Dynamic Response of Axially Moving Beams with Small Non-Idealities at Boundary Supports

Abstract

The nonlinear response of an axially moving Euler-Bernoulli beam having one intermediate simple support and clamped boundaries with small non-idealities is examined. A small parameter representing real physical clearances at ends is used. The clamped support conditions mean that the beam is passing through two frictionless guides and one intermediate simple support placed between two guides. It is assumed that the beam has immovable boundaries at the outer ends. The assumption introduces nonlinearity because of stretching of neutral fibers. It is considered that the beam is axially moving along its length at a harmonically varying velocity about a constant mean value. In this paper variations at the velocities are assumed to be suitably weak. Differential equations governing the motion of the two sides of the beam are derived using variational formulation. A damping term is added into the equations. The method of multiple scales is applied to obtain approximate analytical solutions in this weakly nonlinear system. Non-ideal conditions at boundary supports of the beam are taken into consideration as small parameters at nonlinear equations. Solvability condition for the case of principal parametric resonance is derived, amplitude-phase curves of steady-state responses and stability conditions are investigated. It is found that the effect of small non-ideality at boundary supports has a role on nonlinear response: the width of unstable regions becomes larger with increase of non-ideality value.