FlightGlobal.com
Home
Premium
Archive
Video
Images
Forum
Atlas
Blogs
Jobs
Shop
RSS
Email Newsletters
You are in:
Home
Aviation History
1913
1913 - 1022.PDF
I/UCHT1 Ju,lhu l^ ,u Vhe calculati™ » appeared that it was an easy Z^^X^nF^T ^ **» "d *° '^y were worked' J^HL^ ,ca'culation was made to ascertain if it would be advantageous to flatten out more gradually, and the results are shown U.r„« rt. -i . case'after diving at an incidence of - i° as v^locitV was ra f7 rT PUt ? earHer' namel>r at 2 secs- when the )wlr£ 7SV I Per.,sec> but not quite so hard, giving an inci- tZln L',Td th? P1^! WaS SUPP°sed to maintain th>S angle of 9° fh.fi™ ^ eDd- T^ t0tal vertical fal1 has been »S° "•. and the; final velocity 91 "I,, while the wing stresses have been much less. me!h«l ihh IV «I0U,S'y morc advantag^«s than the former method although the final velocity is less. Later on I shall discuss the exact allowance to be made for this difference of velocity. .hi «?* vg ^en 5Ui Sh°Wn that the vertical fal1 needed to recover desirahWn dep^ded °n 'he method °f diving adopted, it became des.rable to consider whether it was possible to ascertain mathemati- LB5 3OO0 3000 aooo 1000 H 0. k k * 5 8. O cally the /fer/ method of diving so as to give the least vertical fall possible. This has to be done by an application of the calculus of variations. For this purpose it is necessary to be able to express the lift and drift accurately by mathematical formulae. The ordinary formulae for a plane, lift = KSV2 sin 6 cos Sand drift = KSV2 sin-0 + HV-', are very inaccurate for a cambered wing, even if 8 is measured from the neutral axis. They can be made more accurate by giving different values to K for the lift and drift, but even so lead to errors of as much as 10 per cent, or more for very small and large angles of incidence. After considerable labour I have discovered that for the aeroplane BE 2, both lift and drift may be represented with great accuracy in the forms, lift = /sin 3 6 V2, and drift = ds\tP 0V2 + /2V2, when 8 is measured from the neutral axis, and hence = a + 2° 30'. The closeness of agreement may be seen from Fig. 3, where the lift and drift curves are those obtained from the Technical Report (the portions below 20 being filled in by comparison with the diagrams opposite p. 67 of the Report), while the small crosses show points given by the formula?: Lift in lbs. per lb. mass of aeroplane at a velocity of 1 ft. per sec. = '000421 sin 3 (a + 2° 30'), and drift = "0013 sin" (a + 2'30') + •0000179, and consequently the equations of motion become— £Y = g sin 4. - DV= sin3* - HV2 . and^ = g~^ - LV sin 38. at at V where 8 = a + 2° 3o'-and D, H, and L have the values given above multiplied hy g. A prolonged investigation by the calculus of variations gives the following equation to determine the best method of diving : d8 _g sin 8 cos 39_cos 2<p _, sin <p cot (p dt 2V cos20 **. ' , !™*cos* - t^fa*8***^ HL coss3* L V cos. Y cos2e * + _D"Vsir7T»cos2S wh.ch together with the two former equations completely determine the motion when the initial conditions are given u«e™me lJ^f?nnkcing a nUmerical oblation from these equations, certain limitations are necessary. The angle of incidence must not exceed the critical angle of 13* , neither must it be less than -2=30' 1 (i.e., 8=0) for that would give a downward pressure on the planes | IO48 SEPTEMBER 20, 1913. and jerk the pilot out of his seat. Moreover the above formula for the drift does not hold when 8 is negative, as sin:!0 changes sign, and dB ,, may become discontinuous when 8 = o as well as when <j> = o. The numerical solution is very laborious, and when choosing the initial value of 8 one cannot foresee the final result, so that one has to make several abortive calculations before obtaining one leading to approximately the desired conclusion. Fig. 4 shows the results of one calculation, giving a final velocity of 86 "6 ft. per sec. and vertical fall of 125 ft. 00048 00 00 as iooooos The change of the angle of incidence a is also plotted. The aeroplane starts with a velocity of 50 ft. per sec. in a horizontal direction as before. The equations show that the angle of incidence ought initially to be a large negative quantity, but as this is impossible the pilot must put his elevator down sufficiently to bring it to the limit of — 2° 30' and keep it so for f sec. The angle of incidence is then increased, rapidly at first and then more and more gradually, ending with a final value of n£°. In actually performing this manoeuvre the moment of inertia must be taken into account by the pilot. At the beginning the course of the aeroplane is curving rapidly downwards, and the aeroplane which was flying horizontally must
Sign up to
Flight Digital Magazine
Flight Print Magazine
Airline Business Magazine
E-newsletters
RSS
Events
