONG Xinhua, et al. Free and forced vibration analysis of uniform and stepped circular cylindrical shells using a domain decomposition method[J]. Applied acoustics, 2013, 74(3): 425-439. DOI:10.1016/j.apacoust.2012.09.002 (0) [7] PANG Fuzhen, LI Haichao, CHOE K, et al. Free and forced vibration analysis of airtight cylindrical vessels with doubly curved shells of revolution by using Jacobi-Ritz method[J]. Shock and vibration, 2017, 2017: 4538540. (0) [8] LI Haichao, PANG Fuzhen, MIAO Xuhong, et al. A semi-analytical method for vibration analysis of stepped doubly-curved shells of revolution with arbitrary boundary conditions[J]. Thin-walled structures, 2018, 129: 125-144. (0) [9] 汪志强, 李学斌, 黄利华. 基于波传播方法和多元分析的正交各向异性圆柱壳振动特性研究[J]. 振动与冲击, 2018, 37(7): 227-232. WANG Zhiqiang, LI Xuebin, HUANG Lihua. Vibration characteristics of orthotropic circular cylindrical shells based on wave propagation approach and multi-variate analysis[J]. Journal of vibration and shock, 2018, 37(7): 227-232. (0) [10] 薛开, 王久法, 李秋红, 等. Mindlin矩形板在任意弹性边界条件下的振动特性分析[J]. 哈尔滨工程大学学报, 2014(4): 477-481. XUE Kai, WANG Jiufa, LI Qiuhong, et al. Vibration behavior analysis of Mindlin rectangular plates with arbitrary elastic boundary conditions[J]. Journal of Harbin Engineer University, 2014(4): 477-481. (0) [11] 周海军, 李玩幽, 吕秉琳, 等. 弹性支撑及连接边界的多跨曲梁面内自由振动分析[J]. 哈尔滨工程大学学报, 2012(6): 696-701. ZHOU Haijun, LI Wanyou, LYU Binglin, et al. In-plane free vibration analysis of multi-span curved beams with elastic support and connecting boundary conditions[J]. Journal of Harbin Engineer University, 2012(6): 696-701. DOI:10.3969/j.issn.1006-7043.201107007 (0) [12] 庞福振, 李海超, 霍瑞东, 等. 基于Jacobi-Ritz法的旋转组合结构自由振动特性分析[J]. 振动工程学报, 2018, 31(5): 105-114. PANG Fuzhen, LI Haichao, HUO Ruidong, et al. Free Vibration Analysis of Rotating Composite Structures Based on Jacobi-Ritz Method[J]. Journal of Vibration Engineering, 2018, 31(5): 105-114. (0) [13] LI H, PANG F, MIAO Xuhong, et al. Jacobi-Ritz method for free vibration analysis of uniform and stepped circular cylindrical shells with arbitrary boundary conditions:A unified formulation[J]. Computers & mathematics with applications, 2019, 77(2): 427-440. (0) [14] PANG F, LI H, et al. Application of flügge thin shell theory to the solution of free vibration behaviors for spherical-cylindrical-spherical shell:A unified formulation[J]. European journal of mechanics-A/solids, 2019, 74: 381-393. DOI:10.1016/j.euromechsol.2018.12.003 (0) [15] 李鸿晶, 王通, 廖旭. 关于Newmark-β法机理的一种解释[J]. 地震工程与工程振动, 2011, 31(2): 55-62. LI Hongjing, WANG Tong, LIAO Xu. An interpretation on Newmark beta methods in mechanism of numerical analysis[J]. Journal of earthquake engineering and engineering vibration, 2011, 31(2): 55-62. (0) [16] JAFARI A A, BAGHERI M. Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods[J]. Thin-walled structures, 2006, 44(1): 82-90. DOI:10.1016/j.tws.2005.08.008 (0) [17] PELLICANO F. Vibrations of circular cylindrical shells:theory and experiments[J]. Journal of sound and vibration, 2007, 303(1/2): 154-170. (0) [18] 杜敬涛.任意边界条件下结构振动, 封闭声场及其耦合系统建模方法研究[D].哈尔滨: 哈尔滨工程大学, 2009. DU Jingtao. Study on modeling methods for structural vibration, enclosed sound field and their coupling system subject to general boundary conditions[D]. Harbin: Harbin Engineering University, 2009. http://www.wanfangdata.com.cn/details/detail.do?_type=degree&id=Y1655598 (0)