Variational approach to vibrations of frames with inclined members

Abstract

This paper deals with the use of calculus of variations to derive the boundary value problems which describe the dynamical behaviour of two and three-bar frames with inclined members, and with ends and intermediate points elastically restrained. The determination of exact eigenfrequencies and modes in the case of free vibrations is included. A rigorous and complete development is presented. First, a brief description of textbooks and papers previously published is included. Second the variational formulation of the problem is presented. Third, the Hamilton principle is rigorously stated and the corresponding boundary value problem is obtained. Finally, the method of separation of variables is used for the determination of the exact frequencies and mode shapes. In order to obtain an indication of the accuracy of the developed mathematical model, some cases available in the literature have been considered and a special code implementing the finite element method have been developed and used. New results are presented for different geometrical configurations and mechanical parameters.