Date of Award

Spring 1987

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mechanical & Aerospace Engineering


Engineering Mechanics

Committee Director

Chuh Mei

Committee Member

Earl A. Thornton

Committee Member

Stephen G. Cupschalk

Committee Member

Gene Hou

Committee Member

J. M. Dorrepaal


The literature and experiments have shown that nonlinear vibrations produce significant effects in structural analysis, especially the frequency-amplitude-force relation and the analysis of strain. An analysis was developed to predict both the frequency-amplitude-force relation and strains of beam and plate structures. Two finite element methods were developed, namely, the iterative single-mode method (method I) and the multiple-mode method (method II). The harmonic force matrix was developed to analyze nonlinear forced vibrations. Nonlinear free vibration was a special case of the general forced vibration by setting the harmonic force matrix equal to zero. The harmonic force matrix represents the external applied force in matrix form, instead of a vector form, so that the analysis of nonlinear force vibrations can be performed as an eigenvalue problem.

The study showed that the effect of midplane stretching due to large deflection is to increase the nonlinearity. However, the effects of inplane displacements and inertia (IDI) are to reduce nonlinearity. The concentrated force case yields a more severe response than the uniform distributed force case. For beams and plates with end supports restrained from axial movement (immovable case) only the hardening type nonlinearity is observed. For beams with large slenderness ratio (L/R $\le$ 100) with movable end supports, the increase in nonlinearity due to large deflection is partially compensated by a reduction in nonlinearity due to inplane displacement and inertia. This leads to a negligible hardening type nonlinearity, therefore, the small deflection linear solution can be employed. However, for beams with a small slenderness ratio (L/R = 20) and movable end supports, the softening type nonlinearity is found. The effect of the higher modes is more pronounced for the clamped supported beam than the simply supported one. The beam without inplane displacement and inertia (IDI) yields more effect of the higher modes than the one with inplane displacement and inertia. For beams, method I and method II converge into a true deflection shape, provided the number of modes for method II is high enough. Similarly, both method I and method II yield accurate strains provided the number of modes for method II is high enough.