HỘI Cơ HỌC VIỆT NAM ■ VIỆN KHOA HỌC VIỆT NAM « i « I I I BÒ GIÁO DỤC VÀ ĐÀO TẠŨ • BỘ KHOA HỌC CÒNG NGHỆ VÀ MỒI TRƯỜNG • I I I I I T U Y ỂN TẬ P CÔNG TRÌNH KHOA HOC H Ộ I N G H Ị C ơ H Ọ C T O À N Q U O C L Ẩ N T H Í T N ă m PR OC EED ING S OF THE FIFTH NA TIO NA L C ON FE RE NC E ON MECH A NICS (Hà Nội 3 - 5 / XII / 1992) CO HOC DAI CƯƠNG VÀ ỨNG DUNG GENERAL AND APPLIED MECHANICS HÀ N Ộ I - 1993 Proceedings of the Fifth National Conference on Mechanics, Vol. I (38 - 45), 1993 N O N L IN E A R V IBRA T IO N OF A P EN D U L U M W ITH A SU P PO RT IN H A R M O N IC M O TION Dao động phi tuyến của con lắc có điềm treo di động thing điều hòa tùy ý T he nonlinear vibration of a pendulum whose support undergoes arbitrary rectilinear harmonic m otion is stud ied. The main attention is paid to the resonant cases and the stationary vibrations. T h e resonant conditions are explained. The am plitude - frequency curves are plotted for various valu es of param eters and the stability of vibration is investigated, '"he rotating motion of the pendulu m and its stability,are also considered. Let us consider the vibration of a pendulum consisting of a negligible weight rod AM of length i and a load Af of mass m. The pe idulu." ‘Tipport undergoes rectilinear harmonic m otion by m eans of a m echanism shown in Fig. 1 when the '•rank ON of length R rotates around o with a constant angular velocity n and translates slotted bar BA of length L along slides 1.1. We shall tak e the origin of the y axis vertically up. The position of the pendulum w ill be specified by angle (p th at AM makes with vertical axis (Fig. l). The kinetic and potential energies of the pendulum , T and V respectively are: T = — [t2 {P2 + Jỉ2n 2 sin2 fit — 2Rftt<p sin Hí sin(6 + V?)]» J J y = rng[(L + R cosH t) COS 6 - tcostp]. U sing L agrange’s equation, taking into account the damping force, the equation of m otion of vhe pendulum is obtained as where a>2 = gỊ I, h is the damping coefficient. We assum e that R /t and h are sm all and we shall consider small vibrations of the pendulum about th e vertical axis, so that sin V? ~ <p — ị<p3 /ô), cosy? ~ 1 — (<p2 /2). The smallness of the men tioned qu a ntities can be taken into consideration by introducing a sm all dim ensionless parameter e, w hich w ill be set equal to unity in the final results. T hus, w e are led to consider the following equation of m otion: N G U Y ỄN V Ă N ĐẠO Viện Cơ hoc, Viện KtìVN SI. EQUATION OF MOTION (2) (3) 38 where Jĩ , f _ R - , h _ 0 r SB wty c = -7 sin ỏ, D = - 7 COS Í, " 1 — — , 7 = — I I UJ w and a prime denotes the derivative with respect to the dimensionlcsa time r. (4) In the following sections two resonant cases which cause intensive growth of the amplitude of vibration of the pendulum will be studied. §2. PRINCIPAL RESONANCE We consider the case when 7 differs a little from unity. We are interested in finding out what happens close to resonance, that is to say when - 1 is small, namely: 2 - 7 = 1 + 2 (Jjd (5) where A is a detuning parameter. Let us introduce in equation (3) the variables a and rj aa follows ip = a cos 0, <p* = - c n a i n f l , 0 = Tff + ff (6) here the condition a! cos 6 — ar/ sin 0 = 0 is im posed. The equations for new variables w ill be V = •¥>+/) «in0, TdT]' = ¥»+ / ) cos 0, which is a set of equations in standard form with (7) (8) 39 f kị a*)f sin f - o s cos3 9 + c j 1 COS 7 r + D 7aa COS 0 • COS Tff. 6 (9) In the first approximation the right-hand sides of (8) may be replaced by thcừ mean values, regarding a and TJ as constants (l|: 7 o' = - ^ ( h n a + C V sin rỹ), W - - j ( § « + j « ’ + C i a ««•>>)• (10) T h e stationary am plitude a0 and phase rjo are determ ined by hi~ia0 + C~I2 sin rjo = 0, ~ a 0 + ~aị + C~12 cos r?o = 0 . UJẨ 8 A sim ple calculation eliminating T) leads to the response curve equation: W(aot^ ) = 0, w = a ịịhh* + ( S - I + ịa iy } - c ' l ' . (11) ( 12) (13) T his relation is plotted in Fig. 2 for the parameters: R = 2cm , i = IGOcm, ố = o.l&rad, g = 980cm/scc2t h = 2.47 • /li, w = \/<7/ i = 2.47, CƯ2 = 6.125, c = 8.8 1CT3 1. hi = IQ-2 aiid ?. hi = 4 l< r a. Ft?. £ Let U3 now discuss the stability of the possible stationary regimes. To do this we study the eigenvalues of the matrix of coefficients of variational equations of (10): 40 ( I . _ tC 2 ~ 2 ~ I c 0 i *7o c / A . 3 , \ tC t ( a J C O ' , ■ j ( 5 + i ay 1 1 , l n , ° The equation defining these eigenvalues is <V73A3 + * i 2a0hx H J ~ = 0 (14) 8 daQ from w hich we obtain the condition for asym ptotic sta b ility ! ^ > 0 . (15) oan It is noted that function IV (a0 »K2) is p o sitive (negativ e) outside (inside) of resonant curve and equal to tero on it. So, condition (15) shows that the upper branches of resonant curves (hea v y lines) in Fig. 2 correspond to stable statio n ary regim es and the broken lines to unstable ones. §3. PARAMETRIC RESONANCE It is supposed that 7 18 approximately equal to 2, namely (I6) T h e solutio n of equation (3) in this cases is found to be = i c o s ( Ị f + a ) , !f>' = - ^ 7 sin { y + a ). (17) In the first approxim ation equation (3) can be replaced by the averaged ones: *ịV = -ib*ị(hi + D i sin 2 a ) , (18) It is clear that 6 = 0 is a solution of equations (18), but aj we shall show later, it may happen that th is solution is unstable and that the system begins to vibrate spontaneously. The stationary amplitude bo if determined from equation V =r a* = 0 by elimination a: 6s = 8 ( 1 - ^ ) ± 4 ' , v / 0 v ^ ? . ( 19) The response curvet are plotted in Fig. i f o r D = 8 .8 1 0 “ 3 and 1) h\ = 1.65 10~a, 2) h\ =» 1 .7 1 0 " 3. T h i stud y shows that only the positive sign before the radical (19) corresponds to the u y m p totica l stability of nontrivial stationary vibration; and that the solution b =* 0 i< stable outside the resonant curve and unstable inside it. Thus, only the heavy lines (Fig. 3) of th« resonant curve correspond to stability of vibration. 41 3 6 3 8 * y 2 Ft</. ^ §4. ROTATING MOTION OF THE PENDULUM Assuming t h at h and R/Í are small wc con sider the rotating m otion of the p endu la m governed by equa r ion: ỷ + UI2 sin V? - : <P| 0 . (20) w here f (<p, <p, i) = — - Y n 2 cos Of sin((5 + V?). (21) It is supposed th a t the energy of the system considered is high so th a t when e = 0 the p e n d ulu m will be ro ta tin g (equation (20)). W e introduce the variable a and yịỉ (2]: + (JJ) <p = u(a) + ai/(a) COS e = nt + ự>, v(a) = l/y/ã, ị = i/(a) (23) and V? is the solution of the degenerate equ ation £ = 0 if a and 0 are constants; 80 that - a i / 2(a ) sin £ 4* Ui2 s in V? s 0. (24) E q u a t i o n (22 ) i m p ly t h a t £(1 + a COS £) + à sin £ - ix(a) (1 + a COS () =3 0. (25) T h e second equation for ( and à is obtained by sub stitu tin g equation (22) into equation (20): -ai/ịs in £ + + a COS £) + V COS f] â + w3 sin V? = ef(v?, í) (26) 42 here th e subscript ua” or ( )a denotes the derivative w ith respect to the amplitude a. FYom these equation s we get: à = -i-F(<p, <p,t)[ 1 + a COS 0 . : e (27) V- = i/(a) - n + ^ F(v, <fi, t) Bin t, where - A = av sin 3 { + (1 + a COS () [i/flU + a C08 0 + v cos (]• Su b stituting here ỉ/ = l/y/ã we have — = 2ay/a + 0 (\/a • a 3) (28) We shall consider the principal resonant case when the am plitude a takes values close to aQ determ ined by n £2 i/(a 0) = ~ = V ao (29) and use the Jacobie expansions of trigonometric functions in Bessel functions [3]: oo sill (a sin 0 = 2 ^ 2 n - i ( a ) s i n ( 2 n — 1)£, n—i oo co s(asin £ ) = */o(a) -f 2 E ^2rv(a) COS 2n£, n = 1 Jm(a) = ^ ib!(m + fc)! ( f ) ’ m = 0 - 1-2 ksz 0 ' L imiting b y considering the vibration with sm all am plitude a we have in the first approxim ation averaged equations of the form: (30) 2 ^ à = ^ hi/(a) ^ 1+ - " ( 1 2(a i cos \p + a 2 sin v>) ff /Ĩ rị) = i/(a — n + • y n 2 ( a 3 COS + c*4 sin v^), L a V (31) here a 3 = a 4 = (32) ữ l = \.2 ^ ° * ^ ~ 4 ^ 1 + 8*n ^’ <*2 = [ 2(^0 “ ^2 ) + ị + *^3)] co s^> ( i i + J 3) COS 6, (J3 - 3 J i) sin 6 , * - , - ( ! ) * . Stationary regim es of resonant vibrations a* and are determined by equations ả = = 0: 7 y n 2(a Ị cos 0 . + a js in v>.) * /1(1 + ^ ) ,v/ 0 7 y n 2(aJ COS + a j sin ự>.) * /1^1 -f *y j — A 1 2ffa*v/07yn2(ajc08 + aj sin ) = n 7= , (33) 43 Q* = Qi(a = a.), a. 7*^0, n = ~~7 = - — const. The solutions of th ese equations are found in the series: a. = a„ + £Oj + 0(e2), (34) rl>. = i>o + + °(ff ). where a0 satisfies relation (29). Substituting equations (34) into equations (33) we have: ỰÕT0 ^ n 2 ( a l0 COS V>o'+ a 2osin V>o) = /1(1 + * . ~ v 2 / (35) a . = a „ + e a i = a Q + 4 s a ^ n 2 ^ ( a 3o COS \ị>o + c u o s in \ị>„), here a ,0 = Q,(a = ac). The first equation of (35) gives the phase displacement. Then, the correction eai to the stationary amplitude an will be found. We consider now the stability of stationary solution a, and determined by formulae (33). For this purpose we write the variational equations for system (31). Let a = a , 4 - Sa, \p = + brị). By putting these expressions into (31) and linearizing relative to 5a and 6\p we obtain: dtfa- '{(f )„ - f n1( a ) / 01'*-+ (a). € R - sin t/>* - f a 2 COS rp.)ỏ\p> D (36) = {L„ + ^ n 2 [ ( ^ ) u C csự + ( ^ ) n S m V > .]} ía+ £ iỉ + A £ ^ 2( “ ữ3 s*n + Q4 C0S iM i 0- where I(a) = *(a)-n (37) The characteristic equation of this system is of the form À2 + /ÍẰ + G = 0 (38) where * ~ 7 n M [ ( 2 ) . - ( ? ) ] ‘- * - * [ ( 5 ) . * ( ? ) W } - ( ! ) . . G = — sin yp* + Q2 COS ự>.) + 0 ( f 2), and a prime denotes a derivative to a The stability conditions will be (39) K > 0, Ơ > 0. (40) As an exam p le let us consider the case 6 = 0. Then 1 / a2\ a Oj = «4 = 0, = 44 \ ^ n ’ v / a ; ( l - sin ĩị>o ** 2A ( l + , 1 * 0 3 « 1 « . = <i» + j i | n o J t M i a ° ~ n ã Lt form: 3 2 ã~o ( 1 - ^ r ) 8Ìn ỳo *» 2A í 1 + 2 * ) , aná equations (35) ar« of thf form: (41) Because a if small, the first equation of (41) shows that there ire two values of rpo lying on thf first and second quadrants corresponding to two values of cot ĩpo with opposite signs. Therefore the one o f sta t i o n a r y a m p l itu d e c o r r e s p o n d in g to CO* > O i i la r g e t h a n a 0 a n d t h e o the r co r r e s p o n d i n g to CO%$o < 0 is smaller than an. The expressions (39) now zst: K = fc + 0(a2), Ơ = - e ^ n 2 COS ự>o + 0 ( e 2 ) ^ ^ and the stability condition gives: cos 0 O < 0. (43) For the caat 6 = */2 we have a 2 - a 3 = 0, ữi = - + 0(a), a 4 = - Y + o(a), a . = a „ + c ai = « „ - ^ e y n 2 a j sin V'o, J 4 4 j K = eh + 0 (a 2 ), G = J ị R ft2 sin 'Po + 0 (ff2) I and the stability condition is sin 0O > 0. (45) So, in both cases (6 = 0 and s = ff/2) the stationary vibration with sm all am plitude is stab le and that with large amplitude is unstable (see equations (41) and equations (43), (44) and (45)). CONCLUSION 1. The stationary nonlinear vibrations of the pendulum and its stability have been considered. 2. To avoid resonance of the pendulum , the param eters of the system considered should be chosen so that (Jj7 difers from n 3 and J fl2,or f i* n», i n ’ (M) 3. The rotating m otion of the pendulum may occur. U sing the averaging m ethod o f n onlinear m echanics and th« Jacobie expansions, the sm all “vib ration" of the pendulum around the station a ry rotation and its stability have been studied. REFERENCE 1. B o g oliubov N. N ., MitropolBkii Yu. A. A sym p totic m ethods in nonlinear vibrations, M oscow 1974 (in Russian). 2. M oiseev N. N. A sym ptotic m eth ods of nonlinear m echanics, Moscow 1981 (in R u ssian). 3. Sm ừn ov V . I. Course of high m ath em atics, to m 3, pari 2, M oscow 1956 (in R ussian). 45 . động thing điều h a tùy ý T he nonlinear vibration of a pendulum whose support undergoes arbitrary rectilinear harmonic m otion is stud ied. The main attention is paid to the resonant cases and. stationary vibration with sm all am plitude is stab le and that with large amplitude is unstable (see equations (41) and equations (43), (44) and (45)). CONCLUSION 1. The stationary nonlinear vibrations. tcostp]. U sing L agrange’s equation, taking into account the damping force, the equation of m otion of vhe pendulum is obtained as where a& gt;2 = gỊ I, h is the damping coefficient. We assum e that