### Abstract

A finite chain of particles with next-neighbor interactions, undergoing in-plane nonlinear oscillations with fixed-fixed boundary conditions is considered. In the most significant limiting case of low-energy predominantly transverse particle oscillations, the geometric nonlinearity in the system generates a nonlinear acoustic vacuum, whereby the governing equations of motion possess strongly nonlocal nonlinear terms, generated by the near-uniform axial tension, which, in turn, is caused by the fixed boundary conditions. In the continuum limit the nonlinear sonic vacuum exhibits complete absence of any linear acoustics and zero speed of sound (as defined in the classical sense); in that limit the strongly nonlocal terms become a time-dependent "effective speed of sound" for the medium, that is completely tunable with energy. Another unexpected finding is that, under the admitted assumptions, the nonlinear normal modes (NNMs) of the system are exactly identical to those of the corresponding linear chain, so their spatial shapes obey the same orthogonality conditions, which, in turn, can be used to get an uncoupled strongly nonlinear set of modal oscillators with frequencies completely tunable with energy. A rich structure of resonance manifolds of varying dimensions can then be defined, corresponding to resonance interactions between subsets of NNMs. The 1:1 resonance interactions are studied asymptotically and strong energy exchanges between modes are detected, with the most intense corresponding to a limiting phase trajectory. Moreover an additional interesting resonance motion such as time-periodic oscillations with characteristics of both standing (close to the boundaries) and traveling waves (away from them) are detected. In the view of the authors the nonlinear chain studied represents a new class of strongly nonlinear multidimensional mechanical systems whose degenerate dynamics and potential applications, such as shock and vibration absorption and mitigation, are worthy of exploring further.

Original language | English (US) |
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Pages (from-to) | 1742-1762 |

Number of pages | 21 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 74 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2014 |

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### Keywords

- Limiting phase trajectory
- Nonlinear acoustic sonic vacuum
- Nonlinear energy transfer

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*74*(6), 1742-1762. https://doi.org/10.1137/140957949