### Abstract

We study the in-plane damped oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Strong nonlinearity in this system is generated by geometric effects due to the in-plane stretching of the coupling spring elements. The lattice has a finite number of nonlinear transverse standing waves (termed nonlinear normal modes – NNMs), and an equal number of axial linear modes which are nonlinearly coupled to the transverse ones. Nonlinear interactions between the transverse and axial modes under harmonic excitation give rise to unexpected and extreme nonlinear energy exchanges in the lattice. In particular, we directly excite a transverse NNM by harmonic forcing (causing simulataneous indirect excitation of a corresponding axial linear mode due to nonlinear coupling), and identify three energy transfer mechanisms in the lattice. First, we detect the stable response of the directly excited transverse NNM (despite its instability in the absence of forcing), with simultaneous stability of the indirectly excited axial linear mode. Second, by changing the system and forcing parameters we report extreme nonlinear “energy explosions,” whereby, after an initial regime of stability, the directly excited transverse NNM loses stability, leading to abrupt excitation of all transverse and axial modes of the lattice, at all possible wave numbers. This strong instability is triggered by the parametric instability of an indirectly excited axial mode which builds energy until the explosion. This is proved through theoretical analysis. Finally, in other parameter ranges we report intermittent, intense energy transfers from the directly excited transverse NNM to a small set of transverse NNMs with smaller wavelengths, and from the indirectly excited axial mode to a small set of axial modes, but with larger wavelengths. These intermittent energy transfers resemble energy cascades occurring in turbulent flows. Our results show that nonlinear mechanical systems can support extreme energy transfer phenomena, with features resembling “mechanical turbulence”.

Language | English (US) |
---|---|

Pages | 1-20 |

Number of pages | 20 |

Journal | Journal of the Mechanics and Physics of Solids |

Volume | 110 |

DOIs | |

State | Published - Feb 1 2018 |

### Fingerprint

### Keywords

- Energy transfer
- Instability
- Lattice
- Nonlinear acoustics
- Sonic vacuum

### ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

### Cite this

*Journal of the Mechanics and Physics of Solids*,

*110*, 1-20. DOI: 10.1016/j.jmps.2017.09.007

**Extreme nonlinear energy exchanges in a geometrically nonlinear lattice oscillating in the plane.** / Zhang, Zhen; Manevitch, Leonid I.; Smirnov, Valeri; Bergman, Lawrence A.; Vakakis, Alexander F.

Research output: Contribution to journal › Article

*Journal of the Mechanics and Physics of Solids*, vol 110, pp. 1-20. DOI: 10.1016/j.jmps.2017.09.007

}

TY - JOUR

T1 - Extreme nonlinear energy exchanges in a geometrically nonlinear lattice oscillating in the plane

AU - Zhang,Zhen

AU - Manevitch,Leonid I.

AU - Smirnov,Valeri

AU - Bergman,Lawrence A.

AU - Vakakis,Alexander F.

PY - 2018/2/1

Y1 - 2018/2/1

N2 - We study the in-plane damped oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Strong nonlinearity in this system is generated by geometric effects due to the in-plane stretching of the coupling spring elements. The lattice has a finite number of nonlinear transverse standing waves (termed nonlinear normal modes – NNMs), and an equal number of axial linear modes which are nonlinearly coupled to the transverse ones. Nonlinear interactions between the transverse and axial modes under harmonic excitation give rise to unexpected and extreme nonlinear energy exchanges in the lattice. In particular, we directly excite a transverse NNM by harmonic forcing (causing simulataneous indirect excitation of a corresponding axial linear mode due to nonlinear coupling), and identify three energy transfer mechanisms in the lattice. First, we detect the stable response of the directly excited transverse NNM (despite its instability in the absence of forcing), with simultaneous stability of the indirectly excited axial linear mode. Second, by changing the system and forcing parameters we report extreme nonlinear “energy explosions,” whereby, after an initial regime of stability, the directly excited transverse NNM loses stability, leading to abrupt excitation of all transverse and axial modes of the lattice, at all possible wave numbers. This strong instability is triggered by the parametric instability of an indirectly excited axial mode which builds energy until the explosion. This is proved through theoretical analysis. Finally, in other parameter ranges we report intermittent, intense energy transfers from the directly excited transverse NNM to a small set of transverse NNMs with smaller wavelengths, and from the indirectly excited axial mode to a small set of axial modes, but with larger wavelengths. These intermittent energy transfers resemble energy cascades occurring in turbulent flows. Our results show that nonlinear mechanical systems can support extreme energy transfer phenomena, with features resembling “mechanical turbulence”.

AB - We study the in-plane damped oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Strong nonlinearity in this system is generated by geometric effects due to the in-plane stretching of the coupling spring elements. The lattice has a finite number of nonlinear transverse standing waves (termed nonlinear normal modes – NNMs), and an equal number of axial linear modes which are nonlinearly coupled to the transverse ones. Nonlinear interactions between the transverse and axial modes under harmonic excitation give rise to unexpected and extreme nonlinear energy exchanges in the lattice. In particular, we directly excite a transverse NNM by harmonic forcing (causing simulataneous indirect excitation of a corresponding axial linear mode due to nonlinear coupling), and identify three energy transfer mechanisms in the lattice. First, we detect the stable response of the directly excited transverse NNM (despite its instability in the absence of forcing), with simultaneous stability of the indirectly excited axial linear mode. Second, by changing the system and forcing parameters we report extreme nonlinear “energy explosions,” whereby, after an initial regime of stability, the directly excited transverse NNM loses stability, leading to abrupt excitation of all transverse and axial modes of the lattice, at all possible wave numbers. This strong instability is triggered by the parametric instability of an indirectly excited axial mode which builds energy until the explosion. This is proved through theoretical analysis. Finally, in other parameter ranges we report intermittent, intense energy transfers from the directly excited transverse NNM to a small set of transverse NNMs with smaller wavelengths, and from the indirectly excited axial mode to a small set of axial modes, but with larger wavelengths. These intermittent energy transfers resemble energy cascades occurring in turbulent flows. Our results show that nonlinear mechanical systems can support extreme energy transfer phenomena, with features resembling “mechanical turbulence”.

KW - Energy transfer

KW - Instability

KW - Lattice

KW - Nonlinear acoustics

KW - Sonic vacuum

UR - http://www.scopus.com/inward/record.url?scp=85029721602&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85029721602&partnerID=8YFLogxK

U2 - 10.1016/j.jmps.2017.09.007

DO - 10.1016/j.jmps.2017.09.007

M3 - Article

VL - 110

SP - 1

EP - 20

JO - Journal of the Mechanics and Physics of Solids

T2 - Journal of the Mechanics and Physics of Solids

JF - Journal of the Mechanics and Physics of Solids

SN - 0022-5096

ER -