Keywords: free vibration, dynamic analysis, generalized collocation method, domain decomposition, shell of revolution, compound shell.

The analysis of elastic shells has attracted the attention of many researchers, in the last few decades and several studies have been presented for the vibration analysis of such structural elements. The most used numerical tool in carrying out these analyses is probably the finite element method. Although axisymmetric shells can be analyzed using general curvilinear shell finite elements, the use of axisymmetrig (ring) elements with circumferential uncoupling of all dependent variables by means of proper Fourier series expansion is preferred, as it generally leads to much more efficient analysis [1].

Recently, the use of the collocation methods has been adopted in structural mechanics, with a few applications also in the free vibration analysis of cylindrical and conical shells [2]. Although very interesting results can be found in these studies, it is to be noticed that on one hand the use of a shell theory which does not take into account shear deformability seems to be quite restrictive and introduces some further approximation to the model; on the other hand the cases of more complex rotational shells shapes (doubly curved, compound, and closed apex) have not been fully investigated. Therefore, it is the aim of the present paper to make a contribution in the direction of extending the generalized collocation technique to other kinds of shells of revolution, within the frame of a shell theory capable of taking into account both rotary inertias and shear deformations.

The first step of the analysis herein presented is to transform each dependent variable of the problem into a partial Fourier series of harmonic components, which vary along the meridian curve only. These expansions permit taking into account both dynamic equilibrium equations in terms of stress resultants and couples and strain displacements relationships with transverse shear measures, by means of the harmonic amplitudes of the forces and displacements. The material is assumed to be linearly elastic [3]. Incorporating equilibrium, deformation and constitutive equations, leads to the formulation of the dynamic equilibrium of the shell, in terms of harmonic amplitudes of generalized displacements only. The problem is then identified by a system of five 2nd-order linear partial differential equations, in terms of midsurface displacements and rotations, together with a set of five boundary conditions also written in terms of displacements too. This formulation avoids the so-called -point technique [2] and gives rise to the correct formulation of equilibrium and boundary assignments, in correspondence of the apex, for the case of closed rotational shells. Finally, the assumption of harmonic motion with respect to time gives a set of time-invariant ordinary differential equations of dynamic equilibrium with circular frequencies .

An efficient solution technique for this kind of systems is offered by the generalized collocation and passes through five basic steps [4]: Discretization (or collocation) of the space variable (along-meridian coordinate) into a suitable number of grid points. Approximation of dependent variables (mid-surface displacements and rotations), through collocation points values, by an interpolation rule. Approximation of space derivatives using the above mentioned interpolation. Transformation of the original continuous b.v. problem into a set of discrete algebraic systems, each one assigned at an internal (domain) node; boundary conditions being imposed in the domain boundary nodes. Solution of the discretized model which, in this case, typically results in a linear eigenvalue problem.

• Discretization (or collocation) of the space variable (along-meridian coordinate) into a suitable number of grid points.
• Approximation of dependent variables (mid-surface displacements and rotations), through collocation points values, by an interpolation rule.
• Approximation of space derivatives using the above mentioned interpolation.
• Transformation of the original continuous b.v. problem into a set of discrete algebraic systems, each one assigned at an internal (domain) node; boundary conditions being imposed in the domain boundary nodes.
• Solution of the discretized model which, in this case, typically results in a linear eigenvalue problem.

It is evident that crucial to the analysis and to the implementation of the solving codes are the choices of collocation points and interpolation rules. Commonly, Lagrange interpolating polynomials are used in structural mechanics problems treated with the G.D.Q. technique. The effectiveness of the presented procedure is emphasized, by comparing present results from those available in the literature. To verify the reliability of the method, selected examples are treated, evaluating the accuracy of the eigenfrequencies for cases where a closed-form solution is unavailable. Last, it is shown how the case of closed-apex and compound rotational shells, with or without meridional smoothness, can be analyzed easily using a general domain decomposition technique and that good accuracy is achieved.

1

Gould, P.L., "Finite Element Analysis of Shells of Revolution", Pitman Advanced Publishing Program, Boston, Massachusetts, 1984.

2

Bert, C.W., Malik, M., "Free Vibration Analysis of Thin Cylindrical Shells by the Differential Quadrature Method", ASME Journal of Pressure Vessel Technology, 118, 1-12, 1996. doi:10.1115/1.2842156

3

Gould, P.L., "Analysis of Plates and Shells", Prentice Hall, Upper Saddle River, New Jersey, 1999.

4

Bellomo, N., "Nonlinear Models and Problems in Applied Sciences from Differential Quadrature to Generalized Collocation Methods", Mathematical and Computer Modelling, 26(4), 13-34, 1997. doi:10.1016/S0895-7177(97)00142-8

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