CÔNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM VIỆN KHOA HỌC VIỆT NAM ACT A MATHEMATICA VIETNAMICA T O M 5 N " I HÀ NỘI — 1980 ACTA MATHEMÁTICA VIETNAMICA T O M 5, N° 1 (1980) MULTI FREQUENCY EXCITATION OF PARAMETRIC OSCILLATIONS OF DYNAMICAL SYSTEMS N G U Y Ễ N V À N Đ Ạ O Institute of Mechanics, Hanoi L e t US c o n s id e r I h c p a r a m e t ric o s c illa t io n s o f n o n li n e a r s y s te m s d e s c r ib e d b y d i if c r e n l ia l e q u a t io n o f fo rm X + w5 X + E 2 fk (*) sin0k + e Q (X, X) = 0 (0.1) k.1 h e re 0k = v k / + r k, k = 1, a n d v lt v r a re p o s it iv e in c o m m e n s u r a b le n u m b e r s , p k ( X) a r e p o l y n o m i a l s o f x t Q (£, x ) — n o n l i n e a r f u n c t i o n o f X f X . A lo t o f p h y s ic a l a n d t e c h n ic a l p ro b le m s a r c le d to in v e s t ig a t io n o f e q u a tio n ( 0. 1). F o r 'in s t a n c e , o s c illa ti o n o f/a p e n d u lu m w h o s e s u s p e n s io n p o in t a c c o m p lis h e s Ih c c o m p lic a t e d o s c illa t io n s a n d th e t r a n s v e r s a l o s c il la t io n o f b e a m u n d e r the a c t io n o f c o m p lic a t e d lo n g it u d in a l fo r c e a r e d e s c r ib e d b y e q u a t io n ( 0. 1). It is su p p o s e d th a t th e re e x is t s a r e s o n a n t r e la t io n o f ty p e n* Q -p c* Vj + + c* vr = 0 (0.2) w h e r e Q > 0, a n d c * , , c* a r c in t e g e r s . T h e n u m b e r nm is d if f e r e n t f ro m z e ro , b ut so m e o f c* m a y b e d is a p p e a r e d . A r a t h e r s im p le p r o b le m w h e n Q (X , X) ES 33 0. p k (a :) = X h a s b e e n c o n s id e r e d b y Is c h u c h v . v . a n d Ia s c h u c h V . T . [1 ] T h is p a p e r c o n s is ts o f t h re e s e c tio n s . In s e c t io n 1 a g e n e ra l t h e o ry o f c o n s t ru c t io n o f a p p ro x im a t e s o lu t io n s o f e q u a t io n ( 0. 1) b y a s y m p t o t ic m e th o d o f n o n li n e a r o s c illc t io n s is g iv e n . S e c t io n 2 is d e v o te d to in v e s t ig a t io n o f m o n o f re q u e n c y re s o n a n t c a se w h e n th e r e la ti o n n* D. + cm V . = 0 s b ta k e s p la c e . It h a s b e e n f o u n d t h a t th is is th e m o st im p o r ta n t ca s e o f m u lt if r e q u e n c y e x c ita t io n a n d th a t in th e f ir s t a p p ro x im a t io n o n l y th e c o m p o n e n t o f e x t e r n a l e x c it a t io n w it h re s o n a n t f r e q u e n c y (v s) h a s in f lu e n c e o n th e p a r a m e t ric oscillation. T h e influence of other nonresonant com ponents is found on ly in the s e c o n d a p p ro x im a t io n . 21 In section 3 the mullifrequencv resonant case when n* Q + c* Vj + + c* v r = 0 is true wilh more than one of Cj, C2 c, is considered. In this case the influence of exlernal excitalion on parametric, oscillation is found onlv in Ihc* second ap proximation. § 1 . C ON ST RU CT IO N OF A P P R O X IM A T E SO LU TIO N S Following to asvmptotic method of nonlinear mechanics the solution of equation (0.1) is found under the asymptotic expansion [2]: X = Cl C O S (~ 4* I f ) -4- U \(g . z. -f- v |\ 0 ) -Ị- E'U'J ( f l , c, Ỷ “t" where 0 = ( 01. 02 9***9 0r)« and the amplitude a and phase* arc determined from the equations d a ( 1.2) rf/ d / = eAi (a, ty) -f e2Am (fl, V) + e3 + == to — Q -f- eJ5.(a. 'lị') ~j- 6~z?2 ((li V) "4" "T ••• Function “ satisfies the condition - ệ - = ĩì, n * i + c* e, + + c; e, 3 0 al ( 1 .3 ) (1.4) Substituting Ihc expression (1.1) into (0.1) and comparing the coefficienls of E. F.5 wc obtain the following equations «1 à ‘ lỉ\ I o Ồ ~ u ] t » 8-IỈ, QJ- « 2 i o v k 4 - / V i V ^ ■ a + v )2 J Ô 0 j Ồ0 I (h) + Q ) fl -p 2u) li] sin ( I + yf) — (u) — Q ) -— - — 2 CO aB « ir ởlt' r 2 [acos (I -f If)] sin ek — Q[acos (I + \|J), — loasin (; + V|))], C = 1 Ĩ i-v ^ fl-Un *r— . Ờ U-J n 2u) vk — “ — — + 22 VjVk —— ^ h ID u2 = ^ ri O0ka ( | + T|)) k4 ^ j O0j 96u I flS] . n oB] , n . . , n , . ii-Bj • cos (g + \t0 — (1.5) 2.4, /; + + <a, — + aB, — + 2».4i + (n> - a) a A, Ỉ à l + B , òa ồ Cl Ồ A] flip ồ\p aB\ - 2u,aB2 + (u) - Q) iA2 s in ( I + r|>) — 01]> ờtị> cos (I + 1|1) + (1.6) where 22 ÒFĨ R°2 = U) t- ỔX. 2 A, 2 V k-1 i4|C0 s (r 4- T|v) — nB]s in ( ; 4- \ị>) 4- to a2ỉij ỒUi ỒU Ỡ -Iil , ễ) t k h 2u )/li Ỡ0kồơ + 2B] v k k -1 — - + 2 v k «(E + V ) k -1 Ồ0k j ồ2ỈIl òF\ àX' + (d) — Q ) òBi ỜU1 ~p í(J) — Q) . . „ ^ + 2o)JB, - j- -r + ô (; + ip )ô a ÍT Ỉ Ô M (|+ * |> ) ô d - H 1)' 8.4, ÍU , l i F 0 = _ V p k ( x 0) Sin 0k - Q ( x 0. i 0). (1.7) A ty ồ a k=i ô\ị) ô(s+Tị>) x 0 = a co s ( ; + \|ĩ). x 0 = — U) a s in ( I + yp). Let us expand pk(£0), 0(.To, x0) in the Fourier scries. We find P k (x 0) = P k [ a c o s (ỉ + V )l = 2 ,j0km( a ) c o sm ( Ê + V ) ’ Q ( x 0. *o ) = m^o = Q [acos(; -f v>). — uiasin(; + v )]= y \ [fm(a)cosm(l+y) + gm(a)sinm{l+V)), (l.S) here, Dk 27Ĩ "(0) = ề Ỉ 0 P k(acosQ>)dO, 2 7 C o Q(acos<J\ — u)asin<I>)dC\ (1.9) 27T •Okm( Pk(acosO)cos/nC>d<I>, 27T f m(O-) — Q(acosO, — U)asin0)cos77i0do o 20Ĩ g m(fl) = - ỉ — f Q (a c o sO , — a)a sin O )sin /7 70 d <I> . * J • o We shall find the function U\ in expanded form with undefined coefficients “1 = 2 1 u lnci cr sin [n (s + + Cl01 + - + Cr0rl + ylnci cr COS [n(| + lị)) + C,01 -f- ••• + cr0,]j (1.10, Todclermine the unknown cocl'ficicnls Hlnci t , ulnci -c .and functions A'l, B\ WE substitute expressions (1.8) (1.10) into (1.5) and we compare the coefficients of sin [n(l + lị1) + Cj6i + + c,0r], COS [n(; + Ip) + C)01 + C ọ 02 -f + c,.0r]. By subs tituting (1.8) and'(1.10) into (1.5) we got O’ Í4\ r z fa)2 -(*« *>+ 2 ci v')2] Í u lnc, cr sin [*»(£ + V) + - + cr «r] + t>lnc, n*Cl C r i = l Cr cos [n(| + T|0 -f Cifl] + + cr0r] I = (ill — Q ) a — — 2 u ) A j Ỗ1|5 sin (i + ty) (,„ _ Q ) _ 2mflB COS (; + 1|’) — 2 [/m(a)eosni<;+ijO + <7„,(fl)sinm(| + i|>)J— m ^>0 1 2 2 Am (ư) j sin Ịm(| + tị') -}-0,] — sin[m(; + t|') - 0,] I (1.11) i = l m i o § 2. M O N O F R E Q U E N C Y R E S O N A N T C A S E In this section we consider the resonant ease when there exists a resonant relation n*£l + c* Vs = 0 (2. 1) s where /?*, c* , arc integers, 1 < s < r, n*. c* < 0. Comparing with (0. 2) we have c? = c* = = c* = c* . = = 0. 1 2 s- 1 s + 1 c’ 4= <J. s ' T h e la st te r m s in (1 . 11 ) w i ll c o n ta in s in ( I 4- 1|'), c o s ( | + \Ị)) if (m + 1) (I + \|’) 4- 8i = X. On the other hand from (1. 3) we have n*(l + V) + c* 0S = n*vp. Tin- comparison of formulae (2. 2) and (2. 3) gives (2. 2) (2.3) o r m = 4 - 1 -4- a n*. i = 5, 4- 1 = Ơ c , X = ơn*\ị' s _ n ĨÌ m = + 1 d~ . X = ± — lị’. c* c* s s (2. 4) S in cc . Dsm = 0 if m <c 0 a n d n * .c * < 0 , D n* = 0 , th en w e h a v e th e s Sj ■— ■ — ] following terms containing sin ( I + rị>), cos( I + \p) in (1.11) : > D («u - Q ) a + 2 wAi ÔTị) sin(Ê + Tị)) — (u ) _ Q ) - Ì í L l _ 2 taaB1 a Ip = /’i(a)cos(i +rjj) + 3 i(a)sin(! + 'l’) + _ L Z) n 2 D>, n sin i ( |+ r |) ) - — TịỊ +1 Í c* J s s 2 s ' - T - +1 + 0 Ds._Ji sin cos(| + ự) = ( I - H 0 + V cs -J /V \ n* (* + v>) + , V 24 F r o m h e r e w e o b t a in 2 (I)A] + (to — Q ) 0 *- ờ\ị) = 0 l(« o + “ T / ỡ s "• + / ) . n* - 0 n * \ c o s - ^ - T | \ ■ c V ' 1 ** ' c * “ 1 / \ s R • s / 2 toflB j — ((.) n ) — —1 «= ft\p = /*l(fl) + - - / D n* -f- Z) n* + D n* \ sin — — Ap (2 . 5) 2 S . - 1 L - . I s V - 1 s - Cs \ cs cs r s / I f (1) — Q = 0 ( e ) th e n w e h a v e B' - i L f-w + i h ( B H r - + v f ,+D- f . . ) sin f * s s (2.G) uv U) Bv com parison of Iho hiehcr harmonics in (I. n ) we get » M = 0 Cr = « - - 5-(0 V " + 1- c, = « = - ' " <0) " + ' J Z L An(fl) ( i + 5) -(J1U ) + v ,) - ] u lnci = 0 Ci_ , == 0. Ci = l. CI+I = 0 Cr - 0 2 = — Din (a) ■(* s) — (nu) — V i)-] u ln c i _ 0 Ci_) = 0, Ci = - 1 , c - r 0, Cr — 2 = — Ds„(a)fn =/= Ị -4— ĩ “ ) Ịu>” — H u ) - f v s)"l H l n C l = 0 , Cr-1= 0. Cs = 1. Cli+l — 0 Cr — 2 \ _ = ị ọ s J ( f l ) ( n + - - T - ± 1) Ịu)'J — (n il)— Vs) ] u lnC) = 0 CS-1 = 0. C i= —l.c * +l = 0 , C r- 0 2 \ c t > (2 . 7 ) A n d th e r e f o r e w c h a v e J o /.V I s -' i Ĩ o n (a ) s in n ( Ỉ + ^ ) + fn(Cf) COS n ( ; + ^ ) ] 1 " ^ £ > in (g )s in [ n ( | + ^ ) + Bi I + + T ^ ' (nu> + Vi)2 — u>: i = 1 n > 0 i=£=s 25 J __ ^ D\n(o) sin [/?(! -4- ty) — 6i] o 2 / „ \2 2 i = l n > 0 0) - ( n c o — Vi) i=/=s , J__ y - ' D sn (fl) sin [ n ( | + \p) + 9s] 2 n* + vs)“ — or n=7=l H— c n > 0 s + , 1 y 1 Z)s„(a) sin [/1(5 + 1|')- 0 S] /OOX + - - Z - . _ ỳ -ií -1 — ( 2.8 ) n > 0 OJ — (nu) — v s) n «= — IL _ q i 1 c T o d e t e r m in e th e fu n c tio n s ^ 2. H> \vt* c o m p a r e tlie c o e f fic ie n t s o f s in ( ; + \|>) c os ( l + ty) in ( 1 . 6 ) . F ir s t w e w r it e /?•> in th e e x p a n d e d fo r m •K = 2 f^ °in ci c co s Í n( * + 'íO + C101-f ••• “i~ c r0, ] + n ci c r ^ °° n ci (*, s*n f n ^ + V) + c,e, + + Crflr] ị (2.9) where 23T fi°io o = "2jT I* d + 'I’) J R " d ( ỉ o 27T >0 _ 1 f oO 21nc] Cr I 2 0 Zi9 inr= ~ ft COS [ n ( | + ẹ ) + c,e,.+d(|+ \|5) de, der r >0 22nC]«**Cr 0 27T = J R° sin [ n(t + If) + Cj6 + c,0] d(l + T|))de, d0r 0 t Vio I p r m c P f i n t Ct ir» i n Ơ (* _i_ I h \ c i n (b -L. n n i T P c n In (2. 9) the te r m s containing COS (; 4- \|>), sin (I + lị)) correspond to that n and Ci, cr lo r which n(l + ẹ) + c,0,+ + cr0r = ± (i + T|0-+ a On the other hand have n* (l+'p) -4- cmes = n• \p. s From here it follows that /I + 1 = n /1#, Cj = c2 = = Cs-1 = Cs+1 = = Cr = 0, ịi = , a = |i n* lị). c # s Thus we can write R° (2.9) in the form u°2 = c (a, \|>) cos (t + \|>) + s (a, <p) sin (|+t|>) + (2.10) 26 where dots denote the terms which do not contain sin (i + 'lO, COS (t + v) and c (a, rp) = 2 Í ( fi02M!+ + R ° ỉ r ] cos » n# V + (R0?2+ + >sin I* S ia . ip ) = 2 [ a < ° r - H ° £ + ) s in ^ V + i R ' E * - K ° £ - ) c o s i |> / ! > ] ; ROỊi. : _ _ w o 21 21tl + P-n ,C| = 0 C5_1 = o,Cs = Hr* , C»41 = 0 c r = 0 , RO 21 21 , - 1 + JJ. n* , Cl = 0 c.s—l = 0,Cs = ụ ĩ . C&+1 = 0 , Cr = o . = R ° 22 22, 1 -f f-ifl* , Cl = Ò Cs- 1 = 0,Cs = . Cs-fl = rf r — o. /^O ỊX __ 22 22 , — 1 + JA 7Ì , C] = 0 c S _ 1 = 0,Cs = r ? . fs-H = 0 , . . . , Cy = o . (2.1 J) S u b s tit u t in g (2 .1 0 ) in t o (1 .6 ) a n d c o m p a r in g c o e f fic e n t s o f C O S ( I + ty). sin w e o b t a in 2 (i) A 2 + (£2 — U)) a 2a <t> iJ2 + (£2—“>) ■ ò tị' If (0 — 0 = 0(e) then wc have • > 1 D I A a ^ i u ồ = 2 * 1,5 ] + o A i — 1— h Q ỏTp fla a,4j . Ồ Aj - = 4- i j ồ 4- S (a ,\ |) ). B, _ an ? - C(a,x10 . s a n\p a 2 = - Ị - 2ci) 2A\B] -f- CL'l] dBj 0 \|5 4- a B j - Bfi, a\(j = 1 2 ail) [ a , 8 A l A a + B , Ổ.4.1 Btp aB \ + s (a , \|)) (2.12) Thus in the first approximation \vc have / c s X = a COS ( — — es + V) n* (2.13) here a and Tị! arc determ ined by equations : ' . + D . _ da E , e — = — 0 ] (a) + dt 2(1) 4(1) I cos —— \|) , c s dtị) dl - D _ n* \ , - — — 1 s, — — — + 1 I c * f* / s s / « Cl>_ Q + _!_ /-1(a) + - i - / / ) +Z) „* + £ _/>• \ 2aa) 4(oa I s» —— + 1 5* ” — 1 —— + 1 I I • . # « I A c o * c c~ / \ s s s / /2 sill —— rj- . c* (2.14) 27 The refinement of the first approximation is X aco ( — p J -t t W e l V ffn (« )s in n (E + ^ ) - f/ ~ n ( a ) c o n / i( E + ^ ) V n* 5 7 / to* n^ 2 (/I * - l)a.2 I _Ị_ D m ( fl) sin [ n d -M lO + flj ] J ]_ Z)jn( Q )s in [n ( | 4 -ty )— flj] 2 k „>0 ( " W v ,) 9- « r 2 (1)2—(nto—Vj)8 f =f=s i t =5 1 D sn( f l) s i n [ /i (| + I __ ]_ 7),s n (fl)s iD [/? (|4 -t|0 — fls] t (n it )+ \ \s) 2— Cf)2 2 /)= -! + /1 ^ 0 y A .„ ( n )s i n [ /1(E + Tị>)— B,] } (215) * u>2 - (nu)— v s) s « = f = - - V ± i C5 n ^ 0 e.s = (5 + 'J1) V ^ c; c* in which a and \p satisfy the equations (2.14). In the second approximation function X is of form (2.15) but a and \|) arc d e t e r m i n e d b y t h e f o l l o w i n g e q ua ti oi i S = - i - g , (a) + (Z), J l l _|_ J — —— 1 -/ J ,, _ + , ) cos rp - d / 2 CD 4uu , • r * f * c * s s s E2 4 ^ , ồD1 . V a B l . « ,e 2(0" 2Ai Bi + aAj + aB òa dtp d\p dt Cl) - Q + _ £ _ / - l ( a ) + _ I _ ( D „ 4au) s* « c: + 0 + 1 s , — + s (a , yp) , + D 1 s, “ + 1 + 2 a U) ồA ị ỒQ + B, ± iL _ air _ c (a, v>) Ồ\Ị5 1 V . n * )sin-—115-f c? (2.16) A special case Bv making use of the general theory developed in this section we now study the parametric oscillation of nonlinear system of type where Xi are constants. use 01 tne general ineory developed in tnis ^illation of nonlinear system of type r ' Ì + + E I 2 Xi sin 0j + e Q (X, x ) = 0 1-1 ;tants. (2.17) Let us consider the silbharm onic resonant case, for which 2Q — v s = 0, u>* = Q 2 - f eA here A — detuning of frequencies. Bv substituting n* = 2, c* = — 1, P s(x) = XRx,' Ps(acosO) = X5 acosO in (1.8) s we get D,,1 = x5a„ D6,0 = 1K,U = 0, n > 2, 28 and therefore the formulae (2.6) become ,4j = ~ g2(a) + —— (XSG) cos2\p, :U> B] = —Ỉ— fi(a) — xssin 2\|). 2au) 4 U) Following formulae (2.13), in the first approxim ation we have X = acos — (vst + r s) + T|) . here a and \p are determined from equations (2.14): 2 7T = —-— I sin(l) . Ọ(acos<1\ — liiasinO) d o -t— — Xsacos2r|\ dt 2oj7T j 4 lu u 12 7T — UJ — -f -— f cưs(1>. Ợ(acos4>, — masinO) d<|) — Xs dl 2 2aaj7T J 4uj (2.18) (2.19) The refinement of the first approximation is oi' lorm (2.15) wiln I = 0„ X = acos 0S -f H’j -f fc I — /o ( a ) y , /'nConn ( ; + \p) -Ị- g n s in n (I - f v>) n > 2 «*<n«-l> “ r + j i i = l sin 0, + 0, + V'j V i ( v s + V J r Xi sin ị-ị- 0S — ©i - h i = l iH = s V i'(V K — Vi) (2.20) in which a andiị; satisfy equations (1.5). § 3 . MULTI FREQUENCY RESONANT CASE In t h is s e c tio n il is su p p o s e d th at m o re th a n on e o f c * c* a re d if f e r e n t F r o m z e r o , SC) t h a t w e h a v e a r e s o n a n t r e l a t io n n*Q + c* V1 + — “f" c# v r = 0. (3.1) In th is c a se it is e a s y to s h o w t h at th e la s l s u m s in (1.11) d o no t c o n t a in s in (£ + t|>), c o s ( ; + 1Ị)). In f’a c l, th e se te rm s c o r re s p o n d to th e n u m b e r m satisfying r e la t io n + ty) d r 0i = + ( I 4 “ ty) + ^ (3 .2 ) B u i o n th e other h a n d f o ll o w in g (1 .4 ) w e h a v e n*(X "t" V ) H“ C] fli + ••• c r = 29 [...]... o n Received January 22ih, 1979 REFERENCES 1 Ischuch V.V., Iaschuch V.T., Study of equations of Malie type with quasiperiodic coefficients by asymptotic m e t h o d s Asymptotic and qualitative method of nonlinear vibrations, Kiev 1971 (in Russian) 2 Bogoliubov N.N., Mitropolski Iu A., Asymptotic methods in Theory of nonlinear vibrations, Muscou 19G3 (in Russian) 32 . MATHEMÁTICA VIETNAMICA T O M 5, N° 1 (1980) MULTI FREQUENCY EXCITATION OF PARAMETRIC OSCILLATIONS OF DYNAMICAL SYSTEMS N G U Y Ễ N V À N Đ Ạ O Institute of Mechanics, Hanoi L e t US c o n s id e r I h c. than one of Cj, C2 c, is considered. In this case the influence of exlernal excitalion on parametric, oscillation is found onlv in Ihc* second ap proximation. § 1 . C ON ST RU CT IO N OF A P. * )sin-—115-f c? (2.16) A special case Bv making use of the general theory developed in this section we now study the parametric oscillation of nonlinear system of type where Xi are constants. use 01 tne