Nonlinear Dynamics of a High Dimensional Model of a Rotating Euler–Bernoulli Beam Under the Gravity Load

Abstract

Nonlinear dynamic responses of an Euler–Bernoulli beam attached to a rotating rigid hub with a constant angular velocity under the gravity load are investigated. The slope angle of the centroid line of the beam is used to describe its motion, and the nonlinear integropartial differential equation that governs the motion of the rotating hubbeam system is derived using Hamilton's principle. Spatially discretized governing equations are derived using Lagrange's equations based on discretized expressions of kinetic and potential energies of the system, yielding a set of secondorder nonlinear ordinary differential equations with combined parametric and forced harmonic excitations due to the gravity load. The incremental harmonic balance (IHB) method is used to solve for periodic responses of a highdimensional model of the system for which convergence is reached and its perioddoubling bifurcations. The multivariable Floquet theory along with the precise Hsu's method is used to investigate the stability of the periodic responses. Phase portraits and bifurcation points obtained from the IHB method agree very well with those from numerical integration.