Associated equations and their corresponding resonance curve

Authors

  • Nguyen Van Dinh

DOI:

https://doi.org/10.15625/0866-7136/9996

Abstract

In the theory of nonlinear oscillations, in order to identify the resonance curve we usually try to eliminate the diphase Ѳ in the equations of stationary oscillations. We obtain thus a certain frequency-amplitude relationship.

In simple cases when the mentioned equations contain only and linearly the first harmonics (sin Ѳ, cos Ѳ) the elimination of Ѳ is elementary, by using the trigono-metrical identity sin2 Ѳ+ cos2 Ѳ = 1.

In general, high harmonics (sin2 Ѳ, cos2 Ѳ, etc.) are present. Consequently the expressions of sin Ѳ, cos Ѳ are cumbersome or do not exist and the analytical elimination of Ѳ is quite inconvenient or impossible. For this reason, to identify the resonance curve of complicated systems, we use the numerical method.

Below, intending to develop the analytical method, we shall propose a procedure enabling us to transform the "original" complicated equations of stationary oscillations into the so-called associated ones, only and linearly containing sin Ѳ, cos Ѳ. The equivalence of the original and associated equations will be treated and the associated resonance 'curve-that is determined by the associated equations-will be analyzed

The discussion will be restricted to a simple practical case in which, beside sin Ѳ and cos Ѳ, only sin2 Ѳ and cos2 Ѳ are present. Nevertheless, the method proposed and the results obtained can be generalized.

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Published

30-09-1999

How to Cite

Dinh, N. V. (1999). Associated equations and their corresponding resonance curve. Vietnam Journal of Mechanics, 21(3), 147–155. https://doi.org/10.15625/0866-7136/9996

Issue

Section

Research Paper

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