VIBRATION ANALYSES OF OPEN SHELLS OF REVOLUTION

2008

Thesis

Degree Name

Doctor of Philosophy

Program

Mechanical and Materials Engineering

Supervisor

Dr. Anand V. Singh

Dr. Samuel F. Asokanthan

Abstract

A numerical formulation based on the Rayleigh-Ritz method is presented for the static, free vibration and transient response analyses of open shells of revolution such as cylinder, cone and sphere. The shell equations are derived from the three dimensional elasticity equations in the curvilinear coordinates defined on the middle surface of the shell. The equilibrium equation/equation of motion is variationally obtained by the minimization of potential energy under the assumptions of the first order shear deformable shell theory primarily in terms of the parameters on the middle surface. Also, rotary inertia and shear deformation are retained so that reasonably accurate analyses of thin to moderately thick shells can be performed. Based on the need and the complexity of the problem, the shell model will have either one surface patch or multiple patches by dividing the middle surface into quadrilateral segments along the two coordinate lines. Two sets of nodes are defined on each patch; one defines the geometry by two coordinates and the other defines the displacement fields associated with three translational and two rotational components. In the solution stage, two types of polynomials, viz. simple algebraic and Bezier, are considered to define the displacement fields. In the free vibration analysis, the eigen value problem is solved by iteration method. The matrix equation of motion for the forced vibration is solved using two techniques, indirect (combination of mode summation, state-space, and Runge-Kutta algorithm) and direct (Newmark and Wilson-theta) numerical integration methods. To assess the validity and performance of the present method, convergence study is performed first with each of the cases reported. A few classes of problems that are attended in this thesis are open skewed cylindrical panels clamped at the curved edges and free at the straight parallel edges, open skewed cylindrical shells supported only on the middle-third of the straight edges, open shells supported only on portions of the boundaries, and open shells subjected to loads uniformly distributed over the entire surface and on a patch. Wherever applicable, comparisons are made with those results available in the literature and also with those from commercial and in-house developed iii finite element software. In the free vibration cases, parametric studies with respect to shell thickness, subtended angle, and length are performed. The analysis is further extended to conical and spherical shell panels under similar boundary and loading conditions. It is observed that the overall performance of this method is in comparable standards with other methods and offers high degree of simplicity compared to the low order finite elements. For a given order, both simple algebraic and Bezier polynomials give essentially identical results. In many commonly occurring situations, single-patch model proved to be efficient and reliable; however, the need for discretization of the surface is unavoidable in the problems involving partial supports and localized loads. In comparison, the multi-patch model is seen to provide a considerable degree of flexibility, stability and applicability. In the transient response analyses, results from Runge-Kutta, Wilson-theta, and Newmark methods are presented and discussed. Though all these three methods provide very close results in all cases, Newmark method is seen to provide a better performance.

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