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

Toshiyuki Tanaka; Chikara Sato

doi:10.1115/DETC1993-0129

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

Toshiyuki Tanaka, Chikara Sato

Department of Applied Physics and Physico-Informatics

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

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 £ in the time-varying term. This parameter e 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 e. 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
Title of host publication	14th Biennial Conference on Mechanical Vibration and Noise
Subtitle of host publication	Dynamics and Vibration of Time-Varying Systems and Structures
Publisher	American Society of Mechanical Engineers (ASME)
Pages	403-409
Number of pages	7
ISBN (Electronic)	9780791811733
DOIs	https://doi.org/10.1115/DETC1993-0129
Publication status	Published - 1993
Event	ASME 1993 Design Technical Conferences, DETC 1993 - Albuquerque, United States Duration: 1993 Sept 19 → 1993 Sept 22

Publication series

Name	Proceedings of the ASME Design Engineering Technical Conference
Volume	Part F167972-5

Conference

Conference	ASME 1993 Design Technical Conferences, DETC 1993
Country/Territory	United States
City	Albuquerque
Period	93/9/19 → 93/9/22

ASJC Scopus subject areas

Mechanical Engineering
Computer Graphics and Computer-Aided Design
Computer Science Applications
Modelling and Simulation

Access to Document

10.1115/DETC1993-0129

Cite this

Tanaka, T., & Sato, C. (1993). Stability of the second order difference equation with time-varying parameter. In 14th Biennial Conference on Mechanical Vibration and Noise: Dynamics and Vibration of Time-Varying Systems and Structures (pp. 403-409). (Proceedings of the ASME Design Engineering Technical Conference; Vol. Part F167972-5). American Society of Mechanical Engineers (ASME). https://doi.org/10.1115/DETC1993-0129

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

14th Biennial Conference on Mechanical Vibration and Noise: Dynamics and Vibration of Time-Varying Systems and Structures. American Society of Mechanical Engineers (ASME), 1993. p. 403-409 (Proceedings of the ASME Design Engineering Technical Conference; Vol. Part F167972-5).

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

Tanaka, T & Sato, C 1993, Stability of the second order difference equation with time-varying parameter. in 14th Biennial Conference on Mechanical Vibration and Noise: Dynamics and Vibration of Time-Varying Systems and Structures. Proceedings of the ASME Design Engineering Technical Conference, vol. Part F167972-5, American Society of Mechanical Engineers (ASME), pp. 403-409, ASME 1993 Design Technical Conferences, DETC 1993, Albuquerque, United States, 93/9/19. https://doi.org/10.1115/DETC1993-0129

Tanaka T, Sato C. Stability of the second order difference equation with time-varying parameter. In 14th Biennial Conference on Mechanical Vibration and Noise: Dynamics and Vibration of Time-Varying Systems and Structures. American Society of Mechanical Engineers (ASME). 1993. p. 403-409. (Proceedings of the ASME Design Engineering Technical Conference). doi: 10.1115/DETC1993-0129

Tanaka, Toshiyuki ; Sato, Chikara. / Stability of the second order difference equation with time-varying parameter. 14th Biennial Conference on Mechanical Vibration and Noise: Dynamics and Vibration of Time-Varying Systems and Structures. American Society of Mechanical Engineers (ASME), 1993. pp. 403-409 (Proceedings of the ASME Design Engineering Technical Conference).

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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 £ in the time-varying term. This parameter e 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 e. 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.",

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

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