## Abstract

Modal interactions are distinct features of nonlinear systems that can be exploited in applications such as vibration and shock mitigation, targeted (irreversible) energy transfers (TET), and acoustic/stress wave tailoring. For such applications, different types of nonlinearities, e.g. hardening, softening, smooth, non-smooth, material or geometric, have been considered. In this work, we examine the geometric nonlinearity resulting from an initially inclined element consisting of a linear spring and a viscous damper connected in parallel, having an initial angle of inclination, ɸ_{0}. Because of its inclined configuration, this element possesses strong (and doubly tunable with respect to ɸ_{0} and energy) geometrically nonlinear stiffness and damping effects, despite the linear constitutive laws governing its constituent components. First, we consider a single-degree-of-freedom linearly grounded oscillator attached to the nonlinear inclined element. Omitting dissipative effects, we investigate the frequency–energy relation of this system by employing the canonical action-angle transformation and show that, depending on the initial angle of inclination and the energy-level, the resulting nonlinearity can be tuned to be softening, hardening or a combination of both. Next, we explore the efficacy of the geometric nonlinearity to induce strong modal interactions by considering a three-degree-of-freedom lightly damped primary system that is weakly coupled to a single-degree-of-freedom lightly damped attachment with the inclined nonlinear element, subjected to impulsive excitation. Varying ɸ_{0} and the input energy, we demonstrate strong modal energy-exchanges between the modes of the primary system and the nonlinear attachment over broad energy-dependent spans of ɸ_{0}. We show that the passive self-adaptiveness of the nonlinear damping and the hardening–softening geometric nonlinearity can induce narrowband or broadband frequency TET, including high-to-low frequency energy transfers. Interestingly, over a definitive range of ɸ_{0}, these modal interactions may be limited only between the nonlinear mode of the attachment and the highest-frequency linear mode of the primary system, inducing strong high-frequency targeted energy transfer to the primary system.

Original language | English (US) |
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Journal | Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science |

DOIs | |

State | Accepted/In press - Jan 1 2020 |

## Keywords

- geometric nonlinearity
- hardening
- modal energy exchange
- Nonlinear systems
- softening nonlinearity
- targeted energy transfer

## ASJC Scopus subject areas

- Mechanical Engineering