General Formulation for Free Vibration of Elastic Solids with Static Loads and Application to Rotating Tapered Cantilever Beam Vibration 

Abstract

By solving the three one-dimensional (1D) nonlinear dynamic differential equations analytically, it has been proved that unless the nonlinear terms are in first order, a nonlinear dynamic system never has a vibration natural frequency. A simple nonlinear mass-spring system has been invented to demonstrate that vibration frequency is only calculated on a perturbation basis and to demonstrate that an external load may affect frequency too. Then for an elastic solid with nonlinear deformation and with static loads including a rotational angular velocity, a virtual small factor has been introduced to ensure a small deformation, a general formulation to predict vibration frequencies has been developed, which proves that the strain energy and kinematic energy (or the work done by vibration inertial force) are calculated from the linear deformation terms while the work done by external loads is calculated from the second order nonlinear terms that cause stiffening/softening effects on vibration frequency. This may be different to the Rayleigh-Ritz method. Applying the developed formulation to rotating tapered cantilever beams, the simple analytical method has been developed. Validation against FE analysis has been carried out to show that the simple method can predict the out-of-plane vibration frequency of rotating tapered cantilever beams accurately.