Associated equations and their corresponding resonance curve

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.