Nonplanar oscillation of beams under periodic forcing

Abstract

In a study of parametric excitation of planar transverse responses of axially driven strings and rods, Haight and King [S] observed that for certain values of the parameters the excited planar motion in turn parametrically excites a mode which is not in the original plane. The authors also point out that the coupling between the original planar response and the nonplanar motion is due to nonlinearities. After deriving the equations of motion for the case of axially excited rod, the authors discuss the range of frequencies of the periodic forcing for which the nonplanar response occurs and the stability of such a motion. Such nonplanar responses in the context of strings were studied by Murthy and Ramakrishna [7]. They show that a string excited in a direction perpendicular to the axis may, under certain conditions, cause vibrations which are not only in the plane of the driving force. This study has since been extended by several other authors. This paper follows the study initiated in [6], where the whirling motions of nonlinear elastic beams were studied. In this paper the authors study the motion of a simply supported beam with two perpendicular axes of symmetry where the forcing is periodic and planar. The authors approximate the nonlinear hyperbolic equations by a system of coupled ordinary differential equations. These equations are then’ studied for nonplanar response and stability. In [Z] we considered the rotating beam equations of helicopter rotorcraft dynamics. The harmonically forced hyperbolic equations were approximated by a finite element approach using Chebyshev polynomials. A fourth order Runge-Kutta integration scheme was then utilized to study the long-time behavior of the solution of the associated evolution equation. 419 0022-0396/92 $3.00