Shock and Vibration

Volume 2016 (2016), Article ID 9617957, 10 pages

http://dx.doi.org/10.1155/2016/9617957

## Vibration Analysis of Conical Shells by the Improved Fourier Expansion-Based Differential Quadrature Method

College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China

Received 10 August 2015; Accepted 13 September 2015

Academic Editor: Laurent Mevel

Copyright © 2016 Wanyou Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An improved Fourier expansion-based differential quadrature (DQ) algorithm is proposed to study the free vibration behavior of truncated conical shells with different boundary conditions. The original function is expressed as the Fourier cosine series combined with close-form auxiliary functions. Those auxiliary functions are introduced to ensure and accelerate the convergence of series expansion. The grid points are uniformly distributed along the space. The weighting coefficients in the DQ method are easily obtained by the inverse of the coefficient matrix. The derivatives in both the governing equations and the boundaries are discretized by the DQ method. Natural frequencies and modal shapes can be easily obtained by solving the numerical eigenvalue equations. The accuracy and stability of this proposed method are validated against the results in the literature and a very good agreement is observed. The centrosymmetric properties of these newly proposed weighting coefficients are also validated. Studies on the effects of semivertex angle and the ratio of length to radius are reported.

#### 1. Introduction

Conical shells are widely used in various engineering fields, such as aerospace and ship industries. The development of accurate shell theories has been the subject of significant research interest for many years, and a large number of shell theories based on different approximations and assumptions have been proposed. However, more work is focused on the vibration of cylindrical shells compared with the conical shell. Since the conical coordinate system is function of the meridional direction, the equations of motion for conical shells consist of a set of partial differential equations with variable coefficients. Current methods for the free vibration analysis of thin conical shells can be classified as to analytical methods and numerical methods. Saunders et al. [1], Garnet and Kempner [2], Siu [3], and Lim and Liew [4] have studied the free vibration of uniform conical shells by Rayleigh-Ritz method. Ueda [5] analyzed the same problems using the finite element method. Irie et al. [6] studied a conical shell with variable thickness by the transfer matrix method. The DQ method was employed to analyze the free vibration of a uniform conical shell [7]. Jin et al. [8] studied the free and forced vibration of conical shell using the improved Fourier series method by considering the general boundary conditions. The kernel particle (k-p) functions were employed in hybridized form with harmonic functions to study the vibration of the conical shell based on Ritz method [9].

Besides the studies of the isotropic conical shells, laminated and functional graded conical shells have also been fully studied by various methods [10–16]. The differential quadrature (DQ) method was adopted to solve the differential governing equations of the conical shell in those researches [10–13]. For the vibration of the rotating conical shell, in which the centrifuge force should be taken into consideration, the DQ method was also extensively used to study those problems [10, 17]. The reason that the DQ method is widely adopted to study the vibration behavior of the conical shell is the convenience of transforming the partial differential governing equations approximately into a set of linear algebraic governing equations. Imposing the given boundary conditions, the numerical eigenvalue equations for the free vibration of the (rotating or composite) conical shell are derived and solved.

The differential quadrature method is a numerical technique for solving the differential equations. It was first developed by Bellman et al. [18, 19] and their associates in the early 1970s. The DQ method, akin to the conventional integral quadrature method, approximates the derivative of a function at any location by a linear summation of all the functional values along a mesh line. The key procedure in the DQ method applications lies in the determination of the weighting coefficients. Shu [7, 12, 20, 21] proposed two types of weighting coefficients obtained by the polynomials and the truncated Fourier series, among which the Lagrange interpolation functions are widely used for their simplicity explicitness [7, 21]. Although it is well known that Lagrange interpolation functions are limited by the number of interpolation points and severe oscillation may take place if the order is large, the use of the Gauss-Chebyshev points [7, 21] can accelerate the convergence rate of the DQ method. Some works focusing on improving the accuracy and stability of the DQ method are presented by proposing different ways to generate the weighting coefficients and to determine the distribution of grid points [22–24].

The Fourier series with auxiliary functions was first proposed by Li [25] to study the vibration problems of the beam structure. This method is extensively used to study the 2D and 3D structural vibration and vibroacoustic problems [26–30]. The auxiliary functions are introduced to accelerate the convergence and deal with all the possible discontinuities, at the end points or edges, associated with the original Fourier cosine series. This improved Fourier series method becomes a promising method to study the structural vibration problems.

In this paper, the improved Fourier expansion-based differential quadrature method is proposed to solve the free vibration behavior of the truncated conical shell. The weighting coefficients are obtained based on this improved Fourier series in a much easier way. The centrosymmetric properties of these newly proposed weighting coefficients are also validated. The following sections will illustrate the development of this hybrid method, and numerical results are then presented to validate the effectiveness, accuracy, and stability of this current method on predicting the modal characteristics of the conical shell.

#### 2. Theoretical Formulation

For a continuous function defined on with an absolutely integrable derivative, it can be expanded in Fourier cosine series:

The first-order derivative of can be done term-by-term:

The second-order derivative of cannot be obtained term-by-term, which is shown as

These formulations basically tell that while a cosine series can always be differentiated term-by-term, this can be done to a sine series only if . To implement the differential quadrature algorithm, the auxiliary functions are added to traditional Fourier cosine series to cover the discontinuity of the function at the end points and to get the derivatives term-by-term. A function can be expanded as [27]wherewhere and represent the unknown Fourier expansion coefficients. The supplementary functions can be represented as arbitrary continuous functions, regardless of the boundaries. It is easy to verify that and all the other 1st-order and 3rd-order derivatives are identically equal to zero at both ends. The main purpose of introducing these supplementary functions to standard Fourier series is to get the first four derivatives of the Fourier cosine series term-by-term. As an immediate numerical benefit, the Fourier series in (4) will converge uniformly at an accelerated rate.

To implement the differential quadrature method, points are equally distributed on :

The functional values at those grid points can be determined as

The Fourier series is truncated to . Rewrite (7) into the matrix form

The Fourier series coefficients can be obtained by the inverse of the matrixin which

In this proposed method, the number of truncated Fourier series and the number of grid points follow the relation that to ensure is a square matrix to let the inverse be more accurate. Once the constant matrix is determined, the approximated Fourier series coefficients are obtained. When the DQ method was first developed, polynomials were adopted to follow this procedure to generate weighting coefficients which would lead to highly ill condition when is large. The Fourier series, however, show much more stability to derive the coefficients by the inversion of which will be validated in the results section.

The first-order derivatives at those grid points are

Rewrite (11) into the matrix form, in which is the first-order derivative of and is the first-order weighting coefficient matrix of the DQ method: . It is obvious from the above equation that the weighting coefficients of the second- and higher-order derivatives can be completely determined through the same way, which are expressed as in which () is the th-order derivative of and is the th-order weighting coefficient matrix of the DQ method: .

In this paper, only four supplementary functions are added to the Fourier cosine series, which will ensure the first-four-order derivatives to converge at a high rate and to keep stability of the Fourier series. Consequently first-four-order weighting coefficient matrixes can be obtained which are sufficiently enough to study the vibration of a conical shell. Adding more supplementary functions to the Fourier cosine series will give the capability to study the corresponding higher-order partial differential equations.

#### 3. Free Vibration Behavior of a Conical Shell

The free vibration behavior of conical shells has been studied by Shu [7] by the DQ method. This model is adopted again to validate the efficiency, accuracy, and stability of this proposed method.

Consider a conical shell structure with semivertex angle and the radius of the large edge is , as shown in Figure 1. The displacement fields of the conical shell in , , and directions are denoted by , , and , respectively. If the couplings between these three displacement components are ignored, the field functions can be expressed asin which and are the circumferential wave number and the frequency in rad/sec, respectively.