  In this work we obtain an approximate solution of the strongly nonlinear
second order differential equation $\frac{d^{2}u}{dt^{2}}+\omega ^{2}u+\alpha
u^{2}\frac{d^{2}u}{dt^{2}}+\alpha u\left( \frac{du}{dt}\right)^{2}+\beta \omega
^{2}u^{3}=0$, describing the large amplitude free vibrations of a uniform
cantilever beam, by using a method based on the Laplace transform, and the
convolution theorem. By reformulating the initial differential equation as an
integral equation, with the use of an iterative procedure, an approximate
solution of the nonlinear vibration equation can be obtained in any order of
approximation. The iterative approximate solutions are compared with the exact
numerical solution of the vibration equation.
