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The vanderpol's system under external and parametric excitations

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In the present paper, we shall consider the case when these two excitations simultaneously act on the system of interest: the first excitation is external and in the fundamental resonance (or.l-er 1} and the second one is parametric and in the subharmonic resonance of order 1/2. Critical singular points will be used to classify different forms of the resonance curve | : ~ - -1 T~p Journal of Mechanics, NCNST of Vietnam T. XIX, 1997, No 1 (27-33) chi C 0 is a small parameter; a > 0 is the coefficient characterizing the self-excitation; (e, w) (2p, 2w) are intensities, frequencies of the external and parametric excitations, respectively; e > 0, p > 0; cr (0 ::; cr -. (1.13) • - 4p2 Thus, if ( 1.13) is satisfied, the critical part C2 consists of an unique point I,. By rejecting those points satisfying (1.10) but not (1.11), the whole resonance curve C (C, +I,) can be found from the relationship: W(fl,a 2 ) = e2 (Di + D~)- a 2 D~ = 0. (1.14) I* is a nodal point if; D > O, where azw D = ( atlaa• )2 - (azw) ( azw ) 8fl2 8(a2)2 . H D 2 . · a 2 = 1 ±a 1 + f> Thus, the critical region Co is a-clOSed curve~ an "oval" of center(~ =-0,-ci~ = 1). 4 H p2 e* = 4p 2 a~, L, becomes an isolated point, the "inside" loop will either disappear or change into an closed branch. 4 H p 2 2- a 2 - : , the abscissa ~ axis 6 intersects Co. In Fig.1, for fixed values (o- = 0; a= 0.1; p = 0.05) the resonance curves (0)-(5) correspond toe= 0; 0.015; 0.0177; 0.05; 0.1; 0.12 respectively. The curve (0) represents the critical region C 0 • The resonance curve (1) consists of two branches C' and C". Fore~ 0.0177, C' joints to 0 11 at an ordinary singular point J. Increasing e, J disappears and the resonance curve will be of form (3) corresponding to e = 0.05. When e reaches the value e = 0.1, the "inside" loop is reduced to the returning point L,. Increasing e further I,., becomes an isolated point, the resonance curve takes the form(5) corresponding to e = 0.12. 5 Fig.1 29 In Fig. 2, for fixed values (u = ~ ; a= 0.1;p = 0.05) the resonance curves (0)-(7) are plotted for e = 0; 0.04; 0, 0483; 0.05; 0.0516; 0.055; 0.0648; 0.98 respectively. There are ordinary singular points when e"' 0.0483 ore"" 0.0516 (curve (2) and (4)) and new lower loops fore= 0.055; 0.0648. I. is an isolated point for e = 0.08. Fig.2 In Fig. 3, for fixed