Analytical and Experimental Studies on the Large Amplitude Free Vibrations of Variable-Arc-Length Beams

Abstract

Using the finite element method, we investigate large amplitude vibrations of horizontal variable-arc-length beams, considering the effect of large initial static sag deflections due to self-weight. The variability in beam arc-length arises from one end being pinned, and the other end being supported by a frictionless roller at a fixed distance from the pinned end. Using Lagrange’s equation of motion, the large amplitude free vibration equation of motion is derived based on the variational formulation. Included in the formulation are the energy dissipation due to large bending using the exact non-linear expression of curvature and the non-linearity arising from axial force. The non-linear eigenvalue problem is solved by the direct iteration method to obtain the beam’s non-linear frequencies and corresponding mode shapes for specified vibration amplitudes. We also present changes in the frequency of vibration as a function of amplitude, demonstrating the beam non-linearity. A more accurate solution analyzed in the frequency domain of the direct numerical integration method is adopted as an alternative solution. Large amplitude vibration experimental modal analysis was also conducted to complement the analytical results. The measured results were found to be in good agreement with those obtained from two analytical solutions.