An analytical procedure for free vibration analysis of circular cylindrical shells with arbitrary boundary conditions is developed with the employment of the method of reverberationray matrix. Based on the Flügge thin shell theory, the equations of motion are solved and exact solutions of the traveling wave form along the axial direction and the standing wave form along the circumferential direction are obtained. With such a unidirectional traveling wave form solution, the method of reverberationray matrix is introduced to derive a unified and compact form of equation for natural frequencies of circular cylindrical shells with arbitrary boundary conditions. The exact frequency parameters obtained in this paper are validated by comparing with those given by other researchers. The effects of the elastic restraints on the frequency parameters are examined in detail and some novel and useful conclusions are achieved.
Circular cylindrical shells are widely used in engineering structures, such as rockets, aircrafts, submarines, and pipelines. Due to the extensive application, the vibration characteristics of circular cylindrical shells have been a topic of major interest to many researchers. A comprehensive review of various thin shell theories developed before 1973 was presented by Leissa [
A large number of researches on vibration characteristics of circular cylindrical shells with various methods are available in the literature. Chang and Greif [
Recently, Brischetto and Carrera [
As far as the researches of circular cylindrical shells are concerned, most of the existing works are limited to classical boundary conditions. Some efforts are recently made to study the vibration characteristics of circular cylindrical shells with arbitrary boundary conditions. Based on the Flügge thin shell theory, this paper presents a unified and compact formulation for free vibration analysis of circular cylindrical shells with arbitrary boundary conditions including classical ones and nonclassical ones using the method of reverberationray matrix (MRRM). With the analytical solution of the traveling wave form along the axial direction and the standing wave form along the circumferential direction obtained by Yao et al., MRRM is introduced to derive the unified equation for natural frequencies of the circular cylindrical shell with arbitrary boundary conditions. To validate the method presented in this paper, some results for both classical boundary conditions and nonclassical boundary conditions are compared with those found in the published literature. The effects of the elastic restraints on the natural frequencies are examined in detail and some novel and useful conclusions are achieved at the end of this paper.
Consider an isotropic circular cylindrical shell with length
Geometry and the dual local coordinate system for a circular cylindrical shell.
The force and moment resultants acting on the cross section perpendicular to the axial direction in a circular cylindrical shell, shown in Figure
Force and moment resultants in a circular cylindrical shell.
Based on the Flügge thin shell theory, the governing differential equations for free vibration of a circular cylindrical shell are written as
The Fourier transform of an arbitrary physical quantity
Taking the Fourier transforms of (
It can be observed from (
Solutions to the equations of motion of the circular cylindrical shell are generally assumed in the following forms:
By solving (
Therefore, the solutions for the displacement components of an isotropic circular cylindrical shell can be explicitly rewritten as
The rotation of the normal to the middle surface about the circumferential direction is defined as
Substituting (
Substituting (
For an arbitrary circumferential mode number
In the same manner, (
Taking advantage of the unidirectional traveling wave solutions of the circular cylindrical shell obtained in the preceding subsections, the MRRM is introduced to derive the equation of natural frequencies. Since the scattering matrix is related to boundary conditions of the shell, it will be discussed in the first step. Then, the phase matrix and permutation matrix, which are independent of the boundary conditions, are derived. Finally, the reverberationray matrix and the equation for natural frequencies of the circular cylindrical shell are obtained. The formulation mentioned above is presented in detail in the following discussions.
In various boundary conditions including classical ones and nonclassical ones, for instance, the elasticsupport boundary conditions can be expressed in a unified compact form at the local coordinate
Coefficient matrices for common boundary conditions.
BCs 



CE 


FE 


SS 


ES 











Substituting (
Since the phase matrix turns into a unit matrix at the origin of the local coordinate, which is obvious from its definition, (
Similarly, the scattering relation for the other end of the circular cylindrical shell can be obtained as
Assembling the local scattering equations at both ends of the circular cylindrical shell by stacking
The phase relations of harmonic waves in the dual local coordinate system of MRRM provide additional equations for solving the unknown amplitude vectors. Note that the departing wave from one end of the circular cylindrical shell is exactly the arriving wave to the other end, and vice versa. Therefore, the amplitudes of the departing wave and the arriving wave differ with each other by a phase factor. The relations between the amplitudes of the departing wave and the arriving wave are presented as
Assembling both local phase equations results in the global phase equation
A comparison of the global amplitude vectors of the departing waves
Substituting (
To obtain a nontrivial solution of the global amplitude vector of the departing wave
By substituting one of the natural frequencies obtained by solving (
Finally, substituting the global amplitude vector of the departing wave and the arriving wave into the expressions of the displacement components of the circular cylindrical shell, the mode shape corresponding to the natural frequency is obtained.
To begin with, the coefficient matrices
In this subsection, a few wellstudied classical boundary conditions and some nonclassical boundary conditions frequently encountered in practice are taken as calculation examples. The method proposed in this paper is validated by comparing the present numerical results with those previously published in the literature.
Frequency parameters calculated for the three classical boundary conditions are presented in Table
Comparison of the frequency parameter

SSSS  CESS  CECE  

Reference [ 
Reference [ 
Reference [ 
Present  Reference [ 
Reference [ 
Reference [ 
Present  Reference [ 
Reference [ 
Reference [ 
Present  
1  0.016101  0.016103  0.016101  0.016101  0.023974  0.024065  0.024721  0.023930  0.032885  0.033176  0.034879  0.032792 
2  0.009382  0.009388  0.009382  0.009382  0.011225  0.011238  0.011281  0.011215  0.013932  0.013970  0.014052  0.013902 
3  0.022105  0.022108  0.022105  0.022105  0.022310  0.022314  0.022335  0.022309  0.022672  0.022677  0.022725  0.022667 
4  0.042095  0.042097  0.042095  0.042095  0.042139  0.042141  0.042166  0.042139  0.042208  0.042210  0.042271  0.042208 
5  0.068008  0.068008  0.068008  0.068008  0.068024  0.068026  0.068054  0.068025  0.068046  0.068048  0.068116  0.068046 
6  0.099730  0.099730  0.099731  0.099731  0.099738  0.099739  0.099771  0.099739  0.099748  0.099749  0.099823  0.099749 
7  0.137239  0.137239  0.137240  0.137240  0.137244  0.137244  0.137279  0.137245  0.137249  0.137250  0.137328  0.137251 
8  0.180527  0.180528  0.180527  0.180530  0.180531  0.180531  0.180569  0.180533  0.180535  0.180535  0.180617  0.180536 
9  0.229594  0.229594  0.229596  0.229596  0.229596  0.229596  0.229636  0.229599  0.229599  0.229600  0.229684  0.229601 
10  0.284435  0.284436  0.284438  0.284439  0.284437  0.284438  0.284478  0.284440  0.284439  0.284441  0.284526  0.284442 
Subsequently, a comparison of the frequency parameters obtained by the method presented in this paper and those presented by Qu et al. [
Comparison of the frequency parameter









Reference [ 
Present  Reference [ 
Present  Reference [ 
Present  Reference [ 
Present  Reference [ 
Present  Reference [ 
Present  
1  0.021812  0.020328  0.027104  0.025721  0.021669  0.020163  0.033157  0.031534  0.026984  0.025467  0.021527  0.020004 
2  0.011674  0.010301  0.012695  0.011816  0.011471  0.010301  0.013957  0.013881  0.012457  0.011816  0.011272  0.010301 
3  0.022463  0.022211  0.022558  0.02239  0.02244  0.022209  0.022673  0.022667  0.022531  0.022387  0.022418  0.022206 
4  0.042178  0.04212  0.042193  0.042156  0.042175  0.042118  0.042209  0.042208  0.04219  0.042154  0.042172  0.042117 
5  0.068038  0.067936  0.068043  0.068031  0.068037  0.067936  0.068048  0.068046  0.068042  0.06803  0.068037  0.068016 
6  0.099745  0.099736  0.099747  0.099742  0.099745  0.099735  0.099748  0.099749  0.099746  0.099741  0.099745  0.099735 
7  0.137248  0.137243  0.137249  0.137247  0.137248  0.137243  0.13725  0.137251  0.137249  0.137246  0.137248  0.137243 
8  0.180534  0.180532  0.180535  0.180534  0.180534  0.180532  0.180535  0.180536  0.180535  0.180534  0.180534  0.180531 
9  0.229599  0.229598  0.229599  0.229599  0.229599  0.229598  0.229599  0.229601  0.229599  0.229599  0.229599  0.229598 
10  0.284439  0.28444  0.284439  0.284441  0.284439  0.28444  0.284441  0.284442  0.284439  0.284441  0.284439  0.284440 
Tables
In what follows, the effects of each of the axial, circumferential, radial, and torsional stiffness of the elasticsupport on the frequency parameters for the clamped, free, and elasticsupport boundary conditions are, respectively, investigated in the following discussions. The clamped, free, and elasticsupport boundary conditions, respectively, indicate the other three degrees of freedom are clamped, free, and elasticsupported.
Firstly, each of the axial, circumferential, radial, and torsional stiffness is taken as from 10 to 10^{12} N/m (or N/rad for torsional stiffness) while the other three degrees of freedom are assumed to be clamped. The frequency parameters against the nondimensional stiffness
Effect of elasticsupport stiffness on the frequency parameters for the clamped boundary conditions.
Frequency parameters versus axial stiffness
Frequency parameters versus circumferential stiffness
Frequency parameters versus radial stiffness
Frequency parameters versus torsional stiffness
It can be observed from Figure
Besides, Figure
Subsequently, each of the axial, circumferential, radial, and torsional stiffness is taken as from 10 to 10^{12} N/m (or N/rad) while the other three degrees of freedom are assumed to be free. The frequency parameters against the nondimensional stiffness
Effect of elasticsupport stiffness on the frequency parameters for the free boundary conditions.
Frequency parameters versus axial stiffness
Frequency parameters versus circumferential stiffness
Frequency parameters versus radial stiffness
Frequency parameters versus torsional stiffness
Figure
To put it specifically, Figure
Some selected free mode shapes and clamped mode shapes.
Clamped mode shapes
Free mode shapes
It can be observed from Figure
Figure
Besides, it can be observed from Figure
Then, with the other three degrees of freedom elasticsupported, the curves of the frequency parameters against the axial, circumferential, radial, and torsional stiffness are, respectively, presented in Figures
Effect of elasticsupport stiffness on the frequency parameters for the elasticsupport boundary conditions (elasticsupport stiffness
Frequency parameters versus axial stiffness
Frequency parameters versus circumferential stiffness
Frequency parameters versus radial stiffness
Frequency parameters versus torsional stiffness
It can be found from Figure
Figure
It can be observed from Figure
Figure
Finally, curves of frequency parameters varying with the axial, circumferential, radial, and torsional stiffness for different mode numbers of the circular cylindrical shell with the free, clamped, and elasticsupport boundary conditions are shown in Figure
Effect of elasticsupport stiffness on the frequency parameters for the free, the clamped, and the elasticsupport boundary conditions (elasticsupport stiffness
Frequency parameters versus axial stiffness
Frequency parameters versus circumferential stiffness
Frequency parameters versus radial stiffness
Frequency parameters versus torsional stiffness
It can be observed from Figure
Figure
Figure
However, for large radial stiffness, as for mode number (
Figure
An analytical procedure is developed to analyze the free vibration characteristics of circular cylindrical shells with arbitrary boundary conditions. Based on the Flügge thin shell theory, exact solutions of the traveling wave form along the axial direction and the standing wave form along the circumferential direction are obtained. MRRM is introduced to derive the equation of the natural frequencies of a unified and compact form. The agreement of the comparisons in the frequency parameters obtained by MRRM and those presented in the published literature proves the suitability and accuracy of applying MRRM to the study of free vibration of circular cylindrical shells with arbitrary boundary conditions.
Free vibration characteristics of circular cylindrical shells with elasticsupport boundary conditions are investigated by MRRM. Results show that the elasticsupport stiffness effectively affects the frequency parameters in the range of
Consider the following:
Consider the following:
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is financially supported by National Natural Science Foundation of China (Grants nos. 51279038 and 51479041). The authors would like to express their profound thanks for the financial support. The first author would like to sincerely thank Miss Jingjing Yu and Mr. Qingshan Wang for the scientific discussions and suggestions.