REPORT NO SM-137l48
T FORCES PRODUCED BY FUEL OscTr,1ION IN A RECTAIiC-ULAR TANK
By E. W Graham
r
- "1LJLJLJ.I
L_.I
April 13, 1950
Revised:
ApriJ. 16, 1951
ApDroved.:Douglas Aircraft Company, Inc.
SiMLRY
If large quantities of fuel are.carried. in an airplane, and. the fuel tanks are partially .full,' then appreciable forces may be exerted on the
airplane because of fuel oscillation in the tanks.
In stud.yiig the effect of fuel oscillations on airplane dynmc sta-bility it is convenient to replace the fuel b a simple pendulum plus a
. fixed. mass.
For the simple case of a stationary rectangular tank with. small
sinu-soidal fuel oscillations the pendulum characteristics are determined. The
fundamental frequency of the wave motion and the corresponding pressure distribution on the tank are obtained from LPrfQTS
tTffYOdCSf By
inteatjon, forces and moments are obtained so that the axis of the pendu lum can be located and the amplitude of oscillation determined for a given. pendulum mss.
Horizontal forces are also determined by direct use o± Newonts lai, and so related tthe time history Of the free surface shape. This
rela-tion is not restricted, to small amplitudes of wave morela-tion.
It is noted that in general the ouly free surface shapes which recur peiodiclly in rectangular tanks are the. simJ.e sine vav forms..
In Appendix 1 the fundamental frequencies and forces involved in
fluid. oscillation are. given for several simple tank shapes in.add.itiOn to
TRODU.CTIQIT
In tudying the effect of fuel "sloshing" on airplane dnamic stabiiity,
it
is convenient tb replace the fuel mass bya simple pendulum, or by asimple pendulum plus a fixed mass. In order to find the equivaleiit
pend.u-lum it is necessary toknow the period of the fuel motion, and the
maii-tude, dirctio and location of the resltt force.
If all of the fuel mass is assumed. to act as a pendulum then its angular
displacement will be email copezed to the angular dislacezent of the free
surface for the "deep watertt case. It may be convenient to associate a fraction of the fuel mass with the pendulum so that the angular motion of
the endulum approximates the angular motion of the free surface. The.
remajnder of the fuel mass is then considered to be fixed.
The simple ease of a rectangular tank with snall sirusoidal fuel.oscii-ltions is studied here.* Tancs of compie shape coui&be investigated, but the mathemtjcl difficultIes would be much eater,
It is also possible to study free surface shapes other than sinusoidal, but these are In general non-recurring shapes since they are cOosed of Fourier series terms each of rhich has a different time frequency. The
higher frequencies are not in enerl integral multiple's of the basic.
fre-quency.
In the study of large amplitudes of motion nOn-linearities are
intro-duced. These non-linearities occ'ür in the boundary conditions at the free
sface rather than in the basic partial differential eauation (which, is the .Laplae equation), but still add mathematical problems too great to be
considered here.
-*In Appendix 1 the oscillation frequenies and forces are given for
To be strictly accurate in repiesentin the fuel as a pendulum it would be necessy to consider an effective
omeit of
inertia Of the fuel about its center of avity. This effective moment of inertia depends. on the tank shape and is in eeral less than if the fuel rotated as asolid. Since
the
ratio of this fuel moment of inertia to airblare moment of inertia is ordinarilr much less than the ratio of fuel mass to airplane-'where
'vhere 0
The Velocity P6tential and Pressure
Consider a rectangular tank of unit -.iidth with small ambltude fuej.
oscillations which are idetical in all planes parallel to the x - y plane.
H
The velocity potential is1
-.
-ga c0512 iy
cas () cos&- -he)
-
a- csh)
L7t
-k = n-i an integer
g k tanh (k h)
a = maximuni amplitude 0±' the wave
g = acceleration due to avity
h depth of undisturbed iiquid
= length. of tank
x = horizontal. dispiacment y = vertical displacement
t=tinie
6 =
phase angle -.The pressure increment above, atmospheric at any point in 'the liquid,
(neglecting a terfl involving veloity squared) is1
(1)
where the small
.apiitude of
oscillation justifies inteating over theundisturbed. fitrid depth. .
-For the lowest freouency oscillation m = 1 d
cc3/
sin(t)
(5)
cash(Tii)
,&17x
/9&]x9 = za
-
cosh
47
civ (7)= 2g?Z 6,fl(Qt,'E
cos/i
(?Z-ettthg = tan Q', as illustrated
and fitrid
weight-£1 g
I
£9iri.
/. Osrn(c-)
w1
L
ft)
WF = ghJDuth'
-:orizonta1 ForceThe pressure difference between
pothts
at the same elevation on theleft hand arid right hand sides of
the
nk is
Q(=;-
[&}
(3)
The horizontal force exerted on the
tank is
'5; = !tL.=
[ø)
*
(6)
The naxixi.
force corresponds to sin (at=1.0
(,\_ t()
for
(< 1/2,
tanh () ()
d1wi
for
>
1/2, tanh ()
I and(4Qn "
v1/'TrhJ
The above results
are
illustrated in Figure1.
or
Period of the Oscillation
The
period
of the fun e'ta1 oscillation is=
=
rr/
cch(irA)
/
For
<
,coth
h) côh
'q
for ), cöth ()
'.1.and
1, -:-..
(ti1. andLe)
The above resui.ts
are
shown i Figure -,Period
"Shallow
water"
perio(o).
(12).
(-3)
"shallow watert' period
(15)
'deep water" period
(16)
p.
,".l V
T;-14t
tt
-njI-1;-in:-: in:-:
,-11:L4f
4i j
4 I T .j
i:t r 1 -I Li-ij-1!
4 ii": Ii i:i Ti -j j 4 t ' 14
:
I I fl}:flF
'4T
TtLT
:i1: TTT:T1 I';
iT
t jj4ji_
J-11
j
//
+ 1it::.'
"i
tL!
1ii : F :[1l:1T
i
0::
J4L
;et#M
Li
:H
I:
j
-1:
tt LTI14
Tii
LrJ:4t
'4 , ii
-. _-h;ft1!1It
(i El
LFT : 4,f-ft
' .t lii'
:"
:I
4lt't
-p
t 12'1çi 4 F Ii'
L!
' L r4
1jL!.1!1 4:i:L'EL _f1_[
: _Ji'f-
i/
, £tg k±-#'
Ii-p
4-,-'*
:
ht:4fL
i)c3f'
LII F 1 T1 i11 fljLt
' i: : '--Tr(
_L_j
4fnt;:i
r'it;
tJ 'iL'L
-_ L[4i_1
]_
t_1 << -.-r-a
_,' I -1 -, -.-I -ri I t-'
1 i-i L__i±\ : I.., (1:L t
TtTi
1 ALk
T1 IITIT
}f1fr
' I*
4:
i:
I It I --I
1rL
P ILt I I 'f7'0
fl2'1
r91
L
MI11
i! A7
If'c/ç,
jrkt
-LJ
Location o± the
HorizontalForceThe rment of the horizontal force about th
free surace.s
is
1' ,A/j17. - 2 c2 c.-w
a(-7-J
(23)
(21.) N/h=
1ii4L-
ydj
('7)
Substituting from Eq. (6)
vith sin (at E) = 1.0 (ici corresponds tothe maximum horizontal force) gives
= 2..9Ct
j'f COSh/1zf2tth2] dJ(
(.i8)
Inteation by parts gives
rco5h(A) _/-1
cash 7r74
7TI
Dividing M by the
imum horizontal force from (8),vith
sin (at +)
1.0, locates the forceat
= 4
flcash()
Jr
L
sin()
LocatiOn of
the Vertical
Force-The pressure on the bottom of the tank is, from Eq. (2)
e=eL$
AJ
The moment about the left-hand side of the tank sith s.n (at + ) = 1.0
f /-
çsi
/x dx
(22)Coh()
or
,vij
Location of the Reu1tant Force
The maximum
resultant force passesthrough the point defined by Eqs.
(20) and (2k) and
thetangent Of
itsangle
ofinclination to the vertical is
=
ZcLaflh('17)
(25)
The intersection of'the resultant force vith the vertical centerline
of the tank is found by cOmparing similar
trianglesand has the vertical
coordinateTE
iiuJ
9 _/
cash
.21
(26)
The axis of the equivalent simple pendulum is located at
x
=Chracteristcs of the Equivalent SimplePe
ndulum
The period of a simple
pendulim
oflength L is
p =
2 iTc-Ji (A)
-/17
The length of the equivalent
pendulum is then
2.
coh
(2z)
e
tei
If
0,the maximum
angular displacement of thependulum frog the vertical,
is chosen to be
=0' (the approximate free surfae maximum angular
displace-ment) then by reference to Eq. (10) the pendulum weight gust be
-
.Qnh (2)
(29)
P27t1t
'.
(27).
om Eq. (13) the period of the fdientai oscil1tion i the
tank
±s9
(13)
The effective fuel weight We is then
the weight. of
the. fuel contained in a depth = tanh (f-). The remainder of the ±uei isconsidered
as afixed weight.
Sunarizing the pendulun characteristics
(which
are also shown inFigure 2)
Ii It tfl
For shallow ter (-p- smalJ.) rrie cbaracteistcs become
coh
frM
/
anh
(2)
/
ccsh ()
WF
COF9 =
For "deep water" ( large) the charactaristics become
FR e =
L
LL. jJ1__I_
_
t:_iliI
.:::
H
1!L1 i!l:1_iI
_
PMJ/tUAf
/MR9C
i1[!I
/S*/CL
4III !1[tt
L'
L_ti
--j
èJ
. ii_ L -1:I
;L; 2 :: ::.. ; .-.T:-Io4ii-
IP4WPOf&'M
; 1H
t___I
--'_:ç_
r'4.Z .::
__1i'f
2i
-_:
.:
--4---
.-J
_ ---4 ---T --1. -l-H*
r ..:
:
-,-L
''
:'2D,
1 o rI L !'-J
. L --t-, LL 1'
..WX
Wioii
O Y : . :.H'1
We
:rJ r . I ,t:v:;',:.
. :::-
;'_._ r .''_ --r:' .:'. L :i't:1
I;: r:
_ TF -_-:..tr- .
I , I i:::
L
--rLi!ii
I
-: I t I r':_.
ujI-
-TETT--
:Iil1..
:1:
i:
t-t iT:r::::t:Ljt11,1it,,::
1 -TI fl - -, t * IhI ..,i i1 .LL4 ti:g; 11: t-:.
. I T I ::t 1 I: I I' lit iLjI
I 4 t ft 1 I 4 I f -. I I -'14 I 1t i1 1 1 1 r 1 IHorizontal Force Obtained Directly from Newton's Law
By smming the rate of change of horizontal momentum for all fluid
This can also be itten as
=
e/f
a'Y
.
/
ff
dx dy /
(34)
ef[dxdy.
I
or Fh is eGual. to the mass of the fluid
ltiplied b
the horizontal accel-eration of its center of avity.For an incompressible homogeneous fluid in a statiory tank the center
of avity position can vary as the shape of the free surface varies.
The relation between center of avity ositIoz aid free surface shape is
as follows.
Let = (x, t) define the
free
surfacean
let the horizontal velocity of the C.G. of the fluid = V G then,__
'V
--
e;
()
The horizontal acceleration of the C.G. i
a'
P
Lx
clx.aid Fh becomes
dx
This exDression is not
resticted
to small amolitudes of motion.For a simple harmonic motion of the free surface
= f(x) sn?
-,
-I=
and (.36) particleseffc
dxt
(33)
The fuel transferred frorn one side of the tank to t1e other. in a half
cycle is = 2g
dx
and the moment change is M.=2f
' a')(Then the CG. shift for this fuel quantity is
26 =
2LiXdX
2f"dX.
and the rnaxirnu horizotal force is given, by
Ph
This force corresponds to the centrifugal force
produced by rotatIng
1T on an of radius 5 at he angular velocity
o-(39)
(1!.o)
RE.L'iiHE NCES
11.
Lamb,
tEydrodyiamics",, Sixth Edition, Articles 227, 228
Lamb, ".Hydrodynam!cs, Sixth Edition,
Articles:190,257
Lamb, 'IIydrodynaniics", Sixth Edition, 'ArtIcles
191, 257'
Lamb, "Eydrodynm'ics", Sixth Ed±tion,
Article256
Lamb, "Iiyärodynamics", Sixth Edition, Article
259
6.
Smith, Charles C., Jr., "The Effects of Fel Sloshing on theLateral Stability of a Free-Flying Airplane 4od.e1," N.A.C.A. R.M.
#I8c16
Brotm, K., "Laboratory Test of Fuel Sloshing," Report No. Dev.
783,
APPEND DC 1
Fufldseta1 Frequencies and Forces for Liquids Oscillating Th Tanks of Simple Shape
A tabulation of fwidenta1 frequencies nd forces isiven for
several tanks of siniDle shape, most of the information being, obtained.
from Lams "Hydrodynamics." For the first three cases the velocity
potentials are known and the correct free surface forms, oscillation frequenies and forces are given.
Where velocity potentials satisfying all
the boünry
conditionsare riot available, the free surface is assumed. to be a plane. This
corresponds to neglecting the condition that a constant pressure must be maintained over the free surface at all times. For case IV a ve-locity potential is known for this odied prob3em, and the frequency
shown under IV-1--A is then determined, by eqiating the maximum ki±ietic
energy of the fluid to the maximum potential eiergy, and assuming
simie haoaic nioton.
For case V-i-A it is assumed that the hemisphere is divided into a number of half discs by vertical planes parallel to the direction..
of the oscillation This approximatioi permits the use of IV-l-A to
obtain V-i-A by irtegration. -
-When the liquid is replaced 'by a igid body the frequency can be obtained as before. by assumIng simple harmonic motion and equating
the maximum kinetic energy of the "pendulum" to its maximum potential
energy.
The maximum horizontal forces (Fh) pro.Uced. by fluid. osillations
are obtained, from Equation (li.0) after finding the horizontal position
Of the centroid of the transfrred liauid.. The formuias for forces
are tabulated in dimensionless form where W is
!.'.
ratzG'
the total weiit of fiu±d in the tank. The tan Q7 is taken as the ximum vertical displacement of the free surface from its static pos±
tion divided. by half the maximum length of the tank. This maximum vertical d±splacemént occurs at the tank wall. Ecuation (L.0) can be rewritten to give:
The quantity ( "k'' '
is obtained from the actual (or asuxied)
free surface- shape and. the total fltid. weight. Values are. tabü.lated
below for the fire cases:
Case I II III I\T V
(WY-
.2955R
2/z.. 41? 3)?WFtaa9Y
7Th
iz377
6'.Lt
F'
g
Fundamental Frequencies and Forces for Liquids
Oscillating in anks of Simple Shape
*potentjai flow soluti.o for small oscillatlois apProximate soluti on
N.
Configuration Remarks -Freciuency = a r' / (Periodcio)
References for Frequencies IfCtI?.
L-6
Ft::u'
agree with test data( to within
± -5% and ± 10% resPectively.
=
g = * Ref 1tcnhl
ç7T/)
(5o4 /QR
=
,-_--- -
/
/
-
-acceleration due to gravityk- --1
eirono
meiric functions * Ref 3 44J tcfnh(506 1T/tiii Fiee surface a plane
=
Ref i 667-1) Free surface assumed a plane
= /169 (-*)
A) Ref 5 5C0 360 I -A) Larlace equaton satisfiedB) Fluid treoted as rigid body
I
- 0.
IR)
R
fi
surfacei) Free surface assimied a plane
2 = 1.228
= °
(*
/ /6 3
(1.)
1?2)
Ref.6
.565
351507-A) Flow constructed from
B) Fiuidtre:tedasa
rigid body