Free and Forced Vibrations of Fuzzy Structures

Stochastic free and forced vibrations of layered beams are analyzed that result from a single bounded random stiffness parameter whose probability density function is considered to be zero outside of a given interval, i.e., it is a member of a fuzzy set with bounded uncertainty. The relevant properties of natural vibrations of an ensemble of sandwich beams with three perfectly bonded layers under hard hinged support conditions are worked out in detail when a bounded random shear stiffness of the core material is assigned by employing interval mathematics. The main structure of a compound single-span railway bridge, effectively modeled as a two-layer beam, is subjected to a single moving load as well as to a series of repetitive moving loads traveling with constant speed. It serves as a complex example for the resulting forced random vibrations and resonances under the severe condition of an elastic interface slip of bounded random stiffness. In both cases exact homogenization yields a stochastic sixth-order partial differential equation of motion of the layered beam. Light modal damping is considered. The analysis of the illustrative problems is based on the interval representation with a triangular membership function of the stiffness modulus assigned. A short comment provides information on the limits of such triangular membership functions. Membership functions in the form of envelopes of the random natural frequencies, the dynamic magnification factors, and the phase angles in free vibrations are determined. Both, fuzzy peak deflection and acceleration are derived for the forced single-span compound railway bridge subjected to the moving loads. Approximating superposition of modal maxima is considered by standard routines of reliability analysis.