A nonlinear oscillator modeling the transverse vibrations of a rod

This paper has been originated by the motion analysis of a flexible rod whose extreme (pivoted) sections cannot undergo any displacement, but only rotate. First an historical outline of the rod tranverse vibration analysis is provided. The dynamical elastica of such a system can be easy expressed through a linear fourth order PDE solved by a Fourier series: the rod resulting movement is then known by armonical superposition, but the overall vibration rod's period is analytically unexpressed. The authors propose an alternative analytic treatment designing a discrete model, made of a single degree of freedom oscillator, ruled by a nonlinear second order ODE, here solved through the elliptic integrals of first and third kind. This greater complexity is balanced by having found a closed formula for computing the overall vibration period by means of a suitable punctual oscillator.