Nonlinear Modal Interactions of a Cantilever Beam with an Internal Resonance

Nonlinear torsional-flexural modal ;nteractions of a torsionally excited cantilever beam are studied in the case of one-to-one internal resonance. Discrete two DOF model is derived from the equations of motion for the continuous canthwer beam by using the Galerkin mode approx;mat;on method. The method of averaging is used to obtain the approx;mat;on to the response of the discrete model. Steady state constant and periodic solutions are studded by anaiyhcal and numerical approach. A bifurcation anaiysis of the averaged equations is undertaken, and it is shown that torsional-flexural coupled response arises when the torsfonal mode becomes unstable for .some excitation frequencies and ampl;tudes, the coupled response results from energy transfer between the two nonlinearly coupleo modes. The coupled response can lead to amplitude-andphase-modulated response via Hopf bifurcation. 2. EQUATIONS OF MOTION The system analyzed here is shown in Fig.1. In order to perform a nonlinear analysis effectively, it was introduced the following assumptions : (i) A cantilever beam has thin and uniform cross-section with its length(i.e., like “elastica”[l, 21 shape) and its physical properties (e.g., Young’s modulus, shear modulus, mass density etc.) are invariant and homogeneous. (ii) A cantilever beam is satisfied the inextensibility and the unshearabi l i ty condit ions.