Analysis of Dynamic Elastic Stability of Beams under Travelling Vertical Follower Load Systems by the Linearized Finite Displacement Theory

Dynamic stability of flexible beams, of which the deformation is comparatively large, under travelling vertical follower load systems is investigated by applying the fundamental equation, governing the dynamic elastic stability, which is derived by using the linearized finite displacement theory. Firstly, it is shown that the fundamental equation of beams under the above travelling load systems becomes Hill's equation and that parametrically excited unstable coupled vibration occurs. The boundary frequency equations of simple parametric resonance are obtained by Bolotin's method and stability maps of a simply supported beam are given, with account taken of the effects of load mass and damping. Lastly, the behaviors of vibration of the beam in the vicinity of boundary of a main unstable region are estimated by numerical integration.