Journal o f Technical Physics, J. Tech. P h y s 21, 2, 253 -26 5, 1980. Polish Academy o f Sciences, Institute o f Fundamental Technological Research, Warszawa. NONLINEAR OSCILLATIONS OF THE THIRD ORDER SYSTEMS PART m . PARAMETRIC OSCILLATION N G U Y E N V A N D A O (H A N O I) Introductio n "he theory o f the parametric oscillation o f the second-order system has been in- vestgated in a lot o f publications. F o r a long tim e it has played an im portant role in the theory o f nonlinear oscillations. R ecently, in som e problem s o f the dynam ics on e can mee. the param etric oscillation o f the third-order system [7], to the study o f w h ich this charter is devoted (cf. [11, 12]). n the first Section the approxim ate solution o f the m otion eq uation is constructed. The stationary solutions are studied. The second Section is concerned with the stability condition o f the stationary oscilla- tioi. T he R ou th-H urw itz criteria are taken out. The influence o f the C oulom b friction on the param etric oscillation is considered in the Sect. 3. In this case the resonance curve has the d o sed form . n Sect. 4 the influence o f the turbulent friction on the param etric oscillation is studied. T h i friction limits the growth o f the parametric oscillation and causes a considerable chaige in the rigidity o f the system investigated. Under the influence o f the com bination friction (Sect. 5), the resonance curve is simlar in quality to that in the case o f the C oulom b friction. It is o f closed form as w ell. 1. Construction o f A pp ro xim ate S olution Let us consider the parametric oscillation o f the system described by the third order diferential equation o f the form : (1 .) x '+ Ệ x + Q 2x + Ệ Q 2x + e [k x 3 + hx3 + R (x,x ,x )-cxco s y t] = 0, w b re Ệ ,Q ,k ,h ,c ,y are con stan ts and R (x ,x ,x ) is the function characterizing the ncilinear friction. W e assume that there is a resonance relation eA = Q2(l -Tj2), ri=-^Q- T hen, Eq. (1.1) can be written as: Ỉ54 Nguyen Van Dao ^ h e n 1.3) f ( x , X, x) = A X + ỆẠx + k x 3 + hx3 + R (x , X, x ) . A partial tw o -p a ram eters solu tio n o f (1.2) is fo u n d in the series: 1.4) X = ỠCOS + + EU1 ịa , ip, -y/j + e2u2 ịa , ip, -yfj + n w iich us (a , y, 0) are p erio dic fu n c tio n s o f 6 a nd w ith-th e period 2 71, an d a, ip are u ncto ns o f time d ete rm in ed fr om th e set o f e quatio ns: ~ = eAl(a,xp )+£2A 2{ a ,y )+ •••, ; i . 5 ) . ^ = eBL(a, ỳ )+ e2B 2(a, y>) + 1 ) d eterm in e th e functio n s US, A S, B S, first w c calculate: dx y ~dt = 2 d 2x • I A n . CUị \ 2 asm(p + s \A X cos(p—aB1sm(p+ + £ • • 9 y2 I d2ui \ — acoscp+ —yA ! sin(p — yaB1 COSẹ?+ - Ỵ T I + 8 1.6 ) í/3A' y 3 / 3 3 ổ3Mj \ 2 - ^ - 3- = asm 9? + £ I — ~ ^ -yM 1 cosẹ> + fl-Sj SÌ1199 + —^ 3 I + £ •••’ <p = y f + y . S ib stitu tin g E qs. (1.4), (1.6) in to (1.2) a nd co m parin g the coefficients o f e w ith eq ỉegrtes, w e ob ta in : Ô3U1 ô2u1 y 2 dul y 2 I y 2 \ 1.7) + sinọ? = - / o + ữ c o s ^ c c o s /o = / ịa c o s ọ ) , —-yứsinọ?, — -~ ac o sẹ> Ị. N)W, w e expan d th e fu n c tio n / o in the F o u rier series: 0 0 ; i . 8 ) / o = J j ? [qm(a)cosm(p+pm(a)sinm(p], m = 0 Nonlinear oscillations o f the third order systems. Part III 255 lere 2.71 q0 = — Ị fiacosq), — -y ứ sin ọ ? , — ^ị-ứcosọ?! d(p, o ' 271 I .9) qm = — J" /Ịa c o s ẹ ? , — ~-a sin ẹ > , — ^ -aco sọ?! cosmcpdcp, o ' . 271 pm = — Ị / Ị a c o s ạ ) , —-y ứ sin ọ ?, — —-a c o s ^ Ị smmĩpdcp. Ố ' T he function «! satisfying Eq. (1.7) will be fou nd in the form 1.10) ^ [G„(a, yj)cosn<p + B n(a, y)sinm<p] /ith the additional con dition that it contains no resonance terms. It will be seen later hat this condition is equivalent to the following: the function Wi does not contain COS99, in 99. By substituting (1.8), (1.10) into (1.7), w e have1.10) into (1.7), w e have: |yyGn- ! / / „ j s i n A ! 9 5 - ỈƠ„Ị cosnọ? - Ị-^-v41 + y £ a 5 iỊco s 9 9 + Ị^2 ~ ứjBi - y M ij s i n g j = c a c o s ỹ c o s y í 00 - ) (ạmcosrn(p+pmsinm<j - y , (<7 mCoswẹ>+/>msinwẹ>). m = 0 By com parin g the harm onics sinọ?, COS op, one obtains: y Al +yỆaB1 = - - ^ c o s 2 rp+q^ (1.12) y Ệ A i - ^ - a B i = - - ^ - s i n 2 y + P i By com paring the other harm onics, w e get: nguy en van ưao On solv in g Eqs. (1.13), w e ha ve: (1 — 2 ,, -ị- accos2v>j ổ 3, c " = y / y £ . I - y + £/>„ - |-j-r tứ cco s2y > + -^ -a c s i n 2 y I ỗ 3n = n 2 r I t u i Y ' 1— — U( („*_!) Ị^+rl^Ị From (1.12), w e have Q 1 £ < /0 sin99> + .í2</0 cosẹ>> — -Ị-ac COS 2^-* Ạ-OCỆs in 2 y ^ = i 2 ( l 2 + £ 2) ’ ( 1 . 1:) £ < /o c os<p> — í2< /0sinẹ>> + -^-acsin2y> — ^-flccos2y> Bl = a(Ệ2 + ũ 2) ’ wheie ( F ) is th e o perator o f the a veraging function F on tim e. By p uttin g in E q. (1. fo fn m Eqs. (1.3) an d (1.7) a nd calcu latin g, w e h a ve th e fo llo w in g eq u a tio ns o f th e Í approxim ation: (1.10 da £ d t = Ệ2 + Q 2 [ 8 dip ■ E dt = a (i2 + Q*) ĩ-^ -(fc —£Q 2h)a3 — -^ -ứ ccos2y — -^ -^ sin 2 ^ y + i?! j , Ị^— (Ệ2 + & 2) A a + -^-(Ệk + Q*h)a3 + ^ - s i n 2 xp- ac . ■ ~ —ỆQos2tp+R .2 Y w h ec 2Ệ R i = <JR0COSỌỊ>4- — </?0sino9>, y 2Ẽ ' l . n R 2 = — ( R 0c o s (p )-(R 0sinq)'), y i v y 2 R q = Rịacoscp, — |- a s i n (p, — — acoscp Ttus, in the first a p p r o xim a tio n w e h ave a partial so lu tio n o f E q. (1.1) in th e for L 2 Nonlinear oscillations o f the third order systems. Part ỈII 257 iere a and Iff are the solution o f Eqs. (1.16). The refinement o f the first approxim ation is: 9) .Y = a c O S 0 Ệ \ Ệq„—Qnp„+ \-ịnacsm2y)— ~accos2tp jỗ3n ~Ó2{n2-ỉ)(Ệ 2 + n2Q2) - X X C O S /7ọ? + Qnq„ + Ệp„ — ị^-nac cos2ĩf + y -ứcsin 2^Ị<5 3n Q20 ? - ì ) ( Ệ 2 + n2Q 2) sin H(p th a and y> being the solution o f Eq. (1.16). The stationary solution o f the set (1.16) is determ ined from the equations: .2 0) ^ - a 0sin2ĩp + ^ - c o s l i f = -— { k - Ệ Q 2h ) a ị + R y , l y 4 o ~ ~ a 0co$2y.'- sin 2 y = — (Ệ2 + Q 2)Aa0 + -^-(Ệk + Q4h)ao + R 2- 2 y 4 y 4 y By elim inating the phase y), we obtain the equation for the am plitude a0 : .21) W(ao,y) = 0 here 3 ~>QZ .22) W(a0, y) : ỆA+ 4 ka2° + a0( f + Q 2) V 1 + > Rl + 0 2 c 4 Relation (1.21) is plotted in Fig. 1 for the case R = 0, -Ệ = ũ = Ỉ, c* = 0.05, kị, —0.1 and /7* = 0 (curve 1), /7 * = 0.05 (curve 2) and /7* = 0.1 (curve 3). From this cure, it is seen that with increacing h, the maximum of the amplitudes decreases and le nonlinear system b ecom es harder. In Fig. 2 the resonance curves are presented for ie case R = 0, Ệ = o = \, c* = 0.05, /?* = 0.1 and k* = 0 (curve 1), k* = - 0 .0 5 :urve 2), /c* = -0 .1 (curve 3). With decreasing k, the maximum of the amplitude ecreases and the nonlinear system becom es softer. Ịả'* = -Qĩk, /7* — c* — p T c j- F i g . 1 i J o u rn a l T ec hn . Ph ys. 2/80 Nguyen Van Dao B\ contrast with the parametric oscillation in the well known second-order system, ie riiidity of the nonlinear system and the maximum of the amplitudes of oscillation ĩre tepend on the com bination o f the param eters h and k. The system considered is hart system if T = £ k + Q Ah > 0 and a soft one if T < 0. I f Q = ỆQ2h — k is positive, len tie m axim um o f am plitudes decreases w ith increasing Q. 2. S tability o f S tationary O sc illa tio n F ist we shall con sider the stability o f the stationary solution a0 Ỷ 0 o f Eqs. (1.16). ubsttuting in them a = a0+ỗa, y = ip0+ồy) 'iíh í0 , xp0 being the solution of Eqs. (1.20), we have the following variational equations: ỊỊ^-(& -£í22/7)ứẳ + aoỊ— Ị j < 5 c - (Ậ2 + Q 2)Aa0 dt Ệ2 + Q 2 + 2. 1) + *tfỉc+ữ*h)al 2 y cỉôyj - 2 ỗ y > Ị, I Ả-(Ệk + QVi)a0+ Ị * ì Ị ]ôa + ị j { k - Ệ Q 2h ) a l + ~ R ^ ỏ v } - dt Ệ2 + Q 2 Tie characteristic equation of this system is: 2.2 ) yheri 2.3) Ằ2~Z + S = 0 , z = 5 = ễ2 + Q 2 r ~ ( i ' - ỉ ữ 2/ i ) « ỗ + — (0 0 * 1) ' ] . s2a [ 2/1 4Q 2(Ệ2 + Q 2) X y + 2 Q2 Ỵy(Ệk + L)4h) a0 -f 4Í22 (— 2 + - ị ( k 2 + Q 6h2)aị + 3ao(fc- í f i V , ) ( f } + + « ( £ ) ( £ ) ’ +4 í ă í ă i + Ệ2 + w + a 0 \ a 0 Nonlinear oscillations o f the third order systems, Pari III 25! ( 2 .‘ ) rhe expression z can be also written in the form: d W e2a 0 4 Q 2(Ệ2 + Q 2) da0 h er w is o f the form (1.22). C onseq uen tly, the stability condition o f stationary solu tioi is: (2.3 3(k -ỆQ 2h)al + 2(a0R iy < 0, ÔÌV (2.0 ôa0 > 0 . Mow, let us consider a special case o f the stability o f equilibrium a = 0, w h en th sysem (1.16) has the form : ( 2 .') da £ [ 3 r, c „ ca - . _ 1 = ~ỆĨ + Õ T [ 8 4 1ccos2 ^ ~ 2 ’ = ~nr + 5 r [7 (f 2+ fi2^ a + Ậ + . i/r £2 + £ 2 8 <7 dtp h i c ■ n c a t o — flsin2v> — - y - f c o s 2 ^ . (2.r or In this case we put a = ỗữ, y = ^o + ổ y and the variational eq uation s are: d ò a dt EC I y \ = “ Ị 2 c o s 0 + ^ ° / 0 = V + Q2)A 4- -ysin2vj0 - f cos2y>o ỗứ £C • M c —— = 1 ■ ■_ sin(2u’o 4- v) 00, eft 2 y ý Ệ r + ữ 2 (2.0 0 £C 2y (Ệ 2 + Q 2) - (Ệ2 + Q2)A - \/ệ2 + Q2 cos(2tPo + O) ỗ a , 0 = arete V 2Ệ • The second Eq. (2.9) yields: cos(2 tpo + 0) — —- A y i 2 + Q 1 sin(2 Y’o + ớ) = ± ~ \ / c 2—4(Ệ2 + £}2) A 2 am therefore the first Eq. (2.9) is o f the fo rm 1: dòa ~ dt~ ^ ỹ | = f » V - 4 T F ’ + i F ) 2 * í « . Hence, there fo llow s the stability co ndition o f equilibrium a = 0 c M l > 2 ] /ệ 2 + Q 2 8* Nguỵetì Van Dao 2.10) rị1 < 1 , ĩ]2 > 1 H — = = = = , ĩ] = y/2í2. 2í 2 2 | / | 2 + £ 2 2& 2 > / f 2 + í 2 2 In the figures presented the stability conditions are satisfied on the lines in b old face. 3. Th e In fluence o f C oulom b Fric tio n Let us consider the case 3.1) R(x, X , x) = ÌĨQsig n * , vhere h0 is a positive constant, •+ 1 if À '> 0, 3.2) sig n * = — 1 if X < 0, 0 if X = 0. In this case it is easy to verify that 3.3) 71 h0 if Ũ ^ 0 , <i?0sinọ9) = 'o if a = 0 . <i?0cosọ)) = 0 for all a. N o w , Eqs. (1.16), (1.17), (1.22) are o f the form : for a 0: da £ ~dt = Ệ2+ Q 3.4) ; dt a{Ệ2 + Q 2) Ợc — ỆQ2h)a3 ^-ứcos2y — ^£sin2y> — p ~ ^ o | » ± ( p + Q *)Aa+ — (Ệk + Q V i W + ^ s m 2 ụ ,- 3.5) ca f- ", 2 ~ f c o s 2 v + 2 Ệ , Ri — 77 n II R-> = —-ho 71 V - (ỉ a + f f a V + O * L + ị « Í V + ^ h X - $ 3.6) The equation w — 0 yields: ;3.7) r 1 + ~4 Ỵ ị 2 + -Q 2 ) (Ệk* + Q*h*)a 2 + ntf2 + Q'2)a 1 * £ 2 + £ 2 1 -X / 1 2 + Q 2 VÍP V 4 - ? 1 4 f J 2 - 0 ( f / i J|(i 2 2 - A - lls) f l 2 + - ^ - / i S , 7 1 (2 (3.8) ố‘/c yt = /; = e] \ /;* = E^ ° * Í 22 ’ * ~ Í 2 2 ’ 0 Í 3 2 ’ c * = £C Í P V y 2 Q ■ Nonlinear oscillations o f the third order systems. Part III 261 In Fie. 3 the dependence of a0 on i f is presented for the case Ệ = Q = 1, /z* = 0.05, ¥ — —0.1, c* = 0.05 and h0 = 2 .5- 10-3 (curve I), /? 0 = 5 • 10-3 (curve 2). H ere the :sonance curve has a closed form. H owever, only the upper branch lim ited by the ỉrtical taneencies corresponds to the stability o f the stationary cond ition (2.6). T he lerease in h0 leads to the narrowing o f the resonance curve. W ith sufficiently high values f h0, there is no stationary oscillation. T o find the expressions (1.14) first we expan d: 00 4 V I 1 sign sin 09 = — > ——-—— sin(2;?7+ 1) 09. 7r 2m + 1 m — 0 N o w the form ulae (1.9) are o f the form : Po = P i = 4s = 0, i# l , 3 , q1 = ỆAa + ~ kaz, Pl = - A Q a - Ặ h Q 3a3, 4 g3 = - L ka3, p 3 = ~ h ũ 3a3 r\ _ 4/7 o ' o Pin, - 0 , p 2m+1 - ^ i + T r ’ m ^ Therefore, the expressions (1.14) are: Ho = G0 = 0, - - T f S W T W ) ( ^ 0 + ^ - 3 ^ - f « > c o s 2 V + 3i2Ca s m 2 ,, |, " 3 = W t f ‘ + 9Q2) I - ^ - 0 + ■§- (3fr + n3 - 3 £ c a c o s2 y - ĩ c a s i n 2 y I , 2m+1 nQ m (m + l ) [ f 2 + (2m + l ) 2i32] ’ ^ 2m ’ ^ 2m+1 7T Í32w ( w + l ) ( 2 m + 1 )[£ 2 + ( 2m + 1) 2Í 2 2] ’ 2m Nguyen Van Dao 4. The Influence of Turb ulen t Frictio n on P ara m etric O scillatio n N ow, we turn to the study on the case o f the turbulent friction, when R(x, X, x) has 2 form: .1) R(x, x,x) = /?2 x 2sign x , lere h2 is a positive constant. It is easy to see that: ( / ỉo S Ì n ọ ? ) = .2) — ■ 2,71 <R 0coscp) = 0, d therefore Eqs. (1.16) take the form: for a =£ 0: I ? _ 2 h2y a , da dt dtp Ệ2+ Q 2 (ỆQ2h —k)a3 + ~a co s2tp+ -^ -£sin 2 y > + ^ - h 2Qa2 , 4 4 Í2 371 j w = £ dt a(Ệ2 + Q2) ^ 7 ( f 2 4 - £ ? 2) z l a 4 - (£Ả ' + í 24/0 ữ 3 + - ^ - s i n 2y> 2Q Ệ c o s 2 w + -^ -A 2 i3 2a 2| . 4Í2 By com paring with Eq. (1.16), w e have: * , - .4) 3ti hi Qaz. R2 = -Z— h 2 Qza2. 07Z Consequently, the expression (1.22) is: \ 2 U Ỉ 5) W(a0, y ) = U A + ~ k a l\ + q 4 a + ± h Q 2a2o + ^ f - h 2a0 Fig. 4. [...]... 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 , 3 7 ,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...263 Nonlinear oscillations o f the third order systems Part III T h e eq u ation V2 w — 0 y ie ld s : 4 (Ệ 2 + Q 2) ^ k * + Q *h * ) a ° + 37i(Ệ2 + Q 2) h * ũ o ± 1+ v , i 2+ ữ 2 2 ± Ệ2 + Q 7 ^ // A c ì , } h* = J m* 9 Ỉ 2 Q2 ’ 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 ra g u e 196 7 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... m e th o d s ) H C a n d e r e r , N ic h tlin 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 , 2 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 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... 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 3 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) . Warszawa. NONLINEAR OSCILLATIONS OF THE THIRD ORDER SYSTEMS PART m . PARAMETRIC OSCILLATION N G U Y E N V A N D A O (H A N O I) Introductio n "he theory o f the parametric oscillation o f the. with the parametric oscillation in the well known second -order system, ie riiidity of the nonlinear system and the maximum of the amplitudes of oscillation ĩre tepend on the com bination o f the. ’ 0 Í 3 2 ’ c * = £C Í P V y 2 Q ■ Nonlinear oscillations o f the third order systems. Part III 261 In Fie. 3 the dependence of a0 on i f is presented for the case Ệ = Q = 1, /z* = 0.05, ¥ —