Nonlinear transverse vibrations of clamped beams carrying two or three concentrated masses at various locations

Abstract

In a recent work, a discrete model for geometrically nonlinear transverse free constrained vibrations of beams with various end conditions has been developed and validated via comparison with known results corresponding to nonlinear vibration of clamped beams carrying a concentrated mass. It is extended here to continuous beams carrying two or three concentrated masses at various locations and subjected to large vibration amplitudes. The discrete model used is an N-dof (N-Degrees of Freedom) system made of N masses placed at the ends of solid bars connected by springs, presenting the beam flexural rigidity. The large transverse displacements of the bar ends induce a variation in their lengths giving rise to axial forces modelled by longitudinal springs causing nonlinearity. The calculations made allowed application of the semi-analytical model developed previously for nonlinear structural vibration involving three tensors, namely the mass tensor mij, the linear rigidity tensor kij and the nonlinearity tensor bijkl presenting the effect of the change in the bar lengths. The addition of three concentrated masses studied here induces a change in the mass matrix. By application of Hamilton’s principle and spectral analysis in the modal basis, the nonlinear vibration problem is reduced to a nonlinear algebraic system, using an explicit method, developed previously for non-linear structural vibration. This study shows that concentrated masses may be used for practical purposes to shift the resonant frequency; if the three masses locations are appropriately chosen.