### Abstract

This paper deals with the stability of the second order difference equation possessing periodic parameter, which characterizes discrete periodic system. Discrete periodic system corresponding to Mathieu equation is expressed as second order difference equation with small parameter ε in the time-varying term. This parameter ε plays an important role in stability. For the fundamental equation without damping, stability boundary curves are analytically obtained with respect to parameters in the equation by using McLachlan's method, which is based on Floquet's theory and perturbation method. The boundary curves are computed by pursuing periodic solutions on the boundaries and letting secular term zero. The boundary curves are expressed as the power series of ε. When periodic parameter consists of even function of Fourier series, stability boundary curves are obtained. For the fundamental equation with damping, stability criterion is shown in the neighbor of important resonant points. This criterion is obtained by computing points on the boundary curves.

Original language | English |
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Title of host publication | American Society of Mechanical Engineers, Design Engineering Division (Publication) DE |

Editors | Mo Shahinpoor, H.S. Tzou |

Publisher | Publ by ASME |

Pages | 403-411 |

Number of pages | 9 |

Volume | 56 |

ISBN (Print) | 0791811735 |

Publication status | Published - 1993 |

Event | 14th Biennial Conference on Mechanical Vibration and Noise - Albuquerque, NM, USA Duration: 1993 Sep 19 → 1993 Sep 22 |

### Other

Other | 14th Biennial Conference on Mechanical Vibration and Noise |
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City | Albuquerque, NM, USA |

Period | 93/9/19 → 93/9/22 |

### Fingerprint

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*American Society of Mechanical Engineers, Design Engineering Division (Publication) DE*(Vol. 56, pp. 403-411). Publ by ASME.

**Stability of the second order difference equation with time-varying parameter.** / Tanaka, Toshiyuki; Sato, Chikara.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*American Society of Mechanical Engineers, Design Engineering Division (Publication) DE.*vol. 56, Publ by ASME, pp. 403-411, 14th Biennial Conference on Mechanical Vibration and Noise, Albuquerque, NM, USA, 93/9/19.

}

TY - GEN

T1 - Stability of the second order difference equation with time-varying parameter

AU - Tanaka, Toshiyuki

AU - Sato, Chikara

PY - 1993

Y1 - 1993

N2 - This paper deals with the stability of the second order difference equation possessing periodic parameter, which characterizes discrete periodic system. Discrete periodic system corresponding to Mathieu equation is expressed as second order difference equation with small parameter ε in the time-varying term. This parameter ε plays an important role in stability. For the fundamental equation without damping, stability boundary curves are analytically obtained with respect to parameters in the equation by using McLachlan's method, which is based on Floquet's theory and perturbation method. The boundary curves are computed by pursuing periodic solutions on the boundaries and letting secular term zero. The boundary curves are expressed as the power series of ε. When periodic parameter consists of even function of Fourier series, stability boundary curves are obtained. For the fundamental equation with damping, stability criterion is shown in the neighbor of important resonant points. This criterion is obtained by computing points on the boundary curves.

AB - This paper deals with the stability of the second order difference equation possessing periodic parameter, which characterizes discrete periodic system. Discrete periodic system corresponding to Mathieu equation is expressed as second order difference equation with small parameter ε in the time-varying term. This parameter ε plays an important role in stability. For the fundamental equation without damping, stability boundary curves are analytically obtained with respect to parameters in the equation by using McLachlan's method, which is based on Floquet's theory and perturbation method. The boundary curves are computed by pursuing periodic solutions on the boundaries and letting secular term zero. The boundary curves are expressed as the power series of ε. When periodic parameter consists of even function of Fourier series, stability boundary curves are obtained. For the fundamental equation with damping, stability criterion is shown in the neighbor of important resonant points. This criterion is obtained by computing points on the boundary curves.

UR - http://www.scopus.com/inward/record.url?scp=0027801709&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027801709&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0027801709

SN - 0791811735

VL - 56

SP - 403

EP - 411

BT - American Society of Mechanical Engineers, Design Engineering Division (Publication) DE

A2 - Shahinpoor, Mo

A2 - Tzou, H.S.

PB - Publ by ASME

ER -