1、AREPORT No. 762 .THEORETICAL INVESTIGATION OF THE LATERAL OSCILLATIONS OF AN AIRPLANEWITH FREE RUDDER WITH SPECIAL R EFERENCE TO THE EFFECT OF FRICTIONBy HAEEY GREmmEQandLEONARDSTDRNFIELD.SUMMARY ChartiIshowing the variaiiun in dynamic stabiliiy chD=8cw= ac; and so forthCrDHHrk,k.1mm,8eilective incr
2、ement in viscous-damping coefficient .due to solid frictionrudder chorddifbrentisl operator (d/m)hinge momentfictional hinge momentradius of gyration of rudder about hinge axis, dividedby SSmiSpSJlradius of gyration of airplane about vertical axis,divided by semispan .tail lenh divided by wing semis
3、panmaw of airplanemaas of rudderdistance traveled in semispans (2Vt/b)147Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-! .:_. ., that is, the osoilhtionsare limited to a definite amplitude which depends on theProvided by IHSNot for ResaleNo reprodu
4、ction or networking permitted without license from IHS-,-,-THEORETICAL INVESTIGAHON OFfriction. Aerodynamic or viscous damping ofLATER4L OSC!IIk.ATIONSthe rudder,however, crmsesu phase lag that does not change with ampli-tude; hence, if this lag is snilicient, the oscillations willbeunstable-that is
5、, will increase indefinitely.boreasing oscillations due to aerodynamic damping ofthe rudder.lk figure 1 the damping and the frequency ofthe oscillation as represented by values of u and o are shownm functions of the floating-moment and restoring-momentparameters of the rudder.Tlm values shown for u
6、and o are related to the dampingand tho period of the lateral oscillation by the equationsP=6.28/vT=o.69/uwhere the period P is in terms of the number of semispambmgths that the airplane moves for a complete cycle amdthedamping T% refers to the number of semispans the airplanemoves before the oscill
7、ation is damped to one-half its originalamplitude.In figure 1 the control system is asaumed to be fiictionless.For an average value of the airplane radius of gyration“=$) the density ratio employed in figure 1 correspondsto a wing loading of 25 pounds per square foot for an air-plane of 40-foot span
8、 at sea level. A positive value of Ch+corresponds to n stabilizing floating tendency, and a negativevalue of ch corresponds to a stabilizing restoring moment.The magnitude of ch is a measure of the control forcesrequired to deflect the rudder at zero yaw; a Chaof 0.4, forexample, corresponds to 150
9、pounds of pedal force for fulldeflection of a rudder having an area of 25 square feet anda chord of 3 feet at an indicated speed of 100 miles per hour.The oscillation becomes undamped for only positive valuesof OAYand a high degree of aerodynamic balance correspond-ing to small numerictil valu=-O.C8
10、4.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-5Pm%. g -.5 -Increo.shg oscilltions _I .4 i I1 i,3 1.$ I I 1 tII -.2iiI1 i,G1I .,:11 t k/ -. /Dive rgence -. 2/ -.03RedorlGJ-momenf po%meter, Chd-.1FIOUBE3AMfoot of moss balnnca of the rnddor on rnddo
11、rhe atablllty. PM-O.8Z0C,t-o,w; *4,Increosin 9 Osclllo +ions _I.4f%lno,I8-0.1/%1-.f7 -. t. . . . :, -.33 - -1.67 ,1-. -i .3“ . .,. N .x,., . 1, . i c .2 Y. *.i b.,./ -. IDive rqence - -.2-,5 -.4 -.3 -2-.-. I 03Restorihg -mamnf pot-tmeter, ChdFIGURE4.EfIootofrudder domrrlngon boundorgfor fnormningoso
12、lllntlons. z-OWCv. -Ow: /.lw=odOzn.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Increasing oscillotims -.4,It I%6 -0./1 t- .17 - ,3-.33 - -f.67 - -, 1tII .2=. . - .+ . .“ . *. h. -: . -.1:-.: ./ / k.-,1/ Divergence -?-5 -. 4 -. 3 72 -. / 0-.3.-Res
13、tortn g-moment poromatff, Ch *W=O.0222.%,.1 !2 - .- tI1/Lt$ /1! f, .t It :1 / / tl / /, , /+- - .- .1;i,IProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICAL INVESTIGATION OF LATERAL OSCILLATIONS OF AN AIXPLANE WITH FREE RUDDER 153By use of th
14、is relation in conjunction with figure 6, theaction of friction can be explained. If the initial dkturbance/Ch,is very small, the value of effective (?his, according tothe preceding expression, very large and the point represent-ing this value of chwill lie ta the left of the appropriatecurve of fig
15、ure 6. Because this point is in the stable region,the oscillation will damp out completely. If the initial valueof $/Ohf is high enough to place the point on the concave sideof the appropriate curve in figure 6, the motion will beunstable and the amplitude will increase. This increase inmplitude dec
16、reases the numerical value of the effectiveC*Muntil the point on figure 6 moves to the right branch ofthe curve. Any further increase in amplitude is impossiblebecmse it wotid bring the point on figure 6 into the stableregion. If the initial value of /C*is very large, the effectivevalue of ohm is nu
17、merically very small and the point repre-senting it on figure 6 will be to the right of the curve, in thestable region. The amplitude will then decrease and causetho value of obato increase til it equak the vfdue at theright brrmch of the curve.In figure 8 the amplitudes corresponding to both branch
18、esof the curves of figure 6 are plotted against the restoring-moment parameter for two values of the floating-momentparameter. As the condition of aerodynasnic balhce isnppronched, the magnitude of the oscillations increasesmarkedly. When a condition is reached at which the oscil-lations would incre
19、ase without solid friction, they will beunstnble with friction if the initial disturbance is greaterthan that corresponding to the left branch of the curves offigure 6.The region where steady oscillations can occur is boundedon one side by the boundary for increasing oscillations with-out solid fric
20、tion and on the other by the boundary for com-plete damping. The variation of the amplitudes of rudderand yaw oscillations in this (shaded) region is shown infiguroa 9 rmd 10.The amplitudes of the steady oscillation are proportionalto the frictional hinge-moment COOffiCientj a9 shown inappendk B. Th
21、ese omplitud are therefore directlyproportional to the amount of friction and inversely pro-portional to the square of the indicated speed. Overmost of the region the amplitude is extremely small, evenwith relatively large amounts of friction. On a typicalairpkme (appendix B) having parameters. corr
22、espondingto the point shown on figure 9 and with a friction moment of4 foot-pounds, the maximum amplitude of yawing oscillationoccurring when the rudder is freed at 300 rnilea per houramounts to less than 0.5.llffeot of airplane mass characteristics.-As the momen the full line corresponds to n rudde
23、r thecenter of gravity of which is 10 percent of the rudder chordahead of the hinge and the mass of which is about 1 percentof the mass of the airplane. This rudder weight is con-siderably more than usual but could be reduced by increas-ing the distance between rudder hinge and rudder center ofgravi
24、ty.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.36.24.12b120 1,2I/80 f- / .-10 -8 -6Ndder damping PCe*S %6 =10I I I I ITFR-q-LbjStable- yawUnstable - I *-. 5 -.4OVBE8.VarkUOn(a) cA#=oJ.(b) Cy.oaof amplltnde of steady uwfI19tlonapht-1.w.z; =-o.cw.
25、w(th aeredmmnfo Me.uee. ,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.I I I 1/ I I I I I-,5 -.4 -.3 -2 -J/?estof-ing -moment paromeier, gFIOWRE9.-Vnr!ntlonofyawamplltudoofakmdyodllatlom with ooradynamloMom andflcdlng kondomoy./.W- 1.SS%C.,-0.004;
26、 wk=o.4i d$/L71vergence/I1-.5 -,4 -J -,2 -JRestorlnq-mament porme ter, ChFmuR10.-Varlatlon ofrudder rimplltudoofsteady omlllatfonswith aarodwmroloMnnmand flmtlng kondonoy.pk- 1.S62;(V= -0.W; I+PMO.-,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Fxa
27、vm 11.EffW of rdrplnne density and radius of tlon on dynndo stablllty.ci#-o.m4;%H=O.GZ2.4Steady osclllafions7c1+ . 0$ / kcz %._,lf/ /Divergence/“=-.032 -.064-./me+er, Chd oFIGURE12-Effect of!mathercockstntdlltyonboundarleaforfnmeasfngoscllMlons,steadycocfllatlormand dhqenw. IArr=O.9213.IIIITi:,1,.,.
28、,:Ii,:,l:,b,?i.i1ii,)(Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4*00f I.4ITIItIE III t. tI t.-5tecy -Cilfofions o 31I u.+1. 8. I $., I., I 2. 1 I. H.-; ., z 2.ii., .s. 8J bkLc$;o$0.! .? .C%Qa / g ,/ k,/ -. I,/ .,/a71 /”c /d ,/ D/wrgence - -.2-0
29、,076 - /- , f52 ,.,5 4 3 2 -.3-. -, -. -#Restoring-moment poromer,-O.762NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSk I I I I .4I I I1 I . 1 I , a 1 / II I - I I /1 . 1 , 1 l I xl (/ -.IDivergence/t -. 2/-. 3-5 -.4 -3 72 -.1 “ oRestoring -mornenf parameter, G,jFIGURE16.-EffA of - overlmknrn of redder
30、 on dynamia stability. JL.W.0$20;-o.wCONCLUSIONS .The calculations presented in this paper show the existenceof oscillations of constant amplitude in a rudder system hav-ing friction and certain hinge-moment charactmistics. Thecharts presented show the conditions that tend to minimizeor eliminate th
31、ese undesirable oscillations and tire intendedaa a guide to the design of airplanes having rudders with astabilizing floating tendency. The results of these ctia-tions indicate the following conclusions:1. A closely balanced rudder hawing too great a positivefloating tendency will be dynamically uns
32、table if the controlis freed. .2. Under conditions of d-ynamicsbili for a rudder witha positive floating tendency, a continuous osculation of fiedamplitude may be caused by friction in the control system.3. The amplitude of the steady oscillation is proportionalto the amount of friction and, for all
33、 practical pnrposestheoscillation may be eliminated by reducing the friction, pro-vided the aerodynamic balance is not too nearly complete.4. The amplitude of the steady oscillation is inverselyproportional to the square of the indicpted speed.5. e steady oscillation can be eliminated by using aSnfl
34、iciimtly small floating ratio or by mass overbalance ,ofthe rudder.6. A positive floating tendency can be used to compenmtc+for a lack of weathercock stability if the control system isdesigned for small fiction.Flight teats will be necessary to indicate the maximumamount of steady oscillation that i
35、s allowable on an airphme.-LANGHY MEMORIAL AERONAUTICAL LABORATORY,hTmloNALADV160RY COWIITTBE FOR AERONAUTICS,LANGLEY FIELD, VA., urcfi 4, 1943.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-APPENDIX AEQUATIONSOF MOkION FOR THE CASEOF VISCOUSFRICTIO
36、NIN THE RUDDERCONTROLSYSTEMIt was shown in reference 3 that the lateral motion of thecenter of gravity and the rolling motion may be neglected inthe malysis of the lateral oscillations with free rudder. Thenumber of degrees of freedom is thereby reduced to two:namely, angle of yam and rudder deflect
37、ion.The equations of motion are:(2pk.D-D+D+c- +)#+#+ (c.Dpc*J6=o(2P,k:+2,x,Z)D2 C,MDC,$+ (2p,k:DC*.C*8)6=0Substituting #=ilfe and 6=iVeA*in these equations indicates(1)Tho boundq for divergence is obtained by setting 3=0and that for increasing oscillations is found by settingRouths discriminantR=BCE
38、AElW=O (2)Tlm roots of equation (1) can be easily found when equation(2) is sntisfled; in this case the vahma of A correspond tothe undamped oscillation areh= _=_._ . ._._ . . .- -APPENDIX BTREATMENTOF SOLID FRWHON IN CONTROLSYSTEMAPPROXIhlATE RIEYEIODOF CALCULATINGAhlHITUDE9OF STEADY OSCILLATIONSPr
39、evious work (reference 5) has shown that certaindynamical systems can, in the presence. of solid friction,build up constant-amplitude oscillations that would notexist in the absence of friction. This work, however, waslimited to the case of continuous motion of the rudder-thnt is, motion in which th
40、e rudder dow not stop movingduring each cycle. The effect of friction in the case ofdiscontinuous motion has been discussed in the previouslymentioned document by Schairer and Bush of Boeing Z,=* fwhere ChDJf is the value of the viscous damping required-to make R=o minus the vfdue of ChD8 due to aer
41、odynamicdarnping of the rudder. This expression for the amplitudein terms of the amount of solid fiction, the amount ofviscous friction, and the frequency is derived in reference 4.NUhfERICAL EXAIWPLEUBING APPROXIblATE 31EXIIODThe calculation of the amplitude of the steady oscillationdue to solid fr
42、iction will now be made for a speciiic airplanehaving the following characteristics:Pad- L 352 C%- o.2= O.76X:;for the minimum disturbance required to start the oscillation,In this case, it is seen that the steady osciIIation, having omaximum amplitude of less than 0.60, would hardly be per-ceptible
43、 in flight.The period of the steady oscillation is2ar 42.4X600.2138X2 X300 X88=1”42 ecProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICAL INVESTIGATION OF LATERAL OSCILLATIONS OF AN AIRPLANE WITH FRIUlRUDDER 161Wing SemispulsCOMPAIU90N WITH M
44、ORE EXACT CALCULATION OF THE RFFECTOF SOLID FRICTION ON THE MOTIONIn order to check the approximate theory, a step-by-stepculcukdion of the rudder motion following certain initialdisturbances was made for two conditions. The results ofthese calculations me shown in figures 16 and 17. Eachtime the ru
45、dder motion stopped the rudder became lockedby the friction and the subsequent motion was calculatedfor that conditiori until the force on the rudder exeeeded theforce of friction, when the rudder moved back and anotherstep in the calculations was made. The steps in the calcula-tion de thus of two a
46、lternating kinds: rudder-fied motionand rudder-free motion. The motion of the rudder underthese conditions has flat-top peaks as shown in the figuresand also in flight records.Figure 16 shows the motion corresponding to the numeriealexample given in the previous section. The motion withoutsolid fric
47、tion is shown for comparison. An arbitrarily choseninitial dimlacament in vaw was taken. The effect of friction.in causing the motion tovertieal lines on the rightas previously calculatedbuild up is clearly shown. Theof the figure give the amplitudesby the approximate method.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-162 REPORT NO. 762NATIONAII ADVISORY COMMITTEE FOR AERONAUTICSFImwm17. the motion following the small di
