Free vibration of layered circular cylindrical shells of variable thickness is studied using spline function approximation by applying a point collocation method. The shell is made up of uniform layers of isotropic or specially orthotropic materials. The equations of motions in longitudinal, circumferential and transverse displacement components, are derived using extension of Love's first approximation theory. The coupled differential equations are solved using Bickley-type splines of suitable order, which are cubic and quintic, by applying the point collocation method. This results in the generalized eigenvalue problem by combining the suitable boundary conditions. The effect of frequency parameters and the corresponding mode shapes of vibration are studied with different thickness variation coefficients, and other parameters. The thickness variations are assumed to be linear, exponential, and sinusoidal along the axial direction. The results are given graphically and comparisons are made with those results obtained using finite element method.

Circular cylindrical shells are used in various fields like aviation, missiles, ship buildings, and chemical industries. Shells made of composite materials with variable thickness are used increasingly, since composite structures are having high specific stiffness, better damping, and shock absorbing characters over the homogeneous ones. The study of vibrational behavior of such shells is very important. The effect of variation of thickness on frequency parameter of the shell, which is made up of different layered materials, has been studied by very few researchers. Baker and Herrmann [

Mizusawa and Kito [

The present work analyses the flexural free vibration of layered circular cylindrical shells of variable thickness. The equations of motion are derived using Love’s first approximation theory for homogeneous shells. The layers are considered to be thin, elastic, specially orthotropic, or isotropic and assumed to be perfectly bonded together and move without interface slip. Three different thickness variations (linear, exponential, and sinusoidal) are considered along the axial direction of the cylinder. The governing coupled differential equations are obtained in terms of the reference surface displacements which are in longitudinal, circumferential, and transverse directions. Assuming the displacement functions in a separable form, they reduce to a system of ordinary differential equations on a set of displacement functions which are functions of meridional coordinate only. Two sets of boundary conditions are imposed and two types of materials are used to analyse the problem. In general, the equations have no closed form solution, so that the numerical solution techniques have to be resorted to.

The spline function technique is adopted to solve the coupled differential equations which are in three displacement functions. Bickley [

The three displacement functions are approximated using cubic and quintic splines. Collocation with these splines yields a set of field equations which, along with the equations of boundary conditions, reduce to a system of homogeneous simultaneous algebraic equations on assumed spline coefficients which results in a generalized eigenvalue problem. This eigenvalue problem is solved using eigensolution technique to obtain as many frequencies as required, starting from the least. From the eigenvectors, the spline coefficients can be found to construct the mode shapes.

The system of differential equations in terms of longitudinal, circumferential, and transverse displacements components is derived, which characterise the vibration of a thin shell of revolution. The general line of procedure of Ambartsumyan [

Layered circular cylindrical shell of constant thickness: geometry.

In general, the thickness of the

The thickness of the layers is not completely independent. Their dependence is given by

The stress resultants and moment resultants are expressed in terms of the longitudinal, circumferential, and transverse displacements

The differential equations on the displacement functions of (

The parameters are nondimensionalised as

The displacement functions

Convergence study is made for the frequency parameter value to fix the number of knots

Material properties of HSG and SGE.

Material | Density | Young’s Modulus | Young’s Modulus | Shear Modulus | Poisson Ratio |

HSG | 1.5892 | 12.40 | 1.03 | 0.54933 | 0.27 |

SGE | 2.0431 | 5.17 | 1.17 | 0.55060 | 0.25 |

After a number of trials, it is found that the number of knots

Comparative studies are next made for homogeneous cylindrical shells of exponential variation in thickness. Table

Comparison of natural frequencies for free vibration of cylindrical shell of exponential variation in thickness under C–C boundary condition;

Length parameter, | Takahashi et al. [ | Sivadas and Ganesan [ | Present value | |
---|---|---|---|---|

0.4 | 0.6768 | 1.2 | 1.2652 | 1.1522 |

0.6 | 0.6608 | 1.2 | 1.2606 | 1.1632 |

0.8 | 0.6460 | 1.2 | 1.2564 | 1.1816 |

1.0 | 0.6321 | 1.2 | 1.2528 | 1.2037 |

In this work, asymmetric free vibration of layered circular cylindrical shells of variable thickness is studied. Only two-layered materials with HSG and SGE combinations are used in this analysis, and the first three meridional modes are considered in all the analyses that follow.

In Figure

Variation of frequency parameter with relative layer thickness: linear variation in thickness of layers.

The results are not shown here for want of space. For the extreme values of

Figure

Variation of frequency parameter with relative layer thickness: exponential variation in thickness of layers.

Variation of frequency parameter with relative layer thickness: sinusoidal variation in thickness of layers.

Figures

Effect of circumferential node number on frequency parameter for different types of variation in thickness of layers.

The frequency parameter ^{3} Hz) with respect to the length parameter

Effect of length of the shell on frequency parameter for different types of variation in thickness of layers under C–C boundary conditions.

Effect of length of the shell on frequency parameter for different types of variation in thickness of layers under H–H boundary conditions.

In Figure

Effect of taper ratio, coefficient of exponential variation, and coefficient of sinusoidal variation on frequency parameter under C–C boundary conditions.

In Figure

Effect of taper ratio, coefficient of exponential variation and coefficient of sinusoidal variation on frequency parameter under H–H boundary conditions.

The influence of the natural frequencies of the vibration of layered cylindrical shells of variable thickness has been analysed. The materials of the layers, length of the shell, and coefficients of variable thickness affect the frequency. A desired frequency of vibration may be obtained by a proper choice of the relative thickness of the layers, length parameter, and the coefficient of thickness variations. The clamped-clamped (C–C) boundary conditions give rise to higher frequencies in comparison with hinged-hinged (H–H) boundary conditions. The nature of variation in thickness of layers considerably affects the natural frequencies. When the circumferential node number is increased, the frequencies initially decrease and then increase. The effect of increasing the length of the cylinder is a decrease in frequencies, for all kinds of variation in thickness of layers. This study also shows the elegance and usefulness of the spline functions with application of the collocation method for boundary value problems.

The differential operators

The differential operators

This work was supported by Inha University Research grant.