### Abstract

We consider a class of parametrically driven nonlinear oscillators: x+k_{1}x+k_{2}f(x,x)P(Ωt)=0, P(Ωt+2π)=P(Ωt) (*) which can be used to describe, e.g., a pendulum with vibrating length, or the displacements of colliding particle beams in high energy accelerators. Here we study numerically and analytically the subharmonic periodic solution of (*), with frequency 1/ m≅√k_{1}, m=1,2,3,.... In the cases of f(x, x)=x^{3} and f(x,x)=x^{4}, with P(Ωt)=cost, all of these so-called synchronized periodic orbits are obtained numerically, by a new technique, which we refer to here as the indicatrix method. The theory of generalized averaging is then applied to derive highly accurate expressions for these orbits, valid to the second order in k_{2}.

Original language | English |
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Pages (from-to) | 721-728 |

Number of pages | 8 |

Journal | Journal of Applied Mechanics, Transactions ASME |

Volume | 55 |

Issue number | 3 |

Publication status | Published - Sep 1988 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Computational Mechanics
- Mechanics of Materials

### Cite this

*Journal of Applied Mechanics, Transactions ASME*,

*55*(3), 721-728.

**Synchronized periodic solutions of a class of periodically driven nonlinear oscillators.** / Mahmoud, Gamal M.; Bountis, Tassos.

Research output: Contribution to journal › Article

*Journal of Applied Mechanics, Transactions ASME*, vol. 55, no. 3, pp. 721-728.

}

TY - JOUR

T1 - Synchronized periodic solutions of a class of periodically driven nonlinear oscillators

AU - Mahmoud, Gamal M.

AU - Bountis, Tassos

PY - 1988/9

Y1 - 1988/9

N2 - We consider a class of parametrically driven nonlinear oscillators: x+k1x+k2f(x,x)P(Ωt)=0, P(Ωt+2π)=P(Ωt) (*) which can be used to describe, e.g., a pendulum with vibrating length, or the displacements of colliding particle beams in high energy accelerators. Here we study numerically and analytically the subharmonic periodic solution of (*), with frequency 1/ m≅√k1, m=1,2,3,.... In the cases of f(x, x)=x3 and f(x,x)=x4, with P(Ωt)=cost, all of these so-called synchronized periodic orbits are obtained numerically, by a new technique, which we refer to here as the indicatrix method. The theory of generalized averaging is then applied to derive highly accurate expressions for these orbits, valid to the second order in k2.

AB - We consider a class of parametrically driven nonlinear oscillators: x+k1x+k2f(x,x)P(Ωt)=0, P(Ωt+2π)=P(Ωt) (*) which can be used to describe, e.g., a pendulum with vibrating length, or the displacements of colliding particle beams in high energy accelerators. Here we study numerically and analytically the subharmonic periodic solution of (*), with frequency 1/ m≅√k1, m=1,2,3,.... In the cases of f(x, x)=x3 and f(x,x)=x4, with P(Ωt)=cost, all of these so-called synchronized periodic orbits are obtained numerically, by a new technique, which we refer to here as the indicatrix method. The theory of generalized averaging is then applied to derive highly accurate expressions for these orbits, valid to the second order in k2.

UR - http://www.scopus.com/inward/record.url?scp=0024073293&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0024073293&partnerID=8YFLogxK

M3 - Article

VL - 55

SP - 721

EP - 728

JO - Journal of Applied Mechanics, Transactions ASME

JF - Journal of Applied Mechanics, Transactions ASME

SN - 0021-8936

IS - 3

ER -