Nonlinear Dynamics of a Sliding Beam on Two Supports and its Efficacy as a Non-Traditional Isolator

Abstract

This study deals with the nonlinear dynamics associated with large deformations of a beam sliding on two knife-edge supports under external excitation. The beam will be referred to as Gospodnetic-Frisch-Fay beam, after the researchers who reported its static deformation in a closed form. The freedom of the beam to slide on its supports imparts a nonlinear characteristic to the force-deflection response. The restoring elastic force of the beam possesses characteristics similar to those of the roll restoring moment of ships. The Gospodnetic-Frisch-Fay exact solution is given in terms of elliptic functions, and a curve fit of the exact solution up to eleventh-order is constructed to establish the governing equation of motion under external excitation. The dynamic stability of the unperturbed beam is examined for the damped and undamped cases. The undamped case reveals periodic orbits and one homoclinic orbit depending on the initial conditions. The steady state response and transmissibility of the beam are predicted using multiple-scales method and numerical simulation. The transmissibility of the nonlinear beam outperforms linear isolators only at the linear resonance frequency. The response to a sinusoidal excitation at a frequency below the linear natural frequency is numerically estimated for different excitation amplitudes and different values of initial conditions covered by the area of the homoclinic orbit. The safe basins of attraction are plotted for different values of excitation amplitude. It is found that the safe region of operation is reduced as the excitation amplitude increases. Under random excitation, the response probability density function is obtained in a closed form and the response mean energy is found to depend nonlinearly on the excitation spectral density.