FLUTTER SPEED LIMITS OF SUBSONIC WINGS.doc
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8/14/2019 FLUTTER SPEED LIMITS OF SUBSONIC WINGS.doc
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Journal of EngineeringVolume 18 February
2012Number
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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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.++= 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# $low- To satisf# the (utta con&ition'Theo&orsen em%lo#s a boun& "orte*&istribution o"er the airfoil an& a "orte* o"er
the airfoil wake.n or&er to consi&er wake' assume a boun&
"orte* 8 = dx ? at 8 o X ?' an& a she&"orte* 8 ? at 8 o X ?.Then' the "elocit# %otential &ue to "orte* is
= oo X X
Y
X X
Y
tantan
+++=
?8?8
tan Y X X X X
Y X X
oo
oo
Define 8 ooo x X X =+ ?' an& 8' x y x X == ?
Then' += ooo x x X
=+
= oooo
o x x x x
X
The "elocit# %otential can be e*%resse& as
++ = ?8?8
?8tan
x x x x
x x
o
o
=
o
o
xx x
x x
tan
8?
here ' x o x
t is to be note& that the "orte* is mo"ingawa# from the airfoil with "elocit# of 8+?.Therefore' b# ernoulli theorem' the %ressure&ue to the "orte* is
+
= x
V t
p
here
=
o
o
xx
x x
x xtan
x x x
x
o
o
=
x x x x
x ooo =
The %ressure at 8? &ue to the "orte* at 8 o x ?is
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$!TTE/ SPEED M TS $ S!S C ,SProf. Dr. Muhsin J. JweegAss. Prof. Dr. Shokat Al-TornachiEng. Tariq Samir Talib
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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+=
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 ?
. $lutter Equation f The T#%ical SectionMo&el.Equation of Motion- Consi&er the t#%ical section shown inF%60&9:0
The mo&el has a translation s%ring withstiffness ?8 hk an& torsion s%ring' with stiffness
?8 k . These s%rings are attache& to the airfoilat the shear center. Therefore' it is two &egreesof free&om mo&el ?'8 h . An& 8h? is measure&at the shear center 8elastic a*is?.The &ownwar& &is%lacement of an# other %ointon the airfoil is
xh z +=
here 8*? is a &istance measure& from theshear center.The strain energ# an& the kinetic energ# areres%ecti"el# gi"en b#
h ! ! " h +=
dx z .
=
where 8? is the mass %er unite length ofthe airfoil.
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$!TTE/ SPEED M TS $ S!S C ,SProf. Dr. Muhsin J. JweegAss. Prof. Dr. Shokat Al-TornachiEng. Tariq Samir Talib
++= 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-
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=+
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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$!TTE/ SPEED M TS $ S!S C ,SProf. Dr. Muhsin J. JweegAss. Prof. Dr. Shokat Al-TornachiEng. Tariq Samir Talib
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&& :
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show the re&uce& frequenc# 8k
b
V ( .?
an& frequenc# ratio 8
( . ?. $or these
"alues 8f . ' + f . msec?. utwhen the thickness increases to 8. m? thefirst two natural frequencies are equal to8 . ' .? with mass 8. kg?'F%6s0 &
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$!TTE/ SPEED M TS $ S!S C ,SProf. Dr. Muhsin J. JweegAss. Prof. Dr. Shokat Al-TornachiEng. Tariq Samir Talib
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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