### Abstract

The structure of time-dependent resonances arising in the method of time-dependent normal forms (TDNF) for one and two-degrees-of-freedom nonlinear systems with time-periodic coefficients is investigated. For this purpose, the Liapunov-Floquet (L-F) transformation is employed to transform the periodic variational equations into an equivalent form in which the linear system matrix is time-invariant. Both quadratic and cubic nonlinearities are investigated and the associated normal forms are presented. Also, higher-order resonances for the single-degree-of-freedom case are discussed. It is demonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2, ...) solutions. The discussion is limited to the Hamiltonian case (which encompasses all possible resonances for one-degree-of-freedom). Furthermore, it is also shown how a recent symbolic algorithm for computing stability and bifurcation boundaries for time-periodic systems may also be employed to compute the time-dependent resonance sets of zero measure in the parameter space. Unlike classical asymptotic techniques, this method is free from any small parameter restriction on the time-periodic term in the computation of the resonance sets. Two illustrative examples (one and two-degrees-of-freedom) are included.

Original language | English (US) |
---|---|

Pages (from-to) | 35-55 |

Number of pages | 21 |

Journal | Nonlinear Dynamics |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2000 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Computational Mechanics

### Cite this

*Nonlinear Dynamics*,

*23*(1), 35-55. https://doi.org/10.1023/A:1008312424551

**Normal forms and the structure of resonance sets in nonlinear time-periodic systems.** / Butcher, Eric; Sinha, S. C.

Research output: Contribution to journal › Article

*Nonlinear Dynamics*, vol. 23, no. 1, pp. 35-55. https://doi.org/10.1023/A:1008312424551

}

TY - JOUR

T1 - Normal forms and the structure of resonance sets in nonlinear time-periodic systems

AU - Butcher, Eric

AU - Sinha, S. C.

PY - 2000/9

Y1 - 2000/9

N2 - The structure of time-dependent resonances arising in the method of time-dependent normal forms (TDNF) for one and two-degrees-of-freedom nonlinear systems with time-periodic coefficients is investigated. For this purpose, the Liapunov-Floquet (L-F) transformation is employed to transform the periodic variational equations into an equivalent form in which the linear system matrix is time-invariant. Both quadratic and cubic nonlinearities are investigated and the associated normal forms are presented. Also, higher-order resonances for the single-degree-of-freedom case are discussed. It is demonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2, ...) solutions. The discussion is limited to the Hamiltonian case (which encompasses all possible resonances for one-degree-of-freedom). Furthermore, it is also shown how a recent symbolic algorithm for computing stability and bifurcation boundaries for time-periodic systems may also be employed to compute the time-dependent resonance sets of zero measure in the parameter space. Unlike classical asymptotic techniques, this method is free from any small parameter restriction on the time-periodic term in the computation of the resonance sets. Two illustrative examples (one and two-degrees-of-freedom) are included.

AB - The structure of time-dependent resonances arising in the method of time-dependent normal forms (TDNF) for one and two-degrees-of-freedom nonlinear systems with time-periodic coefficients is investigated. For this purpose, the Liapunov-Floquet (L-F) transformation is employed to transform the periodic variational equations into an equivalent form in which the linear system matrix is time-invariant. Both quadratic and cubic nonlinearities are investigated and the associated normal forms are presented. Also, higher-order resonances for the single-degree-of-freedom case are discussed. It is demonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2, ...) solutions. The discussion is limited to the Hamiltonian case (which encompasses all possible resonances for one-degree-of-freedom). Furthermore, it is also shown how a recent symbolic algorithm for computing stability and bifurcation boundaries for time-periodic systems may also be employed to compute the time-dependent resonance sets of zero measure in the parameter space. Unlike classical asymptotic techniques, this method is free from any small parameter restriction on the time-periodic term in the computation of the resonance sets. Two illustrative examples (one and two-degrees-of-freedom) are included.

UR - http://www.scopus.com/inward/record.url?scp=0034273083&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034273083&partnerID=8YFLogxK

U2 - 10.1023/A:1008312424551

DO - 10.1023/A:1008312424551

M3 - Article

VL - 23

SP - 35

EP - 55

JO - Nonlinear Dynamics

JF - Nonlinear Dynamics

SN - 0924-090X

IS - 1

ER -