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 = 0 •

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

ii + 2Dc0jủ + (Oj [l + q(t)]u + ỴU3 = 0 ,

(1.4)

where
q ( t ) = u ( ể , t ) ^ ; (0,
It I

_ tc4E I 1 7I2E A

B

y = 7 — V ; 2D©! = J i .
4 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 = 0 .

(1 .5 )

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

+ 7~
(u -a)

V = M u -a )-to

Iistead of th e function V, function y is used, w ith th e following a lteration:
v=- .
y
F'om ( 2 .2 ), (2.4) y can be calculated

1

(2.4)

.( 2 - 3 )


3

On the la te rạ l o s c illa tio n p r o b le m of...
Using (2.4), (2.5), th e e q u a tio n (2.1) can be w ritte n as
ý + to2y = X .

(2.6)

The solution of (2.6) is
y=

(2.7)

u- a

'2 1 0 )

where vy - in te g ra l c o n sta n t,
(2. 1 1

p 2 = X2 - (02y > 0.

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 . tp = 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 + 1



3. 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 ra 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 2 - in te g r a l c o n sta n ts.

(3.2). T herefore, it can be proved t h a t th is in te g ra l be g e n e ra 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 . 2 )
have the c h a ra 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 = 0 it is periodic, w hen jr0 * 0 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 ra l o s c illa tio n p r o b le m of..

CO

ap

aX

CD

ap

a A.

2

7.48


3,46

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

Fig.3. G ra p h of function ú(u)

ap

aX

2

-0,19

-1,25

Uo

Ử0

a y 1/2

0

1,56

0,19

CO


D a o H u y B ichy N g u y e n D a n g B ic h

6
dh
dt.

4a 2ỴC02
(aẰ,

+

2 a 2Ỵ 4- 3aA.

a p c o s cot)3

co2 + 3aẰ. 4- 2ya 2

aPcos wt)2

(aA. 4-

(co2 4- aX

+

ap

a P sincot.


I 3 C^2
l2
A.
O A
_ A.^
, 2(0^
-r
5 - 6 — - 4 ---p3 a p p 2
p
ap
+

------ -

(4.5)

-

dh
Let ——= 0 only w h e n sin cot = 0 , from (4.4) it can be seen t h a t
dt
f( cos cot) * 0

-1 < coscat < 1 .

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

l
ì
----8- —
1+ 4—
f(i)= — + 1
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
2

CO

2 —> 8 +
+ 68 , or
ap
a 2p 2
p
0)
5
\ a 2p2

2 -< -8
p

-5 -

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 = 0 .
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

2 a y + 3Ả

(ả - p)2

CO2 + 3 a k + 2 a 2y _ k



2

k =-

2

CO

2 CO +

2 a 2y - aA.

CO2 + 3aX, + 2 a Y

a 2y

CO2 + 2 a X + a 2y

((0.)^ ■+■3aA, + 2 a 2y \ ( ờ 2

(2 a 2y -

+ aA .)

(4 .1 7 )

CO2 + 2 a X + a Y

a 2X,


ap

1

2
^
A
2
CO
A.
(O
——+ — + 1 = ---- +

lap

p

pJ

ap

CO,

(4 .1 9 )

p

X.


p «p
» Checking th e conditions (4.10), (4.11), ( 4 . 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 ra p h of th e function h(t) w ith the
identified set of p a ra 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 . 1 ) with
its respective p a ra m e te rs can be considered as the solution of (5 . 1 ).
E x a m p l e 1.
Find the a p p ro x im ate d solution of th e following eq u atio n :
ii - 4 (0,00659 - 0 ,0 3 3 4 1 5 COS 2 t)u = 0 .

(5.2 )

Substitute
intc(
(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 5 }ai the form of
1 1 - 1 Q/IQ7QC

1 0 , 2 8 + 1 , 1 9 COS 2 t
u = l,348735un X ----------- — -----------------14,28+ 1,19 cos 2t '

/5 7 \
1
;

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 ra l o s c illa tio 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 solution of th e following e q uation:
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 ,


1

CO

a(3

aX

2

-1.19

-14.28

a 2y

k

ai

50.62558

-0.00659

0.033415

Fig.6 Graph of function h(t), g(t) with p = 12
(x + p)u 0

1

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 0 X

65,71 + 7 , 4 8 C O S 2t

——

— l ỉ l r . ------- .

Nguyen Van Dao, T ra 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

2.

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.

3.

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 1 - 17.

4.

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.