Effect of Internal Moment Release on the Eigenfrequencies of Combined Linear Systems

Abstract

A method for analyzing the free vibration of complex structural systems, consisting of a simple oscillator attached to a beam with an internal hinge, is presented. A mathematical model possessing important features of singularity functions with their higher order derivatives is proposed to account for the effect of internal hinge and spring force interacting between the oscillator and supporting structure. The particular integral approach together with Laplace transformation is proved to be an efficient alternative to solve the generalized differential equation for the normal modes of dynamically combined systems. Exact vibration frequencies for clamped–pinned and pinned–pinned boundaries are determined. The results are extended to the cases in which the oscillator or internal hinge is removed from the structure. It is shown that the presence of internal hinge does not alter the generalized orthogonality relation for the combined system. The search for the optimal location of the internal hinge, which maximizes the desired natural frequency, is discussed. It is concluded that a combined system with an optimally positioned hinge vibrates with the same natural frequency as an equivalent system without a hinge.