On the Free Vibrations of Variable-Arc-Length Beams: Analytical and Experimental

Abstract

Vibrations of horizontal, variable-arc-length beams considering the effects of large initial static sag deflections due to self weight are investigated. 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, the large amplitude, free-vibration, equation of motion can be derived and simplified for small amplitude motion. These formulations represent small amplitude vibration about the large deformed static sag configurations. A solution is obtained by using the nonlinear finite element method. The vibration eigenvalue problem is solved using inverse iteration techniques to obtain the natural frequencies and corresponding mode shapes. Free and forced vibration experimental studies were conducted. Frequencies and mode shapes are shown for a range of beam specimens to complement the analytical results.