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 7 Equations of motion Mark Voskuijl Semester Delft University of Technology Challenge the future
3 Time schedule Date Time Hours Topic 4 Sep , 2 Unsteady climb 6 Sep , 4 Minimum time to climb 11 Sep , 6 Turning performance 13 Sep , 8 Take off 18 Sep , 10 Landing 20 Sep , 12 Cruise 25 Sep , 14 Equations of motion (wind gradient) 27 Sep , 16 Kepler orbits, gravity, Earth-repeat orbits, sunsynchronous orbits, geostationary satellites 2 Oct , 18 Third-body perturbation, atmospheric drag, solar radiation, thrust 4 Oct , 20 Eclipse, maneuvers 9 Oct , 22 Interplanetary flight 11 Oct , 24 Interplanetary flight 16 Oct , 26 Launcher, ideal vs. real flight, staging, design 18 Oct , 28 Exam practice AE2104 Flight and Orbital Mechanics 2
4 AE2104 Flight and Orbital Mechanics 3
5 AE2104 Flight and Orbital Mechanics 4
6 AE2104 Flight and Orbital Mechanics 5
7 Content Introduction Axis systems and Euler angles Vector / matrix notation Accelerations Forces General equations of motion 3D flight Effect of a wind gradient AE2104 Flight and Orbital Mechanics 6
8 Introduction Newton s laws Newton s laws only hold with respect to a frame of reference which is in absolute rest. This is called an inertial frame of reference Coordinate systems translating uniformly to the frame of reference in absolute rest are also inertial frames of reference A rotating frame of reference is not an inertial frame of reference AE2104 Flight and Orbital Mechanics 7
9 Introduction Objective Derivation of equations of motion General 3 dimensional flight 2 dimensional flight with a wind gradient General approach! AE2104 Flight and Orbital Mechanics 8
10 Content Introduction Axis systems and Euler angles Vector / matrix notation Accelerations Forces General equations of motion 3D flight Effect of a wind gradient AE2104 Flight and Orbital Mechanics 9
11 Axis systems and Euler angles Earth axis system Earth axis system: {E g } 1. X g axis in the horizontal plane, orientation is arbitrary 2. Y g axis in the horizontal plane, orientation: perpendicular to X g 3. Z g axis points downwards AE2104 Flight and Orbital Mechanics 10
12 Axis systems and Euler angles Assumption Assumption 1: the earth is flat 'Centrifugal force' 2 W V C g R h 2 C V W R h g e Example e V 100 [m/s] R 6371 [km] e g [m/s ] h 0 [m] C W (0.016%) Valid assumption AE2104 Flight and Orbital Mechanics 11
13 Axis systems and Euler angles Assumptions Assumption 2: the earth is non-rotating AE2104 Flight and Orbital Mechanics 12
14 AE2104 Flight and Orbital Mechanics 13
15 Axis systems and Euler angles Moving earth axis system Moving earth axis system: {E e } 1. X e parallel to X g axis but attached to c.g. of aircraft 2. Y e parallel to Y g axis but attached to c.g. of aircraft 3. Z e axis points downwards AE2104 Flight and Orbital Mechanics 14
16 Axis systems and Euler angles Body axis system Body axis system: {E b } 1. Origin is fixed to the aircraft c.g. 2. X b lies in plane of symmetry and points towards the nose 3. Y b is perpendicular to the plane of symmetry and is directed to the right wing 4. Z b is perpendicular to X b and Y b AE2104 Flight and Orbital Mechanics 15
17 Axis systems and Euler angles Yaw angle (body axis) AE2104 Flight and Orbital Mechanics 16
18 Axis systems and Euler angles Pitch angle (body axis) AE2104 Flight and Orbital Mechanics 17
19 Axis systems and Euler angles Roll angle (body axis) AE2104 Flight and Orbital Mechanics 18
20 Axis systems and Euler angles Air path axis system Air path axis system: {E a } 1. Origin is fixed to the aircraft c.g. 2. X a lies along the velocity vector 3. Z a taken in the plane of symmetry of the airplane 4. Y a is positive starboard AE2104 Flight and Orbital Mechanics 19
21 Axis systems and Euler angles Azimuth angle (air path axis) AE2104 Flight and Orbital Mechanics 20
22 Axis systems and Euler angles Flight path angle (air path axis system) AE2104 Flight and Orbital Mechanics 21
23 Axis systems and Euler angles Aerodynamic angle of roll (air path axis system) AE2104 Flight and Orbital Mechanics 22
24 Axis systems and Euler angles Summary Four axes systems can be defined Earth Moving Earth Body axes Air path axes Three Euler angles define the orientation of the aircraft (body axes) Yaw Pitch Roll Three Euler angles define the orientation of the aircraft (body axes) Azimuth Flight path Aerodynamic roll The sequence of the Euler angles is very important!!! AE2104 Flight and Orbital Mechanics 23
25 AE2104 Flight and Orbital Mechanics 24
26 Content Introduction Axis systems and Euler angles Vector / matrix notation Accelerations Forces General equations of motion 3D flight Effect of a wind gradient AE2104 Flight and Orbital Mechanics 25
27 Vector / matrix notation Z r xi yj zk z k r P i, j and k are unit vectors along the axes Y i X x AE2104 Flight and Orbital Mechanics 26
28 Vector / matrix notation i r x i y j z k x y z j k x y z E : Row : Column : Square matrix AE2104 Flight and Orbital Mechanics 27
29 Content Introduction Axes systems and Euler angles Vector / matrix notation Accelerations Forces General equations of motion 3D flight Effect of a wind gradient AE2104 Flight and Orbital Mechanics 28
30 Accelerations a dv dt V V 0 0Ea? dv dt a a V 0 0 E V 0 0 E What is the time derivative of the air path axis system? AE2104 Flight and Orbital Mechanics 29
31 Accelerations Time derivative of the air path axis system -k y di 0i z j y k dt x X a dj z i 0 j x k i dt j z j dk yi x j 0 k y dt z k -i z Y a k 0 x z y Z a E 0 0 y x -j x i y E a z x a AE2104 Flight and Orbital Mechanics 30
32 Accelerations dv dt dv dt dv dt a a V 0 0 E V 0 0 E 0 z y V 0 0 E V E 0 y x V V V E a z x a z y a AE2104 Flight and Orbital Mechanics 31
33 Content Introduction Axes systems and Euler angles Vector / matrix notation Accelerations Forces General equations of motion 3D flight Effect of a wind gradient AE2104 Flight and Orbital Mechanics 32
34 Forces F L D T W F T D L E W E a e No sideslip Assume thrust in direction of airspeed vector AE2104 Flight and Orbital Mechanics 33
35 Forces Sideslip angle AE2104 Flight and Orbital Mechanics 34
36 Forces Sideslip angle AE2104 Flight and Orbital Mechanics 35
37 Forces Sideslip angle AE2104 Flight and Orbital Mechanics 36
38 Forces F L D T W F T D L E W E a e Problem: different axis systems Express all forces in 1 axis system AE2104 Flight and Orbital Mechanics 37
39 Forces Transformation matrices {E e } {E a } AE2104 Flight and Orbital Mechanics 38
40 Forces Rotation over azimuth angle () X e X 1 i i cos j 1 j i sin j k 1 1 e k e e e sin e cos i e i 1 j 1 j e Y 1 Y e i1 cos sin 0 ie j 1 sin cos 0 je k 0 0 1k 1 e E T E 1 e AE2104 Flight and Orbital Mechanics 39
41 Forces Rotation over flight path angle () i 2 i 1 X 2 X 1 i i cos k j2 j1 k i sin k sin cos k 1 Z 1 k 2 Z 2 E T E 2 1 cos 0 sin T sin 0 cos AE2104 Flight and Orbital Mechanics 40
42 Forces Rotation over aerodynamic angle of roll () k a j 2 j a Y a Y 2 ia i2 ja j cos k ka j sin k sin cos Y a k 2 Z 2 E T E a T 0 cos sin 0 sin cos AE2104 Flight and Orbital Mechanics 41
43 Forces Transformation matrices {E e } {E a } AE2104 Flight and Orbital Mechanics 42
44 Forces Transformation matrices E T E 1 E T T E 2 E T T T E a E T T T E a e 1 1 E T T T 1 E e e e e a AE2104 Flight and Orbital Mechanics 43
45 Forces Properties of transformation matrices 1 T 1 1 I AE2104 Flight and Orbital Mechanics 44
46 Forces All results combined F L D T W T D 0 LE W 0 0 1E a T T T F T D L E W T T T E a e a T T T cos sin 0 cos 0 sin W Ee W E sin 0 cos 0 sin cos sin cos cos sin sin cos sin cos cos W E W W W E e a a F T D W sin W cos sin L W cos cos Ea AE2104 Flight and Orbital Mechanics 45
47 Equations of motion F ma F T D W sin W cos sin L W cos cos Ea dv a V V V E dt z y a T D W sin mv W cos sin mv L W cos cos mv 3 equations of motion! z y AE2104 Flight and Orbital Mechanics 46
48 Equations of motion Rewrite in traditional form T D W sin mv W cos sin mv z L W cos cos mv y Using transformation matrices to convert x, y and z mv T D W sin mv cos L sin mv L cos W cos AE2104 Flight and Orbital Mechanics 47
49 Equations of motion Conversion x, y, z to d/dt, d/dt, d/dt 0 0 E 1 1 E T T 1 E 1 a cos 0 sin sin 0 cos 0 sin cos cos sin E sin cos sin cos cos a E a AE2104 Flight and Orbital Mechanics 48
50 Equations of motion Conversion x, y, z to d/dt, d/dt, d/dt 0 0 E 2 E T 1 E 2 a sin cos cos sin E 0 cos sin a E a AE2104 Flight and Orbital Mechanics 49
51 Equations of motion Conversion x, y, z to d/dt, d/dt, d/dt E E 3 a 0 0 E 3 E 0 0 a AE2104 Flight and Orbital Mechanics 50
52 Equations of motion Conversion x, y, z to d/dt, d/dt, d/dt x y z E a sin cos sin cos cos cos sin E a AE2104 Flight and Orbital Mechanics 51
53 Content Introduction Axes systems and Euler angles Vector / matrix notation Accelerations Forces General equations of motion 3D flight Effect of a wind gradient AE2104 Flight and Orbital Mechanics 52
54 Equations of motion Final result mv T D W sin mv cos Lsin mv L cos W cos AE2104 Flight and Orbital Mechanics 53
55 Content Introduction Axes systems and Euler angles Vector / matrix notation Accelerations Forces General equations of motion 3D flight Effect of a wind gradient AE2104 Flight and Orbital Mechanics 54
56 Effect of a wind gradient Question An aircraft is flying from A to B over a distance of 1000 nautical miles. The True Airspeed of this aircraft is 120 knots (1kt = 1 nautical mile per hour). The aircraft is experiencing a constant headwind of 20 kts. How long does it take to fly from A to B? A. Less than 10 hours B. 10 hours C. More than 10 hours AE2104 Flight and Orbital Mechanics 55
57 Effect of a wind gradient AE2104 Flight and Orbital Mechanics 56
58 Effect of a wind gradient AE2104 Flight and Orbital Mechanics 57
59 Effect of a wind gradient r V g (ground speed) X a V w (wind speed) V (airspeed) r X g Z g V V V g w AE2104 Flight and Orbital Mechanics 58
60 Effect of a wind gradient Wind is not constant: dv W dt 0 This lecture: only horizontal wind dv W dt f( H) AE2104 Flight and Orbital Mechanics 59
61 Effect of a wind gradient Absolute acceleration a V g So we need to define the velocity first V V V g g w V V E V E g a w g The acceleration can be determined by taking the time derivative a V V V w d d a V E V E dt dt a w g a a w g w g a V E V E V E V E AE2104 Flight and Orbital Mechanics 60
62 a a w g w g a V E V E V E V E Ea Eg What are and??? The ground axis system is at rest E g 0 However, the air path axis system is rotating and translating AE2104 Flight and Orbital Mechanics 61
63 Effect of a wind gradient E a d dt d dt d dt i j k d i 0 i 0 j k dt d j 0 i 0 j 0 k a dt d k i 0 j 0 k dt d i dt 0 0 i 0 0 d E j j E dt 0 0 k 0 0 d k dt a AE2104 Flight and Orbital Mechanics 62
64 Effect of a wind gradient a a w g w g a V E V E V E V E Fill in the results 0 0 a V E V E V E 0 0 Write out a a w g a a w g a V E V E V E Simplify a w g a V V E V E Two axis systems AE2104 Flight and Orbital Mechanics 63
65 Effect of a wind gradient a w g a V V E V E i cos i 0 j sin k g a a a j 0i 1 j 0k g a a a k sin i 0 j cos k g a a a cos 0 sin E g Ea sin 0 cos cos 0 sin a V V E V E sin 0 cos a w a AE2104 Flight and Orbital Mechanics 64
66 Effect of a wind gradient cos 0 sin a V V E V E sin 0 cos a w g 0 a w cos 0 w sin a a V V E V V E w cos 0 w sin a a V V V V E AE2104 Flight and Orbital Mechanics 65
67 Effect of a wind gradient L T X b V X a D X e X g W Z e Z a Z b Assumption: T // V sin 0 cos F T D W i j L W k a a a sin 0 cos a F T D W L W E Z g AE2104 Flight and Orbital Mechanics 66
68 Effect of a wind gradient F mv g sin 0 cos a w cos 0 w sin a F T D W W L E a V V V V E W T D W sin V V g 0 0 W L W cos V Vw sin g w cos AE2104 Flight and Orbital Mechanics 67
69 Effect of a wind gradient H RC V sin s V cos V w AE2104 Flight and Orbital Mechanics 68
70 Effect of a wind gradient AE2104 Flight and Orbital Mechanics 69
71 AE2104 Flight and Orbital Mechanics 70
72 Questions? AE2104 Flight and Orbital Mechanics 71
Flight and Orbital Mechanics
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