Variational Iteration Method for Vibration Problems

8/10/2019 Variational Iteration Method for Vibration Problems

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VARIATIONAL ITERATION METHOD FOR VIBRATION PROBLEMS

Metin O. Kaya

Istanbul Technical University, Faculty of Aeronautics and Astronautics, 3446, Masla!,

Istanbul, Tur!ey

Abstract

In this study, linear"nonlinear, free"forced and da#$ed"unda#$ed vibrations of both one

de%ree of freedo# and continous syste#s are discussed by usin% the &ariational Iteration

Method. Additionally, co##on vibration $roble#s are classified and 'a%ran%e #ulti$liers

are derived for each ty$e of $roble#.

Keywords: &ariational Iteration Method, &IM, &ibration, 'a%ran%e Multi$lier, (onlinear

&ibration

1 I!trod"ct#o!

&ibration of dyna#ical syste#s can be divided into t)o #ain classes li!e discreteand

distributed. The variables in discrete syste#s de$end on ti#e only, )hereas in distributed

syste#s such as bea#s, $lates etc. variables de$end on ti#e and s$ace. Therefore,

e*uations of #otion of discrete syste#s are described by ordinary differential e*uations,

)hile e*uations of #otion of distributed syste#s are described by $artial differential

e*uations + Meirovitch - /.

-


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A considerable a#ount of studies have been #ade in the area of vibration $roble#s.

0everal techni*ues, such as finite ele#ent #ethod, finite difference #ethod, $erturbation

techni*ues, series techni*ues, etc. have been used to handle vibration $roble#s.

The &ariational Iteration Method, &IM, )as first $ro$osed by 1e 2-- and the #ethod has

been a$$lied to investi%ate #any nonlinear $artial differential e*uations, autono#ous and

sin%ular ordinary differential e*uations such that solitary )ave solutions, rational solutions,

co#$acton solutions and other ty$es of solution )ere found by Abdou and 0oli#an -2,

-3. Additonally,1e 4, used &IM to solve linear"nonlinear vibration $roble#s.

In this study 1es studies are e5tended to cover vibration $roble#s )ith da#$in%, forced

vibration and vibration of bea#s. Therefore, here &IMis a$$lied to various vibration

$roble#s includin% vibration of linear"nonlinear, da#$ed"unda#$ed, free"forced vibrations

of one de%ree of freedo# syste#s and bea#s as an e5a#$le of continuous syste#s. The

$rocedure $resented in this $a$er can be si#$ly e5tended to solve #ore co#$le5 vibration

$roble#s such as aeroelasticity, rando# vibrations etc.

$ Var#at#o!a% Iterat#o! Met&od

In order to illustrate the basic conce$ts of &IM, the follo)in% nonlinear $artial differential

e*uation can be considered

+ , / + , / + , / + , /Lu x t Ru x t Nu x t g x t+ + = +-/

)here R is a linear o$erator )hich has $artial derivatives )ith res$ect to x , L is the linear

ti#e derivative o$erator, + , /Nu x t is a nonlinear ter# and + , /g x t is an inho#o%eneous ter#.

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Accordin% to &IM, the follo)in% iteration for#ula can be constructed.

-

7

+ , / + , /

t

n n n n nu x t u x t Lu Ru N u g d + = + + +

+2/

)here is the %eneral 'a%ran%e #ulti$lier )hich can be identified o$ti#ally via

variational theory, nRu and

nN u are considered as restricted variations, i.e. 7nRu = ,

7nN u = .

' D#((ere!t For) o( L O*erator

Accordin% to the L o$erator, a %eneral classification of vibration $roble#s can be #ade as

follo)s

'1 +ase I:

1ere the follo)in% for# of the L o$erator is used

2

2L m

t

=

+3/

8onsiderin% 9*. +3/, 9*. +-/ can be e5$ressed as follo)s

+ , / + , / + , / + , /mu x t Ru x t Nu x t g x t

+ + =&&

+4/

The correction functional of 9*. +4/ can be )ritten as

3


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-

7

+ , / + , /

t

n n n n nu x t u x t mu Ru N u g d + = + + + +:/

Ma!in% the above correction functional stationary, and noticin% that +7/ 7u = , the

follo)in% iteration can be )ritten

-

7

+ , / + , /

t

n n n n nu x t u x t mu Ru N u g d + = + + + +6a/

7+ / + / + / + / + / 7

t

n n t n t nu t m u m u m u d = = = + + = +6b/

)hich yields the follo)in% stationary conditions;

7m= +


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2

2L m c

t t

= +

+/

8onsiderin% 9*. +/, 9*. +-/ can be e5$ressed as follo)s

+ , / + , / + , / + , / + , /mu x t cu x t Ru x t Nu x t g x t + + + =&& & +-7/

1ere 9*. +-7/ denotes vibration )ith da#$in% )here c is the da#$in% coefficient and m

is the #ass.

The correction functional of 9*. +-7/ can be )ritten as

-

7

+ , / + , /

t

n n n n nu x t u x t mu Ru N u g d + = + + + +--/



-

7

+ , / + , /

t

n n n n nu x t u x t mu cu Ru N u g d + = + + + + +-2a/

+ / + / + / + /. + /n n t n t u t m u c m u = = = + + 7

+ / 7

t

nm c u d + = +-2b/

)hich yields the follo)in% stationary conditions

7m c = +-3a/

:


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- 7tm c = + = +-3b/

7t = = +-3c/

8o#binin% 9*s. +-3b/ and +-3c/, 9*s. +-3a/+-3c/ can be re)ritten as follo)s

7m c = +-4a/

- 7tm = = +-4b/

7t = = +-4c/

Therefore, in this case the 'a%ran%e #ulti$lier can be identified as follo)s

+ /- -

ct

mec

= +-:/

A$$ro5i#ate 'a%ran%e #ulti$lier can be obtained si#$ly by e5$andin% 9*. +-:/ as follo)s

2+ /

2 3

2 3

- - - + / + / + /

2> 3>

ct

m c c

e t t t c m m m

+ + +-6/

1ence,

22 3

2 3

-+ / + / + /

2> 3>

c ct t t

m m m

+ + +-


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2

2L m k

t

= +

+-=/

8onsiderin% 9*. +-=/, 9*. +-/ can be e5$ressed as follo)s

+ , / + , / + , / + , / + , /mu x t ku x t Ru x t Nu x t g x t + + + =&& +-/

)hre k is the s$rin% coefficient.

The correction functional of 9*. +-/ can be )ritten as

-

7

+ , / + , /

t

n n n n n nu x t u x t mu ku Ru N u g d + = + + + + +27/



-

7

+ , / + , /

t

n n n n nu x t u x t mu ku Ru N u g d + = + + + + +2-a/

+ / + / + / + / + /n n t n t u t m u m u = = = + 7

+ / 7

t

nm k u d + + = +2-b/


7m k + = +22a/

- 7tm = = +22b/


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7t = = +22c/


[ ]-

sin + /tm

= +23/

)here the circular fre*uency, , is %iven by

k

m= . +24/

A$$ro5i#ate 'a%ran%e #ulti$lier can be obtained si#$ly by e5$andin% 9*. +23/ as follo)s

[ ]2 3- + / + /

sin + /3>

t tt

m m m

+2:/

1ence,

2 3+ / + /

3>

t t

m m

+26/

', +ase IV:


2

2L m c k

t t

= + +

+2


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+ , / + , / + , / + , / + , / + , /mu x t cu x t ku x t Ru x t Nu x t g x t + + + + =&& & +2=/

The correction functional of 9*. +2=/ can be )ritten as

-

7

+ , / + , /

t

n n n n nu x t u x t mu cu ku Ru N u g d + = + + + + + +2/



-

7

+ , / + , /t

n n n n nu x t u x t mu cu ku Ru N u g d + = + + + + + +37a/

+ / + / + / + / + /n n t n t u t m u c m u = = = + + 7

+ / 7

t

nm c k u d + + = +37b/


7m c k + = +3-a/

- 7tm c = + = +3-b/

7t = = +3-c/

8o#binin% 9*s. +3-b/ and +3-c/, 9*s. +3-a/+3-c/ can be re)ritten as follo)s

7m c k + = +32a/

- 7tm = = +32b/


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7t = = +32c/


+ /2

+ / c

tm

Sin t e

m

= +33/

)here

24

2

mk c

m

= +34/

A$$ro5i#ate 'a%ran%e #ulti$lier can be obtained si#$ly by e5$andin% 9*. +34/ as follo)s

2 2+ /

322 2

+ / + / + / -+ /

2 3>

ct

mSin t t c t c k

e tm m m m m m

= + +

+3:/

1ence,

2 23

2 2

+ / + / -+ /

2 3>

t c t c k t

m m m m m

+ +

+36/

, I%%"strat#-e E.a)*%es

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E/AMPLE 1:

In this e5a#$le, a si#$le #asss$rin% syste# that under%oes forced vibration is e5a#ined.

The differential e*uation of #otion of this syste# is %iven by

+ /my ky f t + =&& +3


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Usin% 9*. +4-/ as an initial a$$ro5i#ation, )e %et

( ) ( ) [ ]- 77

-+ / cos sin sin + / cos+ /

t

y t A t B t t F d

= + +

+42/

or

( ) ( ) ( ) ( ) ( )

22

7

22

7

-

coscossincos

++=

tFtFtBtAty +43/

0ince the last ter# in 9*. +43/ auto#atically satisfies the co#$le#entary e*uation, this ter#

)ill not be used. Thus, 9*. +43/ can be si#$lified to

( ) ( ) 7- 2 2cos+ /

+ / cos sin F t

y t A t B t

= + +

+44/

)hich is the %eneral solution of 9*. +3


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( ) ( )- 77

+ / cos sin + / cos+ /

t

y t A t B t t F d = + + +4


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The e5$ression for ny can be )ritten as follo)s

( ) ( )

2 2 4 4 6 6

7 7 7

2 2 2 2

+ /

cos sin - 2 24


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( ) ( )2 2 2 2

- 7 7 72 4 2 4 2

- -cos sin + /

2

ty A t B t F F Cos t F

= + + + +

+::b/

and

( ) ( )2 4 6

2 7 2 4 6 =

-cos siny A t B t F

= + + + + +

( )

+

+

+

=

6

6

4

4

2

27

-cos

tF

+

+

+

+

2

66

7

4

2

2

44

7

6

4

4

2

2

22

7 -


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24

2

mk c

m

= +:/

1ence usin% this 'a%ran%e #ulti$lier, the iteration for#ula can be )ritten as

+ /2

-

7

-+ / + / + / + / + / + / + /

t ct

mn ny t y t e Sin t my cy ky f d

m

+ = + + + +67/

?ividin% both sides of 9*. +:


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Ta!in% this co#$le#entary solution as an initial a$$ro5i#ation, the follo)in% e5$ression is

obtained

+ /2

- 7 7

7

-+ / + / + / cos+ /

t ct

my t y t e Sin t F d

= + +6:/

Inte%ratin% the second ter# of 9*. +6:/, )e %et

2 2

- 7 4 2 2 2 4

+ / 2 + / + /

2+2 -/

Cos t Sin t y t y t

+ = +

+ +

2 2 2 2

4 2 2 2 4 4 2 2 2 4 2

+ / + /

2+2 -/ 2+2 -/ -

t tCos t e Sin t e

++ +

+ + + + +66/

0ince the last t)o ter#s of 9*. +66/ auto#atically satisfy the ho#o%eneous e*uation, they

)ill not be used. The second ter# of 9*. +66/ can be )ritten in a #ore co#$act for# as

follo)s

( ) ( )

2 2

4 2 2 2 4 2 22 2

+ / . 2 . + /

2+2 -/- " 2 "

Cos t Sin t Cos t

+ =

+ + + +6


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( ) ( )( ) ( )

7-

2 22 2

+ /cos sin

- " 2 "

t F Cos t y A t B t e

= + + +

+6/

E/AMPLE ':

In this e5a#$le, transverse vibration of a unifor# bea# )ith si#$ly su$$orted ends is

e5a#ined. The differential e*uation of #otion of this syste# is %iven by

2 42

2 47

y yc

t x

+ =

, 7 x L< < , t @ 7 +


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It is easily noticed that the 'a%ran%e #ulti$lier of this $roble# is

-+ /t

m

= +


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2 4 2 4 = 4 2 4 2

4 = 4+ / sin - + -/

2> 4> +2 />

n n nn

n n

x c t c t c ty t

L L L n L

= + +

L +


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+7, / 7y t = . t @ 7 +=-c/

6+-, / cos

y t t= , t @ 7 +=-d/

+7, / 7xxy t = , t @ 7 +=-e/

-+-, / cos

27xxy t t= , t @ 7 +=-f/

The 'a%ran%e #ulti$lier of this $roble# is

-+ /t

m =

and -m= for this $roble#. 1ence the iteration for#ula can be )ritten as

4 3

-

7

6+ / + / + / +- / + /cos

t

n n xxxxy t y t t y x y x x d + = + + + + +=2/

The co#$le#entary solution of this $roble# that is used as an initial a$$ro5i#ation is

%iven by

< 3 4

7

6+ / cos + /+- cos /

y t x t x x t= + + +=3/

Cy usin% the iteration e5$ression %iven by 9*. +=2/ and the initial a$$ro5i#ation %iven by

9*. +=3/, )e %et

< 3 4 2 3 3

-

6+ / cos + /+- cos / +- /+ 24 -2 + 24 /cos

y t x t x x t x t x x t= + + + + + + +=4a/

2-


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or in co#$act for#

Variational Iteration Method for Vibration Problems

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