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: ~ - -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 < 211") is the dephase between two excitations; eA = e(w 2 - 1} is the detuning parameter (1- own frequency} Introducing slowly varying amplitude and phase (a, 8) by means of formulae: x = a cos'¢, X = -w a sin'¢, and using the averaging method, we obtain for a and . e a=- 2w / 0 , . 1/J = wt + 8, (1.2} e the averaged differential equations: 2 fo = awa(a - 1} + pasin28 + e sin(O- cr), (1.3} e ae = - - g0 , 2w g0 = Aa + pacos 28 + e cos(O- cr). The constant amplitude and phase (a, 8) of the stationary oscillations will be determined from the equations: (1.4} fo = O, 9o = 0 or by their equivalents: f =Jocose- g0 sin8 g = fo sinO+ g0 cos - - - = (p- A)asinO + awa(a 2 -1}cos8- esincr = 0, e = awa(a2 -1}sin8- (p + I A)acosO + ecoscr = 0. By D 0 , D 1 , D 2 and D 0 , D 1 , D 2 we denote following determinants: 27 (1.5} D 0 - D1 =I 2 sin<7 -cosu p- fl o:w(a2 -1) o:w(a I -1),1 o:w(a p+ll.' p D2 = aeDz, 2 - + fl 1) D 2 - I, I {1.6) p- fl aw(a 2 - 1) sin a I -casu · In the ordinary region where Do f'O {1. 7) from {1.5), we can calculate (sine, cos 0) and the ordinary part C 1 of the resonance curve C is given by: W1 (fl , a2) = e2(Di2D2 + D~)- 1 = 0 . a o (1.8) The critical region is characterized by the equality: Do =0. {1.9) It is the resonance curve C0 of the Vanderpol's system subjected only to the parametric excitation 2pxcos 2wt (e = 0). To determine the critical part 02 of the resonance curve, we have to solve the system: Do = 0, under the restrictions: D, = 0, Dz = 0 {1.10) a2 {(p-fl)' +a 2 w2 (a 2 -1) 2} ::0: e2 sin2 a, a 2 { a 2 w2 (a 2 - 1) 2 + (p + fl) 2 } ::0: e2 cos 2 a. (1.11) From (1.10) we obtain a (compatible) point I, of coordinates .6.* = p cos 20'' a2 = _ psin2a 1 • ay'1 + p cos 2a (1.12) for which, the restrictions (1.11) lead to an unique inequality: e2 a2>-. (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 < O, J., is an isolated point and does not belong to the resonance curve. 28 (1.15) 2. Different forms of the resonance curve The equality (1.9) can be written as: 1vp2-f>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 < a 2 - : , Co lies above the abscissa - axis 6.. For very small values e, the resonance curve C consists of two branches: the lower 0' and the upper C". The lower branch G' corresponds to very small values a2 • The upper branch C" consists of two loops, lying respectively "inside" and "outside" Co. These two loops are joined at the critical nodal point I*. Increasing e, 0 11 becomes larger. At certain values e3 1 0 1 joins to 0 11 at ordinary singular point J. As e exceeds ej, J disappears. H e > 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 values a = 0.1; p = 0.12, the curve (0) corresponds to e = 0 and the cesonance curves (1)-(4) correspond toe= 0.06 and u = 0; u = '1r 12 ;u= 1r 7r 6; "= 4 a' 2 3 4 Ll Fig.9 30 • respectively. The resonance curves (1) and (2) have nodal points; the resonance curves (3) and (4) have "inside" closed branches. When a varies, the critical singular point I* moves along 0 0 • This can easily be seen in Fig. 3 as well in Fig. 4; the latter has been drawn for a = 0.1, p = 0.05, e = 0.05 and respectively for 1f' cr=O (1), cr = - 311" 1r -- 4 (2), cr = 2' cr= 4· - z 0 2 0.2 3. System with cubic non-linearity The results obtained can be generalized for the system with cubic non-linearity. In thiS case, the governing differential equation is of the form: x + w2 x = •{ Ax- ~1x 3 + a(1- 4x2 ) :i: + 2px cos 2wt + e cos(wt + cr)}, where ~1 (3.1) is the coefficient of the cubic nonlinearity. The averaged differential equations become: a= -~ !o = 2w -~{awa(a2 -1) + 2w pasin 20 + esin(O- crJ}, (3.2) • • } a0=wgo=- •w { -(1a 2 -A)a+pacos20+ecos(O-cr). 2 2 Stationary oscillations of constant amplitude and phase will be determined from the equations: f = fo cosO- g0 sinO= [(1a2 - A)+ p]asinO + aw(a 2 - 1)acos0- esincr = 0, g = fosinO = awa(a ~ 1)sin0- [(1a - A)- p]acosO + ecoscr = 0. 2 2 (3.3) The critical region is characterized by the equality: (3.4) 31 The coordinates (L:.., a:) of the critical singular point I. will be determined from: fl.,. =')'a~ aw*(a:- 1) + p cos 2u, = -p sin 2u. (3.5) liminating .6..* leads to the equation:· (3.6) H a, p, 1 are relatively small in comparison with unity, we have: - for u= 0, a; = - for cr = 1r, - for sin 2u a~ > = 1, .6.* = 1 +p , 1, .6.* = 1- p , 0, not more one solution 0 < a; < 1 , - for si,n 2a < 0, not more one solution a; > 1 . In Fig. 5, for 0). a2 Fig. 5 Stability conditions To study the stability of the stationary oscillations, we use the variational equations: (4.1) 1ere 6a, 5& are small perturbations of a, 0 respectively. The characteristic equation is of the form: e2 e ap2 + 2w S,p + 4w2 82 = O, (4.2) d stationary oscillations are asymptotically stable if: 8/o 8go 8 1 =a-+-->0 aa ao (4.3) , 32 l The first stability condition is: a 2 1 >-. (4.4) 2 In figs 1-5, the dashed line 8 1 is of equation a 2 = 1 2' and the region lying above 8 1 satisfies (4.4). The second stability condition can be written as: 8f8g 8f8g (4.5) 82 = aa ao - ao aa > 0 or in compact form: (4.6) The latter form is valid for ordinary Stationary oscillations and determines, in the ordinary part C1, stable portions of the resonance curve with vertical tangents at the ends. The critical nodal point is of the same stability character as the ordinary portion considered as containing it. Conclusions The Vanderpol's system subjected simultaneously to external and parametric excitations in the resonances of orders 1 and 1/2 has been examined. By using critical singular points different forms of the resonance curve can be distinguished. This publication is completed with financial support from the National Basic Research Programme in Nat ural Sciences References 1. Yano S., Kohe·Shi. Analytic research on dynamic phenomena of parametrically and self· excited mechanical system. Ingenian -Archiv, 57 (1987) 51-60. 2. Tran Kim Chi, Nguyen Van Dao. Non linear differential equation with self-excited and parametric excitations. Colloquia mathematica societatis Janos Bolyai 62 · Differential equations, Budapest (Hungary), 1991 3. Mitropolskii Yu. A., Nguyen Van Dao. Applied asymptotic methods in nonlinear oscillations Ukrainian Academy of Sciences, National centre for Nat ural Sciences and Technology of Vietnam, Kiev·Hanoi, 1994. 4. Nguyen Van Dinh. Stationary oscillation in critical case. Journal of Mechanics, NCNST of Vietnam T, XVIII, No2, 1996. Received January 9, 1997 HE VANDERPOL DUOI KfCH DONG CUCJNG BlrC VA THONG · & CQNG HUGNG B!c 1 vA. 1/2 s6 Kh

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