Vietnam Journal of Mechanics, N CN ST of Vietnam T. XX, 1998, No 2 (11 - 17) I N T E R A C T I O N B E T W E E N T H E F O R C E D A N D P A R A M E T R I C E X C I T A T I O N S W I T H D I F F E R E N T D E G R E E S O F S M A L L N E S S N g u y e n V a n D a o Vietnam National University, Hanoi A B S T R A C T . The nonlinear system under consideration in this paper has a specification r’hich can be stated as an interaction between the first order of smallness nonresonance larametric excitation and the second order of smallness resonance forced excitation. In he first approximation these excitations have no effect. However, they do interact one /ith another in the second approximation. The equations for the amplitude and phase of oscillation are found by means of the symptotic method. The stationary oscillations and their stability are of special interest. h e e q u a t i o n o f m o t i o n a n d a s y m p t o t i c s o lu t i o n s et us con sid er à no n linear system governed by the d ifferential equation X -f IJ 2 X — e p x c o s u t + e2 [A x — 2h x — /? £ 3 -f r cos(cj£ — 77)], (1 .1) u 2 = l + e2A, (1.2) sm a ll dim en sion less p aram eter, 1 is n atu ral frequency, A is detun ing p a - er, p, h, /3 , r, ĩ], U) are constants and overdots denote differen tiation w ith t to tim e t. /e look for the solution of the equation (1 . 1 ) in the form : X ~ a cos 0 -Ị- eu 1 (a,rp, 0) H- e2u 2 (a, xị), 0) -f . . . , (1.3) 0 = U)t + rị), xit(a,xjj,0) are perio d ic functions w ith period 2n w ith respect h a n g u la r v a ria ble s 0 and Ớ, and a and tỊ) are function s of tim e w hich w ill ,errnined from the equations: ~ = e A ị(a ,xp ) + e 2 A 2 [a,xl>) \ u { L 4 ) ~ = El3ị{a,xp) -I- E 2 B 2 (a,ip ) + . . . . 1 1 ,hese equations Ai(a,rị')), arc periodic functions of the angular variable /ith period 2n. Substituting the expressions (1.3) and (1.4) into the equation (1.1) and co m ing tile coefficient of £ l we obtain — 2iưAi sin ớ — 2oơaB\ COS 6 -f cư2 ^ ' QQ2 + u 1^ — aP cos(0 — 0 ) COS 0. (1-5) tnparing the harmon ics in (1.5) gives: A i = B x = 0, pa «1 2 w 1 cos rị) — - COS(20 — i/>) 3 ( 1.0) Comparing the coefficients of e2 in (1.1) we have — 2 u M 2 sin 0 — 2 u a B i COS 0 + CƯ2 ^ + u2'j = pu 1 COS U)t + Ad COS 6 -f 2hu>asiii6 — /?a3 COS3 Ớ -f rcosỊớ — ( 0 + ^)Ị. (1.7) uating the coefFicients of the first harmonics sinớ and COS 6 in (1.7) we obtain I _ 2 _ p d T yl2 (a, t/>) = - h a - sin 2ip — - ~ s in ( 0 + r7), oOJ Zu) ỉ h ( a ^ ) A p 30 o p , r , , , -f ~ a — cos 2 tp — cos(tp + v), 2 u 12cư3 8 u> 8 w 3 2 wa ( 1.8) ! 1 . Thus, in the second approximation one has X = a cos 0 -f I L 2 w 2 cos xp — - cos(20 — t/>) 3 lere a and tỊ) satisfy the following difTerential equations da dt dxị) dt 2u> 2 tƯ 2 2/iwa H sill 20 -f- r sin(V> -f 7/) , 4 p 2 3/3 ? p 2 r A + - — a + — cos 2xị) - COS(0 + r/) 6 4 4 a (1.9) (1.10) / 0 . 12 t a t i o n a r y s o l u t io n s )enoti Iig c = A + — - - P a l , D = — , H = 2 hw, (2 .1 ) 6 4 4 ve the following equations for stationary values do, 00 satisfying the relations: d-Z = d4 = 0 ; / = „ = 0 , ( 2 .2 ) dt dt ' J y ' v / = H a 0 + D a 0 sin 2xpo + r sin(0 o + Í?), 0 = C a 0 + D a 0 COS 2 0 0 + r cos(0o + *l) ■ Ve tra n sfo rm equations (2.2) into two equivalent ones: / cos Ipo - g sin -00 = { D - C)a0 sin Ipo + H a 0 COS ipo + r sin 77 = 0, / sin 100 + <7 cos V'o = sin 0 0 + [D - f c)a0 COS rpo + r COS r; = 0. o n d itio n for re ality of s in i/>0 and COS 00 is [2 , 3|: sin v>0 = cos xpo = [D2 — (H2 + C 2 ) ] a o r [ i f s i n TỊ — (D — c) COS 77] [.D2 - ( i 7 2 + C 2)]a 0 ” ■ (2.3) (2.4) «2 [(£> - Ơ)2 + / / 2] > r2 sin2 n, (2.5) a.Q [(D + c ) 2 + H 2] > r2 COS2 r]. (2.6) ) Supposing that M = D2 - (//2 + c 2) Ỷ 0, (2.7) ve from equations (2.4): r [H cos 77 - (D + C ) sin rj] (2.8) Climirnating 00 we obtain: lV(ao,w )=0, (2.9) : a 20 [D2 - (//2 + c2)] 2 - r2 [//2 f D2 + c 2 - 2D C COS 2r? - sin 277]. (2.10) 13 b) if M = D2 - (7/2 + c 2) = 0, (2.11) if the resonance curve takes the form c = ±VD2 - IP , Ỉ0aị = 6 ± y ị £ y - 4 h W , i = A +^ , (2.12) 1 by (2.8) one sh o uld have TVi = II cos t] — (D -f c ) sin TJ = 0, 7V2 = H sin TỊ — (D — c) COS r/ = 0, •quivalent-ly, 7VX cos T] -f 7V2 sin 7/ = 0, TVj sin T7 — yv2 COS 77 = 0. ĩse relations give: // = D sin 2 r/, c = Dcos2r 7 . jstitu tin g these values into (2.5) and (2.6) we ob tain the follow ing re strictio n the a m p litu d e at)' -Ỉ > J5 5 • p.13) ite . A s it w ill be seen later, the curve (2 .11) serves as the b o u nd ary of the bilit,y zone. S y s t e m w i t h o u t f r i c t i o n N ow , let us co n side r a special case w hen /1 = 0 and the equations (2 .4 ) have 5 form : { D - C ) a 0 sin ip 0 = -rsinr;, (D -f c)a0 cos tpo — - r COS r/. a) If D - c Ỷ 0 a n d D + C / 0 , then the resonance curve c 1 is d eterm ined the eq uatio n of typ e (2 .10) w ith H — 0: w ự , a l ) = 0, (3.2) lere W { u 2 , a 20) = a 20 { D 2 - c2)2 - r 2 (L>2 -f c2 - 2 D C c o s 2 r /) . (3.3) 14 [n a particular case, when T] = 0 , 7T the resonance curve c 1 degenerates into 1) T h e curve c Ị : D — c (double) 2) T h e curve Cj* : á ị [ D + C ) 2 — r 2 = 0. n i | = ' | — the resonance curve C \ degenerates into 3) T h e curve C j : D — —C (double) 4) The curve C j: a ị(D — c ) 2 — r2 = 0. b) If D — c = 0 (the resonance curve c2), then from (3 .1) we have 0 .a0 sin tpo = —r sin TÌ => sin 77 = 0 =$■ TỊ = 0 , 7T, 2 / } a n cos 00 = —r cos TỊ = ± r =>■ xf>0 = arccos^ ± ~ ^ => afj > r 2 c) If D + c = 0 (the resonance curve c3), then from (3 .1) we have 7T 37T 0 . a 0 c o s 0 0 = — r c o s 77 => COS t] — 0 => T] = — , — , 2 2 / r \ 2 r 2 2 D a 0 sin t />0 = —rsin rj = ± r =>• tpo — arcsin^ ± — — J => a,Q > • S t a b i l i t y o f s t a t i o n a r y o s c il la t io n s W ith the no tatio n (2 .1) the equations (1.10 ) can be w ritten in the form : ịSi — ịjỊ a 2)asin 2xịj -f- r sin(v> 4 - 77 )I , d t 2u> ( 4 1 ) dip e r / \1 a — = C a 4- D a COS 2xị) + r COSI0 4- 77 ) . at 2 u 1 ỉtu d y the s t a b ility of sta tion ary o scillatio n s w ith am p litud e a 0 and phase xpo :rm in ed from eq u ation s (2.2) or (2.4) we introdu ce the v ariation s: a = a - a 0 , ip = Ip — xjjo. s titu tin g these values into (4 .1) we obtain = - — j( // -f D s\n 2t/)0)rt + [‘2D a0 COS 2t/)0 -|- r cos(V>0 + v)} (4-2) = - — j(ơ + c 'a 0 + Dcos2rpo)a - [2 /)a 0 sin 2t/j0 -I' r s ill(-00 + f?)]*/'j> <ỉấ e 2 dt 2 UJ dtj) e 2 (it 2cJ 15 e ơ = - ^ « 0. e 2 e4 aoA 2 + — H* A — — - S = 0, 2 U) 4iol e A is c h a ra c te ristic num bers, The characteristic equation for last two equations is (4.3) i*7+ = a0 [ỉỉ — D sin 200 - rsin(0o + r?)] = 4/iwao > 0, (4.4) s = (H -f D sin 200) [2 D a0 sin 200 + r sin(V>0 + r/)] (4-5) -f [ c + c ' a 0 -f D c o s 2 0 o ) [ ^ ^ ao c o s 2t/'o + rco s(t/>0 + 77)]. T h e exp ression for s can be w ritten as s = a 0 { D 2 - H 2 - c 2 - a ơc c " ) + a 20C ' D COS 2xp0 . (4.6) n (2 .2 ) an d (2.3) it follow s: D a 0 COS 2 0 0 = —C a 0 — r ( c o s 0 0 COS 77 — s i n rpo s in 77) . stitutiing here the expressions cost/>o and sill 00 from (2.8) we obtain L>a0 cos 200 = - C a 0 TTjr— r 7^ r (c - D cos2i]). do\UL — l i L — (JL) s, we have 2 (jD »2 - H 2 - C 2 ) 5 = 2 d o ( D 2 - H 2 - c 2) 2 - 4 a 20 C C ' { D 2 - 7/ 2 - c 2 ) ớ i y - 2 r 2 C C ' + 2 r 2D Ơ COS 2r/ = , <7 d o 5 a i(ẽ n k ^ j S ' (4.7, T h u s, the stability condition of the stationary solutions do and xpo takes the n d W M ^ - > 0, (4.8) ơ ã Q M = n 2 + C 2 - D 2 . (4.9) 3 resotnance curve (w = 0) divid es the plane (d o ,cư) into regions, ill each of ich th e exp ressio n w hag a definite sig n ( + or —). If m oving up along the 16 it line p a ralle l to tlie axis do, we pass from a region w < 0 to a region w > 0, t the p oin t of intersection between the straigh t line and the resonance curve riva tiv e d w /d a o is p ositive. So, this point corresponds to a stable state llation if M > 0 and to an unstable one if M < 0. O n the c o ntrary, if is from a region w > 0 to a region w < 0, then the p oint of intersectio n ponding to a stab le state of oscillatio n if M < 0 and to an un stable one if his w ork was fin a n c ia lly suppo rted by the C o u n c il for N a tu ra l Sciences of im . R e f e r e n c e s litro p o lsk i Y u . A ., N guyen Van D ao. A p p lied asym ptotic m ethods in non - fiear o scilla tio n s. K lu w e r A cad em ic P ub lish e rs, 1997. guyen V a n D ao. In teraction of the elements c h arac te riz ing the q uad ratic o n line arity and forced ex citatio n w ith the other excitatio n s. J. of M echanics o 4, 19 9 7. guyen V a n D ao . Interactio n between the elements ch ara c te rizing the forced nd p a ra m e tric ex citatio ns. V ietn a m Jo urn a l of M ech anics, N o 1, 1998. Received N ovem b e r 15 , 1 9 9 7 T Ư Ơ N G T Á C G IỮ A C Á C KÍCH Đ Ộ NG TH Ô N G s ố V À CƯ Ỡ N G BỨ C CÓ B Ậ C BÉ K H Á C NHAU ự tươ ng tác giữ a kíc h động thông số không cộng hư&ng có độ bé bậc m ột ch dộn g cưỡ ng bức cộng hường có độ bé bậc hai đã được khảo sát. ơ xấp r nh ấ t c á c kích động n à y không gây ra hiệu quả. Song chú ng tương tác lẫ n trong x ấp xỉ thú' ha i. C ác dao động dừng và sự ổn đ ịnh củ a chúng đã được iệt q ua n tâm nghiên cứu. 17 . excitation and the second order of smallness resonance forced excitation. In he first approximation these excitations have no effect. However, they do interact one /ith another in the second approximation. The. second approximation. The equations for the amplitude and phase of oscillation are found by means of the symptotic method. The stationary oscillations and their stability are of special interest. h. teraction of the elements c h arac te riz ing the q uad ratic o n line arity and forced ex citatio n w ith the other excitatio n s. J. of M echanics o 4, 19 9 7. guyen V a n D ao . Interactio n between