Vibration of shells is important for different fields of engineering applications.
Accordingly, many efforts have been made in studying the vibrations of plates
and shells with different scales. Sharma and Mittal (2010)
presented a review on stress and vibration analysis of composite plates. Jayakumar
et al. (2006a) studied multi-layer cylindrical shells under electro-thermo-mechanical
loads. Jayakumar et al. (2006b) also investigated
on piezoelectric cylindrical shells under thermal and pressure loads. They presented
a closed-form solution, utilizing a classical stress formulation approach to
carry out elasto-electro-thermo analysis of generalized plane-strain of a right
circular cylindrical shell. Some researchers have considered the free vibrations
of conical shells due to their use in nozzles. Garnet and
Kemper (1964) analyzed the free vibration of isotropic conical shells using
the Rayleigh Ritz method. Recently, Zhao and Liew (2011)
considered the vibrations of Functional Graded Materials (FGMs) conical panels
and suggested that the effect of thickness on the vibration modes of these structures
was an important factor. Investigation of free vibrations of cylindrical shells
rotating at high speed was performed by Chen et al.
(1993). Irie et al. (1984) also calculated
the natural frequencies of truncated conical shells. Li
et al. (2009) conducted a field study on free and forced vibrations
of truncated conical shells using the Rayleigh Ritz method. Wu
and Lee (2001) applied the Differential Quadrature (DQ) method for studying
free vibrations of conical shells with variable thickness. Generalized Differential
Quadrature (GDQ) method was performed for the first time by Shu
(1996) with square differential correction method for the vibration analysis
of layered isotropic conical shells. Civalek (2007)
used the Discrete Singular Convolution (DSC) to investigate the frequency response
of conical shells. Hu et al. (2002) studied the
vibrations of composite twisted conical shells with respect to the strain tensor.
Sofiyev et al. (2009) studied the vibrations
of orthotropic non-homogeneous conical shells with free boundary conditions.
Tripathi et al. (2007) studied the free vibration
of composite conical shells with random material properties of the finite element
method. However, most of the previous works have been conducted in order to
simply support boundary conditions. Since boundary conditions may have a significant
effect on the response of structural vibrations, in this study, the free vibration
of composite conical shells was investigated under various boundary conditions
using the solution of beam function and Galerkin method.
According to Fig. 1 and 2, the governing
equations of the conical shell with the length of L, thickness of h and radii
of R1 and R2 on the two ends of the cone and half angle
of the head α, based on the approximate extension of Chen
et al. (1993) were:
where, ρ is the average density in the z direction. The resultant forces and moments can be defined as:
|| Top and front view of a truncated conical shell
|| Side view of a truncated conical shell
In the above equation, Aij, Bij and Dij are stiffness coefficients. Strain and curvature in the middle of the shell were as follows:
Using Eq. 1 to 3, the governing equations
can be obtained based on the movement as follows:
In Eq. 4, Lij are derivative operators.
SOLVING THE GOVERNING EQUATIONS
To use Galerkin method for solving the governing equations, displacement field functions must be guessed at first. The field should be set in such a way to satisfy the boundary conditions. Displacement field was proposed as follows:
where A, B and C are fixed parameters and represent the amount of vibrations,
n is the number of half waves along the peripheral, ω is angular frequency
of vibrations and; φ(x) is the meridional function that satisfied boundary
conditions of the geometric scaling.
|| Comparison of frequency parameters for an isotropic conical
shell with a simply support boundary conditions having different vertex
φ(x) function could be determined from the shell and beam theories using
the same boundary conditions. By embedding Eq. 5 in Eq.
4, the residuals Rij could be found. These residuals could be
attained by applying operators Lij on the same approximation functions
in the following way:
The Galerkin method was applied as shown below:
By integrating the past three equations, a 3x3 homogeneous system was found. To reach the non-zero solution of this system, the determinant of its coefficients should be equal to zero. By solving the equations, natural frequencies and corresponding modes of vibrations could be found. Numerical results were presented in order to introduce a dimensionless frequency parameter in the following way:
Table 1 shows the values of frequency parameter for the simply
supported boundary conditions of an isotropic conical shell with different vertex
angles. For comparison, some results Irie et al.
(1984) and Li et al. (2009) are also comprised
in this table.
|| Comparison of frequency parameters for composite conical
As can be seen, there is good agreement between the two sets of results, which
indicates the accuracy and efficiency of the method in studying the vibrations
of conical shells.
Table 2 also shows the possibility of comparing the present
results with the ones by Wu and Lee (2001) as far as
the vibration response of composite conical shells is concerned. In this table,
the frequency parameters of a two-layer conical composite shell with two layers
of non-symmetric cross-ply and with the simply supported boundary conditions
in both ends are presented for different thicknesses. The results confirmed
that the method was suitable for analyzing the vibrations of composite conical
RESULTS AND DISCUSSION
Figure 3 shows that, in the first mode, the frequency parameter
of a two-layered cross-ply asymmetric conical shell with a half 30° cone
angle under different boundary conditions changed relative to the number of
axial half-wave environment. This figure also shows that the frequency of the
shell decreased and then increased for all types of the boundary conditions
with the increase in the number of half-wave. The results are also confirmed
what Wu and Lee (2001), Irie et
al. (1984) and Li et al. (2009) are claimed
using different formulations.
The figure represents the effect of boundary conditions on the vibration behavior
of shells so that the wave number corresponding to the smallest frequency (fundamental
frequency) is different for different boundary conditions.
||Changes in the number of half-wave frequency parameter setting
for composite conical shells under various boundary conditions
According to the curves, the number of half-wave of the fundamental frequency
was equal to 4, 5, 5, 4, 2 and 4 for boundary conditions of FS (Free-Simply
Support), CF (Clamped-Free), CS (Clamped-Simply Support), FF (Free-Free), CC
(Clamped-Clamped) and SS (Simply Support-Simply Support), respectively.
This article showed that the Galerkin method with beam functions can be used well in calculating the natural frequency of the truncated conical shells with different boundary conditions. It can be also concluded that the boundary conditions significantly affected the response of structural vibrations. It was also found that no matter what the boundary conditions were, the frequency of the shell decreased and then increased with the increase in the number of half-wave.