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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.

(2)

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

(3)

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 a

simple 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

(4)

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 a

solid. 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

(5)

-'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)

(6)

where the small

.apiitude of

oscillation justifies inteating over the

undisturbed. 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

£9

iri.

/. Osrn(c-)

w1

L

ft)

WF = ghJ

Duth'

-:orizonta1 Force

The pressure difference between

pothts

at the same elevation on the

left hand arid right hand sides of

the

nk is

Q(=;-

[&}

(3)

The horizontal force exerted on the

tank is

'5; = !tL.=

[ø)

*

(6)

(7)

The naxixi.

force corresponds to sin (at

=1.0

(,\_ t()

for

(< 1/2,

tanh () ()

d

1wi

for

>

1/2, tanh ()

I and

(4Qn "

v1/'TrhJ

The above results

are

illustrated in Figure

1.

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. and

Le)

The above resui.ts

are

shown i Figure -,

Period

"Shallow

water"

perio

(o).

(12).

(-3)

"shallow watert' period

(15)

'deep water" period

(16)

p.

(8)

,".l V

T;-14t

tt

-njI-1;-

in:-: in:-:

,-11:L

4f

4i j

4 I T .

j

i:t r 1 -I L

i-ij-1!

4 ii": Ii i:i Ti -j j 4 t ' 1

4

:

I I fl}:fl

F

'4T

Tt

LT

:i1: TTT:T1 I'

;

i

T

t j

j4ji_

J

-11

j

//

+ 1

it::.'

"i

tL

!

1ii : F :[1l

:1T

i

0::

J4L

;et#M

L

i

:H

I

:

j

-1

:

tt L

TI14

Tii

LrJ

:4t

'4 , i

i

-. _-h

;ft1!1It

(i El

LFT : 4,

f-ft

' .t lii

'

:"

:I

4l

t't

-p

t 12'1çi 4 F I

i'

L!

' L r

4

1jL!.1!1 4:i:L'

EL _f1_[

: _J

i'f-

i/

, £tg k±

-#'

Ii

-p

4-,

-'*

:

ht:4fL

i)c3f'

LII F 1 T1 i11 fljL

t

' i: :

'--Tr

(

_L_

j

4

fnt;:i

r

'it;

tJ 'iL'

L

-_ L[4

i_1

]_

t_1 << -.

-r-a

_,' I -1 -, -.-I -ri I t

-'

1 i-i L__i±\ : I.., (1

:L t

TtTi

1 A

Lk

T1 I

ITIT

}f1fr

' I

*

4:

i:

I It I

--

I

1

rL

P ILt I I 'f

7'0

fl2'1

r91

L

MI11

i! A7

I

f'c/ç,

jrkt

-LJ

(9)

Location o± the

HorizontalForce

The 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 to

the maximum horizontal force) gives

= 2..9Ct

j'f COSh/1zf2tth2] dJ(

(.i8)

Inteation by parts gives

rco5h(A) _/-1

cash 7r74

7T

I

Dividing M by the

imum horizontal force from (8),

vith

sin (at +)

1.0, locates the force

at

= 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

(10)

Location of the Reu1tant Force

The maximum

resultant force passes

through the point defined by Eqs.

(20) and (2k) and

the

tangent Of

its

angle

of

inclination 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

triangles

and has the vertical

coordinate

TE

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

of

length L is

p =

2 iT

c-Ji (A)

-/17

The length of the equivalent

pendulum is then

2.

coh

(2z)

e

tei

If

0,

the maximum

angular displacement of the

pendulum 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

±s

9

(13)

(11)

The effective fuel weight We is then

the weight. of

the. fuel contained in a depth = tanh (f-). The remainder of the ±uei is

considered

as a

fixed weight.

Sunarizing the pendulun characteristics

(which

are also shown in

Figure 2)

Ii It tfl

For shallow ter (-p- smalJ.) rrie cbaracteistcs become

coh

frM

/

anh

(2)

/

ccsh ()

WF

COF

9 =

For "deep water" ( large) the charactaristics become

FR e =

(12)

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:-I

o4ii-

IP4WPOf&'M

; 1

H

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

--r

Li!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 iL

jI

I 4 t ft 1 I 4 I f -. I I -'14 I 1t i1 1 1 1 r 1 I

(13)

Horizontal 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

surface

an

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) particles

effc

dxt

(33)

(14)

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)

(15)

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,

Article

256

Lamb, "Iiyärodynamics", Sixth Edition, Article

259

6.

Smith, Charles C., Jr., "The Effects of Fel Sloshing on the

Lateral 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,

(16)

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

conditions

are 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.

(17)

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

iz

377

6'

.Lt

F'

g

(18)

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' / (Period

cio)

References for Frequencies If

CtI?.

L-

6

F

t::u'

agree with test data( to within

± -5% and ± 10% resPectively.

=

g = * Ref 1

tcnhl

ç7T/)

(5o4 /QR

=

,-_--- -

/

/

-

-acceleration due to gravity

k- --1

eirono

meiric functions * Ref 3 44J tcfnh(506 1T/t

iii Fiee surface a plane

=

Ref i 667

-1) Free surface assumed a plane

= /169 (-*)

A) Ref 5 5C0 360 I

-A) Larlace equaton satisfied

B) Fluid treoted as rigid body

I

- 0.

I

R)

R

fi

surface

i) Free surface assimied a plane

2 = 1.228

= °

(*

/ /6 3

(1.)

1?

2)

Ref.6

.565

351

507-A) Flow constructed from

B) Fiuidtre:tedasa

rigid body

2)E:ital

Cytaty

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