Dynamics cheat sheet

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Algorithm 2 Hohmann rendezvous algorithm, case 1 1: function hohmann rendezvous 1(θ 0 , r a , r b , µ) π 2: θ 0 := θ 0 180 ⊲ convert from degrees to radian 3: a := ra+r b 2 ⊲ Hohmann orbit semi-major axes √ 4: TOF := π 5: θ H := π 6: ω a := a 3 µ ⊲ time of flight on Hohmann orbit ( ( ) ) 3/2 1 − ra+r b 2r b √ µ r 3 a ⊲ required phase angle before starting Hohmann transfer ⊲ angular speed of lower rad/sec 7: ω b := √ µ ⊲ angular speed of higher satellite rad/sec rb 3 8: if θ 0 ≤ θ H then ⊲ adjust initial angle if needed 9: θ 0 := θ 0 + 2π 10: end if 11: wait time := θ 0−θ H ω a−ω b ⊲ how long to wait before starting Hohmann transfer 12: wait time := wait time + TOF ⊲ now ready to go, add Hohmann transfer time 13: return wait time 14: end function An example implementation is below 1 hohmann_rendezvous_1:= proc({ 2 theta::numeric:=0, 3 r1::numeric:=0, 4 r2::numeric:=0, 5 mu::numeric:=3.986*10^5}) 6 local theta0,thetaH,TOF,a,omega1,omega2,wait_time; 7 theta0 := evalf(theta*Pi/180); 8 a := (r1+r2)/2; 9 TOF := Pi*(sqrt(a^3/mu)); 10 omega1 := sqrt(mu/r1^3); 11 omega2 := sqrt(mu/r2^3); 12 thetaH := evalf(Pi*(1-((r1+r2)/(2*r2))^(3/2))); 13 if theta0
18.11.3 Two satellite, separate orbits, rendezvous using bi-elliptic transfer, coplaner In this transfer, the lower (fast satellite) does not have to wait for phase lock as in the case with Hohmann transfer. The transfer can starts immediately. There is a free parameter N that one select depending on fuel cost requiments or any limitiation on the first transfer orbit semi-major axes distance required. One can start with N = 0 and adjust as needed. 85
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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
Page 33 and 34: 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
Page 73 and 74: 73
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: 1 mu = 324859; 2 alt = 1475.776; 3
Page 87 and 88: 1 TOF:=hohmann_rendezvous_2(theta=0
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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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