We use several methods to study the nonlinear modes of one-dimensional
continuous systems with cubic inertia and geometric nonlinearities. I
nvariant manifold and perturbation methods applied to the discretized
system and the method of multiple scales applied to the partial-differ
ential equation and boundary conditions are discussed and their equiva
lence is demonstrated. The method of multiple scales is then applied d
irectly to the partial-differential equation and boundary conditions g
overning several nonlinear beam problems.