DSpace at VNU: On the lateral oscillation problem of beams subjected to axial load

It illustrate?. he “butterfly" effect of the Chaos phenomenon.

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 I L L A T I O N P R O B L E M O F B E A M S

S U B J E C T E D T O A X IA L L O A D

D a o H u y B ic h

V ietnam N atio n a l University

N g u y e n D a n g B ic h

Institute for B u ild in g Science a n d Technology - M in is tr y o f Construction

A b s t r a c t: This paper approaches the problem of la te r a l o s c illa tio n of beams subjected to axial load by m eans of seeking the exact so lu tio n of the lin e a r Math.eu

equation with the periodic function h(t) having a d eterm in ed form

However when h(t) — k + ajcosiot, the equation (1) does not p o s s e s an exact solution but with the v a lu es of param eters k, a lf 0) satisfying som e d e te r m in e d conditions, we can seek an approximated solution Obtained results are s u m m a r iz e d as follows:

The general exact solution for the equation (1) with h(t) h a v in g a determ ined form can be expressed in the form of known m athem atical fu n ctio n s It illustrate? he

“butterfly" effect of the Chaos phenomenon.

The condition and algorithm for finding the approxim ated so lu tio n of the e q in to n (1) with

From obtained results we can discuss about the o scillation of b e a m s.

1 L ateral o s c i ll a t io n o f b e a m s s u b j e c te d to a x i a l lo a d

Fig 1 shows th e oscillation of

a beam h a v in g c o n s ta n t cross

section subjected to axial load P(t)

El, EA, ỊI, t a n d p re p re s e n ts th e

bending a n d axial rigidity, m ass

dam ping coefficient of th e beam

respectively

T U"'1='W

W(x.i)

F ig l A b e a m su b je cted tc axial load

The oscillation of th e beam can be described a s follows [2]

E Iw IV + pw + |IW - EA

W IV w 11 - t h e 4th a n d 2nd o r d e r 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 w - th e 2nd and 1 st order derivatives of w with re sp e c t to t

The b o u n d ary conditions for d isp lacem en t a re w r i t t e n as

1

2 D a o H u y B ich y N g u y e n D a n g B i c h

1 IS a s s u m e d t h a t th e ax ial w ave is negligible a n d U (£, t) is th e d isp la c e m e n t

a t t h e r i g h t e n d of th e b e a m T h e boundary co n d itio n s (1.2) c a n be sa tis fie d w hen w(x It)is set as

Substitute (1.3) in to (1.1) it yields

where

q ( t ) = u ( ể , t ) ^ ; (0, y = 7 — V ; 2D©! = J i

Ii order to in v estig ate the phenomenon in the oscillation of beam s subjected to axial lad s, at first th e following equation should be exam ined

u + C 0 j [ l + q ( t ) ] u = 0 ( 1 5 )

I'q(t) = a coscot, the equation (1.5) leads to the classical M a th ie u ’s equation

I S u j p l e m e n t a r y e q u a t i o n

C m sid er th e follow ing s e t of e q u a tio n s [3]

Ú

V =

u - a

tn v h i a V, u a r e fu n c tio n s of t; x, a, Cl) a re p a r a m e t e r s

A te r V b e in g e lim in a te d from (2.1), (2.2) it yields

( 2 2 )

u - a ( u - a )

Iiste a d of th e fu n ctio n V, fun ctio n y is u sed , w ith th e follow ing a lte ra tio n :

y F'om (2 2), (2.4) y can be calculated

1

On the la te r ạ l o s c i ll a t io n p r o b l e m of 3

Using (2.4), (2.5), th e e q u a tio n (2.1) can be w r i t t e n a s

ý + to2y = X

The solution of (2.6) is

CO

w here cp, A - in te g r a l c o n s ta n ts

E q u atio n (2.3) can be a l t e r n a t e d in to th e form:

— (ú2) - -— ù 2 = -2Ằ.(u - a ) 2 + 2oýí(u - a ) ,

d u u - a

t h a t yields

Ũ2 = -y(u - a f (u - a f - — (u - a ) +

CO

7 - integral co nstan t

After solving (2.9) it can be found t h a t

CO

u - a

x + pcos(cot + \\i)

w here vy - in te g r a l c o n s ta n t,

p 2 = X2 - (02y > 0.

(2.6)

(2.7)

(2.8)

(2.9)

'2 1 0 )

(2 1 1

From (2.5), (2.7), (2.10) it can be in fe rre d th a t:

y = -!T [?i + pcos(a)t + n/)], CD

A _ p

A = - iy tp = Vị/

CO

O u r aim is to find a n y s u p p l e m e n ta r y M a t h ie u ’s e q u a tio n w hich h a s a n exsci solution D iffe re n tia tin g th e e q u a tio n (2.9) w ith r e s p e c t to t we o b tain

ii = - (u - a)[2y(u - o f - 3>t(u - a ) + a)2 j.

and from (2.5) we h a v e

a y + 1 a y + 1

B ased on (2.14), e q u a tio n (2.13) can be w r itte n in th e form

(Ỉ.1 3

(Ĩ.14

0 5

S u b s t it u t e u - a c a lc u la te d i n (2.10), y c a lc u la te d in (2.12) w ith Vị/ - O u t (2.15) it yields

4 D a o H u y B i c h , N g u y e n D a n g B ic h

3 S o l u t i o n a n d c h a r a c t e r i s t i c o f t h e s o l u t i o n

E q u a tio n (2.16) h a s th e following p a r t i c u l a r periodic so lu tio n

u CO2 + aX + a p cos cot

X + p coscot = a +

CO

W hen th e p a r t i c u l a r so lu tio n (3.1) is found, th e g e n e r a l so lu tio n for eq u atio n (2.16) can be e s t im a t e d a s follows

11 = CO2 + CLẤ + aPcoscot

Ằ + (3 coscot c , + c -J o(a)2 + aA, + aPcoscox)~ (a + pcoscox)2dx

in which Cj, C2 - in te g ra l c o n stan ts

From (3.2), th e velocity Ú a n d a c c e le ra tio n ii can be c a lc u la te d

(3.2)

pco3 sin cot (a + pcoscot)2

t

c , + c 2 J

0

(x -f Pcoscox)2dx

(co2 -f aẰ + apcoscox)"

(x + |3cos(ox)2dx

0 (co2 + aX + apcoscoxV

I t is a s s u m e d t h a t w h e n t = 0

u(0) = u 0, ú(0) = ú 0

(3.3)

.(3.4

(3.5)

B ased on th e in itia l c o n d itio n (3.5), from (3.2) a n d (3.3), th e in te g r a l c o n sta n ts

Cj, C2 can be found:

Hence, it can be concluded t h a t (3.2) is th e g e n e ra l so lu tio n for (2.16)

B ased on (3.2), (3.3) th e g r a p h s of th e fu n ctio n s u(t) a n d u(u) with d ifferent

p a r a m e te r s can be p lo tte d a s show n in Figs 2-5

T h ere e x ist c o n s t a n t m a x im a a n d m in im a of t h e fu n c tio n u n d e r th e in te g ra l in (3.2) T h erefo re, it can be proved t h a t th is in te g r a l be g e n e r a liz e d diverse w hen t-»co From Figs 2-5, it can be ob serv ed t h a t th e so lu tio n u(t) e x p re ssed in (3.2) have th e c h a r a c te r is tic of

• D iffusively v a ria b le lim u(t) = 00

t —>00

T he so lu tio n (3.2) d e p e n d s se n sitiv e ly on th e in itia l b o u n d a ry condition

w hen jr0 = 0 it is periodic, w hen jr0 * 0 it h a s th e exceptional characteristic

of th e effect n a m e d “b u t te r f ly ” a s seen in th e “c h a o s ” p h e n o m en o n

On the la te r a l o s c i lla tio n p r o b le m of 5

Fig.2 G ra p h of function u(t)

Fig.3 G ra p h of function ú(u)

Fig.4 G ra p h of function u(t)

4 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 fun ctio n is called p o t e n t i a l of th e e q u a tio n

( x + p cos cot)2 Ằ + P c o s c o t CO2 + (XẰ + a p COS cot

W ith th e following c o n d itio n

th e p o te n tia l fu n ctio n h(t) is c o n tin u o u s a n d periodic

Fro m (4.1) yields

6 D a o H u y B ic h y N g u y e n D a n g B i c h

dh

dt

(aẰ, + a p c o s cot)3 (aA 4- aP co s wt)2 (co2 4- aX + a p c o s c o t ) 2

a P sin c o t (4.3)

t h a t can be r e a r r a n g e d as

dh

d t

I ( c o s 0 ) t ) - —r

-rr-(aA + aPcoscot)3(co2 + aA + a(5 coscot|

(4.4)

n which

f(cos cot) = cos3 cot - 3 — cos2 cot

p

7 -V + 3 —- — + 2

A CO A _ A , (0

- r + - - 5 - - 6 — - 4

dh

L et —— = 0 only w h e n sin cot = 0 , from (4.4) it can be se en t h a t

dt

f( cos cot) * 0 w ith all t such t h a t -1 < coscat < 1

In o rd er to s a tis fy (4.6) th e following p r e lim in a r y r e q u i r e m e n t c a n be used

in which

f(i)=

8 + 0)

CO

a p

CO

(4.6)

I (4.7)

(4.8)

(4.9)

From th e c o n d itio n (4.7) to g e th e r w ith (4.8), (4.9) it y ield s

2

2 — > 8 + p

2 - < - 8 p

CO

a2p2+ 68 - a p , or

- - 5 - + 68 - — - , or

a p

8 5 Ụ + 68 — < 2 — < 8 + J ^ 2 + 68

0)

a p

(4.10)

(4.11)

(4.12)

W hen a n y of th e c o n d itio n s (4.10), (4.11), (4.12) is sa tisfie d , th e p re lim in a ry

re q u ire m e n t (4.7) can be a s s u r e d However, in o rd e r to fully sa tis fy (4.6), th e g raph

of th e function h(t) sh o u ld be p lo tted , in w hich th e se t of p a r a m e t e r satisfied (4.7) is used The c rite rio n for (4.6) b e in g fully sa tisfie d is s e t su ch a s it h a s one m axim um and m in im u m only in a period w h en sin cot =0

To solve th e above m e n tio n e d problem , h(t) is a p p ro x im a te d by g(t) such as both fu n ction s a re c o n tin u o u s a n d periodic

On the la te ra l o s c illa tio n p r o b le m o f . 7

W hen any of t h e c o n d itio n s (4.10), (4.11), (4.12) is s a tis fie d , h(t) a n d g(t) w ould

h ave obtained th e sa m e m a x im a a n d m in im a w h e n sin cot =0 H ence, it can be inferred t h a t th e fu n ctio n h(t) be a p p ro x im a te d by g(t) w h e n t h e i r m a x im a a n d

m inim a are resp ectiv ely eq u al

W hen coscot = -1, we h a v e

2yco2 2 a y + 3 Ả CO2 + 3 a k + 2 a 2y _ k a ( 4 1 4 ) ( ả - p ) 2 X - p (02 + a X - a P

W hen coscot = 1 , we have

2yco2 2 a y + 3Ả CO2 + 3 a Ả + 2 a 2Ỵ = k +

From (4.14) a n d (4.15) it h a s

1 •

a i =

a p

2

CO

k =

-CO

2 CO +

2 a 2y - aA CO2 + 3aX, + 2 a Y

CO2 + 2 a X + a 2y

0.)^ ■+■ 3aA, + 2 a

a 2y

a 2X, CO2 + 2 a X + a Y

2y \ ( ờ 2 + aA )

Based on (4.16), (4.17) it yields

CO

a p

k + a

X

2 - 1

p2

Ằ.

+ a

a p p

—— + — + 1 = +

(4.15)

(4.16)

( 4 1 7 )

(4.18)

( 4 1 9 )

W ith know n v a lu e s of a 1; k a n d CO, th e v a lu e of — a n d — can be found by

solving the set of e q u a tio n (4.18), (4.19)

5 A lg o r ith m fo r f i n d i n g t h e a p p r o x im a t e d s o l u t i o n

Given t h a t th e following e q u a tio n should be solved:

T he following a lg o r ith m for fin d in g its a p p ro x im a te d so lu tio n should be followed:

8 D a o H u y B i c h , N g u y e n D a n g B i c h

• Solving th e 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 o b tain th e v a lu e s of — , —

» C hecking th e co n d itio n s (4.10), (4.11), (4.12 ) If none of th em are satisfied, the a p p ro x im a te d so lu tio n c a n n o t be found by t h is proposed alg o rith m If

th ese conditions a re satisfied we plot th e g r a p h of th e fun ctio n h(t) w ith the identified set of p a r a m e te r s

• If the function h(t) does not posses a m a x im u m a n d a m in im u m only when sin cot — 0, th e a p p ro x im a te d so lu tio n c a n n o t ỒG found by th is proposed algorithm

» If the function h(t) satisfies the abovem entioned condition, form ula (3.1 ) with its respective p a ra m e te r s can be considered as th e solution of (5 1)

E x a m p l e 1.

Find the a p p ro x im a te d solution of th e following e q u a tio n :

ii - 4(0,0 0 6 5 9 - 0 ,0 3 3 4 1 5 COS 2 t) u = 0 (5 2 ) Substitute

intc(<- (4*19), th e r e s u l ts a re

N th the set of p a r a m e t e r s (5.4), condition (4.10) is sa tisfie d

?iom (2.11) a n d (5.4) it can be in fe rre d th a t:

a p = - 1 ,1 9 ; a X = - 1 4 ,2 8 ; cry = 5 0 , 6 2 5 5 8 (5.5)

ỉísed on (5.5), (5.3) th e g r a p h s of h(t), g(t) can be p lo tte d a s show n in Fig 6 Fron olere, it can be show n t h a t th e fun ctio n h(t) h a s only a m a x im a a n d a m inim a whin s n 2 t — 0 The fu nctions h(t), g(t) have id en tica l v alu es of m axim a and

m in n a which a re th e a p p ro x im a tio n of each resp ec tiv e o th e r T h erefo re, it can be

s o l i t i > n o f ( 5 1 )

u = (*■ t P k CO2 + a X + a p c o s cot

}e a p p io x im a te d so lu tio n (5.6) resp ec tiv e to th e p a r a m e t e r s identified in (5 5 }ai the form of

Jib stitu te (5.7) in to (5.2), it is ob serv ed t h a t (5.7) is th e ap p ro x im ate d soliti>nof (5.2)

On the la te r a l o s c i ll a t io n p r o b le m of 9

E x a m p le 2.

Find th e a p p ro x im a te d so lu tio n of th e following e q u a tio n :

ii + 4(0,001783728 -0 ,0 0 7 7 0 2 6 4 9 COS 2t)u = 0 (5.8)

S u b s titu te

CO = 2, k = - 0 , 0 0 1 7 8 3 7 2 8 , a , = 0 , 0 0 7 7 0 2 6 4 9 , '5.9) into (4.18), (4.19), th e r e s u l t s a re

W ith th e se t 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 fe rre d t h a t

a p = 7 ,4 8 ; a X = 6 1 , 7 1 ; a 2y = 9 3 8 , 0 4 (S.11)

B ased on (5.9), (5.10) th e g r a p h s of h(t), g(t) can be p lo tte d as shown ii F,g 7 From th e r e it can be sh o w n t h a t th e function h(t) h a s only a m a x im u n a id a

m in im u m w h e n sin 2t = 0 T h e fu n ctio n s h(t), g(t) h a v e id e n tic a l values of tr a u m a

a n d m in im a, w hich a r e 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 co n d itio n s u(0) = u 0, ủ(0) = u0 = 0 s th e

ap p ro x im a te d so lu tio n of (5.1)

938.04 0.001783728 -0.0'702i49

• K Fig.7 Graph of function h(t), g(t) wih - = 82!

(x + p)u0 CO2 + aX 4- q p COS cot

The a p p r o x im a te d so lu tio n (5.6) resp ec tiv e to t h e p a r a m e t e r s ldeitfi.d in (5.5) h a s th e form of

1

Fig 6 Graph of function h(t), g(t) with p = 12

10 D a o H u y B i c h, N g u y e n D a n g B i c h

_ A Qy1r Q/l 6 5 , 7 1 + 7 , 4 8 C O S 2 t

U 1 = 0,94534u0 X — — — l ỉ l r - ( 5 1 3 \

S u b s titu te (5.13) into (5.8), it is observed t h a t (5.13) is th e a p p ro x im a te d solution of (5.8)

6 D is c u s s io n

In order to satisfy (4.6), th e condition (4.7) p lay s only a role of p re lim in a ry

Ĩ e q u i r e m e n t , b u t it IS p o s s i b l e 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 o r e

c o m p l e x in c a l c u l a t i o n

The accuracy of above m entioned approxim ate m ethod d ep en d s on th e ratio

-p '

From o b tain ed r e s u lts for u(t), the d is p la c e m e n t w(x, t) of b e a m s can be found

A c k n o w l e d g e m e n t T h is resea rch is com pleted w ith th e fin a n c ia l s u p p o r t of the

N ational Council for N a t u r a l Sciences

R e fe r e n c e s

1 Nguyen V an Dao, T r a n Kim Chi, N guyen D ung, “C h a o tic p h e n o m e n o n in a

n o n lin e a r M a th ie u o silla to r”, Proceeding o f the S e v e n th N a tio n a l C ongress on

M echanics, Hanoi, 18-20 December 2002, T l, pp 40 - 49

2 W eidenham m er, F “Biegeschwingugen des S ta b le u n ter axial p ulsierender

Z u fa llsla st" V D I-B rinchte Nr: 101 - 107 1996

3 Dao Huy Bich, Nguyen Đ ang Bich, “On th e m eth o d so lv in g a class of n o n -lin e ar

differen tial e q u a tio n s in m echanics”, Proceedings o f the s ix th N a tio n a l Congress

on M echanics, Hanoi Dec, 1997, pp 1 1 - 17

4 G ra n in o A Korn, T h e re sa M Korn, M a th e m a tic a l h a n d b o o k fo r sc ien tist a n d

engineers, M cGraw-Hill Book Com pany, 1968.