Ah. Nayfeh et Sa. Nayfeh, NONLINEAR NORMAL-MODES OF A CONTINUOUS SYSTEM WITH QUADRATIC NONLINEARITIES, Journal of vibration and acoustics, 117(2), 1995, pp. 199-205
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.