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Nonlinear oscillatory acoustic vacuum. (English) Zbl 1316.74025

Summary: 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 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.

MSC:

74J30 Nonlinear waves in solid mechanics
70K30 Nonlinear resonances for nonlinear problems in mechanics
34E05 Asymptotic expansions of solutions to ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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