Experimental approximate solutions of nonlinear discrete oscillation with parametric excitation

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Abstract

In the analysis of a discrete nonlinear system, one of the important problems is to derive a steady solution for the nonlinear difference equation characterizing the system. However, the study for the solution method of the difference system has not been sufficient, and examples of the methods being applied to the parametric oscillation are few. This paper presents a method which derives the steady solution for the quasi‐linear difference equation containing a periodic parametric excitation in its weakly nonlinear function with small parameter e. In this method the harmonic balance and the averaging methods used in the solution of the continuous nonlinear differential equation are applied to the discrete system. Since the nonlinear term in the equation is bounded by ε the discrete waveform in the steady state is almost sinusoidal. This paper presents numerical examples for the solutions by the two methods, assuming the solution as the discrete oscillation corresponding to the sinusoid of the continuous system. As a result, the same solution is obtained by either method. Comparing the result with the output waveform obtained directly by the numerical calculation for the fundamental equation, the validity of the approximate solution by the proposed method was verified.

Original languageEnglish
Pages (from-to)100-108
Number of pages9
JournalElectronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)
Volume72
Issue number10
DOIs
Publication statusPublished - 1989

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Difference equations
Nonlinear systems
Differential equations

ASJC Scopus subject areas

  • Electrical and Electronic Engineering

Cite this

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title = "Experimental approximate solutions of nonlinear discrete oscillation with parametric excitation",
abstract = "In the analysis of a discrete nonlinear system, one of the important problems is to derive a steady solution for the nonlinear difference equation characterizing the system. However, the study for the solution method of the difference system has not been sufficient, and examples of the methods being applied to the parametric oscillation are few. This paper presents a method which derives the steady solution for the quasi‐linear difference equation containing a periodic parametric excitation in its weakly nonlinear function with small parameter e. In this method the harmonic balance and the averaging methods used in the solution of the continuous nonlinear differential equation are applied to the discrete system. Since the nonlinear term in the equation is bounded by ε the discrete waveform in the steady state is almost sinusoidal. This paper presents numerical examples for the solutions by the two methods, assuming the solution as the discrete oscillation corresponding to the sinusoid of the continuous system. As a result, the same solution is obtained by either method. Comparing the result with the output waveform obtained directly by the numerical calculation for the fundamental equation, the validity of the approximate solution by the proposed method was verified.",
author = "Toshiyuki Tanaka",
year = "1989",
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T1 - Experimental approximate solutions of nonlinear discrete oscillation with parametric excitation

AU - Tanaka, Toshiyuki

PY - 1989

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N2 - In the analysis of a discrete nonlinear system, one of the important problems is to derive a steady solution for the nonlinear difference equation characterizing the system. However, the study for the solution method of the difference system has not been sufficient, and examples of the methods being applied to the parametric oscillation are few. This paper presents a method which derives the steady solution for the quasi‐linear difference equation containing a periodic parametric excitation in its weakly nonlinear function with small parameter e. In this method the harmonic balance and the averaging methods used in the solution of the continuous nonlinear differential equation are applied to the discrete system. Since the nonlinear term in the equation is bounded by ε the discrete waveform in the steady state is almost sinusoidal. This paper presents numerical examples for the solutions by the two methods, assuming the solution as the discrete oscillation corresponding to the sinusoid of the continuous system. As a result, the same solution is obtained by either method. Comparing the result with the output waveform obtained directly by the numerical calculation for the fundamental equation, the validity of the approximate solution by the proposed method was verified.

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