Variational Iteration Method for Vibration Problems
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Transcript of 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$erator, + , /Nu x t is a nonlinear ter# and + , /g x t is an inho#o%eneous ter#.
2
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Accordin% to &IM, the follo)in% iteration for#ula can be constructed.
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7
+ , / + , /
t
n n n n nu x t u x t Lu Ru N u g d + = + + +
+2/
)here is the %eneral 'a%ran%e #ulti$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$ressed as follo)s
+ , / + , / + , / + , /mu x t Ru x t Nu x t g x t
+ + =&&
+4/
The correction functional of 9*. +4/ can be )ritten as
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+ , / + , /
t
n n n n nu x t u x t mu Ru N u g d + = + + + +:/
Ma!in% the above correction functional stationary, and noticin% that +7/ 7u = , the
follo)in% iteration can be )ritten
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7
+ , / + , /
t
n n n n nu x t u x t mu Ru N u g d + = + + + +6a/
7+ / + / + / + / + / 7
t
n n t n t nu t m u m u m u d = = = + + = +6b/
)hich yields the follo)in% stationary conditions;
7m= +
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1ere the follo)in% for# of the L o$erator is used
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#$in% coefficient and m
is the #ass.
The correction functional of 9*. +-7/ can be )ritten as
-
7
+ , / + , /
t
n n n n nu x t u x t mu Ru N u g d + = + + + +--/
Ma!in% the above correction functional stationary, and noticin% that +7/ 7u = , the
follo)in% iteration can be )ritten
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7
+ , / + , /
t
n n n n nu x t u x t mu cu Ru N u g d + = + + + + +-2a/
+ / + / + / + /. + /n n t n t u t m u c m u = = = + + 7
+ / 7
t
nm c u d + = +-2b/
)hich yields the follo)in% stationary conditions
7m c = +-3a/
:
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- 7tm c = + = +-3b/
7t = = +-3c/
8o#binin% 9*s. +-3b/ and +-3c/, 9*s. +-3a/+-3c/ can be re)ritten as follo)s
7m c = +-4a/
- 7tm = = +-4b/
7t = = +-4c/
Therefore, in this case the 'a%ran%e #ulti$lier can be identified as follo)s
+ /- -
ct
mec
= +-:/
A$$ro5i#ate 'a%ran%e #ulti$lier can be obtained si#$ly by e5$andin% 9*. +-:/ as follo)s
2+ /
2 3
2 3
- - - + / + / + /
2> 3>
ct
m c c
e t t t c m m m
+ + +-6/
1ence,
22 3
2 3
-+ / + / + /
2> 3>
c ct t t
m m m
+ + +-
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1ere the follo)in% for# of the L o$erator is used
2
2L m k
t
= +
+-=/
8onsiderin% 9*. +-=/, 9*. +-/ can be e5$ressed as follo)s
+ , / + , / + , / + , / + , /mu x t ku x t Ru x t Nu x t g x t + + + =&& +-/
)hre k is the s$rin% coefficient.
The correction functional of 9*. +-/ can be )ritten as
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7
+ , / + , /
t
n n n n n nu x t u x t mu ku Ru N u g d + = + + + + +27/
Ma!in% the above correction functional stationary, and noticin% that +7/ 7u = , the
follo)in% iteration can be )ritten
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7
+ , / + , /
t
n n n n nu x t u x t mu ku Ru N u g d + = + + + + +2-a/
+ / + / + / + / + /n n t n t u t m u m u = = = + 7
+ / 7
t
nm k u d + + = +2-b/
)hich yields the follo)in% stationary conditions
7m k + = +22a/
- 7tm = = +22b/
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7t = = +22c/
Therefore, in this case the 'a%ran%e #ulti$lier can be identified as follo)s
[ ]-
sin + /tm
= +23/
)here the circular fre*uency, , is %iven by
k
m= . +24/
A$$ro5i#ate 'a%ran%e #ulti$lier can be obtained si#$ly by e5$andin% 9*. +23/ as follo)s
[ ]2 3- + / + /
sin + /3>
t tt
m m m
+2:/
1ence,
2 3+ / + /
3>
t t
m m
+26/
', +ase IV:
1ere the follo)in% for# of the L o$erator is used
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
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7
+ , / + , /
t
n n n n nu x t u x t mu cu ku Ru N u g d + = + + + + + +2/
Ma!in% the above correction functional stationary, and noticin% that +7/ 7u = , the
follo)in% iteration can be )ritten
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+ , / + , /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/
)hich yields the follo)in% stationary conditions
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/
Therefore, in this case the 'a%ran%e #ulti$lier can be identified as follo)s
+ /2
+ / c
tm
Sin t e
m
= +33/
)here
24
2
mk c
m
= +34/
A$$ro5i#ate 'a%ran%e #ulti$lier can be obtained si#$ly by e5$andin% 9*. +34/ as follo)s
2 2+ /
322 2
+ / + / + / -+ /
2 3>
ct
mSin t t c t c k
e tm m m m m m
= + +
+3:/
1ence,
2 23
2 2
+ / + / -+ /
2 3>
t c t c k t
m m m m m
+ +
+36/
, I%%"strat#-e E.a)*%es
-7
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E/AMPLE 1:
In this e5a#$le, a si#$le #asss$rin% syste# that under%oes forced vibration is e5a#ined.
The differential e*uation of #otion of this syste# is %iven by
+ /my ky f t + =&& +3
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Usin% 9*. +4-/ as an initial a$$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#