tailieunhanh - The vanderpol's system under external and parametric excitations

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 ( 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 -. () • - 4p2 Thus, if ( ) is satisfied, the critical part C2 consists of an unique point I,. By rejecting those points satisfying () but not (), the whole resonance curve C (C, +I,) can be found from the relationship: W(fl,a 2 ) = e2 (Di + D~)- a 2 D~ = 0. () 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 , for fixed values (o- = 0; a= ; p = ) the resonance curves (0)-(5) correspond toe= 0; ; ; ; ; respectively. The curve (0) represents the critical region C 0 • The resonance curve (1) consists of two branches C' and C". Fore~ , 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 = . When e reaches the value e = , 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 = . 5 29 In Fig. 2, for fixed values (u = ~ ; a= ;p = ) the resonance curves (0)-(7) are plotted for e = 0; ; 0, 0483; ; ; ; ; respectively. There are ordinary singular points when e"' ore"" (curve (2) and (4)) and new lower loops fore= ; . I. is an isolated point for e = . In Fig. 3, for fixed