VNU JOURNAL OF S C IE N C E , Mathematics - Physics, T.xx, N04, 2004 ON T H E L A T E R A L O S C IL L A T IO N P R O B L E M O F B E A M S S U B J E C T E D TO A X IA L L O A D D a o H u y B ic h Vietnam National University N g u y e n D a n g B ic h Institute for B u ild in g Science an d Technology - M in is tr y o f Construction A b str a c t: This paper approaches the problem of lateral oscillation of beams subjected to axial load by means of seeking the exact solution of the linear Math.eu equation with the periodic function h(t) having a determ ined form ii + h(t)u = • 1) However when h(t) —k + ajcosiot, the equation (1) does not p o sse s an exact solution but with the values of parameters k, a lf 0) satisfying som e d eterm in ed conditions, we can seek an approximated solution Obtained results are s u m m a r ize d as follows: The general exact solution for the equation (1) with h(t) h a vin g a determined form can be expressed in the form of known m athem atical functions It illustrate? he “butterfly" effect of the Chaos phenomenon The condition and algorithm for finding the approximated solution of the eqinton (1) with h(t) = 0)2(k + coscot) 2) From obtained results we can discuss about the oscillation of b eam s Lateral o s c illa tio n o f b ea m s su b jected to a x ia l lo a d Fig shows the oscillation of a beam having c o n sta n t cross section subjected to axial load P(t) El, EA, ỊI, t a n d p re p re se n ts the T U"'1='W W(x.i) bending and axial rigidity, m ass density, length and external F ig l A b e am sub jected tc axial load dam ping coefficient of the beam respectively The oscillation of the beam can be described as follows [2] EIw IV + pw + |IW - W IV w EA u(*,t) + i j ( w n )*dx w 11 = , w 11 - t h e 4th a n d 2nd ord er d e r i v a t i v e s o f w w i t h r e s p e c t t o X, w - th e nd and st order derivatives of w w ith re s p e c t to t The boundary conditions for displacem ent a re w ritte n as (1 ) D a o H u y B ichy N g u y e n D a n g B ic h w(0) = w(í ) = w n(0) = w n( = (1.2) IS a ssu m e d t h a t th e axial wave is negligible a n d U(£, t) is th e d isp lacem en t a t t h e r i g h t end of th e b e am T he boundary conditions (1.2) c an be satisfied when w(x It)is set as w (x,t ) = u ( t ) sin ™ (1.3) Substitute (1.3) in to (1.1) it yields ii + 2Dc0jủ + (Oj [l + q(t)]u + ỴU3 = , (1.4) where q ( t ) = u ( ể , t ) ^ ; (0, It I _ tc4E I 7I2E A B y = — V ; 2D©! = J i ụĩ Ịi (ir Ii order to investigate the phenomenon in the oscillation of beam s subjected to axial lads, at first the following equation should be exam ined u + C0 j [ l + q ( t ) ] u = (1 ) I'q(t) = a coscot, the equation (1.5) leads to the classical M a th ie u ’s equation ii + C02(k + aj cos From (2.5), (2.7), (2.10) it can be in ferred th a t: ( 2.12 y = -!T [?i + pcos(a)t + n/)], CD A _= - p A iy = Vị/ CO O ur aim is to find a n y s u p p le m e n ta r y M a th ie u ’s e q u a tio n which h a s a n exsci solution D iffere n tia tin g th e e q u a tio n (2.9) w ith re sp e c t to t we obtain ii = -(u - a)[2 y(u - o f - 3>t(u - a ) + a)2 j (Ỉ.13 and from (2.5) we h ave ay + (Ĩ.14 ay + B ased on (2.14), e q u a tio n (2.13) can be w ritte n in th e form u u[2y(u - o f - sx{n - a ) + S u b s titu te u - a c alcu late d in (2.10), y c alc u la te d in (2.12) w ith Vị/ - O u t (2.15) it yields D a o H u y B ic h , N g u y e n D a n g B ich ù + co 2yco2 _ 2ctỴ -f 3^ (>, + p cos cot)2 A, + p cos cot co2 + 3aẦ + y a co2 + aX + a p COS cot (2.16) S o lu tio n a n d c h a r a c t e r i s t i c o f th e s o lu t io n E q u atio n (2.16) h a s th e following p a rtic u la r periodic solu tion u CO2 + aX + a p cos cot CO =a + X + p coscot X + p COS cot (3.1) W hen th e p a r t i c u l a r solution (3.1) is found, th e g e n e l solution for equation (2.16) can be e s tim a te d a s follows 11 = CO2 + CLẤ + aPcoscot Ằ + (3coscot c, + c (a + pcoscox)2dx -Jo (a)2 + aA, + aPcoscox)~ (3.2) in which Cj, C - in te g r a l c o n sta n ts From (3.2), th e velocity Ú an d acceleratio n ii can be c alcu late d pco3 sin cot (a + pcoscot )2 ii = CO t (3.3) (co2 -f aẰ + apcoscox)" 3ẰC02 2co4y (^ + pcoscot )3 (x -f Pcoscox)2dx c , + c 2J (x + pcoscot )2 w (0 (x + |3cos(ox)2dx c,+ X + pcoscot (3.4 (co2 + aX + apcoscoxV It is a s s u m e d t h a t w hen t = u( 0) = u 0, ú( 0) = ú (3.5) Based on the in itia l condition (3.5), from (3.2) a n d (3.3), th e in te g l c o n stan ts Cj, C can be found: c = (^ + PK> 0)J + a > i + a p ’ c _ (co2 + q X + qp)ủ0 > + (3 (3.6) Hence, it can be concluded t h a t (3.2) is th e g e n e l so lution for (2.16) Based on (3.2), (3.3) th e g p h s of th e function s u(t) a n d u(u) with different p a m e te rs can be p lo tted as shown in Figs 2-5 T here e x ist c o n s ta n t m ax im a a n d m in im a of th e fu n ctio n u n d e r the integ ral in (3.2) T herefore, it can be proved t h a t th is in te g l be g e n e liz e d diverse w hen t-»co From Figs 2-5, it can be observed t h a t th e solu tion u(t) expressed in (3 ) have the c h a c te ris tic of • Diffusively v a ria b le lim u(t) = 00 t —>00 The so lution (3.2) d e p en d s sen sitively on the in itia l b o u n d a ry condition w hen jr0 = it is periodic, w hen jr0 * it h a s th e e x cep tional characteristic of th e effect n a m e d “b u tte r f ly ” as seen in the “c h a o s” phenom enon 5 On the la te l o s c illa tio n p r o b le m of CO ap aX CD ap a A 7.48 9.69 7.48 9.69 Uo ú„ a y ,/2 Uo Uo a y l/2 0.813 3,46 0.813 3,46 Fig.2 G p h of function u(t) Fig.3 G p h of function ú(u) ap aX -0,19 -1,25 Uo Ử0 a y 1/2 1,56 0,19 CO Fig.4 G p h of function u(t) P o t e n t i a l o f e q u a t i o n (2.16) In e q u a tio n (2.16) th e following function is called p o te n tia l of th e e q u atio n ,V (x + 2yco2 p cos cot)2 2ay + 3X Ằ + P c o s c o t to2 + 3aX + 2ya '2 ^ CO2 + (XẰ + a p COS cot W ith th e following condition X2 - p2 > 0; CO2 + a X + y a > 0, the p o te n tia l function h(t) is c o n tin u o u s a n d periodic From (4.1) yields (4.2) D a o H u y B ichy N g u y e n D a n g B ic h dh dt 4a 2ỴC02 (aẰ, + a 2Ỵ 4- 3aA a p c o s cot)3 co2 + 3aẰ 4- 2ya aPcos wt)2 (aA 4- (co2 4- aX + ap a P sincot (4.3) c o sco t)2 t h a t can be r e a r r a n g e d as dh dt p/ co3a 4p sin cot \ I ( c o s ) t ) - —r rr- (4.4) (aA + aPcoscot) 3(co2 + aA + a(5 coscot| n which A „ co X „ 7-V + —- — + coscot + p2 ap p f(cos cot) = cos cot - —cos2 cot p I C^2 l2 A O A _ A.^ , 2(0^ -r - — - -p3 a p p p ap + - (4.5) - dh Let ——= only w h e n sin cot = , from (4.4) it can be seen t h a t dt f( cos cot) * -1 < coscat < w ith all t such t h a t (4.6) In order to sa tisfy (4.6) th e following p re lim in a ry r e q u ir e m e n t can be usedI (4.7) in which A2 vP ' 8+ V 0)2 '\ ap -4 CO (4.8) ap 2\ ( (02 > Ằ ( Ằ CO f Ằ l ì 8- — 1+ 4— f(i)= — + p ~I u J _p2 a P, (4.9) From th e condition (4.7) to g e th e r w ith (4.8), (4.9) it yields CO —> + + 68 , or ap a 2p p 0) \ a 2p2 -< -8 p -5 - - + „ O)2 68 - - , or ap ’ ap p (4.11) — - Ụ + 68 - — < — < -8 + J - ^ + 68 \ a (3 (4.10) ỵ a p 0) ap (4.12) W hen a n y of th e co nditions (4.10), (4.11), (4.12) is satisfied, th e prelim in ary req u ire m e n t (4.7) can be a s s u re d However, in o rd er to fully satisfy (4.6), th e graph of the function h(t) sh ould be plotted, in which the se t of p a r a m e t e r satisfied (4.7) is used The c riterio n for (4.6) b e ing fully satisfied is se t such as it h a s one m axim um and m inim um only in a period w hen sin cot = To solve th e above m en tio n ed problem , h(t) is a p p ro x im a te d by g(t) such as both functions a re c o n tin u o u s a n d periodic 7 On the lateral o sc illa tio n p ro b le m o f g(t) = k + (4.13) coscot 3.i When any of th e conditions (4.10), (4.11), (4.12) is sa tisfie d , h(t) a n d g(t) w ould have obtained th e sa m e m ax im a a n d m inim a w hen sin cot =0 Hence, it can be inferred t h a t th e function h(t) be a p p ro x im ate d by g(t) w h e n th e ir m axim a a n d minima are respectively equal W hen coscot = -1, we have 2yco2 a y + 3Ả (ả - p)2 CO2 + a k + a 2y _ k a (4 ) (02 + a X - a P X - p W hen coscot = , we h ave 2yco2 { w a y + 3Ả CO2 + a Ả + a 2Ỵ = k + X + p f (4.15) 1• CO2 + a Ằ + aP From (4.14) a n d (4.15) it h a s ap = k =- CO CO + a 2y - aA CO2 + 3aX, + a Y a 2y CO2 + a X + a 2y ((0.)^ ■+■3aA, + a 2y \ ( (2 a 2y - + aA ) (4 ) CO2 + a X + a Y a 2X, CO (4.16) Based on (4.16), (4.17) it yields k+a CO (4.18) X ap p , a X? - p2 V + a Ằ ap ^ A CO A (O ——+ — + = + lap p pJ ap CO, (4 ) p X W ith know n v a lu e s of a 1; k a n d *1 A, the valu e of — a n d — can be found by solving the set of e q u a tio n (4.18), (4.19) A lg o rith m for f in d in g th e a p p r o x im a te d s o lu t io n Given t h a t th e following e q u a tio n should be solved: ii + to2(k + aj coscot)u = , (5.1) The following a lg o rith m for finding its a p p ro x im a te d so lu tion should be followed: D a o H u y B ic h , N g u y e n D a n g B ic h • Solving the set of e q u a tio n (4.18), (4.19) w ith th e v a lu e s of CO2, k, a, given in (5.1), we obtain th e v a lu e s of — , — p «p » Checking th e conditions (4.10), (4.11), ( 12 ) If no ne of th e m a re satisfied, the a p p ro x im ate d so lution c an n o t be found by th is proposed algorithm If these conditions a re satisfied we plot the g p h of th e function h(t) w ith the identified set of p a m e te rs • If the function h(t) does not posses a m ax im u m a n d a m in im u m only when sin cot — 0, the a p p ro x im ate d solution c a n n o t ỒG found by th is proposed algorithm » If the function h(t) satisfies the abovem entioned condition, formula (3 ) with its respective p a m e te rs can be considered as the solution of (5 ) E x a m p l e Find the a p p ro x im ate d solution of th e following eq u atio n : ii - (0,00659 - ,0 3 COS t)u = (5.2 ) Substitute intc( n o f ( ) u = (*■ t P k CO2 + a X + a p cos cot CO + aX + a p Ầ + p coscot (5.6) }e a p p io x im ated solution (5.6) respectiv e to th e p a r a m e t e r s identified in (5 }ai the form of 1 - Q/IQ7QC , + , COS t u = l,348735un X - — -14,28+ 1,19 cos 2t ' /5 \ ; Jibstitute (5.7) into (5.2), it is observed t h a t (5.7) is th e approxim ated soliti>nof (5.2) On the la te l o s c illa tio n p r o b le m of E x a m p le Find th e a p p ro x im a te d solution of th e following e q uation: ii + 4(0,001783728 -0 ,0 7 COS 2t)u = (5.8) S u b s titu te CO = 2, k = - , 0 8 , a , = , 0 7 , '5.9) into (4.18), (4.19), th e r e s u lts a re ^ = 8,25, - = - , 5 p (5.10) ap With th e set of p a r a m e t e r s (5.10), condition (4.10) is satisfied From (2.11) a n d (5.10) it can be in ferred t h a t a p = ,4 ; a X = ,7 ; a 2y = , (S.11) Based on (5.9), (5.10) th e g p h s of h(t), g(t) can be p lo tte d as shown ii F,g From th e re it can be show n t h a t the function h(t) h a s only a m ax im u n a id a m inim um w h e n sin 2t = T he functions h(t), g(t) h a v e id e n tica l values of tra u m a a n d m inim a, w hich a re th e a p p r o x i m a t i o n of each r e s p e c t i v e other Therefo'e it Can be concluded t h a t (3.2) w ith th e conditions u(0) = u 0, ủ(0) = u = s the appro x im ated solu tio n of (5.1) | I J I CO a(3 aX -1.19 -14.28 a 2y k 50.62558 -0.00659 0.033415 Fig.6 Graph of function h(t), g(t) with p = 12 (x + p)u (02 + + aP (0 ap a 7.48 51/1 a2y k a 938.04 0.001783728 -0.0'702i49 • K Fig.7 Graph of function h(t), g(t) wih - = 82! CO2 + aX 4- q p COS cot ?i + pcoscot The a p p ro x im a te d solution (5.6) respective to th e p a r a m e te r s ldeitfi.d in (5.5) has th e form of 10 D a o H u y B i c h , N g u y e n D a n g B ic h _ A Qy1r Q/l U1 = 0,94534u X 65,71 + , C O S 2t —— — l ỉ l r - 61,71 + 7,48 COS t (5 13\ ^ ' S u b s titu te (5.13) into (5.8), it is observed t h a t (5.13) is th e ap p ro x im ate d solution of (5.8) D isc u s sio n In order to satisfy (4.6), the condition (4.7) plays only a role of p re lim in a ry Ĩ e q u ir e m e n t , b u t it IS p o s s ib le to e s t a b l i s h a m o r e p r e c i s e c o n d i t i o n h o w e v e r m ore c o m p le x in c a lc u la t io n The accuracy of above mentioned approxim ate m ethod depends on the ratio p' From obtained re s u lts for u(t), the d isp la c e m e n t w(x, t) of b e am s can be found A c k n o w l e d g e m e n t T his research is com pleted w ith th e fin an c ia l s u p p o rt of the N ational Council for N a tu l Sciences R efe r e n c e s Nguyen Van Dao, T n Kim Chi, N guyen Dung, “C haotic p h e n o m e n o n in a no nlin ear M ath ieu osillator”, Proceeding o f the S e v e n th N a tio n a l Congress on M echanics, Hanoi, 18-20 December 2002 , T l, pp 40 - 49 W eidenham m er, F “Biegeschwingugen des S ta b le u n te r a xial p ulsieren der Z u fa llsla st" V D I-B rinchte Nr: 101 - 107 1996 Dao Huy Bich, Nguyen Đ ang Bich, “On th e m ethod solving a class of non-linear differential eq u atio n s in m echanics”, Proceedings o f the six th N a tio n a l Congress on Mechanics, Hanoi Dec, 1997, pp 1 - 17 G ranino A Korn, T h ere sa M Korn, M a th e m a tic a l h a n d b o o k for scien tist and engineers, M cGraw-Hill Book Company, 1968 ... phenomenon in the oscillation of beam s subjected to axial lads, at first the following equation should be exam ined u + C0 j [ l + q ( t ) ] u = (1 ) I'q(t) = a coscot, the equation (1.5) leads to. .. If these conditions a re satisfied we plot the g p h of th e function h(t) w ith the identified set of p a m e te rs • If the function h(t) does not posses a m ax im u m a n d a m in im u m only... te (5.13) into (5.8), it is observed t h a t (5.13) is th e ap p ro x im ate d solution of (5.8) D isc u s sio n In order to satisfy (4.6), the condition (4.7) plays only a role of p re lim in