Abstract
In this paper an Improved Fourier series method has been employed to study the free vibrations of isotropic homogeneous moderately thick open cylindrical shells with arbitrary subtended angle and general elastic restraints. In this method, regardless of the boundary conditions, each of the displacement components of open shell is invariably expressed as a simple trigonometric series with accelerated and uniform convergence over the solution domain. Distributed elastic restraints are used to specify the elastic boundary conditions along the shell edges and therefore, arbitrary boundary restraints can be achieved by varying the values of spring’s stiffness. All the unknown expansion coefficients are treated as the generalized coordinates and solved using the RayleighRitz technique. A considerable number of new vibration results for isotropic open cylindrical shells with various geometric parameters and boundary conditions are presented. The effects of boundary stiffness, thickness to radius ratio and subtended angle on the vibration characteristics are also discussed in detail.
1. Introduction
Shell structures are widely used in various engineering applications like submarines, rockets, missile, automobiles and aircrafts etc. In these applications the shell structures may be exposed to various dynamic loads under different boundary conditions. These boundary conditions may be classical, elastic, uniform, nonuniform and/or a combination of these. These dynamic loads under various boundary conditions induce structural vibrations which further results in catastrophic structural failures. Many such incidents have been observed in the history. Due to this reason it is very important to study, design and analyze these structural vibrations for reliable, safe, efficient and lasting structural performance. Based on the geometrical shapes the shell structures may be classified into cylindrical, spherical and conical shells, however in the present manuscript only open cylindrical shells are under consideration which are widely used in various engineering applications.
For any structure, modal analysis is performed to study its vibration characteristics. This modal analysis includes the study of natural frequencies and the corresponding mode shapes. This information is of prime importance in order to suppress the vibrations induced in any structure when it is exposed to dynamic loads or excitations. In case of shell structures there are also other geometric parameters like thickness to length ratio, thickness to radius ratio and subtended angle which plays a prominent role in the vibrations, acoustic and safety analysis of these shell structures. For this reason, a lot of research work has been done on the vibration characteristics of shells and various numerical methods have been developed from time to time and used by researchers to deeply analyze the vibrations of shells. A detailed review of various such methods can be found in the Leissa’s book [1]. To the author’s best knowledge, the literature available related to open cylindrical shells as compared to closed shells is very limited. In this manuscript an effort has been put to study the vibrations characteristics of open cylindrical shells therefore it is necessary to highlight some prominent studies related to open shells.
Initially the study of vibration characteristics of cylindrical shells was limited to shallow shells [27]. Later, employing the classical shell theory, Selmane et al. [8], presented a hybrid finite element method for open cylindrical shells. A similar study was performed by Bardell et al. [9]. He used $h$$p$ version of finite element method and studied the isotropic open cylindrical shells. 3D elasticity approach and threedimensional displacement based extremum energy principle was used by Lim et al. [10] to perform the modal analysis of open cylindrical shells. Incorporating the effect of shear deformation and rotary inertia, Price et al. [11] did his research on cylindrical pipes and open shells by employing various shell theories. Zhang et al. [12] used wave propagation technique to investigate the natural frequencies and mode shapes for cylindrical panels. Employing virtual work and d’Alembert’s principle followed by predictorcorrector method, Ribeiro [13] investigated the geometrically nonlinear vibration characteristics of moderately thick shells.
Using first order shear deformation theory, Kandasamy et al. [14] investigated skewed open cylindrical deep shells. Later in another similar study C. Adam [15], addressed nonlinear vibrations of shallow shells with different shear flexibility. Using thin shell theory and discrete singular convolution method, Omer [16] studied the vibration characteristics of laminated conical and cylindrical shells. Similarly, in another study laminated open cylindrical shells were studied by Ribeiro [17] using clamped boundary conditions. Tornabene et al. [18] studied the FGM shell and plate structures using differential quadrature method. In another important research Hadi et.al [19], performed research on shallow cylindrical and delaminated shells for large amplitude vibrations. A 3D higher deformation theory was employed by Khalili et al. [20] to calculate the modal frequencies of circular shells subjected to various classical boundary conditions. Employing Ritz method similar research on different geometrical shell structures subjected to arbitrary boundary conditions were performed by Qatu and Asadi [21]. A lot of other similar important research work on cylindrical shells is given in [2237].
A very important method previously developed for beams [38] and plates [39] is presently a source of attention for researchers and is currently used for studying the vibration characteristics of shells subjected to general boundary conditions. In this manuscript this method has been employed to study the vibration characteristics of moderately thick isotropic homogeneous open cylindrical shells subjected to general elastic boundary conditions.
2. Theoretical formulation
2.1. Model description
Consider an isotropic homogeneous moderately thick open cylindrical shell having uniform thickness $h$, subtended angle $\theta $, radius $R$, and length $L$ as shown in Fig. 1. A cylindrical coordinate system ($x$, $\theta $, $z$) is also shown, in which the $x$ coordinate is taken in the axis of the shell panel and $\theta $ and $z$ represents the circumferential and radial directions respectively. The middle surface displacements are represented by $u$, $v$ and $w$ whereas ${\varphi}_{x}$ and ${\varphi}_{\theta}$ represents the rotation of transverse normal with respect to $\theta $ and $x$ axis respectively.
Three translational springs having stiffnesses (${k}_{u}$, ${k}_{v}$ and ${k}_{w}$) and two rotational springs having stiffnesses (${K}_{x}$ and ${K}_{\theta}$) are introduced along each edge of the cylindrical shell panel to simulate arbitrary boundary conditions. All the classical sets of boundary conditions can easily be achieved by assigning proper stiffness values to the translational and rotational springs. For instance, a clamped boundary (C) is achieved by simply setting the stiffnesses of the entire springs equal to infinite (which is represented by a very large number, 10^{14} N/m). Inversely, a free boundary (F) is gained by setting the stiffnesses of the entire springs equal to zero.
Fig. 1Geometry of open shell
2.2. Energy functional of moderately thick open cylindrical shells
Based on first order shear deformation theory for isotropic homogeneous moderately thick cylindrical shells, the displacement components (${u}_{x}$, ${v}_{\theta}$ and $w$) of the shell in terms of middle surface displacements can be expressed as:
${v}_{\theta}\left(x,\theta ,z,t\right)=v\left(x,\theta ,t\right)+z{\varphi}_{\theta}\left(x,\theta ,t\right),$
$w\left(x,\theta ,z,t\right)={w}_{o}\left(x,\theta ,t\right),$
where $u$, $v$ and ${w}_{o}$ are the middle surface displacements of the shell in the axial, circumferential and radial directions respectively, ${\varphi}_{x}$ and ${\varphi}_{\theta}$ represent the rotations of transverse normal with respect to $\theta $ and $x$axes and $t$ is the time variable. The strain displacement relation for the shell panel in terms of middle surface strains can be expressed as:
where ${\epsilon}_{xx}^{o}$, ${\epsilon}_{\theta \theta}^{o}$ and ${\gamma}_{x\theta}^{o}$ represents the middle surface strains and ${\chi}_{xx}$, ${\chi}_{\theta \theta}$ and ${\chi}_{x\theta}$ represents the curvature changes during deformation for a moderately thick shell panel. For a cylindrical shell panel having constant radius $R$, the middle surface strains and curvature changes are given as:
The transverse shear strains are given by:
According to Hooke’s law the stress strains relations for a moderately thick cylindrical shell are given as:
where ${Q}_{11}=E/1{\mu}^{2}$, ${Q}_{12}=\mu E/1{\mu}^{2}$, ${Q}_{66}=E/2(1+\mu )$, $E$ is the modulus of elasticity and $\mu $ is the Poisson ratio.
The inplane force resultant vector, bending and twisting moment resultant vector and transverse shear force resultant vector is given by:
$Q=\left\{\begin{array}{l}{Q}_{xx}\\ {Q}_{\theta \theta}\end{array}\right\}=\underset{h/2}{\overset{h/2}{\int}}\kappa \left\{\begin{array}{l}{\sigma}_{xz}\\ {\sigma}_{\theta z}\end{array}\right\}dz,$
where ‘$\kappa $’ is the shear correction factor i.e. $\kappa =$ 5/6
The equations relating the force and moment resultants to the strains and curvature changes in the middle surface can be written in matrix form as:
where ${A}_{st}=\underset{h/2}{\overset{h/2}{\int}}{Q}_{st}.dz$, ${B}_{st}=\underset{h/2}{\overset{h/2}{\int}}{Q}_{st}zdz$ and ${D}_{st}=\underset{h/2}{\overset{h/2}{\int}}{Q}_{st}{z}^{2}dz$, $s$, $t=$ 1, 2, 6.
The strain energy $U$ of the open circular cylindrical shell is given by:
$\left.+{Q}_{x}{\gamma}_{xz}+{Q}_{\theta}{\gamma}_{\theta z}\right)Rd\theta dx.$
Substituting Eq. (8) into (9), the strain energy can be expressed as a sum of three parts:
For cylindrical shells:
$\left.+\kappa {A}_{66}{\left({\varphi}_{x}+\frac{\partial {w}_{o}}{\partial x}\right)}^{2}+2{A}_{12}\left(\frac{\partial u}{\partial x}\right)\left(\frac{1}{R}\frac{\partial v}{\partial \theta}+\frac{{w}_{o}}{R}\right)+{A}_{66}{\left(\frac{\partial v}{\partial x}+\frac{1}{R}\frac{\partial u}{\partial \theta}\right)}^{2}\right\}Rd\theta dx,$
$\left.+{D}_{66}{\left(\frac{\partial {\varphi}_{\theta}}{\partial x}+\frac{\partial {\varphi}_{x}}{R\partial \theta}\right)}^{2}\right\}Rd\theta dx,$
$\left.+{B}_{12}\left(\frac{1}{R}\frac{\partial v}{\partial \theta}+\frac{{w}_{o}}{R}\right)\left(\frac{\partial {\varphi}_{x}}{\partial x}\right)+{B}_{66}\left(\frac{\partial v}{\partial x}+\frac{1}{R}\frac{\partial u}{\partial \theta}\right)\left(\frac{\partial {\varphi}_{\theta}}{\partial x}+\frac{1}{R}\frac{\partial {\varphi}_{x}}{\partial \theta}\right)\right\}Rd\theta dx.$
Similarly, the kinetic energy of the open cylindrical shell is given by:
where $\rho $ is density.
Since three groups of translational springs (${k}_{u}$, ${k}_{v}$ and ${k}_{w}$) and two groups of rotational springs (${K}_{x}$ and ${K}_{\theta}$) are attached at each edge of the open cylindrical shell to simulate the arbitrary elastic boundary conditions, therefore the potential or strain energy stored in these elastic springs can be expressed as:
$\left.+{\left({k}_{{\theta}_{\alpha}}^{u}{u}^{2}+{k}_{{\theta}_{\alpha}}^{v}{v}^{2}+{k}_{{\theta}_{\alpha}}^{w}{w}_{o}^{2}+{K}_{{\theta}_{\alpha}}^{x}{\varphi}_{x}^{2}+{K}_{{\theta}_{\alpha}}^{\theta}{\varphi}_{\theta}^{2}\right)}_{\theta =\alpha}\right\}dx$
$+\frac{1}{2}\underset{0}{\overset{\alpha}{\int}}\left\{{\left({k}_{{x}_{o}}^{u}{u}^{2}+{k}_{{x}_{o}}^{v}{v}^{2}+{k}_{{x}_{o}}^{w}{w}_{o}^{2}+{K}_{{x}_{o}}^{x}{\varphi}_{x}^{2}+{K}_{{x}_{o}}^{\theta}{\varphi}_{\theta}^{2}\right)}_{x=0}\right.$
$\left.+{\left({k}_{{x}_{L}}^{u}{u}^{2}+{k}_{{x}_{L}}^{v}{v}^{2}+{k}_{{x}_{L}}^{w}{{w}_{o}}^{2}+{K}_{{x}_{L}}^{x}{\varphi}_{x}^{2}+{K}_{{x}_{L}}^{\theta}{\varphi}_{\theta}^{2}\right)}_{x=L}\right\}Rd\theta .$
After establishing the strain energy and kinetic energy expressions, the Lagrangian expression can be written as:
3. Solution scheme
3.1. Selection of Admissible displacement functions
After establishing the potential energy and kinetic energy expression, the next step is to choose appropriate admissible displacement functions which is of crucial importance in the RayleighRitz procedure. Generally, for shell problems, the admissible functions are often expressed in terms of beam functions under the same boundary conditions. Thus, a specially customized set of beam functions is required for each type of boundary conditions. Instead of the beam functions, one may also use other forms of admissible functions such as orthogonal polynomials. However, the higher order polynomials tend to become numerically unstable due to the computer roundoff errors. This numerical difficulty can be avoided by expressing the displacement functions in the form of a Fourier series expansion because Fourier functions constitute a complete set and exhibit an excellent numerical stability.
In the present study, irrespective of the boundary conditions, each of the displacement function is expressed as a new form of trigonometric expansion with accelerated convergence. Each of displacement and rotation functions of the open cylindrical shell is expanded as:
${w}_{o}=\sum _{m=n=2}^{\infty}{C}_{mn}{\phi}_{m}\left(x\right){\phi}_{n}\left(\theta \right),{\varphi}_{x}=\sum _{m=n=2}^{\infty}{D}_{mn}{\phi}_{m}\left(x\right){\phi}_{n}\left(\theta \right),$
${\varphi}_{\theta}=\sum _{m=n=2}^{\infty}{E}_{mn}{\phi}_{m}\left(x\right){\phi}_{n}\left(\theta \right),$
where:
${\lambda}_{m}\left(x\right)=\frac{m\pi \left(x\right)}{L},{\lambda}_{n}\left(\theta \right)=\frac{n\pi \left(\theta \right)}{\alpha}.$
The sine terms in the Eq. (17) are introduced to overcome the potential discontinuities of the displacement function, along the edges of the shell, when it is periodically extended and sought in the form of trigonometric series expansion. As a result, the Gibbs effect can be eliminated and the convergence of the series expansion can be substantially improved.
3.2. Determination of expansion coefficients
After establishing energy expressions and selecting proper admissible displacement functions, the next step is to find the expansion coefficients in the assumed displacement series. This can be achieved by substituting the assumed displacement fields Eq. (17) in the Eq. (10), (14) and (15) and then minimizing Eq. (16) against all the unknown series expansion coefficients i.e.:
After minimizing the Langrangian against all unknown series expansion coefficients as shown in Eq. (18), we will obtain a series of linear algebraic expressions which can be further expressed in matrix form as:
where $E$ is a vector which contains all the unknown series expansion coefficients, $K$ and $M$ are the stiffness and mass matrices, respectively. $E$, $K$ and $M$ are expressed as:
$K=\left[\begin{array}{lllll}{K}_{uu}& {K}_{uv}& {K}_{uw}& {K}_{u{\psi}_{x}}& {K}_{u{\psi}_{\theta}}\\ {K}_{uv}^{T}& {K}_{vv}& {K}_{vw}& {K}_{v{\psi}_{x}}& {K}_{v{\psi}_{\theta}}\\ {K}_{uw}^{T}& {K}_{vw}^{T}& {K}_{ww}& {K}_{w{\psi}_{x}}& {K}_{w{\psi}_{\theta}}\\ {K}_{u{\psi}_{x}}^{T}& {K}_{v{\psi}_{x}}^{T}& {K}_{w{\psi}_{x}}^{T}& {K}_{{\psi}_{x}{\psi}_{x}}& {K}_{{\psi}_{x}{\psi}_{\theta}}\\ {K}_{u{\psi}_{\theta}}^{T}& {K}_{v{\psi}_{\theta}}^{T}& {K}_{w{\psi}_{\theta}}^{T}& {K}_{{\psi}_{x}{\psi}_{\theta}}^{T}& {K}_{{\psi}_{\theta}{\psi}_{\theta}}\end{array}\right],$
$M=\left[\begin{array}{ccccc}{M}_{uu}& 0& 0& 0& 0\\ 0& {M}_{vv}& 0& 0& 0\\ 0& 0& {M}_{ww}& 0& 0\\ 0& 0& 0& {M}_{{\psi}_{x}{\psi}_{x}}& 0\\ 0& 0& 0& 0& {M}_{{\psi}_{\theta}{\psi}_{\theta}}\end{array}\right].$
For conciseness, the detailed expressions for the stiffness and mass matrices are not shown here.
3.3. Determination of eigen values and eigen vectors
After establishing Eq. (19), the eigenvalues (or natural frequencies) and eigenvectors of moderately thick open cylindrical shell can now be easily and directly determined from solving a standard matrix eigenvalue problem Eq. (19). In the current work, the authors have used MATLAB software to obtain eigen values (natural frequencies) and corresponding eigen vectors. For a given natural frequency, the corresponding eigenvector actually contains the series expansion coefficients which can be used to construct the physical mode shape based on Eq. (17). Although this investigation is focused on the free vibration of open cylindrical shells, the response of the shell panel to an applied load can also be easily obtained by considering the work done by this load in the Lagrangian, eventually leading to a force term on the right side of Eq. (19).
4. Results and discussion
In practical engineering, the study of structure response, when it is subjected to static or dynamic loads is of critical importance. One of such studies is the modal analysis and testing. The modal analysis of any structure includes the identification of resonant frequencies which are subsequently quantified to avoid well known resonance phenomena. In order to find these modes of vibrations or resonant frequencies accurately, researchers have developed various analytical methods for different boundary conditions. For any structure the modes of vibration are highly dependent upon the material properties and boundary conditions. Each mode of vibration is defined by a natural (modal or resonant) frequency, modal damping, and a mode shape. If there is a slight change in material properties or boundary conditions of a structure, its modes of vibration will also change. Since in real scenarios, the material properties and boundary conditions of any structure may vary therefore it is of prime importance to study or estimate these natural frequencies for any change in material properties as well as boundary conditions.
In this section a systematic comparison of the results obtained using the present method and those obtained from ABAQUS is carried out to verify the accuracy, reliability and feasibility of the present method. First of all, the convergence study of the present method is performed. Convergence study is important to check the rationality of hypothetical admissible functions of the displacement fields and also to determine the proper truncated numbers in the calculations to follow. Therefore, for different values of $M$ and $N$ (number of truncation terms) results are calculated for a completely clamped moderately thick open cylindrical shell panel having parameters $\alpha =$60°, $h/R=$0.1, $l/R=$2. For convenience, the fourletter string ‘CCCC’ has been used to refer to clamped boundary condition at edges $x=$0, $\theta =$0, $x=L$ and $\theta =\alpha $, respectively. The clamped boundary condition is easily achieved by assigning a very high stiffness value i.e 1e^{14 }to the boundary springs. Similarly, free boundary conditions can be achieved by assigning zero stiffness value to the restraining springs. In calculations to follow the symbol $S$, $F$ and $E$ will be used to denote the simply supported, free and elastic boundary restraints. For different no. of truncation terms, Table 1 shows first six frequencies (Hz) for open cylindrical shell panel subjected to CCCC boundary conditions.
Table 1First six frequencies (Hz) for completely clamped open cylindrical shell panel (α= 60°, h/R= 0.1, l/R= 2)
$M=N$  Mode sequence  
1  2  3  4  5  6  
2  879.547  1112.311  1463.763  1671.980  2021.887  2140.287 
4  877.588  991.404  1227.954  1228.916  1411.978  1692.215 
6  877.328  988.366  1210.706  1222.908  1402.000  1548.622 
8  877.251  987.671  1208.433  1221.810  1399.761  1540.250 
10  877.223  987.445  1207.782  1221.501  1399.045  1538.153 
12  877.210  987.354  1207.536  1221.390  1398.764  1537.416 
14  877.204  987.312  1207.426  1221.342  1398.635  1537.105 
16  877.201  987.291  1207.372  1221.319  1398.569  1536.956 
18  877.199  987.279  1207.342  1221.307  1398.533  1536.878 
20  877.198  987.271  1207.325  1221.300  1398.512  1536.834 
ABAQUS  877.340  987.040  1207.100  1222.700  1398.600  1537.300 
Table 2First six frequencies (Hz) for open cylindrical shell panel (α= 270°, h/R= 0.2, l/R= 2) subjected to CFCF boundary conditions
$M=N$  Mode sequence  
1  2  3  4  5  6  
2  392.809  399.278  515.197  606.723  752.915  918.837 
4  356.415  363.707  500.894  517.999  600.070  741.259 
6  338.981  345.826  488.791  495.970  578.042  605.890 
8  336.239  336.714  477.231  485.121  573.300  603.493 
10  336.234  336.681  477.000  484.965  571.963  602.551 
12  336.231  336.519  476.880  484.739  569.786  598.987 
14  336.227  336.452  476.808  483.984  568.681  598.606 
16  336.210  336.423  476.761  483.874  568.614  597.998 
18  336.197  336.409  476.729  483.793  568.569  597.745 
20  336.189  336.316  476.706  483.735  568.536  597.610 
It can be seen that the frequency parameter converges very quickly for small number of truncation terms. Furthermore, the results are also in close agreement with those obtained from ABAQUS. Similarly, Table 2 gives first five frequency parameter (Hz) for cylindrical shell panel having geometric parameters ($\alpha =$270°, $h/R=$0.2, $l/R=$2) and subjected to CFCF boundary conditions.
A fast convergence and close agreement with the results obtained from ABAQUS can be seen. Furthermore, the fast convergence of the frequencies can be observed in Fig. 2 which shows convergence of 2nd, 5th and 8th mode frequencies of cylindrical shell panel ($\alpha =$60°, $h/R=$0.1, $l/R=$2) subjected to FFFF boundary conditions and using different number of truncation terms.
It can be observed from table 1 that when the truncated terms change from $M=N=$10 to $M=N=$12, the difference of the frequency parameters does not exceed 0.045 % for the worst case, which is acceptable. More accurate results may be obtained by further truncated numbers, but the computational cost will be increased. Therefore, for the sake of both accuracy and computational cost, the truncated number of the displacement expressions will be uniformly selected as $M=N=$12 in all the following numerical calculations.
Fig. 2Convergence pattern of frequency parameters with no. of terms (M=N)
After studying the convergence of the present method, the accuracy of the method is verified by applying it on cylindrical shell panels subjected to various combinations of classical boundary conditions. The first four nondimensional frequency parameters for a moderately thick cylindrical shell panel having geometric parameters ($\alpha =$180°, $h/R=$0.1, $l/R=$2) subjected to various combinations of classical boundary conditions are presented in Table 3. A good agreement can be observed between the calculated results and those obtained from ABAQUS.
Table 3First four nondimensional frequency parameters Ω=ω×Rρ(1μ2)*E1 for cylindrical shell panel (α= 180°, h/R= 0.1, l/R= 2) subjected to various boundary conditions
BC  Methods  Mode sequence  
1  2  3  4  
CFCF  Present  0.2458  0.2541  0.4127  0.4656 
ABAQUS  0.2448  0.2518  0.4113  0.4655  
CFFF  Present  0.0877  0.0891  0.1872  0.1976 
ABAQUS  0.0861  0.0887  0.1868  0.1959  
FFCC  Present  0.0937  0.1885  0.2876  0.3014 
ABAQUS  0.0906  0.1867  0.2852  0.2951 
After verifying the accuracy of the present method for various combinations of classical boundary conditions, it is important to study the effect of boundary spring stiffnesses on the frequency parameters. Figs. 3 and 4 shows the variation of frequency for 7th, 8th and 9th mode respectively plotted against the spring stiffnesses by varying the stiffnesses of one group of boundary springs from 0 to 10^{14} while keeping the stiffnesses of the other group equal to infinite i.e. 10^{14}.
It can be seen in Fig. 3 that the frequency parameter almost remains at a level when the stiffness of the translational springs in $x$, $\theta $ and $z$ directions is less than 10^{8} and greater than 10^{12 }where as other than this range the frequency parameter increases with increasing stiffness values. Similar phenomena can be observed in Fig. 4 which shows the variation in frequency parameter with increasing stiffness values for rotational springs. It can be observed from figures that the influential range for translational and rotational springs is 10^{8} to 10^{12} and 10^{6} to 10^{10} respectively. Within this range the frequency parameter increases with increasing stiffness values however before and after this influential range the frequency parameters remain at a level. Based on the analysis it can be concluded that stable frequency parameter can be obtained when the stiffnesses for all the restraining springs is more than 10^{12} or less than 10^{6} and also it is also suitable and valid to use the stiffness value 10^{14} to simulate the infinite stiffness in the numerical calculations since the frequency parameter remain at the same level for values greater than equal to 10^{12}. Also, an elastic boundary condition can also be easily defined with any stiffness value between 10^{6} to 10^{12}.
Fig. 3Effect of translational springs stiffness (ku, kv and kw) on the frequency parameter
Fig. 4Effect of rotational springs stiffness (Kx and Kθ) on the frequency parameter
As mentioned earlier the present method can be used to obtain natural frequency parameters for moderately thick open cylindrical shell under general elastic boundary condition regardless of modifying solution algorithm and procedure. The arbitrary boundary conditions including the classical and elastic boundary restraints can easily be achieved by assigning proper stiffness values to the restraining springs as shown in Table 4 where E^{1} and E^{2} represents the two different types of elastic restraints having different stiffness values.
Table 4Corresponding spring’s stiffnesses for different types of boundary conditions.
BC  $x=0$ or $x={L}_{x}$  $\theta =0$ or $\theta =\alpha $  
${k}_{u}$  ${k}_{v}$  ${k}_{w}$  ${K}_{x}$  ${K}_{\theta}$  ${k}_{u}$  ${k}_{v}$  ${k}_{w}$  ${K}_{x}$  ${K}_{\theta}$  
F  0  0  0  0  0  0  0  0  0  0 
C  1e^{14}  1e^{14}  1e^{14}  1e^{14}  1e^{14}  1e^{14}  1e^{14}  1e^{14}  1e^{14}  1e^{14} 
S  0  1e^{14}  1e^{14}  0  1e^{14}  1e^{14}  0  1e^{14}  1e^{14}  0 
E^{1}  1e^{8}  1e^{14}  1e^{14}  1e^{14}  1e^{14}  1e^{8}  1e^{14}  1e^{14}  1e^{14}  1e^{14} 
E^{2}  1e^{14}  1e^{8}  1e^{14}  1e^{14}  1e^{14}  1e^{14}  1e^{8}  1e^{14}  1e^{14}  1e^{14} 
Next, we calculate the frequency parameter for open cylindrical shells using different combinations of classical and elastic boundary restraints. Table 5, 6 and 7 shows the frequency parameters for open cylindrical shell panels having different subtended angles, thickness to radius ratio and subjected to various combinations of classical and elastic boundary conditions.
It can be observed that all the frequencies mentioned in table 5, 6 and 7 are in close agreement with those obtained from ABAQUS. The maximum error for the worst case in the Table 5, 6 and 7 is 0.17 % which is acceptable.
At present, most of the existing techniques available so far to estimate the natural or resonant frequencies are limited to classical boundary conditions (clamped, free, simply supported etc.), however in practical engineering applications the structures are not always subjected to classical boundary conditions rather they may be subjected to elastic boundary conditions. In the present manuscript, the method presented not only helps to accurately estimate these natural frequencies of cylindrical shells subjected to different sets of classical boundary conditions but also accurately predicts these frequencies when such structures are subjected to general elastic boundary conditions. Furthermore, the presented results give an insight of the modes of vibration of these structures having different material properties and subjected to elastic boundary conditions. Moreover, another important contribution of this technique is that this method does not require any changes in procedure or solution algorithms to accommodate different geometries, material properties or boundary conditions. The same solution algorithm or procedure can be used to estimate natural frequencies for different materials and boundary conditions. Different boundary conditions (classical, elastic, uniform & nonuniform) can easily be achieved by simply changing the stiffnesses of the translational and rotational springs attached at the boundaries or edges of the shell structure.
Table 5First four frequencies (Hz) for open cylindrical shell panel (α= 60°, l/R= 2) subjected to different boundary conditions
BC  $h/R$  Methods  Mode sequence  
1  2  3  4  
E^{1}E^{1}E^{1}E^{1}  0.1  Present  78.816  836.356  963.911  1191.668 
ABAQUS  78.704  836.160  963.160  1190.700  
% Error  0.14  0.02  0.08  0.08  
0.2  Present  55.704  1080.569  1326.075  1405.296  
ABAQUS  55.670  1080.100  1325.000  1405.000  
% Error  0.06  0.04  0.08  0.02  
CE^{1}CE^{1}  0.1  Present  870.575  955.476  1195.536  1220.768 
ABAQUS  870.380  954.690  1194.700  1220.300  
% Error  0.02  0.08  0.06  0.03  
0.2  Present  1130.947  1262.642  1432.718  1712.324  
ABAQUS  1130.800  1262.200  1432.500  1712.100  
% Error  0.01  0.04  0.02  0.01  
FE^{1}FE^{1}  0.1  Present  55.660  808.238  830.808  884.843 
ABAQUS  55.654  807.400  830.020  884.020  
% Error  0.01  0.1  0.09  0.09  
0.2  Present  39.395  1027.752  1070.862  1166.300  
ABAQUS  39.365  1027.100  1070.400  1165.800  
% Error  0.07  0.06  0.04  0.04 
Table 6First four frequencies (Hz) for open cylindrical shell panel (α= 90°, l/R= 2) subjected to different boundary conditions
BC  $h/R$  Methods  Mode sequence  
1  2  3  4  
CE^{1}CF  0.1  Present  222.246  440.265  474.896  664.259 
ABAQUS  221.870  439.810  474.620  663.870  
% Error  0.17  0.10  0.06  0.06  
0.2  Present  349.143  591.131  755.025  1024.495  
ABAQUS  349.110  590.980  754.900  1024.100  
% Error  0.01  0.02  0.02  0.04  
CE^{2}CE^{2}  0.1  Present  425.665  539.988  731.036  750.295 
ABAQUS  425.530  539.870  730.790  750.100  
% Error  0.03  0.02  0.03  0.03  
0.2  Present  575.058  657.481  1012.780  1091.955  
ABAQUS  574.800  657.200  1012.500  1091.200  
% Error  0.04  0.04  0.03  0.07  
FE^{1}FE^{1}  0.1  Present  45.480  506.710  522.305  657.038 
ABAQUS  45.425  506.650  522.240  656.910  
% Error  0.12  0.01  0.01  0.02  
0.2  Present  32.143  727.394  755.393  780.742  
ABAQUS  32.136  727.290  755.250  780.560  
% Error  0.05  0.01  0.02  0.02 
Table 7First four frequencies (Hz) for open cylindrical shell panel (α= 270°, l/R= 2) subjected to different boundary conditions
BC  $h/R$  Methods  Mode sequence  
1  2  3  4  
E^{1}FE^{1}F  0.1  Present  29.773  193.010  196.105  316.912 
ABAQUS  29.760  192.970  196.070  316.860  
% Error  0.04  0.02  0.02  0.02  
0.2  Present  21.053  309.710  314.003  423.101  
ABAQUS  21.046  309.650  313.950  423.050  
% Error  0.03  0.02  0.02  0.01  
CE^{2}CE^{2}  0.1  Present  371.725  399.433  449.635  550.224 
ABAQUS  371.660  399.380  449.590  550.090  
% Error  0.02  0.01  0.01  0.02  
0.2  Present  483.725  544.132  587.706  739.569  
ABAQUS  483.690  544.080  587.640  739.410  
% Error  0.01  0.01  0.01  0.02  
CE^{1}CF  0.1  Present  215.061  364.871  413.932  461.280 
ABAQUS  215.000  364.830  413.890  461.240  
% Error  0.03  0.01  0.01  0.01  
0.2  Present  336.439  486.840  557.422  645.701  
ABAQUS  336.400  486.780  557.310  645.560  
% Error  0.01  0.01  0.02  0.02 
5. Conclusions
In this manuscript, an Improved Fourier series method previously developed for beams and plates has been employed to study the vibration characteristics of moderately thick isotropic homogeneous open cylindrical shells subjected to arbitrary elastic boundary conditions. Distributed elastic restraints have been used along the shell edges to achieve the elastic boundary restraints. Irrespective of the boundary conditions, all the displacement components have been presented in the form of simple trigonometric series with accelerated and uniform convergence. All the unknown expansion coefficients have been obtained using RayleighRitz technique. The efficiency, accuracy and reliability of the present method have been fully demonstrated by various numerical examples. All the results obtained have been found in close agreement with those obtained from ABAQUS. The effects of spring stiffnesses, thickness to radius ratio and subtended angle on the vibration characteristics have also been highlighted. In comparison with most existing techniques, the present method does not require any inconvenient formulation or procedural modifications to accommodate different boundary conditions or geometrical shapes. Furthermore, this method can easily be extended to study vibration analysis of different shell plate combinations.
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About this article
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. U1430236) and Natural Science Foundation of Heilongjiang Province of China (No. E2016024)