Mathematics of Flight. Glide Slope II

Transcription

1 Mthemtics of Flight Glide Slope II

2 A glider is specil kind of ircrft tht hs no engine. The Wright brothers perfected the design of the first irplne nd gined piloting experience through series of glider flights from 1900 to With Wilbur Wright t the controls, Dn Tte, left, nd Edwrd C. Huffker lunch the 1901 Wright Glider t Kitty Hwk, N.C. Credit: Librry of Congress, courtesy Ntionl Air nd Spce Museum, Smithsonin Institution

3 During World Wr II, gliders such s the WACO CG-4 were towed loft by C-47 nd C-46 ircrft then cut free to glide over mny miles.

4 If glider is in stedy (constnt velocity nd no ccelertion) descent, it loses ltitude s it trvels. The glider's flight pth is simple stright line, shown in the figure bove. The flight pth intersects the ground t n ngle clled the glide ngle. If we know the distnce flown nd the ltitude chnge, the glide ngle cn be clculted using trigonometry. The tngent tn of the glide ngle is equl to the chnge in height h divided by the distnce flown d: tn() = h / d

5 h d = horizontl distnce flown h = chnge in height = glide ngle d From trigonometry: tn() = h d

6 There re three forces cting on the glider; lift, weight, nd drg. The weight of the glider is given by the symbol "W" nd is directed verticl, towrd the center of the erth. The weight is then perpendiculr to the horizontl red line drwn prllel to the ground nd through the center of grvity. The drg of the glider is designted by "D" nd cts long the flight pth opposing the motion. Lift, designted "L," cts perpendiculr to the flight pth. Using some geometry theorems on ngles, perpendiculr lines, nd prllel lines, we see the glide ngle "" lso defines the ngle between the lift nd the verticl, nd between the drg nd the horizontl.

7 D Verticl L Horizontl W L = Lift D = Drg W = Weight = glide ngle Verticl Eqution: L cos() + D sin() = W Horizontl Eqution: L sin() = D cos() sin() = tn () = D cos() L *Assuming velocity is constnt

8 Assuming tht the forces re blnced (no ccelertion of the glider), we cn write two vector component equtions for the forces. In the verticl direction, the weight (W) is equl to the lift (L) times the cosine (cos) of the glide ngle () plus the drg (D) times the sine (sin) of the glide ngle. L * cos() + D * sin() = W In the horizontl direction, the lift (L) times the sine (sin) of the glide ngle () equls the drg (D) times the cosine (cos) of the glide ngle. L * sin() = D * cos()

9 Lift = pressure fctor x velocity squred x wing re x lift fctor Lift coefficient cl is the rtio of the object s lift to the drg of the perpendiculr flt plte with equl re. Smeton s coefficient k is the drg of 1 squre foot flt plte t 1 mile per hour ccepted vlue =.005, modern ccepted vlue =.00326

10 D Verticl L Horizontl W cl = Lift coefficient cd = Drg coefficient D = cd k V 2 A = cd = tn () L cl k v 2 A cl

11 If we use lgebr to re-rrnge the horizontl force eqution we find tht the drg divided by the lift is equl to the sine of the glide ngle divided by the cosine of the glide ngle. This rtio of trigonometric functions is equl to the tngent of the ngle. D / L = sin() / cos() = tn() We cn use the drg eqution nd the lift eqution to relte the glide ngle to the drg coefficient (cd) nd lift coefficient (cl) tht the Wrights mesured in their wind tunnel tests. D / L = cd / cl = tn()

12 Wright Brothers 1901 Wind Tunnel This is replic of the wind tunnel designed nd built by the Wright Brothers in the fll of 1901 to test irfoil designs. The blower fn, driven by n overhed belt, produced 25 to 35 mph wind for testing the lift of vrious plnes nd curved surfces. Aerodynmic tbles derived from these tests were vitl to the successful design of the Wright 1903 Kitty Hwk irplne. Inside the tunnel is model of Wright lift blnce used to mesure the lift of test surfce. The wind tunnel replic ws constructed under the personl supervision of Orville Wright prior to World Wr II.

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14 During the opertion of the drg blnce the brothers mde mesurements of the effects of wing design on glide ngle through the drg to lift rtio. The inverse of the drg to lift rtio is the L/D rtio which is n efficiency fctor for ircrft design. The higher the L/D, the lower the glide ngle, nd the greter the distnce tht glider cn trvel cross the ground for given chnge in height.

15 With Wilbur Wright t the controls, Dn Tte, left, nd Edwrd C. Huffker lunch the 1901 Wright Glider t Kitty Hwk, N.C. Credit: Librry of Congress, courtesy Ntionl Air nd Spce Museum, Smithsonin Institution

16 Wilbur Wright gliding in The Wrights dded verticl til to their glider to del with the lterl control problems experienced in The more grceful ppernce of the 1902 mchine over the previous gliders is evident in this picture. Credit: Ntionl Air nd Spce Museum, Smithsonin Institution

17 Compre the 1901 nd 1902 Wright gliders Glider Weight Distnce Glide Angle Height lbs 300 feet feet lbs 500 feet 7? For the 1901 Wright Glider: sin() = BC/AB AB = distnce flown = 300 ft = glide ngle BC = height sin() = BC/ 300 ft.1564 = BC/300 ft 46.9 ft = BC B C A

18 Compre the 1901 nd 1902 Wright gliders Glider Weight Distnce Glide Angle Height lbs 300 feet feet lbs 500 feet 7? For the 1902 Wright Glider: sin() = BC/AB AB = distnce flown = 500 ft = glide ngle BC = height sin() = BC/ 500 ft.1219 = BC/500 ft 60.9 ft = BC

19 D Verticl L Horizontl W L = Lift D = Drg W = Weight = glide ngle Verticl Eqution: L cos() + D sin() = W Horizontl Eqution: L sin() = D cos() sin() = tn () = D cos() L

20 For the 1901 glider: L * cos() + D * sin() = W L x D x.1564 = L D = 98 The lift times the sine of the glide ngle is equl to the drg times the cosine of the glide ngle. L * sin() = D * cos() L *.1564 = D * L =.9877D L = D.9877 Now, substitute for D,.9877L (.1564) =

21 More Resources Additionl Resources re vilble online t