NONLINEAR NORMAL-MODES OF A CONTINUOUS SYSTEM WITH QUADRATIC NONLINEARITIES

We use two approaches to determine the nonlinear modes and natural fre quencies of a simply supported Euler-Bernoulli beam resting on an elas tic foundation with distributed quadratic and cubic nonlinearities. In the first approach, we use the method of multiple scales to treat the governing partial-differential equation and boundary conditions direc tly. In the second approach, we use a Galerkin procedure to discretize the system and then determine the normal modes from the discretized e quations by using the method of multiple scales and the invariant mani fold approach. Whereas one- and two-mode discretizations produce erron eous results for continuous systems with quadratic and cubic nonlinear ities, all methods, in the present case, produce the same results beca use the discretization is carried out by using a complete set of basis functions that satisfy the boundary conditions.