Vibration and snapping of a self-contacted beam under prescribed end rotations

Abstract

An initially straight beam is first bent into a circular configuration. Equal rotation angles of opposite direction are then prescribed at the two ends to bend the beam into self contact and eventually snap. We conduct a vibration analysis based on an Eulerian formulation to determine the natural frequencies and the stability of the bent beam. The theory predicts that the rotated beam snaps sideways at the bifurcation point. In experiments, however, the self-contacted beam passes far beyond the bifurcation point and snaps symmetrically by squeezing itself past the center of the two end points to the other side. By looking into the mode shapes of the unstable mode at the bifurcation point, it is believed that the predicted sideway snapping may be prevented by the sliding friction between the contact surfaces, which is not included in the theory. Instead, the beam snaps in a second unstable mode which involves rolling between contact surfaces. We then propose an imperfection analysis in which the bent beam is slanted to one side by a small angle in its initial configuration. In an experiment with a slant angle of 3∘, the deformation follows the load-deflection curve of the imperfect model and snaps sideways near the limit point. This experimental result may be considered as an auxuliary evidence of the existence of the bifurcation point and the associated sideway snapping phenomenon in the perfect model.