flexuraltorsional coupled isik

8/9/2019 FlexuralTorsional Coupled ISIK

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FLEXURAL-TORSIONAL COUPLED VIBRATION ANALYSIS OF A THIN-WALLED CLOSED SECTION COMPOSITE TIMOSHENKO BEAM BY USING THEDIFFERENTIAL TRANSFORM METHOD

Metin O. Kaa an! "#$e "#!e%i&

Istanbul Technical University, Faculty of Aeronautics and Astronautics, 34469, Maslak,Istanbul, Turkey

A'(t&a)t. In this study, a new atheatical techni!ue called the "ifferential Transfor Method#"TM$ is introduced to analyse the free unda%ed vibration of an a&ially loaded, thin'walled closedsection co%osite Tioshenko bea includin( the aterial cou%lin( between the bendin( andtorsional odes of deforation, which is usually %resent in lainated co%osite beas due to %lyorientation) The %artial differential e!uations of otion are derived a%%lyin( the *ailton+s %rinci%leand solved usin( "TM) atural fre!uencies are calculated, related (ra%hics and the ode sha%es are

%lotted) The effects of the bendin('torsion cou%lin( and the a&ial force are investi(ated and the results

are co%ared with the studies in literature)

*. Int&+!,)ti+n

Figure 1.-onfi(uration of an a&iallyloaded co%osite Tioshenko bea

A co%osite thin'walled bea withlen(th L , cross sectional diension Band wall thickness d is shown in Fi().)The (eoetric diensions are assuedto be Bd


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The boundary conditions at /=x and Lx= for !s) #.$'#3$ are as follows

( ) /=+ kEI #4$

( )[ ] /=+ wwkAGwP #$

( )[ ] /=++ GJKPIs #6$A sinusoidal variation of $,# txw , $,# tx and $,# tx with a circular natural fre!uency

is assued and the functions are a%%ro&iated as

( ) ( ) tiexWtxw =, , ( ) ( ) tiextx =, , ( ) ( ) tiextx =, #5$

The followin( nondiensional %araeters can be used to si%lify the e!uations of otion

L

x= , ( )

L

WW = ,

0

0

AL

Ir = , ( )

( )d

d=

6

, ( ) ( )6.

L=

#7$

8ubstitutin( !s)#5$ and #7$ into !s)#4$'#$, the diensionless e!uations of otion are

obtained as follows

/6646

30

66

. =+++ AWAAA #9$

/6

30

66

. =++ BWBWB #./$

/66

30

66

. =++ CCC #..$

where the diensionless coefficients are

.. =A , EIkAGL

EI

rLA

0

0

04

0 =

EI

kAGLA

0

3 = EIK

A =4 kAGP

B = ..

#.0$

00

0

kAG

LB = .3 =B

GJ

PIC s=..

0

0

0 GJ

LIC s=

GJ

KC =3

/. T0e Di11e&entia T&an(1+&% Met0+!

The differential transfor ethod is a transforation techni!ue based on the Taylor seriese&%ansion and is used to obtain analytical solutions of the differential e!uations) In thisethod, certain transforation rules are a%%lied and the (overnin( differential e!uations and

the boundary conditions of the syste are transfored into a set of al(ebraic e!uations inters of the differential transfors of the ori(inal functions and the solution of theseal(ebraic e!uations (ives the desired solution of the %roble)

A function ( )xf , which is analytic in a doain ", can be re%resented by a %ower serieswith a center at /x , any %oint in ") The differential transfor of the function is (iven by

[ ]/

$#

9

.

xx

k

k

dx

xfd

kkF

=

= #.3$

where ( )xf is the ori(inal function and [ ]kF is the transfored function) The inverse

transforation is defined as


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[ ]

=

=/

/ $#$#k

kkFxxxf #.4$

-obinin( !s) #.3$ and #.4$ and e&%ressin( ( )xf by a finite series, we (et

= =

=

m

k xx

k

kk

dx

xfd

k

xx

xf /

/

/

$#

9

$#

$# #.$

*ere, the value of m de%ends on the conver(ence of the natural fre!uencies :.;) Theoresthat are fre!uently used in the transforation %rocedure are introduced in Table . andtheores that are used for boundary conditions are introduced in Table 0 :0;)Ta'e *.


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In the solution ste%, the differential transfor ethod is a%%lied to !s)#9$'#..$) *ere we!uit usin( the bar sybol on , W , and instead, we use , W , )

( )( ) [ ] [ ] ( ) [ ] ( )( ) [ ] /0.0..0.0430.

=+++++++++++ kkkAkWkAkAkkkA #.6$

( )( ) [ ] [ ] ( ) [ ] /..0.0 30. =+++++++ kkBkWBkWkkB #.5$

( )( ) [ ] [ ] ( )( ) [ ] /0.00.0 30. =++++++++ kkkCkCkkkC #.7$

A%%lyin( "TM to !s) #4$2#6$, the boundary conditions are (iven as follows

at /= [ ] [ ] [ ] //// === W #.9$

at .= ( ) [ ] ( ) [ ] /.... 4 =+++++ kkAkk #0/$

( ) [ ] [ ] /... =++ kkwkB #0.$

( ) [ ] ( ) [ ] /.... 3. =+++++ kkCkkC #00$

4. Re(,t( an! Di(),((i+n

In order to validate the co%uted results, an illustrative e&a%le, taken fro =ef :3;, issolved and the results are co%ared with the ones in the sae reference) Additionally, theode sha%es of the bea are %lotted)

>ariation of the first five natural fre!uencies #cou%led and uncou%led$ of the abovee&a%le with res%ect to the a&ial force is introduced in Table 3 and co%ared with the resultsof =ef) :3; and :4;) *ere, it is noticed that the natural fre!uencies decrease as the a&ial forcevaries fro tension ( )4)5=P to co%ression ( )4)5=P ) Additionally,it is seen that thecou%led natural fre!uencies are lower than the uncou%led ones) *owever, the fourth naturalfre!uency becoes less when the bendin('torsion cou%lin( is i(nored)

Ta'e/) atural fre!uencies with res%ect to the a&ial forceNat,&a F&e5,en)ie(

////0300)/0 =r

4)5=P /=P 4)5=P

P&e(ent Re1. 627 P&e(ent8 Re1.6/78 P&e(ent P&e(ent8 Re1.6/78 P&e(ent Re1.627 P&e(ent8 Re1.6/78

4/)95 4/)95 35)./6 35). 3)073 3/)545 3/)5 07)/64 07)/6 0.)975 0.)99

004)09 004)0 .95)650 .95)5 0.5)34. .79)559 .79)7 0./).60 0./).6 .7.)49 .7.)

97)667 97)66 0)66 0)6 90)606 .7)59. .7)7 76).9 76). ..)7.7 ..)9

645)9 645)9 647)49 647)6 645)4.. 647)069 647)3 645)007 645)00 647)/45 647..0)5. ..0)5. 990)757 ' ...9)7 976).99 ' ...3)9/ ...3)9 959)453 '

8atural fre!uencies with cou%lin(

The effects of the a&ial force P and the Tioshenko effect, r, on the first four naturalfre!uencies are introduced in Fi(s) 0#a'd$) ?hen Fi() 0 is e&ained, it is noticed that the naturalfre!uencies decrease with the increasin( rotary inertia %araeter, r, because Tioshenko effectdecreases the natural fre!uencies and this effect is ore doinant on hi(her odes as e&%ected)Additionally, since the fourth ode is torsion, the Tioshenko effect akes a sli(ht chan(e inthe fourth natural fre!uency value)


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- 8 . 0 0 - 4 . 0 0 0 . 0 0 4 . 0 0 8 . 0 0

2 0 . 0 0

2 4 . 0 0

2 8 . 0 0

3 2 . 0 0

3 6 . 0 0

4 0 . 0 0

- 8 . 0 0 - 4 . 0 0 0 . 0 0 4 . 0 0 8 . 0 0

1 8 0 . 0 0

1 8 4 . 0 0

1 8 8 . 0 0

1 9 2 . 0 0

1 9 6 . 0 0

2 0 0 . 0 0

- 8 . 0 0 - 4 . 0 0 0 . 0 0 4 . 0 0 8 . 0 0

5 0 8 . 0 0

5 1 2 . 0 0

5 1 6 . 0 0

5 2 0 . 0 0

5 2 4 . 0 0

5 2 8 . 0 0

- 8 . 0 0 - 4 . 0 0 0 . 0 0 4 . 0 0 8 . 0 0

6 4 8 . 0 0

6 4 8 . 1 0

6 4 8 . 2 0

6 4 8 . 3 0

6 4 8 . 4 0

6 4 8 . 5 0

Figure ".ffect of the Tioshenko effect on the first four natural fre!uencies # , Tioshenko 1, uler$

Mode sha%es of the considered bea under the effect of the co%ressive a&ial force #4)5=P $ are introduced with bendin('torsion cou%lin( in Fi(s) #a'd$) ?hen these fi(ures are

considered, it can be noticed that the first three noral odes are bendin( odes while thefourth noral ode is the fundaental torsion ode)

Fi&(tM+!eS0a=e(

Se)+n!M+!eS0a=e(

*(tNat,&a.F&e5,en)>H#?

-n!Nat,&a.F&e5,en)>H#?

/

&!Nat,&a.F&e5,en)>H#?

2t0Nat,&a.F&e5,en)>H#?

F+&)e >N? F+&)e >N?

F+&)e >N? F+&)e >N?

ww


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Fi$,&e 4.The first four noral ode sha%es of the co%osite bea with bendin('torsion cou%lin(

# ,1 , 1 ,$

Re1e&en)e(

8)*) *o and -)@) -hen, Analysis of eneral lastically nd =estrained on'Unifor ibration #In ress$)

E)=)

flexuraltorsional coupled isik

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