Parametric resonance of a spinning pretwisted beam with time-dependent spinning rate

Abstract

This study deals with the parametric instability of a cantilever pretwisted beam rotating around its longitudinal axis at a time-dependent speed which contains a steady state part and a small periodically fluctuating component. This structural element can be used to model fluted cutting tools such as the twist drill bit and the end milling cutter, etc. Euluer-Bernoulli beam theory and Hamilton’s principle are used to derive the equation of motion which governs the lateral vibration of a spinning pretwisted beam. Rotary inertia, structural viscous damping and a conservative end axial force are included. Then, the Galerkin method is applied to obtain the associated finite element equation of motion. Due to the existence of the Coriolis force, the resulting finite element equation of motion is transformed into a set of first order simultaneous differential equations by a special modal analysis procedure. This set of simultaneous differential equations is solved by the method of multiple scales, yielding the system response and expressions for the boundaries of the unstable regions. Numerical results are presented to demonstrate the effects of pretwist angle, end axial force and steady state part of the spinning speed on the parametric instability regions of the present problem.