Deterministic, stochastic and convex analyses of one- and two-dimensional periodic structures

The periodic structures considered in the dissertation are one-dimensional periodic multi-span beams, and two-dimensional periodic grillages with elastic interior supports. The following specific topics are included: (1) Deterministic Vibration--Exact solutions are obtained for free vibrations of both multi-span beams and grillages, by utilizing the wave propagation concept. The wave motions at the periodic supports/nodes are investigated and the dispersion equations are derived from which the natural frequencies of the periodic structures are determined. The emphasis is placed on the calculation of mode shapes of both types of periodic structures. The general expressions for mode shapes with various boundary conditions are obtained. These mode shapes are used to evaluate the exact dynamic response to a convected harmonic loading. (2) Stochastic Vibration--A multi-span beam under stochastic acoustic loading is considered. The exact analytical expressions for the spectral densities are derived for both displacement and bending moment by using the normal mode approach. Nonlinear vibration of a multi-span beam with axial restraint and initial imperfection are also investigated. In the latter case, the external excitation is idealized as a Gaussian white nose. An expression for the joint probability density function in the generalized coordinates is obtained and used to evaluate the mean square response of a multi-span beam system. (3) Convex Modeling of Uncertain Excitation Field--It is assumed that the parameters of the stochastic excitation field are uncertain and belong to a multi-dimensional convex set. A new approach is developed to determine the multi-dimensional ellipsoidal convex set with a minimum volume. The most and least favorable responses of a multi-span beam are then determined for such a convex set, corresponding to a stochastic acoustic field. The procedure is illustrated in several examples.