FLUTTER SPEED LIMITS OF SUBSONIC WINGS.doc

8/14/2019 FLUTTER SPEED LIMITS OF SUBSONIC WINGS.doc

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Journal of EngineeringVolume 18 February

2012Number

2

FLUTTER SPEED LIMITS OF SUBSONIC WINGSProf. Dr. Muhsin J. Jweeg Ass. Prof. Dr. Shokat Al-Tornachi Eng. Tariq Samir Talib

College of EngineeringAl- ahrain !ni"ersit#

College of Engineering!ni"ersit# of Technolog#

College of Engineering!ni"ersit# of Technolog#

BSTRACT$lutter is a %henomenon resulting from the interaction between aeronamic an& structural

namic forces an& ma# lea& to a &estructi"e instabilit#. The aeronamic forces on an oscillatingairfoil combination of two in&e%en&ent &egrees of free&om ha"e been &etermine&. The %roblemresol"es itself into the solution of certain &efinite integrals' which ha"e been i&entifie& as Theo&orsen

functions. The theor#' being base& on %otential flow an& the (utta con&ition' is fun&amentall#equi"alent to the con"entional wing-section theor# relating to the stea case. The mechanism ofaeronamic instabilit# has been anal#)e& in &etail. An e*act solution' in"ol"ing %otential flow an&the a&o%tion of the (utta con&ition' has been anal#)e& in &etail. The solution is of a sim%le form an&is e*%resse& b# means of an au*iliar# %arameter (. The use of finite element mo&eling technique an&unstea aeronamic mo&eling with the +-, metho& for flutter s%ee& %re&iction was use& on a firectangular an& ta%ere& wing to &etermine the flutter s%ee& boun&aries. To buil& the wing the Ans

. %rogram was use& an& the e*tract "alues were substitute& in the Matlab %rogram which is &esigto &etermine the flutter s%ee& an& then %re&icte& the "arious effects on flutter s%ee&. The %rogramga"e us a%%ro*imatel# i&entical results to the results of the referre& researches. The following wing&esign %arameters were in"estigate& skin shell thickness' material %ro%erties' cross section area for beams' an& changing altitu&e. /esults of these calculations in&icate that structural mo&e sha%e"ariation %la#s a significant role in the &etermination of wing flutter boun&ar#.

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$!TTE/ SPEED M TS $ S!S C ,SProf. Dr. Muhsin J. JweegAss. Prof. Dr. Shokat Al-TornachiEng. Tariq Samir Talib

INTRODUCTIONThe %roblem of oscillating airfoils has

been an im%ortant subect of unstea

aeronamics because of its close link withflutter anal#sis. The sustaine& oscillation is a boun&ar# between con"ergent an& &i"ergentmotions. ence' the s%ee& thus obtaine& is thecritical s%ee&' abo"e which flutter occurs.

&S(de6h%3 7889: &e"elo%e& a co&e forthe com%utation of three-&imensionalaeroelastic %roblems such as wing flutter.&B(*('r%sh ( 7889: n"estigate& the initialmathematical theor# of aeroelasticit# centere&on the canonical %roblem of the flutter

boun&ar# instabilit# en&emic to aircraft thatlimits attainable s%ee& in the subsonic regime.&M(ss%-o B%( #h% 7889: Stu&ie& ametho&olog# to merge state-s%ace time &omainreali)ations of a com%lete numericalaeroser"oelastic mo&el with flight mechanicsequations

UNSTEAD. AEROD.NAMIC FORCESOF T;E T.PICAL SECTION MODEL25

The unstea aeronamic forces arecalculate& base& on the lineari)e& thin -airfoil . n this section' Theo&orsens a%%roachwill be summari)e& an& the flutter anal#sis will be con&ucte& base& on his a%%roach&TheodoreTheodorse


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2012Number

2

.++= h NC

.

.

?

8 xa x

b xbh xbV ++= 8?

# ernoulli theorem' the %ressure is obtaine&as follows

.

==

+

=

t xV

t p

8 ?

An& the force 8%ositi"e &ownwar&? an& the %itching moment 8%ositi"e nose-u%? about theelastic a*is will be e*%resse& as-

dxb pdxb F NC

==.

+

.....

baV hb

8 ?

a xt bbdxa x pb M NC ?8?8

==

+++

..

.

...

abV hbahV b

8?

The Circulator# $low- To satisf# the (utta con&ition'Theo&orsen em%lo#s a boun& "orte*&istribution o"er the airfoil an& a "orte* o"er

the airfoil wake.n or&er to consi&er wake' assume a boun&

"orte* 8 = dx ? at 8 o X ?' an& a she&"orte* 8 ? at 8 o X ?.Then' the "elocit# %otential &ue to "orte* is

= oo X X

Y

X X

Y

tantan

+++=

?8?8

tan Y X X X X

Y X X

oo

oo

Define 8 ooo x X X =+ ?' an& 8' x y x X == ?

Then' += ooo x x X

=+

= oooo

o x x x x

X

The "elocit# %otential can be e*%resse& as

++ = ?8?8

?8tan

x x x x

x x

o

o

=

o

o

xx x

x x

tan

8?

here ' x o x

t is to be note& that the "orte* is mo"ingawa# from the airfoil with "elocit# of 8+?.Therefore' b# ernoulli theorem' the %ressure&ue to the "orte* is

+

= x

V t

p

here

=

o

o

xx

x x

x xtan

x x x

x

o

o

=

x x x x

x ooo =

The %ressure at 8? &ue to the "orte* at 8 o x ?is

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x x x x

x

xV p

oo

o

+

=

x x x x

x xV

oo

o

=

8?

+ =

o

o

x x

x xV

The force on the whole airfoil &ue to a "orte*at ?8 o x will be

= dx pb F

= o

o

o x x

xVb '

The total force can be calculate& b#

integrating with res%ect to ?8 o x

= F F

o

o

o

dx x

xVb

= Similarl#'

= '?8 o xdxa x pb M

= M M

+

+=

dx

x

xa

x

xVb

o

o

o

o

8 ?

t has to be note& that the force an& momentare functions of "orte* strength ?8 .#a%%l#ing (utta con&ition at trailing e&ge the"orte* strength can be &etermine&. The total

"elocit# %otential is..

+++=

htotal

..

?

8 xa x

b xbh xbV +++= # a%%l#ing the (utta con&ition' the followingequation is obtaine&

.

.

?8?8

xb x

xbh

x

xbV x

++

+

=

+

.

??8

8

x

xa xb

$inite. At 8* ? Therefore'

?

8..

=+

=

abbhbV x

x x

Since

x x x x

x o =

The following e*%ression is obtaine& from theabo"e equation.

=

=

o

o

x x

x

x x

oo

o dx x xb

=

++= abbhbV

..

Define

QabhV dx

x

xo

o

o

++=

..

Then' the total force an& moment on the airfoilwill be as follows

= ooo dx

x

xVb F

+=

oo

o

o

o

o

dx x

x

dx x

x

VbQ

+

+=

dx

x

xa

x

xVb M

o

o

o

o

VbCQ = 166


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2012Number

2

+=

aC QVb

8 ?here 8c? is the Theo&orsen function' an& is

&efine& as

+=

o

o

o

o

o

o

dx x

x

dx x

x

C

Assume that the airfoil has a sim%le harmonicmotion

[ ]

++

== oo kxt io

xb s

k io ee

t V

bk

==

here Vt s =

Then' Theo&orsen function is e*%resse& as

+

=

o

ikx

o

o

oiikxt i

o

o

o

dxe x

x

dxeee x

x

C o

o

+=

oikx

o

o

oikx

o

o

dxe x

x

dxe x

x

o

o

8 ?

The Theo&orsen function is frequentl# re%lace& b# sim%le algebraic a%%ro*imation as follows-

ik ik k C ++++= .

--. --.

-.-.?8

8 ?The total force an& moment resulting from thenoncirculator# an& circulator# flows aree*%resse& as

?8.....

k VbQC baV hb F +=

..

... abaVbhbab M +

+

=

here

++= abhV Q

..

f a quasi-stea aeronamic is assume& 8The

aeronamic characteristics of an airfoil whosemotion consists of "ariable linear an& angularmotions are equal' at an# instant of time' to thecharacteristics of the same airfoil mo"ing withconstant linear an& angular "elocities equal toactual instantaneous "alues.?' then C 8k? becomes 8 ?' an& the force an& moment will be

+++= abhV VbbaV hb F QS

.......

+

+

= aVbabaVbhbab M QS

..

...

8 ?

. $lutter Equation f The T#%ical SectionMo&el.Equation of Motion- Consi&er the t#%ical section shown inF%60&9:0

The mo&el has a translation s%ring withstiffness ?8 hk an& torsion s%ring' with stiffness

?8 k . These s%rings are attache& to the airfoilat the shear center. Therefore' it is two &egreesof free&om mo&el ?'8 h . An& 8h? is measure&at the shear center 8elastic a*is?.The &ownwar& &is%lacement of an# other %ointon the airfoil is

xh z +=

here 8*? is a &istance measure& from theshear center.The strain energ# an& the kinetic energ# areres%ecti"el# gi"en b#

h ! ! " h +=

dx z .

=

where 8? is the mass %er unite length ofthe airfoil.

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++= dx x xdxhdxh

....

Define the following.

Mass ?8 = dx# The secon& moment of inertia of the airfoilabout shear center'

== #$ dx x % The first moment of inertia of the airfoil aboutshear center'

== #x xdxS

here ?8 $ is the ra&ius of g#ration an& ?8 x is a &istance from the coor&inate to the masscenter.Then' the kinetic energ# can be written asfollows

....

% h#xh# ++=

The "irtual work &ue to the unsteaaeronamic forces is

{ } +=+== QhQdx xh p zdx p& h

here the force ?8 hQ is %ositi"e &ownwar&an& moment ?8 Q is %ositi"e nose-u%.agranges equations %ro"i&e the equation ofmotion of the airfoil.

( ) ( )'Q'

"

'

" dt d =

.

==+# (

QQh

! ! h

#$#x

#x# h

h

..

..

where 'h' =

=+

..

..

#b M

#b F bh

$

bh

$

x

x

h

where.'''

b$ $ b x x % ! # ! hh ====

The harmonic motions 8t i

o ehh

= ?an& 8

t io e

= ? assume& the equations ofmotion will be-

168


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2012Number

2

=+

M

# F bh

$

bh

x

x

x

h

The unstea aeronamic force an& momentare

++

+=

..

.....

?8

abhV k VbC

baV hb F

+

= i

k C

k ia

k C i

bh

b

++= hh La L Lb

hb

here the re&uce frequenc# is 8 V b

k =

?' an&

'k

C i Lh =

k

C

k

C i L +=

Similarl#'

(

++

+= hh La M b

h La M b M

here

k i M M h

'

==

Then' the equation of motion can be rewrittenas

+

bh

$

bh

$

x

x

h

+++

+

+

+

=

a M La M

La L

La M

L

h

h

hh

h

?8

8 ?

where the mass ratio is &efine& as 8b

#

=

?' 8m? is the airfoil mass %er unit length.

Define 8

=

? an& 8

h ) =

?' then

+

bh

$

)bh

$

x

x

+++

+

+

+

=

a M La M

La L

La M

L

h

h

hh

h

?8

45G MET;OD FOR FLUTTERANAL.SIS25The abo"e flutter equation is e*%resse& in thefollowing matri* form.

[ ] [ ]+=

bh M *

bh ! i+i+i+

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here 8 i+ ! ? is the stiffness matri*' 8 i+ M ?mass matri*' an& 8 i+ * ? is the aeronamicmatri*. ote that the aeronamic is functionof the re&uce& frequenc#' 8k?.

+-g metho& assumes first the artificialstructure &am%ing' 8g?.

i+i+ ! i, ! ?8 +=

This artificial &am%ing in&icates the require&&am%ing for the harmonic motion. Theeigen"alue of the equation of motionre%resents a %oint on the flutter boun&ar# if thecorres%on&ing "alue of 8g? equals the assume&"alue of 8g?.$or a gi"en re&uce& frequenc#' 8

V b

k = ? will

be a com%le* eigen"alue %roblem.

( )[ ] [ ]+=+

bh M *

bh !

i, i+i+i+

8 ?

The Eigen "alue is-

+= i,

$rom this eigen"alue

/e

i=

/e

m

= ,

ANS.S ANAL.SIS OF WINGMODEL25

The wing mo&el anal#sis in the Ans#s %rogram is b# using the suitable element forthe work. The 8Shell ? ma# be use& for skinan& the s%ar web an& the 8eam ? 8D elastic

beam? is use& for the stiffeners in the isotro%iccase

FLUTTER PROGRAM2 5The combination between the 8A SS

. ? an& the 8MATA .? is em%lo#e&. The %rogram is sol"e& b# using the Theo&orsens

theor# with "elocit# &am%ing 8+-g? metho&.The in%uts of %rogram for the wings mo&el are. $rom 8A SS . ? the natural

frequencies are taken.. The static unbalance' frequenc# ratio'

mass ratio' ra&ius of g#ration an& non-&imensional location of airfoil elastica*is.

. Densit# of air at an# altitu&e.

An& the out%uts of %rogram are

- The ben&ing an& torsional mo&e sha%esfor both rectangular an& ta%ere& wingsas shown in $igures a to &.

- The relation between the non-&imensional %arameter 8 k? withstructural &am%ing.

- Calculation of the flutter s%ee&.

RESULTS AND DISCUSSIONS -

RESULTS OF COMPARISON2 5

# using anal#tical an& numericalsolution for the case where 8' r . ' * .' a -.? it is foun& that the results in thework are a%%ro*imatel# equal to the results inreferences as shown table .The following %arameter are to be in"estigate&

E!!e#$ o! $he Ch( 6% 6 W% 6 S?%Th%#? ess The shell thickness is one of the mainim%ortant "ariables in the wing &esigntherefore the effect of "ariation thickness from8. m? to 8. m? was stu&ie& in there&uce& frequenc#' flutter s%ee& an& mass fortwo t#%es of wing 8rectangular wing an&straight-ta%ere& wing?.

Re#$( 6"*(r W% 6T(@*e &7: shows the shell thickness

effects on the "ibration mo&es. $or theconfiguration 8* ? with area 8A mm? an&

thickness 8. m? the first two naturalfrequencies are equal to 8. ' .?' with mass 8. kg?'F%6s0 &/: an&& :

170


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2012Number

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show the re&uce& frequenc# 8k

b

V ( .?

an& frequenc# ratio 8

( . ?. $or these

"alues 8f . ' + f . msec?. utwhen the thickness increases to 8. m? thefirst two natural frequencies are equal to8 . ' .? with mass 8. kg?'F%6s0 &


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the flutter s%ee& is greater with high "alue ofmass. The %ercentage between 8TiA +? an&8 -T? is equal to 8 .?' an& 8TiA +?'8A&".Aluminum? is equal to 8 .?. The

%ercentages &iffer because of the wing mass&ifference.

TAPERED WING$romT(@*e & :'an& when using the

8a&". Aluminum? in the wing &esign withchanging thickness' it is seen that forconfiguration 8* ? with thickness equal to8.m? the first two natural frequencies areequal to 8 .' .? with mass8 . kg?'F%6s0 &9


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is sufficientl# large 8+? where structure&am%ing 8g? changes its sign fromnegati"e to %ositi"e.

The higher wing as%ect ratio &ecreases

the flutter s%ee&' while the increasingof the ta%er ratio increases the flutters%ee&. The flutter s%ee& changeslinearl# with the altitu&e an& it isincrease& with increasing the altitu&e.$lutter %re"ention can be summari)e& b# a&&ing mass or re&istribute mass sothat 8* ? mass balance' increasestorsional stiffness i.e. increase 8?'

ncreasing or &ecreasing 8

h ? if it is

near one 8for fi*e& ?' a&&ing &am%ingto the structure an& require the aircraftto be flown below its critical machnumber.

REFERENCES25

ala (rishnan A.+. 8?' Towar& aMathematical Theor# of Aero elasticit#$light S#stems /esearch Center.!CA.

Darrol Stinton. 8 ?' The Design ofthe Aero%lane. s%. Professional ooks*for&' Englan&.

Massimo ianchin' ,iuse%%euaranta#' Paolo Mantega))a. 8?'State S%ace /e&uce& r&er Mo&els forStatic Aeroelasticit# an& $lightMechanics of $le*ible Aircrafts.De%artment of Engineering' Milano.

tal#.

Mechanical an& Electrical S#stems8 ?. %eration Manual' oeingCommercial Air%lane Com%an#.

Sa&eghi M.' ang# S.' iu) $.' Tsai*. M. 8?' Parallel Com%utation ofing $lutter with a Cou%le& a"ier-StokesCSD Metho&. A AA .

Theo&ore Theo&orsen. 8 ?' ,eneralTheor# of Aeronamic nstabilit# an&

the Mechanism of $lutter. ACA/e%ort o. .

!e&a T. an& Dowell E. . 8 ?'

$lutter Anal#sis !sing onlinearAeronamic $orce. J. of Aircraft'+ol. ' o..

ang T. . 8 ?' $lutter Anal#sis ofa ACA A Airfoil in SmallDisturbance Transonic $low. J. ofAircraft' +ol. .

S+-@o*s25

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FLUTTER SPEED LIMITS OF SUBSONIC WINGS.doc

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