Nonlinear Normal Modes of Highly Flexible Cantilever Beams via Energy-Balance Based Approach

Abstract This work uses an energy balance–based approach to study the steady-state characteristics of autonomous nonlinear systems. The total energy, a scalar quantity, is used to characterize the system. The Hamiltonian of the system is considered an additional constraint. Semi-analytical techniques are used to evaluate steady-state solutions directly. The time variational energy method is used in this approach as it can handle nonlinearity in the time domain without any frequency–time domain transformation. The time variational energy method is validated against numerical integration. The nonlinear normal mode (NNM) motions are realized using the input energy from the frequency–energy plot corresponding to the nonlinear normal mode motion for the time variational energy method. The method is applied to a discrete nonlinear system, a cantilever system with tip nonlinearity, and a thin, highly flexible cantilever beam with a higher aspect ratio for realizing the nonlinear normal mode motion. The gradient-deficient beam element of absolute nodal coordinate formulation models cantilever beams to incorporate large rigid body rotations. The time variational energy method could predict the nonlinear normal mode motions and showed good agreement with the NNM motions obtained using the time variational method. The internal resonances were also predicted accurately using this method.