Journal o f Technical Physics, 20, 4, S it -519 , 1979. Potlsck A cadem y o f Sciences. Institute o f Fundamental T echnological Rtxearch, W arszawa. NONLINEAR OSCILLATIONS OF THIRD ORDER SYSTEMS PART I. AUTONOM OUS SYSTEM S NGUYEN V A N D A O (H A N OI) Introduction A lot of mechanical and physical problems lead to the study of oscillations in the system described by the differential equation of the third order: (0.1) J t +ax+bx + cx = ef(x, X, X, r), where a, b, c are real constants and the function f(x, x9 3c, t) can be expanded in the form: H (0.2) f(x,x,x,t)~ JT elnv,f n(x, x,x). -N It is supposed that the characteristic equation: (0.3) A3 + ứA2-f bX + c = 0 has a pair of cither complex or imaginary roots. We have the following definitions: 1. I f the cha racte ristic Eq . (0.3 ) has a n egative roo t an d a p a ir o f com plex ro ots w ith a negative real part: Ằi = — f, Ằ2 = — *7 + /í2, Ằ3 = —TỊ-iíỉ, then we have the non- critical case. 2. If the characteristic equation has a negative root and a pair of imaginary roots A1 = — f , A2 = iQ> A3 = — /Í3 , then we have the c r it ic a l case. Ia a d d itio n , i f V ^ p ỉqũ , where /7, q are integers and V is the exciting frequency, then we have the critical resonant case. The non-critical case has been investigated in many publications (see, for example [1-6])» but the critical case has not, to the author’s knowledge, been examined hitherto. In this work we study systematically the different kinds of nonlinear oscillations: free oscillation, self-excited oscillation, forced oscillation and the parametric one of the third order system in the critical case, by means of the asymptotic method [8, 9]. In the critical case, in the generated system (fi = 0), there exists the harmonic oscillation depending on two arbitrary parameters a and Y>: X = acos(Q t + y). Wc shall not look for the general solution of Eq. (0.1) depending on three arbitrary constants, instead wc shall construct a family of two-parameters particular solutions, which has a strong stability property. It attracts all solutions close to itself of Eq. (0.1). 7 I. T cch a. Phys. 4/79 512 Nguyen Van Dao The ap p ro x im a te a sym p to tic m ethod is convenie n t fo r the in v e stig a tio n o f stationary o scillatio n s a n d th e ir sta bility. T o assess the v a lid it y o f the theo ry, a series o f experim e n ts on the a n a log com p u ter have been m ade. T h e th e ore tical results show a g o o d agreement w ith the e xperim e nta l ones. T h is c h a p t e r co n s is ts o f five S ections. In the first S ectio n the tw o -p a ram ete rs solu tio n o f Eq. (1 .1 ) is c o n stru cted b y m eans o f the K r u lo v -B o g o liu b o v a s ym p totic m ethod. I n the seco n d S ectio n , the D u ffin g case R - -Ộ X3 is consid e red. It has tu rne d out that the a m p litude o f free o sc illa t ion increases i f ậ > 0 , a n d d ecreases i f /? < 0 . Th e th ir d S e ctio n is d evoted to the stu d y o f the self-excited oscilla tio n in the third -o r d e r system , w hen th e functio n R is o f th e fo r m R = (1 - x 2)x. It has been fo u n d th a t in the system in v estigate d ,' there e x ists a stable lim it cycle. Th e influence o f the C o u lo m b fr ic t io n a n d the turb ule n t one on the self-excited o scilla tio n in the third o rde r system , is exam in ed in Sects. 4 an d 5. It has been p ro ved th a t the a m p litude o f the se lf-e xcited o scilla tio n d ecreases as these fric tio n s increase . In a d d itio n , the self-excited o scillatio n c a n d isap e a r u n d e r the a c tio n o f a stro n g C o u lom b fric tio n . 1. Construction of Two-Parameters Solution. Stability Let u s c o n sid er the fo llow ing n o n lin e ar d iffe rentia l e q u a tio n o f the t h ir d o rde r: (11) 'i' + ỉ'x + ũ 2x + ỉũ 2x = eR(x, X, x), where f , ÍÌ are consta n t s, £ is a s m a ll p aram eters a n d R is a n o n lin e a r fu n ctio n o f X, X, X. W e sha ll fin d a so lu tio n o f E q. 1.1 in the fo rm : (1.2 ) X = acosẹ>+ eUịia, 9? ) + e2u2(a, <p)+ cp = íĩt + yĩ' H ere a an d y are the fu n ctio n s satisfyin g the fo llo w in g d ifferentia l e q u a tio n s: (1.3 ) S u b s titu tin g the expressio n s (1.2 ), (1.3) into (1 .1) and c o m p a rin g the coefficients o f e w ith e qu a l degre es, w e h a v e : (1 .4) 2í2(QaBị —ỈAỵịsinqĩ—líĩiỉa B i + í2 ^ i)c o s 9 ? + (1.5) R0 = i?(acosẹ>,-í2 asin<p,-í22đcos7>)‘ T o fin d the u n k n o w n fu n ctions A l , B i, Uị fro m (1 .4 ), w c first e xpand Rq in th e F o u rie r series: Nonlinear oscillations o f third order systems. Part I 513 00 (1.6 ) Ko = £ [g ,(a ) c o s « 9>+/i„(a)sinM (p], /7=0 £0(0) = J- f Rodv - <^o>> 0 2n g„(<j) - I R0cosn<pd<p = 2 < /ỉ0c0SH95>, 71 0 2n /?„(ớ) = — f R0sinn(pd(p — 2 (R0sinnọĩ>, 71 0 in w hich /J = 1,2, and < ) is the averaged operator on time. The function ;/x is also found in the series: (1.7) ií, = ^ [M1B1(a)cosw ^ 4 i>im(«)sin;/i<iO, Ml with the condition that it does not contain tbc term s having a zero denominator. Dy substitutin g (1.6 ), (1.7) into (1.4 ) \vc o b tain : (1.8) 0 * 2 ( 1 - m 2) [(Ệuim+mQvln)cosm<p+(ỉvim-m íìu lm)smm<p]- m 00 —2 ũ (ữA l + ỉaBị)cos<p+ĩũ(a^B1 -Ệ A^sinip = ^ [g„(a)co&n<p+hn(a)smn<p]. «» 0 B y c o m pa rin g the s in 9?, COS9? terms in (1.8 ) w e have: iQịŨA. + aỆB,) = - £ t(a), 2 i3 (-^ 1+fli251) = A,(a). H ence one obta in s: A < \ _ £ M ứ) + % i ( a) _ í</? o sin ọ 7 > + i3 < ^ 0cos(p> ( ) lW 2£(f2+ £ 2) a tf’+ fl2) » _ £ < -RoS in < p > -£<*oC O S 9>> lV ' 2ứ a ( í J + í 3 2) i 2a ( f J + £ J) T h e c o m p a riso n o f the oth er h a rm o n ics yie ld s: i 2 2 ( l -W 1 2) ( f w , B + w i 2 & lln ) = g m( a ) , £ 2(1 - m 2) (-rní2uỉm + Ệvlm) = /fm(a ), m # 1. H e n ce it fo llow s: ỉ g m ( à ) - m í ì h m {a) Í22(l —m2) (ỉ2+m2ữ2) ’ ih m(à)+ m ữ gm(a) tfJ(l —ma) ( |J+mĩÃ2) 514 Nguyett Van Dao Thus, in the first approximation we have: (1 .1 1 ) X = aco s(.Q /H-yO , where a aud xp arc the solution of the equations: d a _ í < / ?0 sinẹ>>-l í?<y?0 Cos <p> ( 1. 12) dtp _ Q(R0sin(py — Ệ(R0cos(py d t e Q a ( Ệ 2 + Q 2) T h e refinement of the first a p p roxim a tio n is : (I n ) r = /7rvYc(.Q/+ ,,,) + V { h m-inQh„)cosm(Qt+ỳ) + (Ậli,„+mQgm)únin[Qt+' Z j " ú 2(l —/«*)(£* +/»*£?*) #/* =» 0 m*\ T h e c a lcu lation s o f higher ap p roxim a tio n s present n o difficultie s, but are rath er 1. and «irc n o t re p rod u ced here. W e can w rite the first equation o f the system (1.12 ) as: (1.14) 6 0 (a ). dt Obviously, the stability condition of the stationary solution a = a0y &(ơ0) = 0 (1.14) is of the form: (1.15) 0 '(ao) < 0. 2. The Duffing Case L e t u s c o n sid e r the case i? (x , X, X) = —ậ x 3. N o w , E q. (1 .1) has the fo rm : (2 .1) X + Ệx+ ữ2i + Ỉ Q 2x+ ePx* = 0 . I n this case w e have: (2.2) R0 = —/?ứ3cos3<p = — Ỵ ứ3 (3 cos 9?+cos 3ẹ?), a n d th e refore g t( a ) = - J 0 a 3 , h ^ a ) = 0 , (2.3 ) w \ _ w JJ , x 3Ệpa2 M a) ~ 8 + • Bl{a) - + ■ E q u a tio n s (1 .12 ) are o f the form : (2 4Ì ^ = _L/?*/73 _ £ /?*ỵj2 ứ* dt 2 dt ~ 2 Q P ' " 4 (f a + f l a) ■ Nonlinear oscillations o f third order systems. Part I 515 B y in tegrating Eqs. (2.4) w ith the in itia l v a lues (/0 , a0) we o b tain : (2 .5 ) a2 = - CL = a - / ? * ( / - / 0) al Í ft* V = V o -yfí- l n l - ^ ơ - / o ) . H e n ce it follo w s that (see F ig . 1): 1. I f /? > 0 then the am plitude o f the o scilla tio n increases fo r t > f0 fro m a = a0 to a ~+ 0 0 . 2. I f /5 < 0 then the am plitud e o f oscilla tio n decreases from a = (ỈQ to a 0. T h us, 1he n o n linear term has a great in fluence on the fo rm o f the response curv e o f free o scilla tio n . T o c h eck the v a lid ity o f the th e o retical analysis, an a n a lo g -com p u tcr analysis has been carried out. T ile exam ple s o f the w ave fon n s obtained o n the analo £ - computer for E q . (2.1), with the param eters Ệ = 10, i2 = 1, are shown in Fig. 2. In this figu re, the first w ave (a) shows the case p - 0 , the second (b)—(ỉ *-= - JO, and the third (c)—ỊÌ = 10. T h e figure conta in s all types o f osc illa tio ns that arc predicted by the th eory. a „ 0.3 0.2 Fig . 1. F ig. 2. 3. Self-Excited Oscillation In this S e c tio n we s h a ll study the V a n der P o l case: (3.1 ) R(xt X, x) = (1 —x2)x. I t is easy to see th at: (R0sin 9?) = — < /? o C o s < p > = 0 , (3.2) 516 (3 .2 ) Nguyen Van Dao gn = 0, 'in, hm = 0 , m # 1 , 3 , Q Hị = —aQ T h e refo re, the equa tio n s o f the first a p p roxim atio n are: (3.3) X = flcos<p, (3.4) da h ĩ ■* 20FTQ2) 4 - t ). E q u a tion (3 .4) has a statio n ary solutio n ứ0 = 2. T h e sta b ility co n d ition (1.15) for this so lutio n is satisfied. F ig . 3. T h e re finem ent o f the first a p p ro xim atio n is o f the fo rm : (3.5) X = 2cosỢ2/ + y0)+ 4 (g2 + 9Q1) [3 cos 3(^r + Vo) - -jjSin3(flf + Vo) j . T o v e rify th e th eoretical resu lts, the o rig in a l Eq . (1.1 ) w ith R fro m (3.1 ), is m odelle d o n the an a lo g -com p u ter M E D A 4 1 -T C fo r the case ỉ = Q = 1, e = 0 .1 . T h e oscillacio n d ia g ram s are presented in F ig . 3 : displacem ent-tim e an d F ig . 4 : phase pictu re . T h e e xperim e nta l re sults agree w ell w ith the theoretic al ones. “x 2 Nonlinear oscillations o f third order systems. Part Ị 517 4. Influence of Coulomb Friction on Sclf-Exdtcd Oscillation In th is Section the influence o f the C o u lom b fric tio n (4.1) Rf = /?0s ig n x \ h0 > 0 on the self-e xcited Van der Pol o sc illatio n is exam ined. In this case the m o tio n equatio n is o f the form : (4 .2) ‘x +ỆX + Q2X + ỆQ2x+ £ (x 2- l ) x - f £/j0sign.v = 0 T o fin d the solu tio n o f Eq. (4.2), we use the theory represented in Sect. 1. Ta k ing in to account <sinọ? sig n s in ẹ?) = 2/tt, < c o s <p sig n sin <p> = 0 , we have in the firs t a p p roxim a tio n : (4 .3) A* = a cos <p, (4.4) da dt d(p di CỆ ỉ 2+ n 2 ị ( ' i i v = Q eQ ___________ . 1 - - - 1 2(Ỉ2 + Q2) V 4 F r o m E q. (4 .4), it is seen that by the a p p e a ran c e o f the Co u lo m b fric tio n , the o rig in .V = X = 0 is not still an e q u ilib riu m o f the syste m considered. T o c le ar up the in fluence o f the C o u lo m b fric tio n on the self-excited o scilla tio n , w e investigate the statio n ary state w ith the am plitud e determ ined b y the equation (4 .5) A( 1 - A 2) = /4 = - y > .0. T h is equation can be solved g ra p h ica lly b y c o n sid e rin g the poin t o f in tersectio n o f the cu b ic curv e y t = A(i-A2) and the starig ht lin e y 2 = Iho/jtii. F ig u r e 5 leads to the follow in g con c lu sio ns: 1. W h e n h0 = 0 in the system consid ered, th ere exist two stationary states corresp o n d  ing to A ị = 0 (unsta b le) and A 2 = 1 (self-excited o s cilla tio n ). 2. W ith in cre asin g h0 (0 < ỌìqItcQ) < 1 / 3 ^ 3 ) , A l in creases and A2 decreases. 518 Nguyen Van Dao E q u a tio n (4.4) can be w ritten in th e fo r m : cỉ/Ấ £ Ẽ (4-6) ^ r = - 2(ỉ * + Q2) (A ~A^ (A ~ Ẩĩ) = e0(A), w here A3 < A ì < A2y (A3 < 0 ). It is easy to see th at 0 'C ^ i) > 0> <P'(A2) < the refo re the statio n ary state c o rre s p o n d in g to A l is unstable a n d the state A 2 ii (se lf-e xcited o scilla tio n ). O bviously , in th is cas e the C o u lo m b fric tio n decreases the tude o f the self-excited o scillatio n . 3. W ith h igh valu e s o f h0(h0 > (QrtỊ3 j / 3 )) the self-e xcited o sc illa tio n disap pea T h u s, depend in g on its values, the C o u lo m b frictio n c a n either decrease o r exti the self-e xcited oscilla tio n . 5. Influence of TarbuleDt Friction on Self-Excited Oscillation N o w , we go o ver to the study o f the o s c illa t io n d escrib e d b y the e q u a tio n : (5.1 ) x + Ệ x + & x + Ệữ2x + e (x 2- l ) i + e h 2x2signx = 0 , w h ere in the Jast term ch aracterizes the t u rbu le n t frictio n . B y m a k ing use o f the m pre sen ted in P ar. 1 we obtain the fo llo w in g equ a tio n s o f the first a p p ro x im atio n : * = acoscp, d A EỆ ~ dt = (5 .2 ) d(p dt = Q 2 (Ệ2 + Q2) e Q A{\ A * ) - y - Q h 2A* ]• A = 2(ề2 + Q2) Considering in the first quadrant of the plane (y, A) the points of intersection o c u b ic c u rve y x = A ( \-A2) and the p a ra b o la )*2 = (ỉ6/3n)íĩlĩ2A2 (F ig . 6) one can that: F i g . 6 . 1. I n the system described by E q . (5.1 ) there a lw a y s exists the u n sta b le station* state A = 0. 2. W h e n in cre a sin g the coefficient h2 fro m z e ro, the a m p litude o f self-excited o s c illa ti decreases. I n con trast to the case w ith the C o u lo m b fric tio n , the self-excited o scilla ti here is n o t extin guished com pletely. Nonlinear oscillations o f third order systems. Part I 519 References 1. z . O s in ski, Vibra tions o f an one-degree o ffre edom system w ith non -linear inte rnal fric tio n and relaxa tio n , Proc. Inter. Conf. on N on-Linear Oscillations, T. Ill, K iev 1963. 2. z. Osinski, G. B o y a d uev, The vibra tio ns o f the system with iion-linear friction and relaxation with slowly variable coefficients, Proc. 4th C onf. on N on-L inear O scillations, Prague 1967. 3. H. R. S rir a n g a r a ja n , p. Srin iv asan, Application o f ultra spherical polynomials to forced oscillations o f a third order non-linear system, J. Sound V ibr, 36, 4, 1974. 4. H. R. S rir a n g ara ja n , p. Sriniv asan , Ultra spherical polynom ials approach to the study o f third-order non-linear system s, J. Sound V ibr., 40, 2, 1975. 5. A. T o n d l, Notes on the solution o f forced oscillations o f a third-order non-linear systcm> J. Sound Vibr., 37, 2, 1974. 6. A. T o n d l, Additional note on a third-order system, J. Sound Vibr., 47, 1, 1976. 7. z. Osinski, N guyen V an Dao, Parametric oscillation o f an uniform beam in a rheological model, Proc. 2nd N ational Conf. on M echanics, H anoi 1977. 8. N . N. B o g o l iu b o v , Yu. a . M i tro p o l s k y , Asym ptotic m ethods in the theory o f non-linear oscillations, Moscow, 1963. 9. N guyen V an D ao, Fundamental methods o f non-linear oscillations, Hanoi 1969. ]0. H . K aud e rer, Nichtlineare Mecfiartik, Berlin 1958. Streszczenie NIELINIOWE DRGANIA U K LAD 0W TRZECIEGO RZẸDƯ CZẸỐC I. ƯKLADY AUTONOM ICZNE Rozpatrzono drgania nieliniowego ukladu opisanego rów naniera rózniczkowyra zwyczajnym trzecicgo rzẹdu. Przyjẹto, ze u klad podstawow y ma rownanie charakterystyczne o jednym rzeczywistym i dvvổch urojonych pierw iastkach. w niniejszej, picrwszcj czẹáci pracy zbadano digania ukladu autonomicznego. R ozpatrzono nielinio- w oSỎ funkeji restytucyjnej w postaci funkeji Duffinga, charakterystyki tlumienia w postaci tarcia Coulomba i tarcia turbulentnego. Zbadano drgania samowzbudne przy rốznych rodzajach tarcia. p e 3 K> M c H E JIH H E H H L IE KO JIEBA H R H C H C T H M T P E T b E rO nOPflJXKA M A C Tb I. A B T O H O M H B IE C H C T E M L I PaCC M aT pH BaiO Tai KOJieỐaHHfl He/IHHCHHOH CRCTCMbI OHHCyCMOH c DOMOLUbJO OỔbCKHOBCHHOrO A H Ộ Ộ e- peHUHajibH oro ypoBHCHHH TpeT bero n o p a n n a . n p c A n o jia r a c T o r, *rro xapaKTCpHCTH^ecKoc ypaBHCHHe HMeeT OAHH AeftCTBHTCJTbHWA H JJB a MHHMbIX KOpHfl. 3TO T C J iynail H23BAH KpHTJTOCKHM. B HaCTOHmeỒ, nepBoft uacm paốOTbi HCCJieAyioTCH KOJie6aHHH aBTOHOMHOỈí CHCTCMbi. PaccMaTpHBaioTca HCJiHHeữH0CTb ộyHKUHH BOCraHOBJieHHH B BJtfle ộyHKUHH ilK>4>4>HHa H XapaKTCpHCTHKH ACMnỘHpOBaHHH THUa TpCHHH K yjio H a H T yp6yjieHTH oro TpcHKH. HccjieAyioTCH caM 0B03ổy>KtuicMi>ic KOiieổaHHH ỊỤIH pa3JDFiHbix THnoB TpeHHH. Received October 17, 1978. . Physics, 20, 4, S it - 519 , 19 79. Potlsck A cadem y o f Sciences. Institute o f Fundamental T echnological Rtxearch, W arszawa. NONLINEAR OSCILLATIONS OF THIRD ORDER SYSTEMS PART I. AUTONOM OUS. u a tio n s (1 .12 ) are o f the form : (2 4Ì ^ = _L/?*/73 _ £ /?*ỵj2 ứ* dt 2 dt ~ 2 Q P ' " 4 (f a + f l a) ■ Nonlinear oscillations o f third order systems. Part I 515 B y in tegrating. ones. “x 2 Nonlinear oscillations o f third order systems. Part Ị 517 4. Influence of Coulomb Friction on Sclf-Exdtcd Oscillation In th is Section the influence o f the C o u lom b fric tio n (4 .1) Rf