...,,,-, ~;:,-~" k~ ~,.-," ~,";,~','",!.',,~:/~." " ~:-'<"k c r-,~,.,"..;, ~!,,:, ~,,;.,:,5!..~ :.?~ ',l:-' :=~,! l~'.y.a~,![:)j,v.~-hh. ~:~ :.C~'.i.'J P'-~,' ~ :-,'..*-'....,:..b,;..~.,,,,":./:%.:y% '_,',,,. ii ~. & gl No. 88 (~L627) A.R.C. Technica Repo~ "...... :s~%-:x;7', /!.c,/.2', L:.: i:.. :..;!!i "~ '!~. ~,",~... tl... (".:'--'":,"..,!"--: [! MNSTRY OF SUPPLY AERONAUT[CAL RESEARCH COUNC][L REPORTS AND MEMORANDA A~rcm o S~ L~n d g ~n ea~: Greun. d Lo~& when 8?inn~ngoup the Whee~a ~t Teuchod ~n J W BLNKHORN, B.Sc., A.F.R.AmS. Crown Copyright Reserved LONDON : HER MAJESTY'S STATONERY OFFCE 95 2 FOUR S H L L N G S NET
N~"D[O~EL. AERONAUTCAL EST,O,~L~9~3>~~: N Aircraft Landing Gear. Ground Loads when Spinning-up the Wheels at Touch-down J. W. BLNK~O~N, B.Sc., A.F.R.AE.S. COMMUNCATED BY THE PRNCPAL DRECTOR OF SCENTFC RESEARCH (AR) M~STRY OF SJPPLY Reports and Memoranda No. 88* June, 948 Summary. The investigation covers all combinations of landing speed, coefficient of friction between the tyre and the ground, and harshness of landing, for any type of pneumatic tyre and wheel unit. Particular attention has been given to landing speeds between and 1 m.p.h., coefficients of friction from to 2., and landings giving vertical wheel accelerations of lg, 2g, 3g, 4g. t was found that for any landing, the vertical reaction at any wheel which has just finished spinning-up increases with increase in the moment of inertia of the wheel and tyre unit, and the landing speed, and decreases with increase in the free tyre radius, the aircraft weight, tile time to reach the maximum vertical wheel reaction, and the coefficient of friction between the tyre and the ground. t should be noted, of course, that there is a relation between the moment of inertia, tile free tyre radius and the aircraft weight,--in general the free tyre radius and the moment of inertia will increase with aircraft weight. For any wheel and tyre unit it is shown that there are various combinations of landing speed and coefficient of friction which will cause the wheel spinning up to iust cease at the same instant as the maximum vertical wheel reaction is reached, and except for very gentle landings, the maximum value of # required is usually much less than 1.. Figs. 1 to 7, together with the notation given in Section 8, are self-explanatory and expand the above observations. 1. ~troduction.--t is well-known that wheel drag loads are caused at touch down by spinning-up the landing gear wheels from rest to the speed corresponding to the landing speed of the aircraft. A limited investigation into the magnitude of these drag loads and the associated vertical loads was made some years ago in S.M.E. Department, Royal Aircraft Establishment. For various reasons it is now~desirable to extend this investigation. 2. Range of ~ vestigation.--the ground loads arising when spinning-up wheels at touch down have been determined in a manner which can be applied to any type of wheel and pneumatic tyre equipment fitted to undercarriages incorporating oleo-pneumatic or spring-type shock absorbers. The results of this investigation can be applied to aircraft landing at any forward speed on a surface giving any coefficient of friction between the tyre and the gronnd and for any practicable severity of landing, i.e., very light to very heavy. e R.A.E. Tech. Note mech. Eng. No. 16--received 17th July, 1948. (222) A
. Method of Analysis.--Considering one wiaeel: R Vertical reaction at any instant R, Unit static load at landing weight which is equal to aircraft weight + number of main wheels for main undercarriage, and to static load with aircraft at rest for auxiliary tail or nose unit Z s Coefficient of friction between tyre and ground Vertical reaction factor at any instant = R/R Vertical reaction factor when spinning-up ceases ~,,, Maximum vertical reaction factor reached during any landing V Landing speed of aircraft w~ Angular velocity of wheel at any instant r Free radius of tyre r~ Radius of tyre at any instant, i.e., height of axle above the ground re t tm Effective rolling radius of tyre at any instant, i.e., = V/wi Moment of inertia of complete wheel and tyre unit about its axle Time measured from instant of touch down Time to reach maximum vertical reaction t, Time taken to spin up the wheel T g Dynamic tyre rate (deflection per unit load) Acceleration due to gravity. Then at any instant during the spinning-up period we have: dwi ~Rri = - g dt Therefore, f f* Rri dt = van. g................ At time t = t, when spinning-up ceases, i.e., no slipping between the tyre and the equation (1) becomes ground, f ~ Rri dt -- V g re i.e., # = V grer fi r, dt.. (2) 2
Before an explicit expression for equation (2) can be obtained, substitutions will have to be made for re, r~. and 2 in terms of time t and known constants. These substitutions can be made by the following assumptions: (i) X = L,:sin [(~/2). (t/tin)] has been found to be approximately correct for a large number of landing gear units on which performance tests have been made. These units had the usual oleo-pneumatic or spring type of shock absorber, and further examination of the validity of this equation would have to be made if an unusual type of shock absorber were used. (ii) deflection is directly proportional to the vertical load, so that re= r-- ½RT = r -- ½~.R1T = r -- ½ZKr and ri - r -- 2Kr where K = R~T Table 1 shows the value of K for a selection of tyres in the ' intermediate ' range and in the Ministry of Supply standard tyre ranges. These show that K is of the same order for all these tyre ranges and varies from about.1 to.16, the general tendency being for K to increase as the tyre size increases. Therefore, equation (2) becomes V =,.... (3) f E =E t= o t~t~ The integral in equation (3) becomes t t s t t s L,, sin d -- 2,n2K sin s d.~t = --= ~?11 t2?l t t s t t s ---- 2m _2 -- cos t ~" "' 2"2K ~" 1 -- 2 t. --= --= tm t m - E, -. os 1.,2K t~ 2 ~, + _ As 2 2~m {(i V (~)]) ~mk sin_ As 2 2~L ~ ( 2 ", L,, /~s As 2
Equation (3) may therefore be written r2rlgtr~ ~ -- --E(1- ~/{1- (~mm)}) ]L,ng ( Sin- ~s ~s 2 2 ;%, " " Equation (4), then, defines the relationship between the various factors at the end of the spinning-up period, and it will be seen that the left-hand side of this equation is non-dimensional. Fig. 1 shows the variation of i~/(v/r"rlgt,,~) with ~, for values of Z,,~ ' 1, 2, 3, 4 respectively, K being taken =.15, these curves being obtained, of course, by substitution in the right-hand side of equation (4). Similar curves for K =.1 and K =.2 were practically identical with those for K =.15 and to avoid confusion these have not been shown. The larger the value of K the slightly further from the origin the curves moved. From Fig. 1 any specific set of conditions can be examined, for K =.15 as follows, using lb fl sec ~nits unless otherwise stated, g being taken = 32.2't/sect: - (i) Figs. 2, 3, 4, 5 show, for 2m = 1, 2, 3, 4 respectively, the variation of Z, with/~, for values of /r2r~t,~ =.,.75 and. 1, and values of V=, 1, and 1 m.p.h. For nose wheel units L, will usually be much greater than 4. for very heavy landings, but curves for ally value of % can be added to Figs. 2, 3, 4, 5. Table 2 shows the values of /r~r~t,,, for various wheel and tyre units, assuming tm=.1 sec which is approximately correct for all the oleo-pneumatic or spring shock absorber units on which drop tests have been made. (ii) For,u = 1., Fig. 6 shows the variation of Z,/Z,,, with V, for various values of ~,,~ and /r~r~tm. Similar curves may be obtained for various values of/~. (iii) n Fig. 7 is shown, for various values of /r~r~tm and ~... the variation of ~ with V for landings in which,l = %, i.e., for landings in which spinning-up ceases at the same instant as the maximum vertical reaction is reached. The value of,% varies for any shock absorber unit according to the vertical velocity of descent at touch down. For dry grass surfaces # is about. but increasing wetness will decrease this considerably. Dry concrete normally gives about 1., although for spinning-up conditions it is possible that this vahe may be exceeded. 4. Exampies.--To demonstrate the use of Fig. 1 for any particular case we will consider two hypothetical examples. (i) An aircraft fitted with two single-wheel main undercarriage units having 17.--18 in. patterned tread tyres at a pressure of lb/in? lands at a weight of 28,2 lb at a speed of 92 m.p.h, on a runway having a coefficient of friction with the tyre of.75. The landing is rather severe, giving a maximum vertical acceleration of 2.5g at the main wheels. We can now estimate as follows, the vertical reaction factor X, at the main wheels when spinning up just ceases. n Fig. 1, we know ~, r, R1, V'and g. Also can be determined (= 568 lb ft 2) and t,~ estimated from drop tests -- say tm=.11 sec,, compared with the overall average value of.1 sec assumed when obtaining curves based on Fig. 1. Therefore, _.75 (f/~2--~l) (V/gt-t,~)- (568/2"12 X 14,1)(134.8/32.2.11)= 2.21, for which, interpolating for % = 2.5 in Fig. 1, the corresponding value of ~ is equal to 1.97. 4
The time t, taken to spin up the wheels is given by 1.97 : 2"5 sin (2!~1 ) from which ts =.64 sec. (ii) The above aircraft is fitted with a tail wheel unit having a 7.--1. in. patterned tread tyre at a pressure of lb/in. 2, and the static tail wheel load at the landing weight of 28,2 lb is 2,8 lb. The tail wheel touches down a few seconds after the main wheels, the landing speed being then reduced to 75 m.p.h, and the maximum vertical acceleration at the tail wheel unit is 1.5g. n Fig. 1 we again know or can assume the values of s, r, R1, V, g,, and tin--say t,~ from drop test results is equal to.95 see. /,.75 Therefore, 4.2, (/r2r1)(v/gt,n) = (15" 12/1"12 X 2,8) (11/32.2 "95) for which, interpolating for 2,,, = 1.5 in Fig. 1, the corresponding value of 2, is = 1.1. The time ts taken to spin up the wheel is given by:-- from which ts = O. sec. 222) 5 A*
- R1T TABLE 1 Values of K for Various Ranges of s Range Size in. ~i'ree, radius r of tyre in. pressure lb/in. ~ Static load R1 lb Dynamic tyre rate T in. per lb X 1 * K - r ' ntermediate' 6.--6½- 7.--7½ 7.--1~ 9"75--12 15.--16 17.--18 8"53 9"95 12.13 15-63 22.23 "5 1, 1,3 1,6 1,3 1,7 2,1 1,8 2,3 2,85O 2, 3, 4, 6, 8, 1, 8, 11, 14,1 1"1 8.17 6.88 1"6 8"49 7.11 7.12 5:99 5"18 8.1 6.58 4"73 4.81 3. 3.14 4.32 3.23 2.65.121.127.131-143 "149 "153 "19.116.122.143 "1 "146-1.137 "i41.144.146.146 Ministry of Supply Standard Range ( lb/in. ~) 19 6.--9 21 6.75--1 23 x 7.--1 7-75--11 32 1.5--14 37 X 13--15 42 15.--16 48 X 18--18 9"7 1-65 11 "5 12"55 15"95 18"65 21 " 23"88 1,3 1,7 2,1 1,6 2,1 2,5 1,9 2,5 3,1 2,3 3, 3, 4, 5,2 6;41 5,985 7, 9,1 7, 9, 12,1 11,7 14, 16,2 8.57 6-69 5-48 7-18 5.58 4.57 7.8 5.52 4.52 6.51 5~6 4.14 5. 4"13 3-39 4"56 3"63 2"99 4"13 3"23 2"64 3"23 2-73 2.36.122.122-122.11.11.11.122.122.122.122.122.122.136.136.136.146.146.146 "151.152.152.159 "1.1 Ministry of Supply Standard Range (- lb/in. 2) 22 6.75--11 26 7.75--13 9--15 11 "2 13" 15" 2,71 4.53 3,1 3.94 3,6 3.44 3, 4. 4,2 3.55 4, 3.17 5,1 3.53 6, 3. 6, 2.61.11.112.112.123-116.119.12.12.12
TABLE 1 (Cont.) Range Size in. Free radius r of tyre in. pressure ib/in. 2 Static load R1 lb Dynamic tyre rate T in. per lb 1 ~ K -- RT f Ministry of Supply / Standard Range (- lb/in, e) 34 1.75--16 13.--17 16--18 52 18.5--21 64 22.5--26 17.5 19"8 22"6 26.1 31 "9 7,1 8,3 9, 1, 12,75 13,8 14, 16,7 19,2 19, 22, 26, 28, 33, 38, 3"16 2"69 2"34 2"72 2"32 2-2 2.37 2-3 1 "77 2"9 1 "79 1 "56 1 " 1-41 1 "23.132.132-132.141.142.142.1-1.1.155-155.155.143.148.149 Ministry of Supply Standard Range (- lb/in. ~) 2 5.--11 24 6---13 27 7.--14½ 31 8.75--15½ 36 1.75--16} 41 12.75--18 48 15---21 56 17.--24 66 2.75--28 1. 11 "9 13.75 15"65 17-8 2 "5 24 27 "8 32.75 9O 7O 8O 7O 8O 9O 2, 2,8 3,2 3,4 3,885 4,3 4,8 5,485 6,1 6,82 7,7 8,6 1, 11,1 12,2 13, 15,5 17,2 19, 21, 24, 26,2 29, 33,2 37, 42, 47, 3"97 3"57 3" 3-3"18 2.85 3"5 2"51 2.39 2"65 2.34 2"9 2"13 1 "93 1.76 1.96 1.75 1-58 1.67 1.47 1.32 1.43 1.26 1.12 1.18 1"4-93.11.98.13-14.13.14.18.1 '17.115.116.116.12.12.12.133-133.132.132.132.132.134.134 "134.1 "1.1 7 (222) A2*
TABLE 2 Value of /r~rlt,n for Various Wheel and lyre Units (t,,, = O. 1 sec) Note The values of have been determined from the expression where =.6 M ~,r~ + CMT(R 2+ 1.St, 2) mw MT R weight of wheel weight of tyre and inner tube rim radius = (r + r )12 free tyre radius " 1 C = (r - -- "95 for tyres with thin treads and 1.1 for tyres with thick or patterned treads. Range Size in. lb ft = Pressure lb/in. ~ Static Load R1 ]b ~'~Rltm (per sec) ' ntermediate ' 6"--6½* 7"--1¼ 9"75--12" 15'--16 17---18 2"63 15"1 52.2 274. 568.5 1, 1,3 1,6 1,8 2,3 2,8 2, 3, 4, 6, 8, 1, 8, 11, 14,1.51.39-32..63.52.11.81.64.133.1..148.11-89 Minist W of Supply Standard Range ( lb/in. ~) 27 x 8.75--12 37 13--15 48 18.--18.6 114.4 472. 2,8 3,6 4,4 5,985 7, 9,1 11,7 14, 16,2.86.67.55.79.63.52.11.85.73 Ministry of Supply Standard Range (- lb/in. =) 24 7.--12 26 7.75--13 22"3 39 "4 3,22 3,7 4,2 3, 4,2 4,.68.58.51.93.79.68
TABLE 2 (Cont.) Range Size in. lb t s Pressure lb/in. ~ Static Load R' lb r2 R l t,,~ (per sec) Ministry of Supply Standard Range (- lb/in. 2) x 9--15 32 x 1--15 x 16--18 64 22.5--26 65" 87 "3 376. 1,985 5,1 6, 6, 6, 7, 8, 14, 16,7 19,2 28, 33, 38,.82.69 '..83.72.63-74.63-55.99-84.73 Ministry of Supply Standard Range (- lb/in. ") 26 6.--14 33 x 9.75--16 i! 43 x 13.5--19 L 28"7 92"9 281.5 4, 4,. 5,1 8,1 9,175 1,2 15,2 17,2 19,2 O. 61 O- 54 O. 48 O. 59 O. 52 O" 57 O. 57-51 - * Smooth tread. All other tyres have patterned treads. 9
X~ l.c 2 u C~ O '~ LANDNG SPEED =SOm.pJ'~. t -~.loo;s "_ ";" gll t~ ",'~ L~. 8.. D X~ 1. z Z Z E l.fl i_ Q Z d z \,5 LANE~NG 5PEE.O = O~.p.h. ~-._. ----------~o.ovsl z l~ ~ o.ozsj P- O 1 Z o u w _1 U n" m > \ 'K=O-15 -O "5 LANDNG SPEED -- 15Or~.h. ~ ~ :S 1 3. ~ R~ ~m 5 15 ~ P-5 O.~ -O -5 r ~ R, V ~...b..,-, FiG. 2. Variation of ~, ~4th ~, for Xm = 1. FG. 1. Variation of ~, with # (S/r'R.) (V/g. t,.)"
.2 8 LANDNG SPEED : r o O ~ " 12":"5 l.~ ~o O.O?_SJ LANDNG SPEED ~O,-.p.h ---------._.a_ D.ZSJ O >',5 =3 3"1~51 3-t5 L 3"1 7 _]: LANDNG 5P F. F. D = O,',-,.p.h. D-?.Sfl L.~NDNG SPE~'D = lo~.p.~ 2%.c, P_ y_ %J%5.5 \ LANDNG SPEED = 1 ~.p,h. ~ 7 ~- 3"O?-~.P_, 't -5 ~ t.o -5 L.ANb,Nc, 5PEED= 1SO ~.p.h. FG. 3. Variation of 1, with y, for lm = 2. o-5 j~.o 1.5 FG. 4. Variation of ~, with y, for,%,, = 3.
'~xe, i.o ">xs.5 _ j l "~-5 /..--- / / '75 9.i2_5] T O '>, LANDNG SPEED =,'~.p.h,,, i 3OlSJ 1.,',,, ~ ">,m "5 / f j l / / f 4 ] -751 ---- r~ R., t~. D.Ogsj X~:z Fo 2. LANDNG 5 P ~. E ~ )'~5j ~s ' O'5 f.,-r~1 / j l f l'-"'- j----._..---.195 ] "75 r = R~ ~oo O'O~5J.4- _ANDNC~ 5P~EO = 1m.h o:ol~j " 1. / f J / / "~ 1.5. 5 f O-O~_5J FG. 5. Variation of & with F, for l,, = 4. o ~lo tbo ts v \ J FG. 6. Variation of ;~s/~m with V for ff = 1-.
~. # ~'Xm: }lab t' / t D-7. ~ ~r R~ m O-2. ~ 2. # )xm'2 t' F i.~5) 2" # ' i 3"t~51 >o~51 x 3"a5] ~t~ ~' # >-,~ = 4 - lo ~!,_ANDN~ SPEE[~ V (m,p~.) FG. 7. Variation of ~ with V for,~ =,~,,,. (222) Wt. 5-6 K9 1/52 F.M. & S 13 PRNTED N GREAT BRTAN
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