Transfer of Energy From High- to Low-Frequency Modes in the Bending-Torsion Oscillation of Cantilever Beams

We investigate the nonlinear bending-torsion response of a cantilever beam to a transverse harmonic excitation, where the forcing frequency is near the natural frequency of the first torsional mode. We analyze the case where the first in-plane bending mode is activated by a nonresonant mechanism. We use the method of time-averaged Lagrangian and virtual work to determine the equations governing the modulations of the phases and amplitudes of the interacting modes. These equations are then used to investigate the nonlinear behavior of limit-cycle oscillations of the beam as the excitation amplitude is slowly varied. As an example, we consider the response of an aluminum beam for which the natural frequency of the first in-plane bending mode is f_v1 ≈ 5.7 Hz and the natural frequency of the first torsional mode is f_ϕ1 ≈ 138.9 Hz.