Flight and Orbital Mechanics
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1 Flight and Orbital Mechanics Lecture slides Challenge the future 1
2 Flight and Orbital Mechanics Lecture hours 3, 4 Minimum time to climb Mark Voskuijl Semester Delft University of Technology Challenge the future
3 Content Introduction Summary previous lecture Performance diagram Optimal climb Solution low speed aircraft Energy height Solution high speed aircraft (M>1) Example exam question Summary AE2104 Flight and Orbital Mechanics 2
4 Content Introduction Summary previous lecture Performance diagram Optimal climb Solution low speed aircraft Energy height Solution high speed aircraft (M>1) Example exam question Summary AE2104 Flight and Orbital Mechanics 3
5 Introduction Question What is the most efficient way (minimum time) to go from take-off at sea-level to Mach 1.5 at 15,000 m? A. Climb at airspeed for max. At 15,000 m accelerate to Mach 2 B. Climb at airspeed for max RC. At 15,000 m accelerate to Mach 2 C. Climb at at airspeed for for max max RC to RC 11,000 to 11,000 m. Descent m. and accelerate Descent and to Mach accelerate 1.2 at 9,000 to Mach m, climb 1.2 at and 9,000 accelerate m, to climb Mach and 2 at accelerate 11,000 m. Decelerated to Mach 2 at climb 11,000 to 15,000 m. m, Mach Decelerated 1.5 climb to 15,000 m, Mach 1.5 D. Accelerate at sea-level to Mach 1.5, climb to 15,000 m at airspeed for max RC AE2104 Flight and Orbital Mechanics 4
6 Introduction Solution Altitude [m] M = Subsonic M = 2 M = 1.2 (Supersonic) 0 AE2104 Flight and Orbital Mechanics 5
7 Content Introduction Summary previous lecture Performance diagram Optimal climb Solution low speed aircraft Energy height Solution high speed aircraft (M>1) Example exam question Summary AE2104 Flight and Orbital Mechanics 6
8 Summary previous lecture A typical climb (civil subsonic aircraft) is performed at constant indicated airspeed and at a constant power setting. Therefore, the true airspeed is actually increasing. Since airspeed is not constant, it is an unsteady climbing flight The climb is almost a straight line. It is therefore a quasi-rectilinear flight Corresponding equations of motion: W dv T D W sin g dt L W AE2104 Flight and Orbital Mechanics 7
9 Summary previous lecture The rate of climb in an unsteady climb is smaller than in a steady climb because part of the excess energy is used to accelerate RC RC st 1 V dv 1 g dh dv ; 0 dh You must be able to derive this ratio from the equations of motion You must be able to calculate this ratio (see example exam question) For more background information: read Ruijgrok Elements of airplane performance section 14.2 AE2104 Flight and Orbital Mechanics 8
10 Content Introduction Summary previous lecture Performance diagram Optimal climb Energy height Solution low speed aircraft Solution high speed aircraft (M>1) Example exam question Summary AE2104 Flight and Orbital Mechanics 9
11 P Performance diagram P a W P r RC steady P a P r V AE2104 Flight and Orbital Mechanics 10
12 Aerodynamic forces (year 1) Lift Drag polar C D C D 0 C 2 L Ae AE2104 Flight and Orbital Mechanics 11
13 Aerodynamic forces (year 1) Lift L W 1 2 C V S W C L 2 W 2 1 S V L 2 Drag C D C D 0 C 2 L Ae 1 2 D C V S D C 1 L 1 2 D C D V S V S Ae W D C D V S V S S V Ae W D C D V S Ae V S 2 So, one part of the drag decreases (!) with airspeed (1/V 2 ) and one part increases with airspeed (V 2 ) AE2104 Flight and Orbital Mechanics 12
14 Aerodynamic forces (year 1) Drag as a function of airspeed Aircraft are quite unique in the sense that drag increases when airspeed decreases! D D D D0 C D V S 2 2 W Di Ae 0 i V S AE2104 Flight and Orbital Mechanics 13
15 AE2104 Flight and Orbital Mechanics 14
16 P P a Pa Pr W RC steady P r V V 1 V 2 V 3 V 4 RC steady Determine magnitude of difference and divide by aircraft weight (W) V V 1 V 2 V 3 AE2104 Flight and Orbital Mechanics 15
17 Performance diagram How does it change with altitude? Note: this sheet is from the first year lecture AE2104 Flight and Orbital Mechanics 16
18 P P a Pa Pr W RC steady P r V 3 V 4 V RC steady H 1 H 2 V AE2104 Flight and Orbital Mechanics 17
19 Performance diagram Rate of climb as a function of airspeed and altitude AE2104 Flight and Orbital Mechanics 18
20 Performance diagram Rate of climb as a function of airspeed and altitude H RC = 6 m/s RC = 2 m/s V AE2104 Flight and Orbital Mechanics 19
21 Content Introduction Summary previous lecture Performance diagram Optimal climb Energy height Solution low speed aircraft Solution high speed aircraft (M>1) Example exam question Summary AE2104 Flight and Orbital Mechanics 20
22 Optimal climb During climb at constant indicated airspeed, the aircraft is accelerating. V(H) is fixed Easy flight technique for the pilot But is this optimal??? What is optimal? AE2104 Flight and Orbital Mechanics 21
23 Optimal climb What is optimal? Minimum time to climb (time) Minimum fuel consumed during climb (fuel) Distance covered during climb (distance) Noise during climb (noise) AE2104 Flight and Orbital Mechanics 22
24 Optimal climb What is optimal? Minimum time to climb (time) Minimum fuel consumed during climb (fuel) Distance covered during climb (distance) Noise during climb (noise) We will focus on the first option, but the other three are optimal as well AE2104 Flight and Orbital Mechanics 23
25 Optimal climb Objective: to find the true airspeed as a function of altitude that yields the minimum time to climb dh RC dt dt t dt H H 2 1 dh RC dh RC Time t is not minimal if the integrand is minimal at every altitude H because the term dv/dh is in the integrand Variational calculus is necessary RC RC steady 1 1 V dv g dh 0 t H H 2 1 V dv 1 g0 dh RC steady dh AE2104 Flight and Orbital Mechanics 24
26 Variational calculus? t H H 2 1 V dv 1 g0 dh RC steady dh Minimize integrand: RC steady maximum dv/dh <0 dv/dh<0 RC steady max At H = 0, RC st is maximal at V = 100 The integrand can be minimized more by choosing (dv/dh) 1 < 0 Choosing maximum RC steady at sea level At seems altitude like a H good > 0 starting no point optimum V can be chosen Conclusion: No local optimum but global optimum (consider complete flight path) AE2104 Flight and Orbital Mechanics 25
27 Optimal climb H H 2 t 1 V dv 1 g0 dh RC steady dh Solution Simplify (low speed aircraft) Energy method (high subsonic / supersonic aircraft) AE2104 Flight and Orbital Mechanics 26
28 Content Introduction Summary previous lecture Performance diagram Optimal climb Solution low speed aircraft Energy height Solution high speed aircraft (M>1) Example exam question Summary AE2104 Flight and Orbital Mechanics 27
29 Low speed aircraft AE2104 Flight and Orbital Mechanics 28
30 Low speed aircraft t H dh RC H 0 0 V dv 1 g0 dh RC st dh Assumption: Low speed, low altitude dv/dh is small t H 0 dh RC st Choose maximum steady rate of climb at each altitude AE2104 Flight and Orbital Mechanics 29
31 Low speed aircraft W dv T D W sin g dt Pa Pr W dv RC V W g dt Pa Pr RC W steady P H 1 H RC steady 2 H V V AE2104 Flight and Orbital Mechanics 30
32 Low speed aircraft Solution propeller aircraft RC P a st P P a r W can be assumed constant as function of airspeed for propeller aircraft 3 V RC P C C Ae opt st max r,min L, opt D0 W S C L, opt Note: this optimal C L was determined in the first year lectures! P a Corresponding V V V constant because V V e i i e 0 constant P r V P r,min AE2104 Flight and Orbital Mechanics 31
33 Low speed aircraft Summary Conclusion: if the pilot selects the airspeed for the maximum steady rate of climb at the ground and if he / she keeps the indicated airspeed constant, then the climb will be optimal Optimal in this case means minimum time to climb (assuming dv/dh is small) AE2104 Flight and Orbital Mechanics 32
34 Low speed aircraft Story World War I AE2104 Flight and Orbital Mechanics 33
35 Low speed aircraft Story World War I Allied airfoils (British / French) versus German airfoils Thin airfoils have low C L,max Therefore, German aircraft were able to fly slower British / French) versus German airfoils AE2104 Flight and Orbital Mechanics 34
36 Low speed aircraft Story World War I P r P a Numerical example C 3C Ae L D o CD A 6 e 0.8 C V L, opt E V 1.26 (large!) ground W 2 1 S C 0 L, opt V min (thick profiles) V min (thin profiles) AE2104 Flight and Orbital Mechanics 35
37 Content Introduction Summary previous lecture Performance diagram Optimal climb Solution low speed aircraft Energy height Solution high speed aircraft (M>1) Example exam question Summary AE2104 Flight and Orbital Mechanics 36
38 AE2104 Flight and Orbital Mechanics 37
39 Energy height Concept to solve the minimum time to climb problem Altitude and flight speed are rapidly interchangeable (exchange of kinetic energy and potential energy) Increasing the total energy is much slower because it depends on the excess power 1 2 1W 2 Energy 2mV mgh 2 V WH g H Line of constant energy V AE2104 Flight and Orbital Mechanics 38
40 Energy height H H e e Energy W 2 V H 2g D B A C AE2104 Flight and Orbital Mechanics 39
41 Energy height H Large energy Small energy Optimal combination of V and H to increase the energy So: Minimum time to climb problem approximated by another problem: find V opt (H) at which minimum time to total energy V AE2104 Flight and Orbital Mechanics 40
42 Energy height Energy method Energy height E mg H mv W E WH 2 V g H e 0 E H W 2 V 2g dhe dh 1 dv dt dt 2g dt dhe Pa Pr dt W RC steady Equation of Motion W dv T D W sin g dt 0 1 dv sin T D g dt W 0 V dv Pa P V r sin g dt W dv dh Pa Pr 2g dt dt W 0 AE2104 Flight and Orbital Mechanics 41
43 Minimum time to climb dh dt e RC s dt dh RC e st t H e2 H e1 dh RC e st (time to energy height) AE2104 Flight and Orbital Mechanics 42
44 Optimum path subsonic aircraft (for each constant energy height line, find the maximum steady rate of climb) AE2104 Flight and Orbital Mechanics 43
45 Content Introduction Summary previous lecture Performance diagram Optimal climb Solution low speed aircraft Energy height Solution high speed aircraft (M>1) Example exam question Summary AE2104 Flight and Orbital Mechanics 44
46 Solution high speed aircraft Excess power, h 1 <h 2 <h 3 h 1 ft/sec h 2 h 3 Altitude h Mach AE2104 Flight and Orbital Mechanics 45
47 Supersonic aircraft Dip in excess power is caused by transonic drag rise: 0 C C M k M C 2 D D L supersonic aircraft High subsonic aircraft AE2104 Flight and Orbital Mechanics 46
48 Solution high speed aircraft AE2104 Flight and Orbital Mechanics 47
49 Solution high speed aircraft Example performance Mig-29: Climb from sea level to 6000 m in less than 1 minute AE2104 Flight and Orbital Mechanics 48
50 Content Introduction Summary previous lecture Performance diagram Optimal climb Solution low speed aircraft Energy height Solution high speed aircraft (M>1) Example exam question Summary AE2104 Flight and Orbital Mechanics 49
51 Example exam question AE2104 Flight and Orbital Mechanics 50
52 Example exam question AE2104 Flight and Orbital Mechanics 51
53 Content Introduction Summary previous lecture Performance diagram Optimal climb Solution low speed aircraft Energy height Solution high speed aircraft (M>1) Example exam question Summary AE2104 Flight and Orbital Mechanics 52
54 Summary AE2104 Flight and Orbital Mechanics 53
55 Questions? AE2104 Flight and Orbital Mechanics 54
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