A nonlinear partial differential equation which describes the natural free-free vibrations of beamlike structures is derived, and then solved approximately by reduction to a nonlinear ordinary differential equation through the use of the Duffing and Ritz-Kantorovich methods. This ordinary differential equation, which is a nonlinear eigenvalue problem, is then solved by perturbation and shooting methods. The solutions show that higher modes of vibration are not possible for the nonlinear vibration.