Dynamics cheat sheet

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14.2.3 Derivation for τ = Iω in 3D using principle axes The above derivation simplifies now since we will be using principle axes. In this case, all cross products of moments of inertia vanish. ⎛ ⎞ I xx 0 0 I = ⎜ 0 I yy 0 ⎟ ⎝ ⎠ 0 0 I zz Hence ⎡ ⎤ A A { ⎛ }} ⎞ ⎛ ⎞{ ⎛ ⎞ { ⎛ }} ⎞ ⎛ ⎞{ τ = d I xx 0 0 ω x ω x I xx 0 0 ω x dt ⎜ 0 I yy 0 ⎟ ⎜ω y ⎟ + ⎝ ⎠ ⎝ ⎠ ⎜ω y ⎟ ⎝ ⎠ × ⎜ 0 I yy 0 ⎟ ⎜ω y ⎟ ⎝ ⎠ ⎝ ⎠ ⎢ ⎣ 0 0 I zz ω z ⎥ ⎦ ω z 0 0 I zz ω z ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ I xx 0 0 α x ω x I xx ω x = ⎜ 0 I yy 0 ⎟ ⎜α y ⎟ ⎝ ⎠ ⎝ ⎠ + ⎜ω y ⎟ ⎝ ⎠ × ⎜I yy ω y ⎟ ⎝ ⎠ 0 0 I zz α z ω z I zz ω z ⎛ ⎞ ∣ ∣ ∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣ I xx α x i j k = ⎜I yy α y ⎟ ⎝ ⎠ + det ω x ω y ω z I zz α z I xx ω x I yy ω y I zz ω z ⎛ ⎞ ⎛ ⎞ I xx α x ω y (I zz ω z ) − ω z (I yy ω y ) = ⎜I yy α y ⎟ ⎝ ⎠ + ⎜−ω x (I zz ω z ) + ω z (I xx ω x ) ⎟ ⎝ ⎠ I zz α z ω x (I yy ω y ) − ω y (I xx ω x ) ⎛ ⎞ ⎛ ⎞ I xx α x ω y ω z (I zz − I yy ) = ⎜I yy α y ⎟ ⎝ ⎠ + ⎜ω x ω z (I xx − I zz ) ⎟ ⎝ ⎠ I zz α z ω x ω y (I yy − I xx ) So, we can see how much simpler it became when using principle axes. Compare the above to ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ I xx I xy I xz α x ω y (I zx ω x + I yz ω y + I zz ω z ) − ω z (I yx ω x + I yy ω y + I yz ω z ) ⎜I yx I yy I yz ⎟ ⎜α y ⎟ ⎝ ⎠ ⎝ ⎠ + ⎜ω x (I zx ω x + I yz ω y + I zz ω z ) − ω z (I xx ω x + I xy ω y + I xz ω z ) ⎟ ⎝ ⎠ I zx I yz I zz α z ω x (I yx ω x + I yy ω y + I yz ω z ) − ω y (I xx ω x + I xy ω y + I xz ω z ) So, always use principle axes for the body fixed coordinates system! 62
15 Wheel spinning precession x MgL L y z Mg Wheel spinning around y-axis, hanging from ceiling Weight create a torque which changes the angular momentum Torque vector representation y Important: This analysis is valid only for LARGE y H Angular momentum x z y H y I yy y p y t Angular momentum change due to the torque applied After torque applied for t z This is what wheel will look like after del T time due to precesses, notice that SPIN angular momentum changes in the direction of the torque y Time it takes for the wheel to precesses one full cycle T p 2 p H z Angular momentum after t t H aftert x H I yy y t x I yy y p y H t H t But H t I yy y p , hence I yy y p hence p I yy y MgL I yy y Nasser M. Abbasi Precesses.vsd 6/1/2011 Therefore, precession velocity p is MgL I yy y 16 References 1. Structural <strong>Dynamics</strong> 5th edition. Mario Paz, William Leigh 63
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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 ⎧ ⎫ ⎡ ⎨
Page 11 and 12: Hence The first eigen vector is Sim
Page 13 and 14: The transformation from modal coord
Page 15 and 16: Where F n = 2 T F 0 = 2 T ∫ T 0
Page 17 and 18: Let z = Re { Ze iωt} , z ′ = Re
Page 19 and 20: Now, u = Re { Ue iωt} , u ′ = Re
Page 21 and 22: Hence at t = 0 the phase of the res
Page 23 and 24: The following table gives the solut
Page 25 and 26: 5.3 no loading (free) mu ′′ + c
Page 27 and 28: 5.5 Impulse sin function Input is g
Page 29 and 30: Solution to single degree of freedo
Page 31 and 32: 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: 14.2.2 Derivation for τ = Iω in 3
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 and 86: 18.11.3 Two satellite, separate orb
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
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Image from http://www.americanflyer
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From FAA pilot’s handbook I do no
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zero-lift angle The angle of attack
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Image from http://www.aerospaceweb.
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