Associated equations and their corresponding resonance curve

Nguyen Van Dinh

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.


Full Text:

PDF


DOI: https://doi.org/10.15625/0866-7136/9996 Display counter: Abstract : 36 views. PDF : 28 views.

Refbacks

  • There are currently no refbacks.


Copyright (c) 1999 Vietnam Academy of Science and Technology

 

                      

Editorial Office of Vietnam Journal of Mechanics

3rd Floor, A16 Building, 18B Hoang Quoc Viet Street, Cau Giay District, Hanoi, Vietnam

Tel: (+84) 24 3791 7103

Email: vjmech@vjs.ac.vn