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The cubic non - linearity of order e: In the second approxim ation

In the present paper, intending to develop the results obtained in [4, 5, 6, 7], we examine the class of systems with the cubic restoring non-linearity. This non-linearity introduces a lot. of terms into the equations for stationary oscillations and makes thus difficulty revealing its own effects. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. :n, 1999, No 4 (203 - 212) THE CUBIC N ON -LINEARITY OF ORDER IN THE SEC ON D APPROX IM AT ION e: NGUYEN VAN DINH Institute of Mechanics It is well-known that, for almost quasi-linear systems, t he oscillation can qualitatively and quantitatively - be determined in the first approximation (order e:}. However, for certain systems and even for qualitative information, the calculus must be performed in the second approximation (order e: 2 ). Some of mentioned systems have been considered in [1, 3, 3]. Especially, in [4, 5, 6, 7], a systematic study has been devoted to a whole class of systems that having the quadratic restoring non-linearity as an element of order c. There. the author has particularly concentrated attention on the effect of the quadratic non-linearity in the second approximation. It has been shown that although, in the first approximation, the non-linearity interested does not express any influence (on a family of harmonic oscillations with arbitrary constant amplitude and initial dephase and with frequency equal to the own frequency) it may play an important-even decisive - role in the second approximation. For instance, under certa.in condition (very weak friction} and in "combination" wi~h certain other elements (excitations) of the same order e:, the quadratic non-linearity may produce intense oscillations of parametric type. In the present paper, intending to develop the results obtained in [4, 5, 6, 7], we examine the c.lass of systems with the cubic restoring non-linearity. This non-linearity introduces a lot. of terms into the equations for stationary oscillations and makes thus difficulty revealing its own effects. In order to analyse the role of the cubic nonlinearity in the second approximation, we use the so-called simplified equations obtained by estimating and neglecting certain terms of order higher tha,n e:2 . The asymptotic method [8] is applied. Typical system are treated and .