#### Date of Award

Summer 2000

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mechanical Engineering - (Ph.D.)

#### Department

Mechanical Engineering

#### First Advisor

Bernard Koplik

#### Second Advisor

Rong-Yaw Chen

#### Third Advisor

Benedict C. Sun

#### Fourth Advisor

John Tavantzis

#### Fifth Advisor

A. C. Ugural

#### Abstract

Based on Hamilton's principle and the theory of elasticity, both linear and nonlinear governing equations of motion are derived for elastic spherical shells. Most of the analysis is focused on the free axisymmetric vibrations of homogeneous and sandwich shallow spherical shells. The nonlinear behavior, which allows for small deflections, is also discussed.

The linear vibrations of homogeneous and sandwich shallow spherical shells with a hole at the center are investigated with emphasis placed on the effect of thickness-shear deformation and rotatory inertia. The effects of curvature and the size of the central hole are studied for free axisymmetric motions of spherical shells with various types of edge conditions. Two tracers are introduced to identify the transverse- shear deformation and rotatory inertia terms in the homogeneous shell and in the core of the sandwich shell. The refined model is derived which includes these two effects while the classical model is formulated by setting the tracers equal to zero. The face layers of the sandwich shell are taken to be membranes. Equations governing the axisymmetric motion of shallow shells are deduced yielding a system of three second-order differential equations in terms of the displacements and time. The numerical solution of the governing equations is obtained by using the *Shooting Method*, in which the natural frequency is assumed to be the same in all displacements. By introducing dimensionless -ratios related to the elastic and geometric properties of the shell, the equations appear in a form which is readily adaptable for solutions of homogeneous and sandwich shallow spherical shells as well as circular plates.

Numerical results for the linear vibrations of a sandwich shell reveal that the effect of thickness-shear deformation and rotatory inertia must be considered in calculating the natural frequencies. For both homogeneous and sandwich shells, the curvature has a dramatic effect on the increase of the fundamental frequencies. The fundamental frequencies increase with the increase of the size of the central hole for all of the boundary conditions investigated for the refined model. For the classical model, the frequency increase with size of hole is limited to the case of the clamped outer boundary conditions.

Numerical results show that for the nonlinear vibrations of the homogeneous shell with relatively small curvature, the magnitude of the deflection should be small in order to get, the natural frequency.

On the other hand, the *Finite Difference Method*, which is employed to solve both the linear and nonlinear cases, shows that the natural frequency may not be the same for all displacements.

#### Recommended Citation

Wong, Wai-Kwong, "Axisymetric vibrations of homogeneous and sandwich spherical shells with a hole at the center" (2000). *Dissertations*. 424.

https://digitalcommons.njit.edu/dissertations/424