Non-Linear Free Vibrations of a Rotating Beam Carrying a Tip Mass With Rotary Inertia

Abstract

The non-linear, for each of the first three modes, of planar, large amplitude flexural free vibrations of a beam clamped with an angle to a rigid rotating hub and carrying a tip mass with rotary inertia are investigated. The shear deformation and rotary inertia effects are assumed to be negligible, but account is taken of axial inertia, non-linear curvature and the inextensibility condition. The Lagrangian dynamics in conjunction with the assumed mode method, assuming constant hub rotation speed, is utilized in deriving the non-linear unimodal temporal problem. The time transformation method is employed to obtain an approximate solution to the frequency-amplitude relation of the beam-mass free vibration, since the order of the nonlinear terms is not small which includes static and inertial geometric stiffening as well as inertial softening terms. Results in non-dimensional form are presented graphically, for the effect beam root-attachment angle, hub radius and the attached inertia element ratio on the variation of the natural frequency with vibration amplitude.