Geometrically nonlinear transverse vibrations of Bernoulli-Euler beams carrying a finite number of masses and taking into account their rotatory inertia

The objective of this paper is to establish the theoretical formulation of the problem of nonlinear transverse vibrations of Bernoulli-Euler beams carrying a finite number of masses at arbitrary positions, with general end conditions. The generality of the approach is based on use of translational and rotational springs at both ends, allowing examination of all possible combinations of classical beam end conditions, as well as elastic restraints. The method used is based Hamilton's principle and spectral analysis for nonlinear free vibrations exhibiting large displacement amplitudes. The problem is reduced to solution of a nonlinear algebraic system using numerical or analytical methods. This has been previously applied to nonlinear transverse vibrations of continuous structures such as beams, plates and shells, to nonlinear longitudinal vibrations of 2-dof and multi-dof systems and to nonlinear transverse vibrations of N-dof systems. The nonlinear algebraic system has been solved using an approximate explicit method developed previously (The so-called second formulation) leading to the nonlinear fundamental mode shape of beams carrying a finite number of masses and to the corresponding backbone curves.