NON-LINEAR VIBRATIONS OF A BEAM ON AN ELASTIC FOUNDATION

The non-linear vibrations of an elastic beam resting on a non-linear tensionless Winkler foundation subjected to a concentrated load at the centre is presented in this paper. Since the foundation is assumed to be tensionless, the beam may lift off the foundation and there exists different regions namely contact and non-contact regions. Since the contact regions are not known in advance, the problem appears as a non-linear one even though there is no non-linear term in the foundation model. In this case, the calculation of the roots of a non-linear equation is needed to obtain contact lengths. The perturbation technique is used to solve the non-linear governing equation associated with the problem. Using this technique, the non-linear problem is reduced to the solution of a set of linearized partial differential equations. The lift-off points and the displacements are obtained in linear and non-linear cases, and the variation of these points with respect to various parameters are presented. It is concluded that the contact length varies with the magnitude of the load because of the non-linearity.