Dynamics cheat sheet

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Algorithm 3 Hohmann rendezvous algorithm, case 2 1: function hohmann rendezvous 2(θ 0 , r 1 , r 2 , N, µ) π 2: θ 0 := θ 0 180 ( ⊲ convert from degrees to radian ( ) ) 3/2 3: θ H := π 1 − r1 +r 2 2r 2 ⊲ Find Hohmann ideal phase angle before transfer 4: if θ 0 = θ H and N = 0 then ⊲ adjust for special case 5: a := r 2+r 1 2 (√ 6: TOF := π ) a 3 µ 7: else 8: ω 2 := √ µ r2 3 9: t 2 := (2π−θ 0)+2πN ω 2 10: a 1 := rt+r 1 2 11: a 2 := rt+r 2 2 12: TOF := π (√ a 3 1 µ + a3 2 µ ) ⊲ angular speed of slower satellite in rad/sec ⊲ find time of light of the slower satellite ⊲ time of flight for the fast satellite 13: r t := solve (t 2 = T OF ) for r t ⊲ Solve numerically for r t 14: end if 15: return TOF 16: end function An example implementation is below 1 hohmann_rendezvous_2:= proc({ 2 theta::numeric:=0, 3 r1::numeric:=0, 4 r2::numeric:=0, 5 N::nonnegint:=0, 6 mu::numeric:=3.986*10^5}) 7 local theta0,thetaH,TOF; 8 theta0 := theta*Pi/180; 9 thetaH := Pi*(1-((r1+r2)/(2*r2))^(3/2)); 10 if theta0 = thetaH and N = 0 then 11 proc() 12 local a:=(r1+r2)/2; 13 TOF:= Pi*(sqrt(a^3/mu)); 14 end proc() 15 else 16 proc() 17 local t2,a1,a2,rt,omega2; 18 omega2 := sqrt(mu/r2^3); 19 t2 := ((2*Pi-theta0)+2*Pi*N)/omega2; 20 a1 := (rt+r1)/2; 21 a2 := (rt+r2)/2; 22 TOF := Pi*(sqrt(a1^3/mu)+sqrt(a2^3/mu)); 23 rt := op(select(is, [solve(t2=TOF,rt)], real)); 24 end proc() 25 fi; 26 eval(TOF); 27 end proc: And calling the above for two different cases gives 86
1 TOF:=hohmann_rendezvous_2(theta=0,r1=6678,r2=6878,N=0): 2 evalf(TOF/(60*60)); #in hrs 3 1.576892101 4 TOF:=hohmann_rendezvous_2(theta=160,r1=6678,r2=6878,N=1): 5 evalf(TOF/(60*60)); #in hrs 6 2.452943266 87
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Dynamics cheat sheet Nasser M. Abba
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18.7.2 Bi-Elliptic transfer orbit .
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are the natural frequencies of the
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Similarly, µ 2 is found ⎧ ⎫T
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Or in full form and ⎧ ⎫ ⎡ ⎨
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Hence The first eigen vector is Sim
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The transformation from modal coord
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Where F n = 2 T F 0 = 2 T ∫ T 0
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Let z = Re { Ze iωt} , z ′ = Re
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Now, u = Re { Ue iωt} , u ′ = Re
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Hence at t = 0 the phase of the res
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The following table gives the solut
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5.3 no loading (free) mu ′′ + c
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5.5 Impulse sin function Input is g
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Solution to single degree of freedo
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7 Some common formulas These defini
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8 Derivation of the solutions of th
Page 35 and 36: 8.2.2 under-damped ζ < 1 The gener
Page 37 and 38: When ϖ ≈ ω but less than ω, le
Page 39 and 40: Hence the full solution is u p = p
Page 41 and 42: 8.4 Response to impulsive loading 8
Page 43 and 44: where A = B = F 0 m 2ω √ ξ 2
Page 45 and 46: where ∆ is very small positive qu
Page 47 and 48: at t = t 1 ˙u (t 1 ) = −ξωe
Page 49 and 50: 8.5 references 1. Vibration analysi
Page 51 and 52: 11 Formulas sin 2 x 1cos2x 2 cos 2
Page 53 and 54: 12 Velocity and acceleration diagra
Page 55 and 56: 12.3 spring pendulum with block mov
Page 57 and 58: 13 Velocity and acceleration of rig
Page 59 and 60: 14.2 3D Not Using vehicle dynamics
Page 61 and 62: 14.2.2 Derivation for τ = Iω in 3
Page 63 and 64: 15 Wheel spinning precession x Mg
Page 65 and 66: 18 Astrodynamics 18.1 Ellipse main
Page 67 and 68: magnitude of ⃗r period T mean sat
Page 69 and 70: 18.3 Flight path angle for ellipse
Page 71 and 72: This diagram below from Orbital mec
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Page 75 and 76: 18.7.3 semi-tangential elliptical t
Page 77 and 78: 18.8.1 Rocket equation with plane c
Page 79 and 80: 18.10 interplanetary transfer orbit
Page 81 and 82: 2. Find speed of second planet V 2
Page 83 and 84: 1 mu = 324859; 2 alt = 1475.776; 3
Page 85: 18.11.3 Two satellite, separate orb
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Page 91 and 92: 18.14 Orbit changing by low contiuo
Page 93 and 94: C r below is the core chord of the
Page 95 and 96: 1 C L = ∂C L ∂α α = C Lα α
Page 97 and 98: 1 2 3 C Lwb = ∂C L wb ∂α wb α
Page 99 and 100: 2. stall from http://en.wikipedia.o
Page 101 and 102: http://en.wikipedia.org/wiki/Flight
Page 103 and 104: http://www.grc.nasa.gov/WWW/k-12/UE
Page 105 and 106: http://en.wikipedia.org/wiki/Lift_c
Page 107 and 108: http://adamone.rchomepage.com/cg_ca
Page 109 and 110: From http://www.solar-city.net/2010
Page 111 and 112: Page 184, flight dynamics principle
Page 113 and 114: Image from http://www.americanflyer
Page 115 and 116: From FAA pilot’s handbook I do no
Page 117 and 118: zero-lift angle The angle of attack
Page 119: Image from http://www.aerospaceweb.
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