Enforcing Linear Dynamics Through the Addition of Nonlinearity

The current trend of developing more slender structures is increasing the importance of nonlinearities in engineering design, which, in turn, gives rise to complicated dynamical phenomena. In this study, we evidence the somewhat paradoxical result that adding purposefully nonlinearity to an already nonlinear structure renders the behavior more linear. Isochronicity, i.e., the invariance of natural frequencies with respect to oscillation amplitude, and the force-displacement proportionality are two key properties of linear systems that are lost for nonlinear systems. The objective of this research is to investigate how these properties can be enforced in a nonlinear system through the addition of nonlinearity. To this end, we exploit the nonlinear normal modes theory to derive simple rules, yet applicable to real structures, for the compensation of nonlinear effects. The developments are illustrated using numerical experiments on a cantilever beam possessing a geometrically nonlinear boundary condition.