In [ 1], a preliminary study on the so-called critical stationary oscillations has been proposed. In the present paper, some additional remarks on tlie second stability condition are of our interest. It will be shown that the compact form of the mentioned condition can be established for ordinary as well as for critical stationary oscillations. | Journal of Mechanics, NCNST of Vietnam T. XIX, 1997, No 2 (15 - 20) STABILITY OF THE CRITICAL STATIONARY OSCILLATIONS NGUYEN VAN DINH Institute of Mecham"cs 1 Han01: Vt"etnam In [1], a preliminary study on the so-called critical stationary oscillations has been proposed. In the present paper, some additional remarks on tlie second stability condition are of our interest. It will be shown that the compact form of the mentioned condition can be established for ordinary as well as for critical stationary oscillations. 1. The two parts of the resonance curve Stationary oscillations of the quasilinear oscillating system examined in [1] are determined from the equations: {!} 8 = {A(~,a)u+B(~,a)v-E(~,a)} 8 =0, {g} 8 = { G(~, a)u + H(~, a)v- K(~, a)} 8 () = 0, where: f, g are polynomials of (~, a, u, v), linear relative to (u, v); the subscript "8" indicates that u, v must be substituted by u(O) =sinO, v(O) =cosO; other notations have been explained in [1]. In the planeR(~, a), the resonance curve Cis defined as the ensemble of representing points a) whose ordinate a is the amplitude of the stationary oscillations corresponding to the detuning parameter abscissa 1::. I(~, In general, C consists of two parts: the ordinary C 1 and the critical C2 . C, lies in the ordinary region R, : Do(~, a) f. 0; it is given by the relationship: • )=Di(~,a)+D§(~,a) W( 1 u, a D (~,a) 5 1 = 0. To obtain (} we have imposed the trigonometrical condition u 2 u(~, a)= D,(~, a)/ Do(~, a), v(~, a) = + v2 = 1 on the D2(~, a)/ Do(~, a), () expressions: () which are the solutions in R, of the equations () con!idered as the algebraic ones of two unknowns u, v (~, a play the role of parameters). :Do(~, a) = 0; it consists of critical representing points a.). If the matrix [Do] is assumed to be of rank 1, C2 is determined by the compatibility conditions () and the trigonometrical restrictions (): C 2 lies in the critical region (curve) R 2 I.(~., D 0 .
