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3 ‘Ii76 00071 4973 A

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—.. —

NATIONAL ADVISORY COMMITTEE

FOR AERONAUTICS

TECHNICAL P40TIl

No. 1193

EFFECT OF PRODUCT OF INERTIA ON LATERAL STABILITY

By Leonard Sternfield

Langley Memorial Aeronautical Laboratory

Langley Field, Va.

Washington

March 1947
.

.*?

● ☛ ✎

A

of the

on the

NATIONALADVISORYCOMMITTEEFOR AERONAUTICS

TECHNICALNOTE NO, 1193

——..

~~E(jT ~~ ~~()~U~TOFJN~~IA ~ LAT~J&ST~I~~

By LeonardStem.fieM.

theoreticalinvestigationwas made to determinethe effect

product-of-inertiaterm in the lateral equationsof motion

I.ateral-stability boundaries. The productof inertia

resultsfrom the inclinationof the princi~allongitudinalaxis of

the airpianeto the flightpath,

The results of the calculationsindicatedthat the productof-ine~ttatermsshouldbe includedin the lateralequationsof

motion to determtnethe lateralstabiii,tyof an ai.rplanegThe

value of the directional-stabilityderivative Cn requiredfor

B *

stability,as d.eterrnined. from the calculationswhich includethe

productof inertiacausedby the inclinationof the principal

.

m

longitudinalaxis above the flightpath, is considerablyless than

the va?.ue pretlfctedby calculationsneglectziygthe productrofinertiaterms. LAW

?

1P ,pwfw d!.

INTRODUCTION

Recentflighttestsand testsin the Langleyfree-flighttunnel

indicateda discrepancybetweenthe observed.and calculatedconditions

for dynamiclateralstability. It was suggestedthat the discrepancy

might “bethe resultof neglectingterms jn the equationsof motion

involvingthe productof inertia,

The theoreticalinvestigationsof tinelateralstabilityof airylanes undertakenin the past have for the most part neglectedthe

effectof the inclinationof the principal.longitudinalaxis of the

airplaneto the flight path on tineOScilla:tGry-Stibility boundary.

(Seereferences1 to 3~) In the appendixof reference1, Zimmerman

mentionsthat the results of supplementarycalculationsimd.icated.

*4, that to neglectthe angularityof the principalaxis to the flight

yath did not seriousiyaffectthe oscillatory-stabilityboundaryin

the noml-flight range and gave slightlyconservativeresults. The ,* .

resultsof reference1, consequently,are based on lateralequations
.“.

*=. p

2 NACA ~ ?#O* 1193

of motionfrom which the effectof the incllnalxhnof the principal

axis to the flightyath are omitted,namely,the termsIncludingthe

productGf inertia.

For the rangeof airplaneparameters.investigatedby Zimmerman,

the product-of-inertiaterms probablyhave a smallstabilizing

effecton the oscillatory-stabilityboundary, However,the effect

of productof inertiaon the resultsOY stabilitycalculations

coveringthe range of’parametersof present-dayairplanesdesigned

for high-speed,high-altitudeflight- %hat is, high valuesof the

relative-densityfactor v, a largeincreasein the effectiveilihedralparameter ClB causedby the use of swept-backwings,and

the changein tke mass distributionof the airplane- has not been

hvestigatc?d.

In the presentinvestigation,calculationswere made to determine

the effectof the productiof inertiaon the requirementsfor lateral

stability. The resultsof the computationswere plottedas a function

of the directional-stabilityderivative C Ecndthe effective-

%

dihedralparameter Cl .

P

-v’

o

9

b

s

w

m

%

v

%

o

SYMBOLS

airspeedsfeet per second

mass densityof air, slugsper cubicfoot

dynamicpressure,

wing span,feet

wing area,square

poundsper squarefoot ‘L@

()

feet

wei@.tof atrpl.ane,pounds

mass, slugs (w/g)

accelerationof gravity,feet per secondper second

() relative-densityfactor .J’!L

\pSb

radiusof gyraticn aboutprincipallongitudinalaxis,feet
.*

NACA ‘TNNo. 11.93

w.

,

,* .

*

.%

.

. ,.

,

.% .

radiusof gyrationabout prtncipalnormaiaxis,feet

moment-of-inertia

axis

moment-of-inertia

moment-of-inertia

moment-of-in~rtia

path

prod.uct-of-iner tia

ml axisnorm?.

(

(

(

(

Coefficient

coefficient

coefficient

coefficient

aboutprincipallongitudinal

aboutprincipalnormalaxis

aboutflight-path

..

aboutaxis normal

coefficientwith respectto flightpath

to flightpath

Mf.t coefficient(J$ co. 7)

rolling-momentcoefficient@.CQLQ%+?..!i~

yawing-moment

lateral-force

angleof bank,

coefficient

(

Qteral forc~

.

\

qs )

radians

azimuthangle,radians

angle of sideslip,radians

yawingangularvelocity,radiansyer second (&~/dt)

ro~.lingangularvelocity,radiansper second (d~/dt)

effective-dihedralderivative,rate of changeof

rolling-moment coefficientwith angleof ~idesli~,

per~aaiy (%ia~)

directional-stabilitydorivativejrate of changeof

yawhg-manentcoefficientwith wgle of sidesl.ip,

per rad~an ~Cn/M)

3

,.
b

Cy

P

(2*

r

c1

P

Cn

P

c

2

r

CY

P

Cy

r

t

II

A

n

7

6

NA5ATN No. U93

.G

lateral-forcederivative.rate of clmuzeof lateral:forcet

c~eff’~cj,ent

fla.mping-in-yaw

coefficient

per radian

with ar@& of sideslip,-peradian (8CY13$

derivative;rate of changeof yawtng-moment

with yawing-angular-velocity factor,

dmping-in-rol.lderivatd.vejrate of changeof roll.ing-mnent

coefficientwith rolling-a@fla-V310cityfactor,

rate of changeof yawing-momentcoefficientwith rollingarq@ar-ve16cityfactor,yer radian (%//8$:)

rate of changeof rolling-momentcoefficientwith yawingw.guiar-veloci.ty factor,per radian (a@i*)

rate of changeof lateral-fcrcecoefficientwitlnrollingangular-velocityfactor,per radien’(M$*)

\

rate of changeof lateral-forcecoefficientwith yawing”

angular-velocityfactorsper r.adlan@cy/a#

WE> secmds

differentialoperator (d/dt~

angleof sw~epba.ck,degrees

angleof attackof ~rinci~~,ll.ongitudlna.l axis of airplane$

positivewhen principalaxis is abc~eflightpath,degrees

(seefig. 1]

angle of flightpath to horizontal,positivein a climb,

degrees (seefig. l)

apgle‘betweenrefereaceaxis end korfzmtal: positivewhen

referenceaxis is abcv3horizent.a.1,, degrees (seefig. 1)

-i. .“

,.

● ✎

✎ ✎

.

.

. .

,,

. .
TNNo. 1193 5

R

anglebetweenreferenceaxis end principalaxis,po@sitive

when

(see

RouthtS

referenceaxis is aboveprincipalaxis,degrees

fig. 1)

discrimfnant

EQUATIONSOF MOTICN

The linearj,zedeauat~ons of motion. referredto the axes in

figure1, used to cal~ulatethe spiral-’andoscillatory-stability

boundariesfor any flightcondition,are:

Rolling

‘%-

_.i.. --

.. 1.. :.-: .:;- \

.-~ —---- . . .. . . . ____ . . .- -pizo-xo)’ ‘ *] -Cl,, =o sin ~cos~D2+CZ6D ~

l.— —1

ILIzo COS2V + I

) I

.x sin2qD2-C D*

o ‘$

Sideslipping

_,~. “’. >.

~-I.I (

Z. ) ‘1 xsinqcos~D2+Cn, l)@-C~

0,

-. ‘$,’ ‘n@ ‘o

,

.
and

.

The symbols Ixa and

principalaxes of the

Inclineds% an angle

aboutthe fli~ht--qath

T

NACATN NO- 1193

=20 ra,presentthe moments@’ i~~ta aboutthe

.

airplane. If’the longitudinalprincipalaxts is

~ to the flightpath,the mommts of Znertla

cixis and the axis nomal to the flj.p$htpath are:

●✌

✎

.

‘.

,m.

8. ,.
J

NACA TN Noo 1193 7

. .

When ,f%oe‘t is substituted,for @, V~e~t for ~, ma poeht

-..

for p in the equationswrittenin detemimnt form, k must be a

root of the equation

where

.*
NACA ~ NO. 1193

The conditionsnecessaryto obtatnthe oscillatory-stability

bounda~ are that the coefficientsA, B, C, and E must be positive

aad Routh~sdiscrimi.nant,R = BCE - AE2 - B%’, must equal zero,

TJm spiral.-stability boundaryis fmndby setting F = O. The

completelystalleregionis thereforeboundedby the curves R = O

and l!’= O, whichare plottedas a functionof the directionalstabilityderivative Cn

P

and the effective-dihedralderivative Cz s

B

StabilityDerivativesand Mass Characteristics

Calculationswere made showingthe effectof the product-ofinertiatermson the oscillatory-s-taljility boundaryfor a hypothetical

supersonicfighterairplane,an experimentalfighterairplane,and

a model.flown in the Langleyfree-flighttunnel, The valuesof the

stabilityderl~ativesand mass characteristicsof the two airplanes

and the model are given in table1, The con-tributionof the tail

to the derivativesCn and CZr (seereference2) was included

P

in the characteristicsof the experimentalfighterairplaneand the

free-flight-tunnelmodelbut was neglectedin the calculationsof

the hypotheticalwpersonic fighterairplane.

RESULTSAND IXE3CUSS1ON

The results of the investigationare presentedin a seriesof

figureswhich show the oscillatory-and spiral-stabilityboundaries

as a htndbn of Cn and Cl . The solid Xlu O curve&’ figure 2

P P

representsthe oscillatory-stabilityboundaryfor landing?’Ilg!it.

for a bd~otheticalsupomonic fi~tor airp”lanewhich h~s itS

principalaxis or fuselageinclined~o above the flightpathbut

the productof inertia IXZ is assumedto he zero. The dashed

curve in this figureis the R = O boundaryfor the same condition

with the product-of-inertiatermsincludedin the equationeof

motion. A comparisonof both curvesshowsa large stabilizingshift

in the oscillatory-stabilityboundaryfor the case in which the product

of inertiais takenintoaccourh, As C7 is increased,the value

P

of Cn requiredfor oscillatorystabilityas determinedby the

P

calculationsincludlng Ixz is considerablyless than the value

ddermined by the calculationsneglecting In,. The spiralstabilityboundaryplotted.in figure2 and in subsequentfigures

appliesto both sets of’calculationssincethisboundaryis not a

●✌

✎

,’.

.’
NAdA ~NOC 1193 9

J.

,

functfcmof the productof inertia. This resultdoes not mean,

however,that the rate of divergenceor convergenceof the spiral

motionwill be the same inasmuchas the stabilityequationis

differentin each case. The effectof 1X2 on the R = O

boundaryfor the seineairplanein cruisingflightis presentedin

f’igjtq~e 3. The SO1.Mcurverepresentsthe case of the @.ncipal

longitudinalaxis inclined~o eitherabove or below the flightpath

but neglectsthe effectof Ixz. The other two R = O boundaries

thatwere calculatedincludedthe effectof productof inertia.

For the case in which the principalaxis is incllnedabove the

flightpath, 1X2 = -0.00178, the stableregionis increased;

ktnereasfor we case in which the principalaxis is inclinedbelow

the flightpath, IXZ = 0.00178, the stableregionis reduced.

The experimentalfighterairplaneand the free-flight-tunnel

model were testedin flightwith the principalaxis inclined16°

and 10°, respectively>above the flightpath. The 5.itialcalculationsmade on the assumptionthat In = O (thesolidcurvesin

figs. h and ~) indicatedthat the airplaneand model would be

unstablein flightfor -thecombinationof Cn

P

and Cl denotedby

s ,.

.. “

.

.“-

““

,-

the circled pointsin figures4 and ~. The flight-test”results,

however,showedthat the experimentalfighterairplanewas stable

and the free-flight-tunnelmodelwas marginallystable. Subsequent

calculationswhich includedthe product-of’-inertia termsIndicated

that the R = O boundaryfor the experimentalfighterairplane

increasedthe stableregionto includethe combinationof Cn

B

and CZ testedin flightand that the R = O boundaryfor the

B

free-fl~ght-tunnelmodel was shiftedvery close to the marginally

stabletest point.

In general,the inclinationof the principallongitudinalaxis

above the flightyath causesa stabilizingshift in the oscillatorystabilityboundarybut the extentof the shift”isa functionof

otherairplaneparameterswhich are stillto be investigated.

The’solidcurvesof figures2 to 5 may be consideredto

representthe R = O boundaryfor the actualflightconditionsof

an airplanewhich has its principallongitudinalaxis in line with

the flightpath (Ixz= O) providedthe wings are set at an angle

of incidenceto the fuselageto obtainthe lift coefficientdesired

for flight. The dashedcurveon each figurewould, therefore,

representthe R = O boundaryfor the same flightconditionsas the

solidcurvebut with the wings set at a differentangleof incidence
sincethe yrincipalaxis is inclinedto the

principalaxis is inclinedabove the flight

NACA TN No. 1193

fllghtpath. If the

path, the angle of

incideicewhich correspondsto the dashedcurveis smallerthan the

angle of incidencecorrespondingto the solidcurve. If the

prin.ci~alaxis is inclinedbelow the flightpathlhowever>the w~g

incidenceis largerthan the wing incidenceof the airplanein which

the prjncinalaxis coincideswith the flightpath. For e=plej

the solidcurveof figure2 W the R = O boundaryfor an airplane

with the wings set at an angleof 17° to the fuselageand the

princi~alaxis in linewith the flightpatii;whereasthe dashed

curve 5s the R = O boundaryfor the sameairplanewith the wings

set at a 10° angle of incidenceand the principalaxis inclined5°

shove tileflightpath, In figure3, the wing incidenceiS 7° for

the solidcurve,50 for the case ot the principalaxis above the

flightpath,amd 9° for the case of the principalaxisbelow the

flightpath, The solidcurvesof figures4 ends are the R =0

boundariesfor the atrpleneand model designedwith a wing incidence

of 1..6°and 10°, res~ecti.vely;for both cases the das~ ‘d cmve

representsan airplanewith a wing incidenceof 0°~ This interpretationappliedto the solidcurvesof figures4 and 5 is only

approximatelytrue sincethe angleof attackat the tail is zero if

the wing is at 16° or 10° incidenceend the principalaxis coincides

with the line of flight;whereasthe calculationsincludethe effect

of the tail at em angleof attackof 1.6° and 10° on the stability

derivativesC

%

and Ct ●

r

A comparisonof the solidemd dashedcurvesin eachfigure

clearlyindicatesthe increasein the oscillatory-stabilityregf.cm

for an airplanedesi.gnedwith a wing set at 0° incidence,thereby

necessitatingthe inclinationof the principalaxis above the

flightpath to otitainthe desiredlift coefficient.

CCNCLUD3N~REMARKS

The results.of the amlysi.smade to investigatethe effectof the

productof inertiaon the lateral~tabilitybowbries emphasl.zethe

necessityof includingthe product~f-inertiaterms in the lateral

equationsof motionto determinethe lateralstabilityof an airplane.

The calculationsIndicatethat the inclinationof the principal

longitudinalaxis abovethe flightpath causesa stabilizingshiftin

the oscillatory--stability boundary.

LangleyMemorialAeronauticalLaboratory

NationalAdviwxcyC?ommitteefor Aeronautics

&ngleyFfeld, Va.8 August13, 1946

,.

. .

.*.

.-:
. NAC,ATN NO. 3.193 .

n

.

J. REFERENCES

1. Zimmerman,CharlesH,: An Analysisof LateralStabilityin

Power-Off’Fl,ightwith Chartsfor Use in Design, NACA

Rep.No. 99, 1.937,

2. Bamher,MillardJ,: Effectof Some Present-DayAirplaneDesign

Wends on Requirementsfor LateralStability,NACATN No. ?314,

~gt$~,

3. Bryant,L. W., and.l?ugsley,A, G.? The LateralStabilityof

HighlyLoaded.Aeroplanes, R,& M*No* 1.840) BritishA.R.Cm,

1938●

..
12 NACA TN No. 1193 .

,

TABLWI

SIWILITY DERIVATIVESMD MASSCHUWTESISTICSUSED~ STABILITYCALCULATI~

Hypothetical euperamicExpe*talr~ghter a-laneFrea-fli@t-tunnel !ncdo

fighter airplane

condition Lending Oruimlng Landing Landing

?4/s,Ib/eqft 80 80 35 1.85

b,ft. 20 20 34 3&

0,slugs/cu ft 0.00238 0.0002 0.Ou.23a o.0023E

~,~~~~ec 264 1465 173 44

% 1 0.372 1 OA

P 54 620 13 6,33

kXO,ft 2.02 2.02 5,28 0.51

so,l-t 9.64 9.6b 8.03 1.29

CZP,perr~dlan -o.lsq’ -0.197 -0.3 -0.17

CZ,perraWm 0.25 0.0929 0.25-0.04c 0.142-0.o%zn

r ‘p(tail) B(tail)

~P,perrad~an -0.0198 -0.00732 -0.029-0.04C%(tail) -0.0406 -0.099c~

B(tail)

c~ ,Pm radian -1.47C -1.47C

r % (tail) ?3(tail)

‘O.086- 0.twl~ -0.0131- 1.2C

p(tall) %(m)

:Yp,k= radian o 0 0 0

Jyr,perradian o 0 0 0

+B, Per radian-1.33cn

p(tall) -

1.33c%(tai1) .oJ3 . 2a34~(tii1) ,-13.o~4- 1.76cnp(mi11

k

13gee:e~, -0.25 -0.25 -0.02 0

,,deg o 0 0 -9

, deg 60 63 35 42

NATIOMLADVISORY

cmQ1l’EEFORAEsommc9
>

* NACA TN No. 1193 Fig. 1

. %

,-

.

,J.,

/

+

z

\

\

\

NATIONAL ADVISORY

COMMITTEE FOR AERONAUTICS

“. .
0

f= o

-,0+ -.08 -./3 -./6

Stohle

/

1

-.20

-—. _

>4%2 Lr

RZO,omI/t!uwy-~+QbIII+yhowdary

~=OJ8pwu/-s+ibi!/fy boundary

-24 -.28 732

NATIONAL ADVISORY

Effec7t Ye - dihea(rul derl w+l ve ~ C2P CoNmmm mmuncs

:36

!2

3

●

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.
.

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+,,

b

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.,

0

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T

Q=L7

?=0

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——. . .

-.00}76 z

——— . (90/ 78 –2

Q=~ ascl[latoty-~ tibdjtyboum?iwy

D

}-af&+v

#

,rTiiTT ,T,T ~T +?=LJ

-.04

figure 3. – La#era/- st..b~l~ty bowxtorjes for cruJsIng fl~gh f for

‘ hypot~et}cal Supersonm fighter oqplme. w
●

✎

h

.

-.

.

,,. &

.
.4

.2

0

X5

f =0

J

i

42 T/6 720 :24 :32 :36

NATIONAL ADVISORY

Effec+l ve. dihedral der I vu +1ve ~ Czp CONMTTEE RIRAERDWWCS

F/gu re $. — La+erol - ~jabjlify boundaries for Iand/ng

fl~gh+ for free- fl}gh+- tunnel model.

2!

w1-.

yl

u-l