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

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

(1) with
h(t) = 0)2(k +

coscot) .

2)

From obtained results we can discuss about the oscillation of b eam s.

1. 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. 1 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 = 0 ,

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 2 nd and 1 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

(1 1 )


2

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( 0 = 0.

(1.2)

1 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 = 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 coswhere

k =^ ị,

(1.6)

a1 = - ặ .

Cl)

( 1 .7 )

CO

I. S u jp le m e n ta r y e q u a t io n
C m sider th e following se t of e q u atio n s [3 ]
V + V2 = A.(u - a ) - CO2 ,
V=

(2.1)

Ú
u- a

(2 . 2 )

tn v h i a V, u a re fu n ctio n s of t; x, a, Cl) a re p a ra m e te r s .
A te r V b e in g e lim in a te d from (2.1), (2.2) it yields
ii
u - a

2Ú2

,

+ 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)

+ A cos(ot + (p),
CO

w here cp, A - in te g r a l c o n sta n ts.
Equation (2.3) can be a lte r n a te d into th e form:
d I . o\
4.2 „ \2 .o..2
—
(ú2) ----- -— ù 2 = -2Ằ.(u - a ) 2 + 2oýí(u
- a),
du

( 2 .8 )

u - a

t h a t yields
CO
Ũ2 = -y(u - a f (u - a f - — (u - a ) +
Y
Y

(2.9)

7 - integral co n stan t.
After solving (2.9) it can be found t h a t
CO
x. + pcos(cot + \\i)

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

(Ĩ.14

ay + 1

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 ) +

.

0 .5

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


4

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 + 2 y a 2
co2 + aX + a p COS cot

(2.16)

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.

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
0

(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

0 (co2 + aX + apcoscoxV

It is a s s u m e d t h a t w hen t = 0
u( 0) = u 0, ú( 0) = ú 0 .

(3.5)

Based on the in itia l condition (3.5), from (3.2) a n d (3.3), th e in te g ra l c o n stan ts
Cj, C 2 can be found:
c =

(^ + PK>
0)J + a > i + a p ’

c _ (co2 + q X + qp)ủ0
2

>. + (3

(3.6)

Hence, it can be concluded t h a t (3.2) is th e g e n e ra l so lution for (2.16).
Based on (3.2), (3.3) th e g ra p h s of th e function s u(t) a n d u(u) with different
p a ra 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 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

9.69

2

7.48

9.69

Uo

ú„

a y ,/2

Uo

Uo

a y l/2

0

0.813

3,46

0

0.813

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

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 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 + 2 a X + y a 2 > 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

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.

(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 4 sin cot

\

I ( c o s 0 ) t ) ----------------------------- —r ----------------------------------rr-

(4.4)

(aA. + aPcoscot) 3(co2 + aA. + a(5 coscot|

n which
A.
„ co X „
7-V + 3 —- — + 2 coscot +
p2
ap p

f(cos cot) = cos 3 cot - 3 —cos2 cot
p

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

(4.7)
in which
A2
vP

'

8+

V

0)2 '\
ap

1 -4

CO

(4.8)

ap

2\
(
(02 > Ằ (
Ằ
2
. CO
f
Ằ

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 -

- +

„
O)2
68 - - , or
ap ’

ap

p

(4.11)

—

8 - 5 Ụ + 68 - — < 2 — < -8 + J - ^ 2 + 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 = 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

a

(4 1 4 )

(02 + a X - a P

X. - p

W hen coscot = 1 , we h ave
2yco2
{ w

2 a y + 3Ả

CO2 + 3 a Ả. + 2 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

ai =

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,

CO

(4.16)

Based on (4.16), (4.17) it yields
k+a
CO

(4.18)

X

ap

p

, a

X?

1

2 -

p2

V
+ a
Ằ.

ap

1

2
^
A
2
CO
A.
(O
——+ — + 1 = ---- +

lap

p

pJ

ap

CO,

(4 .1 9 )

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

2

— can be found by

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

5. 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 = 0 ,

(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

8

• 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), ( 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(<-

CO= 2, k = - 0,00659, a , = 0,033415,
(4*19), th e re s u lts a re
A.

CO2

“ = 12, ^ - = - 3,361344538 .
p

(5.3)

(5.4)

a p

N th the set of p a r a m e te r s (5.4), condition (4.10) is satisfied .
?iom (2.11) a n d (5.4) it can be in ferred th a t:
a p = - 1,19; 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 ra p h s of h(t), g(t) can be p lo tte d as show n in Fig. 6 .
Fron olere, it can be show n t h a t the function h(t) h a s only a m ax im a and a m inim a
whin s n 2 t — 0 . The fu nction s h(t), g(t) hav e id e n tic a l v a lu e s of m axim a and
m inna which a re the a p p ro x im a tio n of each respective o th er. T herefore, it can be
c o n lidid th a t (3.2) with
th e conditions u(0) = u o,ú(0) = ủ 0 = 0 is th e approx im ated
s o l i t i > n o f

( 5 . 1 )

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 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 ,

'5.9)

into (4.18), (4.19), th e r e s u lts a re
^ = 8,25, - = - 0 , 5 3 5 5 9 4 6 7 2 .
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 = 7 ,4 8 ; a X = 6 1 ,7 1 ; a 2y = 9 3 8 , 0 4 .

(S.11)

Based on (5.9), (5.10) th e g ra p h s of h(t), g(t) can be p lo tte d as shown ii F,g 7
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 = 0. 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 0 =
0 s the
appro x im ated solu tio n of (5.1)

|
I

J

I

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

(02 +

+ aP

(0

ap

a..

2

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 0 X

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

——

— l ỉ l r . ------- .

61,71 + 7,48 COS 2 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).

6.

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 ra l Sciences.

R efe r e n c e s
1.

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.