V o l u m e 5 , N U M B E R HANOI 1993 Proceedings of the NCST of Vietnam, Vol. 5, No. 2 (1993) (3-20) M echanics SO M E PROBL EM S OF N O N -L IN E A R OSCILLATIONS IN SYST E M S WITH LARG E STATIC DEFLE CTIO N OF ELASTIC ELEM ENTS N guyen Van Dao Institute of Applied Mechanics, Hochiminh City Summary. In this work two following problems have been examined: 1) The non-linear oscillations of electro-mechanical systems with limited power supply and large static deflection of the elastic elements. 2) The interaction between the self-excited and parametric oscillations and also between the self-excited and forced ones in the non-linear systems with large static deflection of the elastic elements when the mechanisms exciting these oscillations coexist. In both problems there is a common feature characterized by the fact that the nonlin- eaxity of the system under consideration depends on the parameters of elastic elements and theừ static deflection and by the appearance of the non-linear terms with different degrees of smallness in the equations of motion. Stationary oscillations and theừ stability have been paid special attention. N o n -lin e ar o scilla tio n s in sy stem s w ith large sta tic deflection of e lastic elem en ts have b een exa m in ed in [l ]. T h e sp ecificity o f these s yste m s is: Their hard ness essen tially d epe nd s on b oth th e p ara m eters of th e ela stic elem e nt and its s ta tic d eflectio n. T his featu re lea ds to th e ch an ge o f the a m p litu d e curve and the stab ility on it. In th is w ork so m e rela te d prob lem s w ill be in vestig ated: T he s yste m w ith lim ited p ow er su pp ly an d the in tera ction o f n on -lin ear oscillatio ns. I - NONLINEAR OSCILLATIONS OF THE SYSTEM WITH LARGE STATIC DEFLECTION OF THE ELASTIC ELEMENTS AND LIMITED POWER SUPPLY In th is p art th e n o n -lin ea r oscillatio ns of a m ach in e w ith r otating u nba lan ce and large s ta tic d eflection o f th e n on -lin ear spring and lim ited pow er su pp ly are co nsidered. T he eq u ation s of m o tio n o f th e sys te m un d er con sid eration are d ifferen t w ith tho se of cla ssic al p rob lem [2] by th e a ppe aranc e o f th e n on -lin ea r term s w ith differen t d eg rees o f 4 NGUYEN VAN DAO sm a lln ess. Th is feature lead s to the dep en dence o f the hardn ess o f the sy stem not only on the p aram eters o f the ela stic elem ent b ut also on its sta tic deflection. T h e re su lts obtained are d ifferent in b oth qu ality and q uan tity w ith th ose ob ta in ed by K o non enk o V . o. [2]. 1 . E q u a t io n s o f m o ti o n F igu re 1 illu str ates a m achine w ith a pair of cou nterro ta ting ro tors of equal u nbalance (so th a t horizontal co m p on ents of the centrifugal force v ectors ca nc el), isolated from the floor by n on -lin ea r springs and d ashp ots w ith d am ping coefficient h0. 1 V 0 SL ỷ ? f X Fig. 1 T h e sp rin gs su pp ortin g the m ass are assum ed to be negligible in m ass w ith a n on linear ch ara cte ristic function : /(u ) = cou + 0ou3; (L l-1) w here cn is a po sitive c on sta n t, /30 is either p ositive (hard ch ara cte ristic) or n egative (soft ch a ra cter istic). T he d eform ation of the spring in th e sta tic equ ilibrium p osition is A , and th e sprin g force c(, A ■+■ i (lA 3 is equal to the g ravita tiona l force m 0g a cting on the m ass: C0A + £ 0A 3 = m0 g, w here m„ = m l + m is defined as the sum of th e m ain m ass m i and the ro ta tin g u nbalance m asse s m, t h a t is the to tal m ass sup por ted by the sp rin gs. T he d isp la cem en t X is m easu red from th e sta t ic equilibriu m p ositio n w ith I chosen to be p ositiv e in the upw ard d ir ection . A ll q u a ntities - force, velocity, and accelera tion - are also p ositiv e in the upw ard direction . T h e sy stem under co nsid eration has tw o d egrees o f freedo m and th e generalized co or din ates X and ip c om p letely define its p osition . T h e k inetic energy of the syste m under co nsid eratio n is ^ 1 o m T = - m X 4- — V 2 1 2 2 m) Im = X -h r COS <p, Zm = r sin V?. H ence SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 5 T = - m 0 i 2 — mrxip sin <p H— /<p2, / = m r2. (1.1.3) 2 2 For the p oten tia l energy, the reference can be chosen at th e level o f the s tatic equ i lib riu m p o sition: Ư = — (A — x)2 + — (A — x)4 *f rnt)gx 4- mgr COS (p. (1-1-4) 2 4 T he L a sgran ge’s eq u atio ns give I<p = mrx sin (p + mgr sin <pt m0 X + c0 X + /?013 -f 3/30 A2x — 3/Ju Ax2 = mr^ sin + mrip2 COS V?. Taking into account the driving moment L(<p) and the frictions H(<p), k0x we have the follow ing eq ua tion s o f m otion : Iỷ = L(yi>) — H[<p) -+- mrxsin <p + m^r sin V?, T71()X + c0i - f / i 0i + /?ni 3 4- 3/90 A2x — 3)ổn Ax2 = mr^> sin Ip -f mr<p2 COS (p. Supposing that A is rather large and X is enough small, so that £tlx3 is a small quantity of second degree (s2), while 0it Ax2 is of first degree (5), where £ is a small positive p aram eter. O b viou sly, in this case # ,A 2X is fin ite. It is assu m ed a lso th a t — 1, ^ 1. T he friction forces, the forces YTiTtf? COS <p, m0 / m r< p sin y ? a n d t h e m o m e n t s mrxsin<pf mgr simp a re s u p p o s e d t o b e sm a ll q u a n tie s o f e2. T h u s, we h ave th e follow in g eq uations of m otion : w here <p = e 2 [ A / 1 (ý> ) + < 7(1 + 9 ) s i n sơ ] , ĩ + w2 ĩ = í 7 I 2 + £2 [pỷ sin ip + p<p2 COS ip - hi — Ị3x2) 2 _ mr 2 L A 2 /3,, f p = —- , e h = t c7 = -IS— c p = zf" > m«, m„ m„ m 0 ^ , e2Mi(yỉ>) = ị [£((£>) - #(¥?)], e2? = m„ / mr T (1 . 1 .8) T h e eq ua tio ns (1.1.7) are different w ith those in K on onen ko V . o . w ork [2] by the a pp eara nce o f the q ua dratic term 57X2 and by th e d egrees of sm alln ess o f th e term s. T h ese e qu a tio ns c hara cte rise th e s ystem s w ith weak e xcitation and large s ta tic d eflection . 2. So lutio n We lim it ou rselve s by con sid erin g th e m otion in th e reson ance r egio n, w h e Te the frequ ency u o f th e free o scillation is near to the frequ ency n = <p o f the forced oscillation s. W e sh all find th e solu tion o f e qua tion s (1.1.7) in the series [3] X = a cos(<p + 0) + eui(a, \ịỉ, <p) + £2U2(a, t/>, <p) + £3 . NGUYEN VAN DAO w h e r e u t (a, d o n o t c o n ta in th e first h a r m o n ic s COs\pi sin t/>, \ị) = <p + Ỡ a n d a r e p e rio d ic f u n c tio n s o f a n d (p w i th p e r i o d 27T, a n d a , 6 a re f u n c t i o n s s a tisf y in g t h e e q u a tio n s à = £w4i(a, Ớ) + e2w42 (a,0) + . # = UJ — n -f £#1 (a, 9) 4" e2B 2(a, Ỡ) 4" dip (1.2 .2 ) n = dt T he first e q u a tio n of (1.1.7) is th en á n = e2[Mi(n) +q(x + g) sin <p] (1.2.3) To de te rm in e th e un kn ow n functions Ay, Bx, ut we differentiate th e expression (1.2.1) an d s u bs titu te it in to (1.1.7). We have: ( du Ì X = —CLUJ sinrp + e< — aB 1 sin + Ảỵ COS 0 H- w — + n —— >-h 9 í „ ổu, <9u. <9u, i + £ I - a i ?2 sin t/> -f i42 cos + i4i ~ - L + n + X = — auj COS ^ 4- (cư — n ) — - — 2aa>i? i] cos \ịỉ — f(u» — n)a + 2cưj4i I L a# J L ƠỚ . ^2u i 2 ^ U1 1 + 2 w n - ^ - + w ^ - } + sin + n 20 + 2w n 5V'3^ <9^>2 d\ịỉd<p d\ị)2 i 4- é 2 Ị ^(cj — n) —^ — 2 a u B 2 COS xp — Ị(u> — 0 ) a -— 2 + 2(iM2 sin \ịỉ+ + {Ax^Ệ± + 5 1 -Q0 - - aSỉ) C0S - ( 2A 1 jBi + aAl-g-^- + a 5 l ^ i') *in V»+ +2^ f e +2^ l ầ +2n5i0 * +2“Si9 ^ (w n)^ + + (w - n ) cMi <3u r>2^2u2 d2U, 2^2u- _ L Q2 — ^2. + 2cjQ — — + ur — — <90 da d<p2 diọdrịì d\Ị) 2 + e (1.2.4) S u b stitu tin g the ex pre ssions (1.2.4) in to the eq uation (1*1.7) an d c om p aring the co efficients of e a nd e 2 . . . w e ob tain: + / d d \2 2 \n d i + ” i k ) u ‘ + w u - •+* Ị(cư — n ) ~~Ệ q ’ ~~ 2a u Bi j COS rp — Ị(cư — n j a - ^ 1- - f 2cư Ai sin xp = 7a 2 COS2 t/>, / a a \2 , ( n a 5 + “ a v;) “ ’ + " “ >+ + — n) — 2aw B2| cos t/> — (w — sin ip — + 2 70Uj COS v> — /3a3 cos3 1/1 + sin Ip + pfi2 COS ip, (1.2.5) (1.2.6) where R(0, Bi) = R(Ah 0) = 0. Comparing the coefficients of the harmonics in (1.2.5) we have (w - “ 2au) B 1 - °> (u; — n ) a — ^ 4- 2a ;A 1 = 0, SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 7 dỡ A /_ ổ Ổ \ 2 2 2 _ 2 /. ( n - — + U1 U1 = cos V* <9yp dip Solving these equations yields X i= 0 , B i = 0, Ui = ^ 7(1 - jc os 2V >). (1.2.7) Comparing the coefficients of the first harmonics sinự>, cosV> in the equation (1.2.6) Wfc have (w - n ) - 2ow £2 = g 2 - f - - + pH 2 COS Ớ, (w - n ) a ^ “ + 2a>i42 = —hauj — p H 2 sin Ỡ. 0 6 F rom th ese e qu a tio ns on e ob ta in s: ha pCi2 A2 = —— —— sin 6 ) 2 u + n _ 1/3- 5 7 2 \ 2 pft2 „ B 2 = — [ Ẹ ~~ n ) a “ 7 * r ~ cos Ớ. 4w \2 3 CƯ / (w + n)fl T h e eq u atio n for d eter m in atio n u3 is /_<9 Ổ \ 2 o / 7 2 /?\ -I r ~ k +UJ- k ) u’ + w u’ = - ( ò + ĩ ) a co sH - ( 1.2 . 8 ) H ence “’ = ĨS ? ( 2 + ắ ) ° 3'°,3'í- (l2-9) Thus, in the resonance zone n « u; we have the following equations in the second approximation X = a cos(<p + 0) + « ( p 2 - 6^2 c o s 2^ ) . (1.2.1 0 ) W'here n, a and 0 satisfy the equations d ii e2 [w . 1 2 • Ả ^ = — — — (u>/ia + p f i2 sin ớ ), (1.2.11)— = - — 77 [U)tia + DM sin £7 2u>0v đo _ 1 f n ff2p fi2 — = “ We - w -= COS dy9 n V 2a»a « i) , where w' = a, + ỉ r a2’ “ = 4^"S£- (L2-12) The stationary solution of the equations (1.2.11) is determined from the relations dCl _ da dỡ ^ d<p d<p dtp A/i(n) + -<7tc>2asin0 = 0, /la w 4* pH 2 s in 6 = 0, (1.2 .13) _ 2 P^2 w e — n - e —— COS 6 = 0. 2ua Eliminating the phase 6 from the last two equations of (1.2.13) we obtain w (a 2, n ) = 0, (1.2.1 4 ) where W(a2,n) = u 2a2 \eAh2 -f- 4(u;e - n)2] - e Ap2n 4. (1.2.IS) In the resonance zone n « U/, the equation (1.2.14) gives approximately 8 NGUYEN VAN DAO n 2 = u>2 + e2cxa2 ± e2 y j h2(jj2 . (1.2.16) From this relation it follows that the non-linear oscillation has: - a hard c hara cte ristic (F ig. 2a) if Q > 0 or if c0 > 7/?0A 2, (1.2.17) - a soft ch ara cteristic (F ig . 2b) if Co < 7/? 0A 2, (1.2.18) - a linear cha racteristic (F ig. 2c) if c# = 7/?0A 2. (1.2.19) E lim in ating 6 from the first two eq uations of (1.2.13) w e have L (n) - 5(H) = 0, (1.2.20) where S(ũ)=H(í)) + ~ h y . (1.2.21) The equation (1.2.14) is similar to that in the system with ideal power supply [1]. The difference is th a t n should be satisfied the relation (1.2.20) which can be solved graphically SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC as shown in Fig. 9 Fig. 3 3. S ta b ility o f statio n a ry oscillations Equations (1.2.13) can give some stationary values n = O 0 , CL = a0, 6 = 0o . To study the stability of these values we introduce a perturbation of n, a and 0: ỐH = Q — n 0) ổa=a — aof Ỏ6 = 6 —0Q. 10 NGUYEN VAN DAO We denote the right-hand sides of the equations (1.2.11) by $ n (n, a, 0), $ 12(Ấ, a, 6)J a, 0) respectively. Below, the derivatives will be calculated at the stationary values of n, a and 6 which satisfy the relations (1.2.13). We have the following variational equations: ^ - = bll6ĩì + bll6a + bliSe> ^ = fc„5n + fc„5 a + 6,3*0, (1.3.1) dip — - = b^ỏĩì + b^ỏa + ay? where - d- ề r - . w - - £ ( i ( n ) - f f ( n ) ] an in ’ dn L ' ' ' 3 < & n / i „ w 3 <3 $ 1 1 m n u ; 3 2 3a 2/n 3a’ 13 /fi3 )a ’ L d $ 12 , < 3 $ 1 2 /in . an m„n2 ’ ” aa 2m„n ’ f1-3.2) 6„ = - £ ( * « - n ) a , i 4l = £ ± i i = J j ( n - 2 w e) , ■ d _ Ổ$13 >lo <90 2mon -^13 1 a / \ o a r ' t M 1-" - 1 - " The characteristic equation of the system (1.3.1) is A3 + DịX2 -f Đ2^ Dz == 0» where Di = -(6 n 4- 6ai + 6SJ , ^ 2 =: ^11^33 ^73^33 ““ ^33^33 ^13^31 ““ ^13^31» •^3 = ^11^33^33 ■+* ^13^31^53 + ^13^23^31 ” ^13^31^33* The Routh-Hurwitz’s criterium of stability is D i > 0, .D3 > 0, D1D2 — ^3 ^ 0- (1.3.3) We have D' z=\ ~ m - {L3-4) m ứ n iw As usually, it is supposed that -^-L(n) is negative and — i/(n) is positive, so that TV is afi ail negative. Hence, Di is always positive. The second stability condition D3 > 0 as shown by Kononenko [2] is the most impor tant one. This condition is equivalent to the inequality ( 6 , A , - M M) ^ j * ĩ i ( n . M ) < 0 (1.3.5) SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 11 w here (ft, a (n ),0 (n )) and a (n), 6 {n ) are found from th e last two e qu ation s of (1.2.13); namely *h = £ f [ m - s (fi)} It is easy to verify that 2 dW ^77 "" ^33^33 ~ ơ £ fl2 ’ where w is of the form (1.2.15) and Ơ2 is a positive constant. Now, the stability condition (1.3.5) can be represented in the form . i [ i ( n ) - S(n)] < 0. (1.3.6) dW It is noted that — - > 0 is the stability condition of stationary oscillation when n is a da2 dW / given constant. is positive on the heavy branches of the resonant curve. The sign of da2 the derivative G = J y [ L ( n - s ( n ) j (1.3.7) can be obtained by considering the relative positions of the graphs L{n) and 5(0). For the case of the system with a hard characteristic (Fig. 3) it is clear that G is dW negative at points Rị, R2 and R3i so that the points Rị and i?3, where -—r is positive, da correspond to the stability of stationary oscillations. The point R2 corresponds to the instability of stationary oscillations, where — z is negative. pa* In comparison with a system with an ideal energy source [l], the unstable branch of the resonance curve remains the same. But the jump phenomenon occurs in a different maner. As n is increased the amplitude of oscillation will follow the solid arrows and the ju m p in th e a m p litu de w ill take place from p to Q. W ith a decrease in frequ en cy n the amplitude will follow the dashed arrows and the jump will be from T to u . The points o f collap se p and T are th e p oints of c ontac t o f the ch aracteristic L(n) and the fu nc tion s Sin)- For the case of the system having a soft characteristic (Fig. 4), the part of the reso nance curve indicated by the dashed (heavy) line corresponds to the instability (stability) o f s ta tio n ary o scilla tio n s, provided the frequency n is a given c on sta nt. O n th is part dW f d w \ ’ ~ 2 < ( ^ 2 > ° j - SiZn °f t ^ie d erivative G (1.3.7) d epen ds on th e slo pe o f the 0 a0 characteristic, i.e. on the quantity — L(n). It is necessary to distinguish two cases: afi 1) w hen th e ch ar acteristic is steep , i.e . -jprL(Cl) has a la rge a bsolu te value (F ig .4.) ail 2) w h en the ch ar acteristic is gen tly slop in g, i.e. has a sm all a bso lute value . aw (Fig. 5). In th e first ca se th e deriva tive G (1.3.7) w ill be n eg ativ e on the par ts P U , P T and T Q ( F ig .4 ). T h erefore, the sta bility con ditio n (3 .6) w ill no t be sa tisfied on P T , w here div dW Tp-J < 0, but it will be satisfied on PU and QT; where > 0. [...]... d istin g u ish in g feature w h ich is ch a rac ter iz ed by th e fact th a t its n on-lin ea r ity (hardness, so ftn ess) d ep en d s on th e p a ra m e ter s o f e la stic e le m e n t and its s ta tic deflection Nam ely, w h en th e initial sy s te m SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 13 has a hard characteristic, the resonance curve may belong to a soft type... n o n - lin e a r it y essentially d e p e n d s n o t only on th e characteristic o f th e ela s tic e le m e n t b ut also on th e s ta tic deflection T h e ch aracter m en tion ed has significant influence on th e in tera ction o f s e l f - e x c i t e d oscillations w ith th e forced ones, in both quantity and quality SOME PROBLEMS OF NON-LINEAR OS< 'ILLATIONS IN SYSTEMS WITH LARGE STATIC ... ta tic d efle c tio n are necessary to be in v estiga ted carefully T h is work was su p p o rted in part by the N ational B asic Research P rogram in N atural S c ie n c e s REFERENCES 1 Nguyen Van Dao Nonlinear oscillations in systems with large static deflection of elastic elem ents Journal of M echanics, N C ST VN, No 4, 1993 2 K ononenko V o V ibrating system w ith a limited power supply M oscow... F L E C T IO N In this part th e a tte n tio n is concentrated on1 s tu d y in g th e c o n d itio n s under w h ich the resonance regim es o f oscillations occur, on explainin g th e role o f n o n - lin e a r fa cto rs in th e form ation of resonance situ a tio n s of the s y stem s w ith large s t a t i c d eflectio n of e la stic e lem en ts [l ] T h e d istin gu ishin g feature of these s y s... s ta b ility T h e sta tio n a r y a m p litu d e a0 and phase V are determ in ed by th e relations ’o e2 / a2 \ e2E (HI.2.1) , « /-w e a0 , = t'E - J — -— 1 - s i n \/»0 , a ( w + t/) SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 19 w here I \ _ e2°‘ 2 F ro m th e s e e q u a t io n s o n e o b ta in s tg * = f a i f e l L , ( i n 2 2 ) W ( a 0 , v ) = 0, Ị 4 Ị1 - u,e(a)]2... ( I I I 1 1 ) in th e form : I = a COS(ut + \ị> + 5Ui(a, (p, S) + e2u2(a, < 6) + ., ) p, (III.1.2) w here u, are p eriod ic fu n ctio n s of < and 0 with period 2tt which do not contain the first p harm on ics sin0, COS 6, (III.1.3) d\ị) It T h e in ten sive interaction of m echanism s of generation of the oscillations of different natu... 3 )0 +« 2 V tư 1 ea ( I/ A - — COS 2 v>) • / (IL1.11) SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 15 2 S ta tio n a r y o s c illa tio n s a n d th eir s ta b ility E q u ation s (II.1.11) have a trivial solution a = 0 T h e non trivial station ary am p litud e da „ dip drp a 0 an d p h a se rp0 are d e te r m in e d b y e q u a tio n s — = 0 = 0 or dt ~ ' dt - / ơa2... problem s w h en two m ech a n ism s excitin g the self-su sta in ed oscilla tio n and param etric one c o e x ist in one sy stem Following the assu m p ­ tion s in [l], th e s e o scillation s are weak T h ey appear only in the second approxim ation of the s o lu tio n and their interaction is w eak, also In co m p a r iso n w ith th e classical problem on th e interaction b etw een self-excited and... In this paper it is assumed th a t D E Ỷ 0 T h e question is stated as follows: w h a t h ap p en s in th e s y stem (III.1.1) when two m ech a n ism s of generation of se lf-e x c ite d oscillation and forced one c o e x is t? We will be specially interested in the s ta tion a ry o scillation s and their stability U sing the a s y m p to tic m eth od [1 ] we find the approxim ate so lutio n s of. .. the elastic elem en t and A is its static 9lf deflection A s y s te m w ith hard springs may b ecom e a less hard or soft one 2 T h e m e n tio n e d in te r e s tin g fe a tu re affects on b o th th e q u a lity an d q u a n tity o f th e oscillating p h e n om en a of the classical problems [2, 4, 5] 3 T h r o u g h s o m e p r o b l e m s e x a m i n e d in t h i s p a p e r o n e c a n s e e t h . regimes of oscillations occur, on explaining the role of non-linear factors in the formation of resonance situations of the systems with large static deflection of elastic elements [l]. The distinguishing. its non-linearity (hardness, softness) depends on the parameters of elastic element and its static deflection. Namely, when the initial system SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS. significant influence on the interaction of self-excited oscillations with the forced ones, in both quantity and quality. SOME PROBLEMS OF NON-LINEAR OS< 'ILLATIONS IN SYSTEMS WITH LARGE STATIC