Experimental Determination of Airplane Moments of Inertia using Compound Pendulum Method, Lecture notes of Acting

Download Experimental Determination of Airplane Moments of Inertia using Compound Pendulum Method and more Lecture notes Acting in PDF only on Docsity! . ) P 4 . 4 a c a z ‘J 1 ;, I NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS i 4 ; 4 ( P TECHNICALNOTE No. 1629 THE EXPERIMENTAL DETERMLNATION OF THE MOMENTS OF INERTIA OF AIRPLANES BY A “SIMPLIFIED COMPOUND -PENDULUM METHOD By William Gracey Langley Memorial Aeronautical Laboratory Langley Field, Va. =qljm’Jpj7 Washington June 1948 AHWE T13NRM!ML L!ERARY AFL 2811 t . ,- . / —-. I ,.. .-) TECH LIBRARY KAFB. NM . . NATIONAL ADVISORY COMMITTEE TECHNICAL NOTE NO. 01444’99 FOR AERolwTIcs 1629 TEE Exl?ER~ DETERMNATIOl? OF THE MOMERTS OF IMERTIA OF AIRPLANES BY A SIMPLIFIED COMPOUND-PENDULUMKE!J!HOD By William Gracey SUMMARY 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. & ~$”- @-’..——...—--——--. ... .—- -— ---- -——..—.—.- 4 P % MA IV 1A ‘G lCL , density of air acceleration 0? gravity additional mass virtual moment of addltlon&l mxnent mment of inmtia nmpent of inertia Subscripts: EJxp experimental Calc calculated . inertia of airplane of of of inertia miw gear about axis of rotation steel bar about midpoint APPLIC&JION OF STANDARD PENDULUM METHOD 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 ,9 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 ‘rodis 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. .— —— ...— , . Ihammzch as the center of gravity of the airplane is ths origin of the axes about whtch mments of inertia are determined, its location must he found prior to the swinging tests. The tr~.wrse locati~ of the center of gratity is assured to lie in the plane of symmtry. The horizontal and vertical locations, however, must be determined by experiment. The plumib-linesuspension.~thod employed for this “ determination consists essentially in suspending the airplane in two or nme positiou in the XZ=pl.aneand locating the intersection of the projections of the plumb line from the pint of suspension. These “ plumb lines, detemnined by nmans of a transit, apply to the entire suspension system, so that corrections must be tie for the moment applied by the gear. The suspension of the airplane in different attitudes is accomplished by rmnrting “theair@ne on the samq swinging gem uped for the tests of the Y-axis. For this reason the cradle which supports the airplane is made longer than would be required to support the airplane in a level attitude. Nose+own end tail+iown attitudes are obtained by sliding the airplane forward-and rearward along the cradle. Obviotily, the angular displacement between the two posi~ions should be as large as possible for an accurate detarmi– nation of the center of gravity. In practice, the total ~spkce~nt must be kept less than 30° because of the danger involved. The experimmtal data obtained from the swinging tests provide a . measure of the mxmnt of inertia of the complete pendulum, consisting of ths airplane and the swinging gear, about the sxis of oscillation. The moment of inertia of the airplane about the axis of oscillation is then obtained by subtracting the rmmnt of inertia pf the gear about this ~~S . For the.compound pendulum, in which case the body axis of the airplane is remmed from the axis of oscillation, the moment of inertia must be transferred to the airplane axis by an additional comput&ion. For this transfer of axes, the mass which must be con- sidered as operating about the axis of oscillation includes not only the mass of the airplane but also the mass of the external air dis- turbed by the motion of the airplane (so+mlled “additiona,l+uass effect”). The quantity remaining after the transfer of axes is called the virtual moment of inertia and includes the mxmgnts of inertia of the airplane structure, of the air entrapped within the structure, snd of the additional mass about the airplane ads. Since the true moment of inertia of the airplane consists only of the mxnents of inertia of the structure and the entrapped air, the mmmt of inertia of the additional mass abut the body axis (called additional mmmt of inertia) must be evaluated and subtracted from the virtual nmmmt of inertia by a further set of computations. Thts additioti mment of inertia is determined by two factors, namely, the dimensions of the projected areas of the various components of the airplane acting about the body axis and the coefficients of additional nmngnt of inertia obtained from tests of flat plates. The formulas and coefficients employed in these calculations may be found in references 3, 6, , Snd 7. -.-— ----- .— - .—-———— .. ... . -.. .—-.-. — .—z .. ———. — —____ 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: IV located, the airplane is for each of the three ems. Iv is calculated in each case from for the bifilar torsional pendulum, and %=m&@+vp+MA)z2-lG (1) (2) 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, M =:+ V8P+(V-V8)P =;+VP . (“3) 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 —. — ———-. — NACA TN No. 1629 9 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. D~ OF THE SIMPLIFIED COMPOUND-PEKDULO?4 The development of the simplified compound-pendulum directly on the test procedure described in reference ?. MEI!HOD method is based Silmllystated. 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 (4) (5) 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 (6) . — . —— __ -—-.. __ -.—— _ —.——. ..—. ..— —.——. —.— . 10 . NACA TN NCI. 1629 w substitution of equation (6) in equations (4) and (5) @elds. (w2S + WvsZ’s)Ts2 IVS = 4Y(2 - ( ) -%:+vP+M& ( )W2L -1-W’LZ’T TT2 IVL = ( ) -% ~+ Vp+MAtT2 41f2 From the relation ~ = 2s + A2 (Where A2 is the difference between the two Suspmmion lengths), equation (8) may be expressed as (7) (8) - 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 8qUS.tiOIU3beconws . 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 ● advisable. This ~asurement can be made with sufficient accuracy by ._. —.,— —— -- ——— —- NACA TN No. 1629 weighing the airplane in a level attitude of mommts. When mounted on the swinging orientid so that its center of gravity is IL and applying the ~inciple gear, the airplane should be directly above the center line of the cradle. If the center of gravity has been accurately located, the cradle will be level. The accuracy .of the measurement of the suspension length by m%ne’ of the simplified method is seen from equation (10) to depend to a large extent on the precision of the periods of oscillation. In the past, the period had been determined %y timing 50 complete oscillation by mans of an ordinary stQp watch. The watch was operated manually when the center of the gear was observed to Pam a vertical reference. As a Bsns of improving the precision of these measurements, a mne accurate, automatic timing system was devised. The improvement in accuracy was accomplished by employing an electronic timing mechanism which had been previously developed for use with an ultra~gh-speed camera. The automtic feature of the system consisted of an electrical , meanE for starting the clock snd stopping it again at the end of 50 oscillations. The actuatir used to operate the clock is a mercury contact switch nwnted at the center of the cradle. From a of IV was accuracy of of both Z cursory examination of the simplified method the precision thought to be influenced to an appreciable extent by the (JVp + M , for this quantity enters into the calculation and IV. Actually, as win be shown in.the section entitled “PRECISION?,”the errors in this quantity tend to csncel, so that a given error in (Vp + l@ will produce a smiler error in Iv as computed by the simplified &thod than would be incurred if the standard ~thod were employed. The difference in the length of the two suspensions should, of course, be made as large as is practical. The distance between a ‘ reference point on the airplane (or cradle) and any fixed point directly below or above is then foiuuifor each suspension. As the difference in, the sus~neion lengths involves only two dimensions, loth of which can be determined easily and with good precision, ths value of AZ can be determined very accurately. In order to determine from actual e~rimmt the precision which could be expected with the simpl.i fied compound-pendulum methodj swinging tests were conducted on a low+ing nrmoplane weighing 6358 pounds. From , these tests the moments of inertia about the X- and Y-axes were deter- mined by both the standard end the simplified methods. As the results obtained by the older method were to be used as the standsrd for .-.— —. — 14 NACA TN NO. 1629 appreciated, however, that an accuracy of 1/8 inch In the determination of the centemf-gavity location by the standard method will seldom he reall zed in testing low+d.ng nnnoplenes. If the error is as nmch as 1/2 inch, which is not at all unconmon for this type of airplane, tlm individual error woul~ be 0.46 percent instead of 0.13 percent and the sum of the errors would-then le O.81 percent. !l?heprecision of Iv - as determined by the standard method is seen, therefore, to be largely dependent on the accuracy of the suspension length. The precision of IV as detemined by the simplified mthod, on the otbr hand, depends for the most pert on the accuracy of the periods. If the error in timing had been 0.001 second, for example, the error contributed%y the priods would have been 0.50 percent for ths simplified nwthod. An error of 0.001 second in the calculation of Iv by the standard mthod, however, would produce an error of only 0.15 percent. As noted previously, errors in (~~ + @ tend to cancel when the simplified mthcd is employed. A given error in this quantity, consequently,produces a som3what smaller error in the final resuits obtdned with the simplified mthod than is yroduced in the moment of Inertia computed by the standard mthod. The results of this analysis we in agreement with the results of the swinging exp3rim3nts in showing the over-all precision of the two methods to be essentially the seine. This conclusion applies, of course, only when the error in the suspension length for the standard method is no greater then that assured herein. . . EVACUATION OF MEJ?HODS Several advantages may be realized in the use of the simplified COllQOUIld-~IldUhllllmethod. The 3m3t Important advantage is the elimination of the necessity for suspending the airplane in the unusual attitudes required for the cente~f-gravity determination by the standard pndulum method. This feature of the ~thod not only avoids very serious handling difficulties %ut alSO m.inimlzesthe hazard involved in swinging an airplane. Furthermore, as the cradle need be only long enough to support the airplane in a level attitude, the size of the swi~ing apparatus may be reduced. For special cases, for example, when the airplane is equippsd with suitably located lifting lugs, the supporting cradle may be discarded entirely. Suspending the alrplene from these points of attacmnt would introduce en additional simplification in that the nrxnentof inertia of the suspension rods can be readily estimated; the need for finding the moment of inertia of the swinging gear by experiment would thus be eliminated. The fact that the entire . — .———.- . —-.——..—.— ..-. .—. ___ . .——_ _ , . NACA test only much 15 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. CONCLUSIONS 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. 2. The precision of the results obtained in the application of the simplified mthoil is equal to that obtained from the standsxd pendulum n&hod. 3. ‘l!he simplifiedmethodyermits a reduction ti exper~ntal technique, test apparatus, and time required to perform the tests. , Lan@eyM&orial Aeronautical Laborato~—. National Advisory Committie for Aeronautics ~ey Field, Vs., January 14, lg48 . . . ..— ——.— .—. ..— — ..-—— .—. - . 16 COMPUTATIONS FOR The following are the data determining the virtual moments low+fing nmnoplane. X*8 .- The experimental NACA TN NO. 16@ Al?mN-Drx SWINGING TESTS OF AIRHAME and computationswhich were used for of inertia about the X- and Y-axBs of a data for this axis Short suspension w, lb***?****.***== 63% wt, lb . . . . . . . . . . . . . 450.1 W, lb . . . . . . . . . . . . . . 6808.1 2,ft. . . ● , ● . . . . ● . . . 10.172 ~*,ft . . ● ● . ● ● * .* , . . 12.1o1 . are Long suspension 6358 456.4 6814.4 11.376 13.093 L,ft. . . . . . . . . . . . . . 10.300-’ 11.491 T,sec . . . . . . . . . . . . . 3 ●9379’ “ 4.0900 Vp, slugs ● . . . ● ● . . . ● . . 1.37 1.37 MA,slugs .2. . ● . . . . . . . . 0.99 0.99 l&slu&ft ● . . . . ...** 2360 2793 By the standami pendulum method~ IV “iscalculated as follows: IVS = ( 6&)9.1x(3.9379)2 xlo.300 _ 6358 399479 ) ~~+ 1.37+ 0.99 (10.172)2-2360. IVS . 4475 slug-feet sqyare . IVL = 6814.4 X (4.0900)2 x 11.491_ 39●479 ( ) .%+ 1.37 + 0.99 (11.376)2 -2793. *VL = h486 slug-feet square 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● (Vp + h~ and IVL, . solving the equations for ~) and Iv being the -——. I -. . 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: .[6358+,03j(3.w,,2+,6R13q Q- (4.u@)2x4~.lf10. nl<4.4n3)2(63W3 .w94W.~4.%4)+39.4W -7 b“ [ 1 (,*17“)6358 (4.4~3)2<4.1200 )2-@.95&3~7’99~+i 37 and 6828,1 X (4.1203)2 X 9+171 ITS = 39,47’9 -(::i’’.$”053)2)2’3434 I 20 NACA TN ~rO. 1629 . 1. Green, M. W.: Measurement of the l&nents of Inertia of Full Scale Airplanes. NACA TN No. 265, 1927.. 2. Millerj M. P.: An Accurate Ibthod of l%suring the Mmmts of Inertia of Airplanes. NACA‘J3i?io. 351, 1~30. - 3. Soul:, Hartley A., and Miller, hrvel F.: ‘Ths Experimental Determination of the Rkxrents of. Inertia of Airplanes. NACARep. No. 467, 1933. h. Gates, S. B.: The ‘Determination of the hkments bf Inertia of Aeropl.anes. R. & M. Ho. 1415, British A.R.C., 1932. 5. Pobedonoszeff, J.: An Experimental Y!bthcd of Determination of the Moments of ~rtia of Aeroplanes. Rep. No. ‘201, Trans. CAHI ‘ (lbecm?), 1935. 6. Gr~cey, Wi~iam: The Additional-Mass Ef feet of Plates as Determined ~\ by Experirn9nts. NACARep. No. 707, 1941.—_ -“( 7. h%.lvestuto, Frank S., Jr., and Gale, ~wrenc~- J.: Formlas for Additional Mass Corrections to the Mxasmts of Inertia of Airplanes . NACA‘IN ~0 . 1187, 1947. # . . \ .—— ---- . . ..— —.—-— —.———-. —- F Axi 8 . \ Iv Iv Ehmpewlon (:) (alu&rt2) (rman value) (Blu&ftP) ~tiatlon of lV from Iv (mm value) (~rcent) x Short 10 .lp Long ~ IL .376 :E } 4481 0.12 Y Short ~ 9.064 &?461 E .851 }- .05 x“ Short ~ ‘lo.lp LOng u.381 Y Short 9.053 k~ u .852 simplified IMthcxi 1 8470 I .09 \ #- 24 NACA TN No. 1629 Figure l.- Air@ane and swinging gear arranged for the determination of the moment of inertia about the X-axis by the compound-pendulum method. . . , , — -——— . ..— —— -——— ,4 25 =%= LC“’”‘0’“-”am . . . .,. - FQure .2---A.lrplane and swiz@ing gear ~ranged Ya the IM-=tion of the moment of tiertia ~%out the Y--@-s ‘by tie mmpmnd-p-en~- method. r . . .’ :: _.. . ——..——————— ————. ~.- .-— —.— —.. — :--————- --- — -— ..__ .NACATN ~iO. 1629 ,, “:. ” IT “ r r his or osolllation (colncidontuithZ-axis) J e 1- 4 ,:.: ,’ d ‘% ,, D ! . . . .. . .< “- L C.g. of alrplan ... .. .< ,.- ..--. . . + --- .1 I Figure 3.- Airplane and swin@ggear arranged fortiedetertimtion of the’’moment of tier.tia almut the Z-axis Qy the ibtiilat torsion + ‘. -.pend~mrnethod. ~ ~ “... - - ‘ ~ . . . . . . ~ . . —-