Warner, E P

English

NASA Technical Reports Server

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No. 3 -
-.
-s cm THE TEEORYCF THE MXWEiROMETTLR.
.
‘by
8 ,.- 3. P. Warner,Chie5- Physicist,
AarodymmicalZdoratory, lf.A.C.A.
La@eg Field,Va.
,,
‘i?asllingtm,
.** 1%30.
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https://ntrs.nasa.gov/search.jsp?R=19930080810 2020-03-17T05:02:59+00:00Z
RYTESON TfiETi3N31Eu!?!1WEACkWXERGMETER.
By Edward P. Warner.
The accelerometerswhichha.e beenproposedfor use on
airplanOSfall intotwo classes. ‘&s first class,representedby
theRAE. instrumentand the me on whichthe staffof the Nat50n-
al AdvisoryCommi*tiee5s now working,has .ssi=gla deflecting
member. This deflectingXQembe= #4~r :n~s p.~~cirtiz gla:~-- --
the FLA.E.aooelerometer,a fIat steels-pringti the AdvisoryCom-
mittee~s. The secondclass,the oziir epresentativeof whick SMO%”
airplaaaaccaleromekers %s t- o= designedby Dr. Z@m, has a
nmber of plungershelduy againststopsby springsof varying
; JiWw@h cm W springsof the same stirengt> and lmded with vary-
i
“’idg weights. Iicreasa aocelerat~oncause8tinesprirtgsto leava
their stogs,one afterano*hsr,and to make mmtaot #.th a sheet
of paperwhiohmoves below them with a normal clearams of only a
few thousandthsof am inob. The s-pr~ngsin the Z&m ins+xunent
are-helical,bu% theymight ‘beflat withoutc&3n@ng the operation
of the inotrzmeat.Aa ace~lerometerof tkiskizxlhas ths a&/aMt-
age of a very quick responsato shocksOS briefduration,but it
makes a W3cogti.nuousraoord,givingindicationoxily of the values
between.which the aooeleration lay at q given instaxit.E there
are enoughspringsa closeapproximationto the mums of aooe3era-
tion can be Se.ved, ‘k% this is essent%ally inferior to an instru-
ment givinga oontinuousrecord directlyif any satZsfactmy re-
suitscan be scouredfrm suohafi inshx.nerz$.
The best
eter is to discuss
m4$-
wq of qproachtng the theoryof the accekxcL-
firstthe O-bvimsinherentErrorsaniithemeans
for reducingthem to a xnbim’tm ~ t~hm to take up the eqaatiom of
motiond deducafrom them the lag in reqotiing to S&O&, the
dampi~ of the sensitivemmker, azd the possibleerrors due ‘~
resonancewith the vibrationsof the atrplaraengize.
FL2t springaooelezozetersof ei%hsrthe firstor the see-
ond classare subjectto errors a-ising from acoeler=tionsalong
the axis& the s@.ng superimposedozztkosewhich it is intexided
to measureat rightanglesto the axis and from angularacoeh3ra-
tionsof tileairplane. Furthermore,all aocsiemmetsrsare af-
fectedazzdgiva inaccuratereadingsif they are not placedexactZy
a% the centerof grEwitFof the airp~ane(assumi~ Mat i% 5s t>
.
accelerationof the center of gravitywhioh is wante~and not, for
specialseason,that of some,otherpart of the machirs). The er-
rors from this cause,la turn,may be divide&intotwo groups.
First, if the bstmment is pl~oedin a planepassi~ throughthe
centerof gravit~and psrpendiculsr*O the axis alongw52iohzhe
acceleratiou is %0 be aeasured,but m% &t the Sea”- of gravi%y
itselx, any
to a l~ar
reoordsd as
measurement
e.ngukraooeleratim of the i&pku39 will mrse~ond
aocelaration of the accelerometerspr%ngand “wilibe
suoh. Second,M the aocelermete: lies on the axis of
any SW@aX veloc%tyabut either
will produoe a centeipe%d accelerationwhioh
the instrwnent.
Takingup thesesouraesof erro= in
of the otheraxes
will be recordedbF
ths order in which
they WGre firstmentim%l, the firs$ is the effectof linearac2el-
erationsalongthe axis of the spring. ti trsatirgthi8 it til be
assuned,tc startwltb,
straightand that it is
-4”
the% the defbct l.ngmsmbor is zxmnally
homogeneousin ma%rial, constmt iz sec-
tion,and suhnittedto en unvaryinglosd.
nearlytruefor the N.A.G.A. instrmmnt, a
from the truth for the R.A-E.type,but it
This asevmptionis very
considerabledistance
will be shownlater
that the distr5but50n of load s30ngthe springhas very littleef-
feet on the errordue to longitudinalacceleration.
In a fIat spring,2s?sw be the weight-pa=tit l.enb~k,
x the dist~ce from the free em of the spring,a& & the acoel-
emkion normal to the plane of the spzingin t-s of g (takingthe
‘accelerationHas includingthe compo~nt of gravitynorma3to the
plane of the spring, so that it correspondsto the totalair foroe
actingon the airplanea@ preheatingit from fallingwith an
accelerationg dowmwards). Then the bendhg momentdue to normal
aoceleratiozzis:
.
M&5$&_
Integrati.ngtwice,the defleotio?aat w point is fcmmito be:
AS regardsthe bendingdue to longitudinalaooelerations
of the spring,themomentarm of an e&nen% of masa at w poiat
with rsspaotto an axis of bandingat any otherpoint ie equalto
the
The
differerioe”betweenthe normaldeflectionsat the two points.
bendingmoment is then:
% ‘ J : %’”(’0- ‘)&’
whereq is the I.ongitudinalaccalerat+.on(including,my component
-5-
of gravit~actingalong*hat axis) 3% tsrmscf g. Subst5tu+ingthe
ex~ession givenabovefor y. and y,
nrop~ingthe
ing twicem
subscript,sinoe
have:
‘o
cm hava any
@x4 ~?
—-. -
435
vahs, and iz2tegrat-
4.-/68
r
SubstitutingP for x in the expreeeionsfor y= and yy, it appears
that the iiefle.ctionsof t-heftee aJIMof the springdue to aocelera-.
tions in the two Mrectlons are
The ratio & is a oonstant for slg
%x
to the staticdeflectioaof the free
of defections is then
72k. y=
70 $.p
given inst.ment,and is equal
end of the spring. The ratio
..
,twice:
-x? 42i +w.~(~+- \
= EI
The deflectionat the free ezxi,measured
2!J
&d
relativeto the base,is
Proce6dingin the’s.smemanneras for a distributedload,
.
.,=.!? g@.!x-3?Q’k}
EI 12 120 24
l?hedeflectionat the freeend, due to longitudinalac-
celeration=,ia
.W2G+5 2
%’0 = ~212 l E
Then
The errcr due iiolongitudinal acceleration
-.
is thereforealxmtl% greaterthan in the easeof a
Y5
Sn this case
uniformlyd.is’
tributedloading. These casesare the extremeant~theses of each
other and the true ;alueof the errorin eitherthe R,LE. or the
,
IV.A.C.A. instrument will lie soaewhere betweenthe two values
founl,as both theseaccelerometershave a tendencyto concentrate
the activemass near thefree end of the springs.
ql?~
atiuuof thatmath{ne whenpul?.i~out of a divemay b as great
as L92g. The conditionsassumedin this prob?.emwerermiulyse-
vere, and g may be tek,eaa themax.inmm8,00s1ezat iul alcmgthe
springaxis to whichthe accalerczceterwill be subnitted. la the
R.A.E.aocehmmeter en aecale~ation of thisme~” tudewould j?ro -
duce an errorof +5,43$ or -4.E@ $Z the deMminat5-on~f the nor-
aa.1acceleration,or a maximumerro;b02 .23g for the laygzS% noz-
mal aoceleration so faz-ra cedG2. ‘kid is ccu;smerablygreater
than the sensitivityof the instrument,and shows thatit is not
safe to raly on its iniicatiom to ~$ith~n 0.Ig.
In the li..LC.A.acoelsrometeras originallycissigned
(notyet triedout) /’ is 12.7 cm.,and b o is 0.006c+ The
maximumerrordue to a longitudinalaccelerationof g under these
conditionswcnfldbe 0.05%,the plv.saad ainas errors‘Miag~rac-
t ically
ation.
tory in
anglee
leoted
identioal. This is 0.90Zgfor thomeximunnormalacoeler- ,
1$ is evidentthat in this instmme.nt,if it is SatiSfac-
otherrespects,the errcrsdue to
tc the one to be measuredwill be
withFerfeot3afety.
accelerationsat right
small
Znstrunentssuchas the one designed
helicalsprings,are 02 courseq~fi.teftee from
enoughto be neg-
by u?. Zahm,u&ing
any errorsdue to
lozgitudind.asoelerations.
.
As has alreadybeennoted,the assumptionof a unifozm
distributionof weightalongthe sprizgdoes not aeoordciosely
@th thefacts. 7- it 1% ass=ed, as an alternative,thatall
the weightis concentratedat the freeend of tho ~rirlg,the hend-
ingmmwnt due to the normalaccelerationis:
>Sims any changein
of the Springgivesrise to a
Pial!b
defh=ticn perpetiicukr ta the plane
changein the deflectiondue to
forcesacting.parsllelto thatplans,thereis a secondary eZfect
WhichE0difi8SF=. If,for example,~ aots towardsthe free end
of the spring,so that y= aridyx are in the ssmedirection,~he
inoreaseof y due to the adait ion of y= will itselfproduosa
furtherincrease,and *he %ota.1~ffectof longitudinalacoelera-
tion will be greater than &ha%gives by the f‘tistapproximateion
written alxwe. H the two defleotionsare opposed,on the other
hard , the actualvalueof ~y w%U be less C= thatgivenby
the approximateformula. ~se effectscan be dlowed for by sub-
stituting y~, the total defleotio~ for y= in the aboveeqoatioa,
writing
In the glassfjbar imt~nt devisedani used at the
%yal AircraftEstablis&ent ~ is
.05 to .08om. For an acceleration
qring 3 would thereforebe .05.
1.3 om. and ~ o rangesfrom
of g alcmg the SXis of the
Zhe a,ccele20metermay be
—
&
plaoexiwith the qxis of the springcoincidingeitherwith the X
or the X-axisof the a5rplane. The accelerationsalongthe Y-axis
certainlyneverexceedg, whereasthe computationof the behav-
ior of a JM2 dur$nga loop* showsVIE% the longitudinaldeceler-
* Fcrcesin Dive and Loop:BulletinAirplaneEngineering
Department,U.S.A.,-hum, 1918.
-9-
Effectof .Angu.larAccelerations.
I
The effectof angulw accderationcapmarsin two ways.
In the firstplaca,the spring,no matterwhere it may be placed,
is affectedby the angularaccelera~ionas such. Secondly,if the
originof oo-ordinatesin the springdoes not coincidswith the .
centerof gravityof the airpbnc ailangularaccelerationabout
thb C.G.wili give a Enear acceleratimtc the zgriw.
The origin will be takenat the base of the springas a
first assmption, beingshiftedlaterto amore convenientend
logicallychosenlooation. An angularaccelerat~onof ~ radians
per sec.per sec.,the baseof the eprhg behg assunedto re-
main stationary,imposesupon every elementof lm@h A x a load
x.lr&.w.d x
t3
wher~w is the weightper unit length.
from the base is, integratingfrom the
the point in qyestion,
The shearat a distancex
free end of the spring to
Ga = sx x.~.w.ti - - ka.w ~ J?Z4 )J—--T-- - ‘—% 2
and the beadingmorcentis
X3
Y3 )-ka.Pi~.~z~ - -Ma=_
g -%- T 3
Integratingtwicemore,
Ya = ka.w
gEI
X5 $3=2 &=3 >
(.— +7 - —
120 12
!l!hedeflectionat the free end is then
-lo=-
The direoterror arisingfrom a~gul,arWC0~8ra%ionsis
thereforedi~ectlyproportionalto the lengthOY the spring,ani
the R.A-E.instrument,with itsvery shortspringwould seam to
have amsrked advantxe in thisparticular. It is, however,evi-
dbnt that ajudioious locationof ths originof co-grdinataswith
raspecttc the C.Q. of the airpl~ ‘AIZ intr@u@ titiaraec3l83-
ations,resultingfrom angular aooelerations,whichwill ooun-
terbalanoathe direct e~fectof the aooeleratedrotationalmotion.
k normalacceleratimreqpired toproduoe adeflec-
tioaequtvalentto thatproduoedbytha aqq2a.rsocslmationka
would be of the magnitude
where ~ $s e~ressed in terms03 ft. per sac.
thent.evidenthat, if the centerof gra’’ityof
in the pkrm of the s@ng &
therewill be no deflect%onof
angularaccelerations,tha tm
suohaccelerations.appeesjust
is COn0&ltrat8dat the tip Of
u
15 of its 2ength
par sea. It is
the airpl~ lies
from its base,
tha free end of the spring due to
mannarsin whioh the effeotsof
canoellingeaoh othar. If the waight
the 8Ering,insteadof being uni-
foz@v distztbuteda.lo~its whole length,the free end shouldob-
viouslybe at the C.G. Compromisingbetweenthe two conditicxls
it maybe saidthat,foz the accelerometersnow in use, the bcm-
tionof thamountingshou.ld~ such thatthe C.G. lies from75%
to 8* of the way out alongthe sprtng,
~ the C.G. is ~t at the poi~t thus defirsd%isre axe_
as has alresdy been pointedout, two posstblesorts of error. The
first of theseis the errordue to angularaccelerationwhen the
C.G. lies in the exis of the spriug. 3y properlychoosingthe or-
igin in the springthe directeffectof angukr accaleration can
be eliminated,and the totaleffeet can ‘bereducedto that of a
linearaccelerationgiven (in termsof g) by the expression:
wherad is the distancefrom the centerof ~r.avityof the airplane
to a pqint in the s-pringand 75~ of its lengthfrom the base.
The aaal.ysisof the ‘lcopproblerq~!alreadymentioned,
showedthat,under the conditions&sumed, the angularaccelera-
tion aboutthe Y-sxishas a maximm value of 10.!5radiaasper sec. .
.
per sec.at the instantwhea the elevatorwas pulledup, t-hatit
ESUS to 2.4 rad &ansper sm. per sec.in 0.3 second, and t’hat
it never risesabove1.0 rad. per sec.per sec.after0.9 sec.un-
til the loop is completed. The aseumptiolq~ada b this tiyeis,
that the elevatori’spulledup instantaneouslyis, of course,nuch “
too severe,and it is probablethat 6.5 zad.per sec.per sec. is
the largestaccelerationin pitch that an airplanewould ever
have to undergg. Experimentson the rollingmomentdue to ths
aileronssuggestthat the accelerationaboutthe X-axishas a
maximuavalue,on small and media-sized airplanes,of abo-at5
rd. per sec. per sec.
~ ~, ~, d dz be the projectionson the three axes
of the distancefrom tha originof co-ordinatesin the .$pri~to
W28F
%he aenterof gravityof the &%rpl&%, it is evident%hat them~i-
ma errordue to accelerationin roll is 0.15gwhen~ is 1 foot,
.
and thatthe similarerrorarisingfrom the pitchingmotionis
0.20gwhen% is Zfoot. Correctionsc=be made,usingestimat-
ed valuesfor the en@ar accelerations,which Gan be reliedwon
to reducethesaesrorsby abotit6@, Iaorder that the normal
acceleration,thus oorreoted,m~ be accuratewi.this0.05g,the
value of ~ Zntistbe 18SS than 7*O and ~ mot not ezuoedlW.
The other source:oferroris the centripstalaccelera-
tion dm to anguiarvelocity. In the usual case,wherethe ac-
celeration along the Z-axis are being measured,cantx3.?etaLac-
celerationsarisewheneves
tiozaif the activemass of
C.G. The accelerat%oais,
%=
The loopanalysisshoweda
there is ~ roUing or ~ltch~% mo-
ths imtruent Zs aboveor belowthe
of course,
iens per sec.?.ocourring0.6 eeo.~+terthe slevatorwas pdileduy.
Sincethe theoreticaltimefor c~pleting the 10CP was abouta
thiraless than actualmeasurementshowsto be sequired,thisms-
imum is about5% to~ high, and the trwemaximummay ‘betakenas
1.2 rad.per see. Sinoeit is reportad that an drpl~ Oan bS
rolledonto its badk in 3 sec., the maxirmmrollingvelocitymust
be about1.5 rad. per see. The ~rrorwhen ~ is 1 foot wouldthea
be o.o?g. As in the caseof angularaccelerations,ap~oximata
correctionscan be made,and the errorreducedby at least50~.
To keey the correcte&valueof the rsrmalaocelmatim withh 0.05 g
-13-
03 the truth,d= must not exceed16”,
realize,and it willusuallybe found
mounting an acoals20mete2
The dynem%c8 of
a subsequentreport.
is directZy
that tbe best plaoefor
abcwa or belowthe C9G.
will be treated in
,


Notes on the theory of the accelerometer

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