Solution to Vibrations of Double-Beam Systems under General Boundary Conditions

Abstract

The problem of vibrations of double-beam systems is of technological interest and has been extensively studied in the literature. However, a general solution is yet to be developed for the general boundary conditions. In this paper, a general solution of the free and forced vibrations of double-beam systems under classical, nonclassical, and mixed boundary conditions is derived. The obtained mode-shape solutions are exact and explicit. The characteristic frequency equation of a double-beam system is of order-4 and order-8 for the classical boundary conditions and the nonclassical boundary conditions, respectively. The solution of forced vibrations is developed with the classical modal expansion technique. In addition to the theory, applications of the proposed method are illustrated by numerical examples. The obtained frequencies for special problems are compared with the literature, and the agreement between the proposed method and the literature is remarkably good. This shows that the proposed shape function method may be applied reliably to determine the transverse vibration mode shapes and natural frequencies of the double-beam system under the general boundary conditions. The solutions of free and forced vibrations of the double-beam systems presented in the paper could easily be programmed into a computer for vibration analyses.