A semi-analytical solution in time domain for evaluating the nonlinear normal modes of a cantilever beam with a tip nonlinearity

In this work, a semi-analytical approach called the time variational method is used to evaluate the system’s nonlinear normal modes in the time domain. A flexible cantilever beam with a cubic nonlinear spring supported at the tip is modeled using the Euler–Bernoulli beam element in the classical finite element method and the 3D gradient-deficient beam element in the Absolute Nodal Coordinate Formulation. The time variational method approach is used to extract the nonlinear normal mode for both modeling strategies, which is then compared with the nonlinear normal mode of the Euler–Bernoulli beam model computed using the harmonic balance method and is in good agreement. Furthermore, the Absolute Nodal Coordinate Formulation was able to predict additional internal resonances that were not present in the Finite Element Method’s nonlinear normal mode.