Large Amplitude Forced Vibrations of Restrained Beams Resting on Elastic Point Supports

The present paper concerns the study of geometrically non-linear forced vibrations of beams resting on two different types of springs: rotational and translational. Assuming that the motion is harmonic, the displacement is extended as a series of spatial functions determined by solving the linear problem. Hamilton’s principle and spectral analysis are used to reduce the problem to a non-linear algebraic system solved using a previously developed approximate method. The effects of the nature of the added springs and their location on the non-linear behaviour of the beam are examined. A multimode approach is used in the forced case to obtain results over a wide range of vibration amplitudes. This leads to examining the non-linear forced dynamic response for different positions of each spring and different levels of excitations. Following a parametric study, the non-linear forced mode shapes and their associated bending moments are presented for different levels of excitations and for different vibration amplitudes to give an estimation of the stress distribution over the beam length.