Geometrically Nonlinear Analysis Of A Flexible Beam With Rotating Motion

The nonlinear vibration of beams is a typical problem of nonlinear one. To meet the demands of high rotating speeds, fast response and accurate positioning for industrial applications such as the design of helicopters, robotic arms, satellite appendages, and turbine blades, thorough investigation for dynamic characteristics of such system is needed by using suitable mathematical models with proper simplifying assumptions concerning the geometric nonlinearity.But the traditional models adopt the small deformation assumption in the structural dynamics. In this paper a refined geometrically nonlinear model for a rotating slender flexible beam is developed. With rotating coordinate system, the dynamics of a rigid-flexible coupled system are analyzed for a central rigid body with rotating motion and a flexible viscoelastic beam. The axial deformation, the transverse deformation and the cross section rotation of the beam are taken into account. The geometric nonlinearity of the coupling system is discussed by Green strain formula and the material of the beam obeys the Kelvin-Voigt constitutive relation. A set of dynamic equations is established by utilizing the differential element method. Then the Galerkin method is used to discretized the equations and the governing equations are obtained.Through neglecting the axial motion, a simplified model, suitable for studying the transverse vibration, is presented. The governing differential equation is strongly nonlinear, so various approximate analytical methods based on perturbation theory for a weakly nonlinear system can't be used directly. By using the improved L-P method, different from the known, the first order approximate solution which has fair accuracy is obtained.Then the equations of the motion are solved numerically using the fourth order Runge-Kutta method. Graphical results are presented to show the influence of centrifugal stiffening effect, viscoelasticity, geometric nonlinearity. Numerical results indicate that the system is stable in case of high rotating speed and that the viscoelastic damping decreases the amplitude of the axial and the transverse deformation.