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A simplified compound pendulum method for determining the moments of inertia of an airplane about the X- and Y-axes. The method involves oscillating the airplane as a compound pendulum and suspending it as a bifilar torsional pendulum to obtain the moment of inertia about the Z-axis. The document also discusses the importance of accurate measurement of moments of inertia for airplane stability and control studies.
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NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
i 4 ; 4 (P
No. 1629
By William Gracey
Langley Memorial Aeronautical Laboratory Langley Field, Va.
=qljm’Jpj
Washington
June 1948
AHWE
T13NRM!ML L!ERARY
AFL 2811
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TECH LIBRARY KAFB. NM
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FOR AERolwTIcs
TEE Exl?ER~ DETERMNATIOl? OF THE MOMERTS
OF IMERTIA OF AIRPLANES BY A SIMPLIFIED
COMPOUND-PENDULUM KE!J!HOD
By William Gracey
A simplified compound-pendulummethod for the experimmtal determination of the mmnts of inertia of airpknes a%out the X– and Y-axes is described. The method is developed as a nblification of the stsndard pendulum EMthod reported previously (NACA Rep. No. 467). ‘ A brief review of the older method is included to form a basis for discussion of the simplified mthod.
The simplified method eliminates the necessity for determining the cente=f+gravi ty location of the airplane and the suspension length by direct mmaurement. The suspension length (and hence, the vertical location of the center of gravity of the airplane) is found from the swinging experiments by determining the period of oscil- lation for two suspensions, measuring the difference between the two suspension lengths, and solving the equations for the two suspensions simultaneously for one of the suspnsion lengths. The mmkmt of inertia of the airplane is then computed in accordance with the stiard procedure.
The moments of inertia of an airplane and of a simple body were determined by both the standard and the simplified nethods. The results of these tests show that the precision of the data obtained by the two Bthods is very nesxly equal.
The several advantages which can be realized in the application of the new mthod are discussed. The hazardous aspects of this type of test, for example, are to a large extent eliminated because of the fact that the complete test program can be conducted with the airplane in a level attitude. In addition, the expmhmrtal technique,“test apparatus, snd ti- requim”d to perfozm the tests are reduced. Because of these advantages, the possible application of the rmthod to the testing of large airplanes is noted.
teetlng the type of airplane (biplanes end parasol nmoplanes ) in use at the t!m the method was developed. With^ the advent of^ low-wing monoplanes, however, the application of the plumb-line auspenston method becam increasingly difficult and, as a consequence, the pre– cislon of the experiments decreased appreciably. In an effort to overcome these difficulties the NACA has developed the procedure suggested by the British into a complete and valid -thod by Ming full account of the various factors (buoyancy, entrapped air, and ambient air) which must be considered for oscillations occurring in an air wdium. This mthod has not only proved satisfactory for testing low-wing monoplanes but has also provided a much simpler procedtie which can be advsntageousl.yapplied to all types of airplsnes.
The need for another method for the experimmtal detertination of the nmnents of inertia of airplanes has been accentuated recently in connection with stability and control studies of large airplanes and heavy missiles. The purpose of thLs paper is to present the simplified pendulum method as a possible solution to ‘Ais problem.
Al
D
d.
T
T
sYMBo&’
weight of airplane
weight of swinging gear
weight of pendulum (w + w’)
distance from axis of oscillation airplane (suspension length)
distance from sxis of oscillation swinging gear
distance from axis of oscillation pendulum (pendulum length)
difference between two suspension
length of bifilars
distance between blfilars
period of oscillation
total volume of airplane
volume of ai”rplanestructure
to center of gravity of
to center of gravity of
to center of gravity of
lengths
_ _... .— -..—---^ ___^..^. ...—^ ..——^ -..-—^ —^ ——^ .. —-^ .-~^ -
P
%
MA
IV
1A
lCL
, density of air
acceleration 0? gravity
additional mass
virtual moment of
addltlon&l mxnent
mment of inmtia
nmpen t of inertia
Subscripts:
EJxp experimental
Calc (^) calculated .
inertia of airplane
of
of
of
inertia
miw gear about axis of rotation
steel bar about midpoint
In accordance with the proced~e outlined in reference 3 the moments of inertia of an airplane are determined aboub the three body
axes: ~ly, the X-axis, parallel to the thrust axis in the plane of symmetry, the Y-axis, perpendicular to the plsne of s-try, and the
Z-ads, perpendicular to the thrust line in the plane of’symmetry. The moments of inertia about the X- and Y-axes me obtained by oscill- ating the airplane as a compound pendulum; whereas the moment of
inertia alout the Z-axis is obtained by suspending the airplane as a bifilar torsional pendulum. (^) For the X- and Y+xxms, the axis of oscil- lation Is parallel to the body axis; for the ‘Z-axis,the axis of rotation snd the body axis sre coincident.
Because of the practical difficulty of finding suitable att.aclumnt points on the airplane structure for suspending the airplane during the stinging expwiments, it has been found necessary to employ a rigid
supporting apparatus, general~ termed the “swinging gear .“ When used as a compound pendulum, the swinging gear consists of a rectangular framework suspended from two knife edges by a system of tie rods (figs. 1 and 2). The arrangement of Lie rods is nmdified in the case of the torsional pendulum by the addition of two vertical rods with unl-fersaljoints at the lower ends. (^) A rigid spacer ‘rod is mounted between the two universal joints in order to mintain the same dishance between the ~ertical rods (bifilars) when the pendulum is oscillating (fig. 3). The ~mmts of inertia of the swinging gear ere determined experimentally by swinging the gesr as an hdepenflen-tpendulum; the center of gravity of the gesr is foun~ by computation.
.— ——^ ...—
6 NACA TN No. (^1629) L
After the center of gravity has been _ at two different suspension lengths The virtual mment of inertia the following equations:
located, the airplane is for each of the three ems. Iv is calculated in each case from
for the bifilar torsional pendulum, and
(1)
for the compound psndulum.
Because the tests me conducted in air, the weight of the airplane which must be considered as contributing to the restoring mment of the pendulum is the virtual weight, that is, the true (or vacuum) weight less the buoyancy of the structure. As the quantity which is detemlned when the airplane is weighed in air is also the virtual weight, the weighing results can be applied directly in the preceding equations.
In transferring the mwent of inertia from the axis of rotation to the body exis, however, the true mass of t@ airplane must be considered. The true mass of the airplane was shown in refemmwe 3 to consist of two items: the mass of the airplsne structure end the mass of the air entrapped within the structure. The true mass is obtained by correctiti the virtual mai3s w/g for the effect of buoyancy and adding the mass of entrapped air; thus,
where Vs is the volume of the stiucture and V is the total volum
of the airpkne. The quantity ~ + Vp , therefore, represents the
true &ss of the airplane.
b
i
—. —^ —z ———-.^ —
.
.
.
The additional+na8s factor MA) which mst also be taken Into account in transferring the moment of inertia to the airplane axis, is computed from a consideration of the projected area of that pat of the airplane normal to the mtion of the pendulum. Details of the procedure emplopd in these computations My be found in references 3 and 7. For the X-axis, the pro~ected area includes the side area of the fuselage end the vertical tail surfaces. In the case of the Y-axis, the frontal area of th3 airplane is ordinarily so small.that the additional=mass correction for this axfs cm he neglected. The center of the additional mass is as-d to coincide with the center of gravity of the airplane; for this reason the suspension length of the additional mass is the same as that of the eirplane.
The virtual nment of inertia about the Z-exis is found inmB– diately upm substitution of the pendulum characteristics in equation (1). In the case of tlm X- and Y-sxes, Vp and MA me first calculated snd Iv is determined by substitution of these values in equation (2). A^ check computation is then made by^ solvi equations for the two suspensions simultaneously, IV and ~~? MA) being the unbowns. Swinging the airplane at two suspensions,. therefore, not only provides a reasure of ths prscision of the experi- ments but is also useful as a means of checking the computed values of tti q~tity (Vp + MA).
It will be seen from equation (2) that the characteristics which must be evaluated for dbtemninfhg the moments of inertia about the X- and Y-sxes are the weight, the suspension length, the period of oscillation, and the quantity (Vp + ~~. The weight of the airplane can be mmsured very accurately without tifficulty. Similarly, by taking the man of ~ or mme oscillations, the period can be detemined with good precision. Furthermore, if reasonable care is exercised in computing the airplane volume and projected areas, sufficiently accurate values of (VP + MA) can ordinarily be obtained. Actually, relatively large inaccuracies can be tolerated in evaluating this Item, because the mgnitude of the combined effects of the entrapped and anibientair is small in relation to the measured mmnt of inertia. It was shown in reference 3, for example, that en error of as mch as 10 ~rcent in the computation of the mass of the entrapped air and the additional mass contributes an error of only O.8 percent in the mmnent of inertia about the X-axis and only O.3 percent in that about the Y-axis. These estimates were based on the type of airplane in existence during the early 1930Ss. F’or modem, mre dense airplanes, the effects of the entrapped air and the additional mass will represent an even smaller percentage of the final results.
In contrast to the other three items, the measurement of the suspension length, that is, the distance between the axis of oscil- lation and the center of gravity of the airplane, is both difficult
..-——... - .———-~— .—. —..-—— — -- .. —.—.__—
NACA TN No. (^1629 )
directly and with good accuracy. These advantages, couyled with the fact that no transposition of axes is necessary in the case of the torsional pendulum, account for the higher precision ordinarily o%tained for the moments of inertia about the Z-axis.
The development of the simplified compound-pendulum directly on the test procedure described in reference ?.
method is based Silmlly stated. t the method consists ~^ determining the period bf oscfition^ ~o~ two^ ‘ suspensions, measuring the clifference between the two suspension lengths, and solv@g the eq~tions of the two suspensions simultaneously for one of the suspension lengths. (^) The solution of these equations determines the vertical location of the center of gravity of the airplane immediately. The tirtual moment of inertia is then found by inserting the suspension length in the appropriate originel equation and proceeding with the compu- tations in the manner outlined in reference 3.
The eqmtion required for the solution of the suspension length is derived by the application of eqpation (2). (^) When the airpl@e is tested at two suspension lengths, the equations for the two suspensions become
where the subscripts S emd L refer to the short and long suspensions, respectively. (^).
From the principle of moments, the pendulum length may be expressed in terms of the moments of the airplane snd of the swinging gear about the axis of rotation; thus,
W-L+ W’z’ L. w
. — (^). —— __ -—-.. __ (^) -.—— _ —.——. ..—. ..—
—.——. —.— (^).
w substitution of equation (6) in equations (4) and (5) @elds .
( )
W2L -1-W’LZ’T TT IVL = ( )^ -%
~+ Vp+MAtT 41f
From the relation ~ = 2s + A2 (Where^ A2^ is the difference between the two Suspmmion lengths), equation (8) may be expressed as
The mmnt of inertia of the airplane about its course, the same for both long and short suspensions
body axis iS, of so that IVL = ITS.
The suspension length for the short suspension can, therefore, be found by solving equations (7) and (9) simultaneously. The solution of these
.
From the value of 2S ‘fOUnd m this ~:, ths pendululnlength msy be calculated from equation (6) and ths virtual moment of inertia detemimd by the solution of equation (4). .,
Although a lmowledge of the longitudinal location of the airplane center of gravi@ is not required for calculating the moments of inertia, the determination of tlds location prior to the swinging experiments is
l
advisable. This ~asurement can be made with sufficient accuracy by
._. —.,— —— -- ———^ —-
12 NACA TN No. 1629
comparison, extre= care was exercised in locating the center of gravity of the airplane and in masuring the suspension length. In addition to the measuremmts required for the application of the standhrd meth~d, the distance fioma point on the wing to a reference mint directly above waE found for each suspension. Values of AZ were obtained as the difference between two such measure~nts.
The computations employed for these tests are given in the appendix. The results of these computations are sumarlzed in table I.
These results show the computed values of the suspension length to check the measured values to within 0.011 foot or slightly more than 1/8 inch. The precision of the standard method, as shown by the agree~nt between the two values obtained by this method, is regarded aa unusually good for this type of airplane. The precision of the simplified method, as based on the deviations of the test results from the man value obtained with the standard mthod, is seen to be almst the same as that of the standard mthod.
. (^) In spite of the good agreemnt in the results of the airplane tests, it was felt that the two methods should be compared independently against a third staniard. Swinging tests were therefore conducted with a solid steel bar, the mmmt of inertia of which could be accurately calculated. These tests differed from the airplane tests in that the center of gravity did not have to be determined experimentally, the suspam ion length could be mmsured directly, and the quanti- ties (Vp )
=MA ti 1A could be neglected. The dimnsions of the . bar chosen for the tests were 1~ inches by k inches by 18 feet . 9* incws; the Wight was 423.3 pounds. Although the mass of the bar was small compared to that of an airplane, the suspension lengths and periods were of the same order SE those of the usual airplarm test. The moment of inertia of the bar about its center llne ICL as determined in each case is presented in table II.
The computed values of the suspension length are shown to agree with the reasured values withig 0.006, 0 .O@, and O .007 foot (less than 1/8 inch in each case). The precision of the virtual nmmmts of inertia, as defined by the deviations from the computed value, is of the same order for both standard and simplified methods.
The precision with which the moments of inertia about the airplane axes can be found depends on three items: (^) (1) the precision of the nmasured mmmnt of inertia about the axis of oscillation, (2) the precision of the evaluation of the entrapped air end the additional.
4
.
— — (^). .. ___ —.—
.
l
.
.
mEV3sin and (3)
tranqwsing the compound-pendulum results to the airplane axes, the precision in the computation of the additional nrxmentof inertia. The relative magnitude of the precision of these items for each of the airplane axes was estimated in reference 3. On the basis of this analysis, the ove~ precision of the true mments of inertia was shown to be 22.5 percent for the X-axis, 21.3 percent for the Y-axis, and ~.8 prcent for the -s.
The sum of the precision of the first two preceding items defines the precision of the virtual mment of inertia of the airplane about its axis. The yecision of the tirtual moments of inertia obtained by the standerd mthod was estimated in reference 3 to be less than q percent for the X- and Y-ems. This^ estimate^ of^ precision^ represents the accumulated errors in the measummmt of the weight, the period, the suspension length ( including, of course, the error in the cente~ of-gavity location), “and the quantity (VP + MA). For the simplified method the precision of the virtual moment of in&tia depends for the most pert on the errors in the weight, the two periods of oscillation, the difference in the suspension lengtlm, W the qmtfty (VP + MA) l (The Pndulum characteristics of the s@@ng gear are assumed herein to be determined with neg@jible error. )
As a means of evaluating the relative precision of the standard and simplified xethods, computations‘were made to dete@ne to what extent each of,the individual errors would affect the virtual moment of inertia as calculated hy each mthod. For this analysis the error in tie weight measuremmt was estimated to be 5 pounds, that for the suspension length 0.01 foot (1/8 in.), and that for At O .00~ foot (l/1.6in.)’. The probable error of the periods of oscillations was computed to be less than ti .0005 second. The value of Vp was assumed to be accurate to within 10 percent; the additional mass (^) ‘& for the case considered, was negligible. The competitions were made by use of the data from the tests of the airplane about the Y-axis. (See ap~ndix. ) In each computation one of the variables was changed by the amount noted; for the evaluation of the period error in the simplified mthod, the two periods were changed in opposite directions. The results of these calculations axe given in table III.
On the basis of the estimated errors used in these calculations, the precision of the virtual nnment of inertia is shown to be 0.43 per- cent for the standmd method and 0.55 yercent for the simplified mthod.
The* -inch error aseumd for the suspension length was chosen because of the ~ement in the results of the airplane tested in the present investigation and because of the accuracy with which the center of gravity could be located on the type of airplane (biplanes and so forth) for which the mathod was developed. It should be
—— —.. ..— —— —.—— —.—
,
.
test only much
program can be conducted vlth the airplane in a level attitude not provides a simpler method but also makes possible the testing of larger end heavier airplanes.
The precision of the measuremmts has already been noted to improve as the suspension length te decreased. ~ simplified mthod perndts the use of shorter suspension lengths for low=wing mmoplanes because the necessity for sighting the center of gravity of the airplane in measuring the suspension length by the stsndsxd method is avoided.
By eliminating the procedure for determining the cente=f~avlty locatlon and the suspension lengthby direct ~asurement, the total tinm required for finding the momenta of inertia about the three exes may be reduced considerably. (^) If it is desired h check the results obtained by the simplified ~thod, the airplane may be swung at a third suspension length. (^) The additional. tim requlmd for the third suspension would be of little consequence compared with the tiw saved by elimi- nating the cente~f-gratity end suspensio~length procedures.
The results of comparative tests of the standard and simplified methods have shown that *he several advantages of the simplified method can be realized without sacrificing the precision of the final results.
A simplified compound-pendulummethod which eliminates the necessity for determining the cente~f~vity location of the airplane and the suspension length by direct measuremmt has been developed as a modifi– cation of ths standard method described in NACA Rep. No. 467. The
. following conclusions are indicated: 1. The method canbe successmly applied to the determination of the moments of inertia about the X- and Y+xes of airplsnes.
, Lan@eyM&orial—. Aeronautical Laborato~ National Advisory Committie for Aeronautics ~ey Field, Vs., January^ 14,^ lg
.. . ..— ——.— .—. ..— — ..-—— .—. -.
The following are the data determining the virtual moments low+fing nmnoplane.
X*8 .- The experimental
Al?mN-Drx
and computations which were used for of inertia about the X- and Y-axBs of a
data for this axis
Short suspension
w, lb?****.== 63% wt, lb............. 450. W, lb.............. 6808. 2,ft... l , l.... l... 10. ~,ft.. l l. l l * . ,.. (^) 12.1o.
are
Long suspension
6358
L,ft.............. (^) 10.300-’ 11. T,sec............. (^3) l9379’ “ 4. Vp, slugs l... l l... l.. (^) 1.37 1. MA,slugs .2.. l........ 0.99^ 0. l&slu&ft l.... ...** (^2360 )
By the standami pendulum method~ IV “is calculated as follows:
6&)9.1x(3.9379)2 xlo.300 _ 6358 (^399479) )
IVS. 4475 slug-feet sqyare.
6814.4 X (4.0900)2 x 11.491_
The value of Iv, that is, the average of
is 4481 slu~eet square.
A“check value of ~ is obtained by the two suspensions simultaneously, unklowns l
(Vp^ +
h~ and^ IVL,
. solving the equations for ~) and Iv being the
-——.
I
-.
... —.—. — ——_ _
The value of IVL
RW1.11.- The
will, of course, be the S- as IvS..
experimental data for this axis sre
Short suspension (^) Img suspension
w, lb............... wi, lb............... W, lb............... l,ft......... l. l l l. Zt>fte. l. l l l. l.. l... L,ft............... T,.sec............... Vp, slugs l.......... l. MA,e@3s a............ I&slug-ft............
By the standard pendulum method,
Iv ia calculated
as ‘follows:
6828.1 x (4.1200)2 x 9.182^
39.479 (^) ( )
8461 slug-feet square
6828.1 x (4.4773)2 x W.970 _ ( )
39.479 (^) l
Therefore,
slug-feet square
the yalue of Iv is 8465 slug-feet square.
The check value of IV (^) is found from the equations:
I~s = 8574 -vp(g.064)
IVL = (^) 8696- Vp(m .851)
Vp = 1.
IV= ITS = IVL. 8453 sl~eetsq~e
.
-. (^) _—... (^) — —— —-—.
.
1
i
The egreamant between the a~era~ and the oheck values ie 0.14 percent.
BY the simplified pendulum methcd, Iv is calculated as fo~OwS:
b“
[ 1 (,*17“)
6358 (4.4~3)2<4.1200 )2-@.95&3~7’99~+i 37
and
39,47’9 (^) -(::i’’.$”053)2)2’
I
