Ị N )W , w e e x p a n d t h e f u n c t i o n / o in t h e F o u r i e r ser ie s: 00 ; i ) / o = J j ? m= [ qm( a ) c o s m ( p + p m(a )s i nm (p ] , - / o + ữcos^ ccos Nonlinear oscillations o f the third order systems Part III 255 lere 2.71 q0 = — Ị f i a c o s q ) , — - y ứ s i n ọ ? , — ^ ị - ứ c o s ọ ? ! d(p, o ) ' 271 I qm = — J" / Ị a c o s ẹ ? , — ~ - a s i n ẹ > , — ^ - a c o s ọ ? ! cosmcpdcp, o ' 271 pm= — Ị / Ị a c o s ) , — - y ứ s i n ọ ? , — —- a c o s ^ Ị smmĩpdcp Ố ' T h e f u n c t i o n « ! s a t i s f y i n g E q ( ) w i l l b e f o u n d in t h e f o r m ^ 1 ) [G„(a, yj)cosn
+/>ms ( p + p msinm ) m = B y c o m p a r i n g t h e h a r m o n i c s sinọ?, COS op, o n e o b t a i n s : y Al +yỆaB1 = - - ^ c o s rp+q^ ( 12) y Ệ A i-^ -a B i = - - ^ - s in 2y + P i B y c o m p a r in g th e o th e r h a rm o n ics, w e get: n g u y en van ưao On s o l v i n g E q s (1 ), w e h a v e: (1 , , - - ị - a c c o s v > j ổ 3, — c " = y -y / y £ I |I - j - r t ứ c ct ou si2Yy >' 1+— -— ^ - U( a c s i n y I ỗ 3n + £/>„ - n = r („*_!) Ị^+rl^Ị From ( 1 ), w e h a v e Q £ < / sin99> + í < / c o sẹ > > — - Ị - a c COS ^ - * ^ Ạ-OCỆs i n y i ( l + £ 2) = ’ ( :) £ < / o c os
— í < / sinẹ>> + - ^ -a c s in y > — ^ -flc c o s2 y > Bl = a ( Ệ + ũ 2) ’ w h e i e ( F ) i s t h e o p e r a t o r o f t h e a v e r a g i n g f u n c t i o n F o n t i m e B y p u t t i n g i n E q (1 fo f n m E q s ( ) a n d ( ) a n d c a l c u l a t i n g , w e h a v e t h e f o l l o w i n g e q u a t i o n s o f t h e Í a p p r o x im a tio n : da dt = £ ĩ-^ -(fc —£ Q 2h ) a — -^ -ứ cc o s2y — -^ -^ s in ^ y + i? ! j , Ệ2 + Q [ ( 1.10 dip dt = ■ E a ( i + Q*) Ị^— ( Ệ + & 2) A a + - ^ - ( Ệ k + Q * h ) a + ^ - s i n xp- ac ■~ — Ệ Q o s t p + R Y w hec 2Ệ R i = < JR0 COSỌỊ>4- — < /?0 sino9>, y 'l.n 2Ẽ R = — ( R c o s ( p ) - ( R sinq)'), y i v y R q = R ị a c o s c p , — | - a s i n (p, — — acoscp T tu s, in th e first a p p r o x im a t i o n w e h a v e a p a r tia l s o lu t io n o f E q (1 ) in t h e fo r L Nonlinear oscillations o f the third order systems Part ỈII 257 iere a and Iff are th e s o l u t i o n o f E q s ( 1 ) T h e r e f in e m e n t o f t h e first a p p r o x i m a t i o n i s : Ệ Ệq„—Q n p „ + \ - ị n a c s m y ) — ~ a c c o s t p j ỗ 3n 9) Y = a c OS - ~Ó2{ n - ỉ ) ( Ệ + n 2Q 2) \ X Qnq„ + Ệp„ — ị ^ - n a c cos2ĩf + y - ứ c s i n ^ Ị < 3n X C O S /7ọ? + s i n H(p Q 20 ? - ì ) ( Ệ + n 2Q 2) th a a n d y> b e i n g t h e s o l u t i o n o f E q (1 ) T h e sta tion ary s o lu tio n o f th e se t ( 1 ) is d e t e r m i n e d f r o m t h e e q u a t i o n s : ^ - a sin2ĩp + ^ - c o s l i f ly = - — { k - Ệ Q 2h ) a ị + R y , o ) ~ ~ a c o $2 y ' 2y s i n y = — (Ệ2 + Q 2) A a + - ^ - ( Ệ k + Q 4h)ao + R y 4y By e l i m i n a t i n g t h e p h a s e y), w e o b t a i n t h e e q u a t i o n fo r t h e a m p l i t u d e a0 : W(ao , y ) = ) h ere 2 ) W ( a , y) : ~>QZ Ệ A + ka2° + a 0( f + Q 2) V + > Rl + c 02 R e la tio n ( ) is p l o t t e d in F ig fo r th e c a s e R = 0, -Ệ = ũ = Ỉ, c* = , kị, —0.1 and / * = (curve 1), /7 * = 0.05 (curve 2) and / 7* = 0.1 (curve 3) F rom this cure, it is s e e n that with increacing h, the m axim u m o f the am plitudes decreases and l e nonlinear s y s t e m b e c o m e s h a rd er I n F ig t h e r e s o n a n c e c u r v e s are p r e s e n t e d f o r i e case R = 0, Ệ = o :urve 2), /c* = - = \, c* = 0 , /?* = 0.1 a n d k * = (c u r v e 1), k* = (curve 3) With decreasing k , the m axim um o f the am plitude e c r e a s e s a n d t h e n o n l i n e a r s y s t e m b e c o m e s s o fte r F i J o u r n a l T e c h n P h y s /8 -0 ig Ịả'* = - Q ĩ k , / * — c * — p T c j- Nguyen Van Dao B \ contrast with the parametric oscillation in the well k n ow n second-order system , ie riiidity o f the nonlin ear system and the m ax im u m o f the am plitudes o f oscillation ĩr e te p e n d o n t h e c o m b i n a t i o n o f th e p a r a m e t e r s h a n d k T h e s y s t e m c o n s i d e r e d is h art s y s t e m i f T = £ k + Q Ah > a n d a s o f t o n e i f T < I f Q = l e n tie m a x i m u m ỆQ2h — k is p o s i t i v e , o f a m p l i t u d e s d e c r e a s e s w i t h i n c r e a s i n g Q S t a b ilit y o f S ta tio n a ry O s c illa t io n F i s t w e shall c o n s i d e r t h e s ta b ilit y o f t h e s t a t i o n a r y s o l u t i o n a0 Ỷ o f E q s ( ) ubsttuting in them a = a0 + ỗ a , 'iíh í0 , y = ip0 + y ) xp0 being the solu tion o f Eqs (1.20), we have the follow ing variational eq u ation s: ỊỊ^ -(& -£í22/7)ứẳ + aoỊ— Ị j < c - (Ậ2 + Q 2) A a + Ệ 2+ Q dt 1) + *tfỉc+ữ*h)al - 2y ỗ y > Ị, cỉôyj dt I Ả - ( Ệ k + Q V i ) a + Ị * ì Ị ] a + ị j { k - Ệ Q 2h ) a l + ~ R ^ ỏ v } - Ệ2 + Q Tie characteristic eq u ation o f this system is: 2 ) Ằ2 ~ Z  + S = 0, yheri z = ễ2+ Q 2.3) r ~ ( i ' - ỉ ữ 2/ i ) « ỗ + — s 2a = [ /1 Q 2(Ệ2 + Q 2) X y ( 0 * 1) ' ] Ỵ y ( Ệ k + L)4h) a0 - f 4Í 22 ( — + - ị ( k + Q 6h 2) a ị + + 2Q + Ệ2 + w 3ao(fc- í f i V , ) ( f } + + « ( £ ) ( £ ) +’ +4 ía 0ă í\ a ă0 i Nonlinear oscillations o f the third order systems , Pari III rh e e x p r e ssio n z c a n b e a l s o w r i t t e n in t h e f o r m : ( ‘ ) w her 25! is o f t h e f o r m e 2a dW Q ( Ệ + Q 2) da0 (1 2 ) C o n s e q u e n t ly , th e sta b ility c o n d it io n o f s ta tio n a r y so lu t i o i is: (2.3 ( k - Ệ Q 2h ) a l + ( a 0R i y < 0, Ơ ÌV (2.0 ơa0 > M o w , let u s c o n s i d e r a s p e c i a l c a s e o f t h e s t a b i l i t y o f e q u i l i b r i u m a = 0, w h e n th sysem (1 ) h a s th e fo r m : da £ [ i/ r = ~ỆĨ £2 + + Õ £ T [ 88 c r, ca - _ „ 1c c o s ^ ~ ’ ( ') dtp — - cya- ft c o s 2o ^ — ỗữ, y = ^o + ổ y a n d th e va ria tio n a l e q u a tio n s are: EC = + I y \ Ị cos “ + ^°/ (2.r V + Q2)A 4- -ysin 2vj0 - f cos2y>o = ỗứ or —— = eft ( 2.0 £C • M c - ■■_ 2yýỆr+ ữ £C = a rete 00, (Ệ + Q 2) A - \/ ệ + Q c o s ( t Po + O) ỗ a , - 2y ( Ệ + Q 2) sin (2 u ’o 4- v ) V 2Ệ • T h e se c o n d E q (2 ) y ie ld s: c o s ( tpo + 0) — —- A y i + Q sin (2 Y’o + ớ) = ± ~ \ / c — ( Ệ + £} 2) A a m t h e r e f o r e t h e first E q ( ) is o f t h e f o r m 1: dò a ~ d t~ ^ ỹ | = f » V - T F ’ + iF ) * í« H en ce, th ere f o llo w s th e sta b ility c o n d it io n o f eq u ilib r iu m a = c Ml > 8* ]/ệ2 + Q Nguỵetì Van Dao 2.10) rị1 < , ĩ]2 > H - — = = = = , 2í 2 | / | + £ ĩ] = y/2í2 2& > /f 2+ í 22 In th e fig u r e s p r e s e n t e d t h e s t a b i li t y c o n d i t i o n s a r e s a tis fie d o n t h e l i n e s in b o l d fa c e T h e In flu e n c e o f C o u lo m b F r ic t io n L e t u s c o n s i d e r th e c a s e R ( x , X , x) = ÌĨQ s i g n * , 3.1 ) vh ere h0 is a p o s i t i v e c o n s t a n t , •+1 ) sig n * = if À '> 0, —1 if X < 0, if X = In th is c a s e i t is e a s y t o v e r if y t h a t 71 ——-——sin(2;?7+ 1) 09 m —0 2m + t h e f o r m u l a e ( ) a r e o f t h e f o r m : Po = P i = 4s = , q = ỆA a + ~ Pl = 3, k a z, - A Q a - Ặ h Q 3a 3, g = - L k a 3, P in, - i # l , r\ 0, p3 = ~ _ p 2m+1 - h ũ 3a 4/7 o ' ^ i + + ^ T r ’ m ^ o T h e r e f o r e , th e e x p r e s s i o n s ( ) a r e : H o = G0 = , - " = W 2m+1 2m+1 - T fS W T W ) ( t f ‘ + 9Q 2) I - - ^ ^ + ■ § - (3fr + n Q m ( m + l ) [ f + ( m + l ) 2i 2] ’ ^ 2m T Í3 2w ( w + l ) ( m + ) [ £ + ( m + ) 2Í 2] ’ - ^ - f « > c o s V + i Ca s m , , | , n3 - £ ca co s2 y - ĩc a sin y I , ’ ^ 2m Nguyen Van Dao T h e In flu en ce o f T u rb u le n t F ric tio n on P a m e tric O s c illa tio n N o w , w e turn to th e s tu d y o n th e c a s e o f th e t u r b u le n t f r ic t io n , w h e n R ( x , X , x) h a s fo r m : R ( x , x , x ) = /?2 x 2s i g n x , 1) lere h2 is a p o s it iv e c o n s t a n t It is e a s y t o s e e th a t: I (/ỉo S Ìn ọ ? ) = — 2) ■ < R c os c p) 2,71 = ? _2 h2y a , 0, d th e r e fo r e E q s (1 ) ta k e th e f o r m : fo r a =£ 0: da dt (Ệ Q2h —k ) a + ~ a c o s t p + - ^ - £ s i n y > + ^ - h Q a , 4 Í2 371j Ệ2 + Q dtp w = £ dt a( Ệ2 + Q 2) ^ ( f - £ ? 2) z l a - (£Ả ' + í / ữ + - ^ - s i n y> 2Q 4Í2 Ệ c o s w + - ^ - A i 2a 2| B y c o m p a r i n g w ith E q ( 1 ) , w e h a v e : * , - ti hi Q a z .4) R = -Z — h Q za 07Z C o n s e q u e n t l y , th e e x p r e s s io n ( 2 ) is: \ 5) W(a0 , y ) = U A + ~ k a l \ + q a UỈ + ± h Q 2a 2o + ^ f - h 2a0 F ig 263 Nonlinear oscillations o f the third order systems Part III T h e eq u ation V2 w — y ie ld s : (Ệ + Q 2) ^ k * + Q *h * ) a ° + 37i(Ệ2 + Q 2) h * ũ o ± 1+ v , i 2+ ữ 2 ± Ệ2 + Q ^ì // A c , } h*2 = J m Ỉ* Q2 ’ V ' - ỆQ2h2 a Q(ỆQ2h!t.—k^)a0 + j7 l y 2Q ' T h i s r e la tio n is p lo t t e d in F i g f o r t h e c a s e Ệ = Q = 1, A* = 0 , fc* = —0 , = 0 a n d / * = 0.01 (c u r v e ), / * = (cu r v e 2) H e r e th e c h a n g e o f th e c o e ffic ie n t le a d s n o t o n l y t o th e c h a n g e in t h e m a x i m u m o f t h e a m p lit u d e s o f o s c illa t io n but t o th e c h a n g e in r ig id ity o f t h e s y s t e m c o n s id e r e d T h e in c r e a s e in h2 le a d s t o a d e ase in th e m a x i m u m o f t h e a m p l i t u d e s a n d th e s y s t e m b e c o m e s h ard er T h e In flu en ce o f C o m b in a tio n F ric tio n on th& P a m e tric O sc illa tio n In th is S e c t io n w e s t u d y t h e in f l u e n c e o n th e p a r a m e tr ic o s c i l l a t i o n o f th e c o m b i n a n fr ic tio n : R ( x , X, x) = (h0 + h x 2) s i g n x 1) N o w , E q s ( 1 ) are o f t h e f o r m : fo r ữ / 0: da ~dt = dy dt e ~ Ệ 2+ Í 2 | i ( f f l ^ - f c ) a > + ^ - c o s 2v + ^ f s i n 2v > + ^ ( *0 + j A ca E 4Í2 a(Ệ2 + Q 2) w ) £ c o s 2y + I + —- h + ~ h Q 2a 71 Nguyen Van Dao Th: e x p r e s s io n (1 2 ) is: w — ịặA + ị k a ị J + Ì Q / i + ị / i Q 3a 20 + ^ - h + ^ - h Q 2a0\ -.3) Th: e q u a tio n th a t y ie ld s th e r e la t io n s h ip b e t w e e n th e a m p l i t u d e o f o s c illa t io n a n d t h e Lcitirg fr e q u e n c y w ill b e : r ) = 1+ AQ ht (Ệ + Q 2) 371 Ệ2 + Q ± v h p ' \ / ^ 2+ Q ) c l - 7i(Ệ2 + Q 2) a + (Ệ Q 2h * - k * ) a 20 + ~ Ệ Q 2/ĩ* a + - ^ - h * n 7ian In this c a s e th e r e s p o n s e c u r v e is c lo s e d t o o (se e F ig 5) fo r Ệ = Q = , /7 * = 0 , * = - , c* = 0 , h i = 5* " 3, h*2 = 0 R eferences z )SIKSKI, Vibrations o f an one-degree o f freedom system with non-linear internal friction and relaxa tion P ro ce ed in g s o f In te rn a tio n a l C o n fe re n c e o n N o n -lin e a r O s c illa tio n s, I I I , K ie v 1963 z )SINSKI, G B o y a d jie v , The vibrations o f the system with non-linear friction and relaxation with slo\ly variable coefficients, P ro c 4th C o n fe re n c e o n N o n - L in e a r O s c illa tio n s, P g u e 196 H V S r i r a n g a r a j a n , p S r in iv a s a n , Application o f ultraspherical polynomials to forced oscillations o f third order non-linear system, J S o u n d V ib r , , 4, 1974 L H V S r i r a n g a r a j a n , p S r in iv a s a n , Ultraspherical polynomials approach to the study o f third-order nonlinear systems , J S o u n d V ib r , 40, 2, 19 75 > A ONDL, Notes on the solution o f forced oscillations o f a third-order non-linear system, J S o u n d V ib r , ,2 , 1974 j A T o n d l , A d d itio n a l n o te o n a K z ♦ s in s k i, th ird -o r d e r s y s t e m , J S o u n d Vibr., 47, 1, 1976 N g u y e n v a n D a o , Parametric oscillation o f an uniform beam in a Theological model,P ro c 2nc N a tio n a l C o n fe re n c e o n M e c h a n ic s , H a n o i 19 77 } N 'í B o g o liu b o v , Y u a M it r o p o ls k y , Asymptotic methods in the theory o f non-linear oscillations, M o c o w 1963 K Nguyen van r* Dao, F u n d a m e n ta l m e th o d s ) H C a n d e r e r , N i c h t l i n e a r e M e c h a n ik , o f n o n -lin e a r Berlin o s c illa tio n s , H a n o i 1969 1958; N g 'Y E n v a n D a o , Non-linear oscillations o f the third order systems Part / Autonomous systems, J re ch n P h y s , 0, 4, 1979 N g ' Y E n v a n D a o , N o n - l in e a r o s c illa tio n s o f th e t h i r d o r d e r s y s te m s P a r t I I N o n - a u t o n o m o u s s y s te m s , J re ch n P h y s , , 1, 1980 S tre szcze n ie N IE L N IO W E D R G A N IA U K L A D W T R Z E C IE G O R Z Ẹ D U C Z Ẹ ắ C I I I P A R A M E T R Y C Z N E D R G A N IA N iie js z a p r a c '1 sta n o w i czẹ sc I I I p c [11] i [12] R o z p a t rz o n o w n ie j p a m e try czn e d rg a n ia n ie lin io *go u ia d u trzeciego rzẹ d u W ^ n a c z o n o p rz y b liz o n e ro z w iạ z a n ie ró vvn a n ia ru c h u o z p o d a n o w a ru n k i stateczno sci s ta c jo n a rĩgo rc w i^ z a n ia Z b d a n o w p ly w ta rc ia k u lo m b o w s k ie g o i tu rb u le n tn e g o n a d rg a n ia p ara m etry czn e R o z p a t rz o n o kze d g a n ia w p rz y p a d k u k o m b in a c y jn e g o ta rcia N o n lin e a r o sc illa tio n s o f th e th ir d o rd er systems Part 111 265 p e K) M e HEJIH H EH HLIE KOJIEEAHHil CH C TEM T P E T L E rO nO PflJIK A ^ ỈA C T L I I I IIA P A M E T P M ^ E C K J iE K O J IE B A H H H HacTOHiuan paooT a c o cra B jifleT TpeTLK) ^lâCTL p aốoT [1 ] H [ ] PaccMOTpeHbi B H e ổ n a p a M eT p ii' He K O Jie6aH H H H ejniH eH H O H CHCTeMbi T p e T b e r o n o p H A K a O npeA ejieH O n p H ố jm > K e H H e p e m e H n e y p aB H eH H H H n p u B e A C H b i VCJIOBHH y c T O H H H B o c m noH apH oro pem eHHH I i c c ji e f lo B a H o BjiH H H H e K y jio H O B C K o ro H T y p G y jie H T H o ro TpeHHH H a n a p a M e T p m e c K H e K O Jie6aH H H P a c c M O T p e H b i TOH