Dynamics cheat sheet

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$my Latex and Tex4ht cheat sheet$

5.4 impulse F 0 δ (t) loading ⎧ ⎨ −iω roots ζ = 0 ⎩ +iω u ′′ + ω 2 u = 0 ζ < 1 ζ = 1 ζ > 1 transient { }} { u (t) = u(0) cos ωt + u′ (0) ⎧ ω ⎨ −ξω + iω √ 1 − ξ 2 roots ⎩ −ξω − iω √ 1 − ξ 2 steady state { }} { sin ωt + F 0 sin ωt mω transient steady state { ( }} ){ { ( }} ){ u (t) = e −ξωt u(0) cos ω d t + u′ (0) + u(0)ξω sin ω d t + e −ξωt F0 sin ω d t ⎧ ω d mω d ⎨ −ω roots ⎩ −ω u (t) = (u (0) (1 + ωt) + u ′ (0) t) e −ωt + F 0t ⎧ ⎨ λ 1 = −ωξ + ω √ ξ 2 − 1 roots ⎩ λ 2 = −ωξ − ω √ ξ 2 − 1 ⎧ u (t) = Ae ⎪⎨ λ1ωt + Be λ2ωt + ⎪⎩ A = u′ (0)−u(0)λ 2 2ω √ ξ 2 −1 B = −u′ (0)+u(0)λ 1 2ω √ ξ 2 −1 m e−ωt F 0 m 2ω √ ξ 2 −1 eλ 1ωt − The impulse response can be implemented in Mathematica as parms = {m -> 10, c -> 1.2, k -> 4.3, a -> 1}; tf = TransferFunctionModel[a/(m s^2 + c s + k) /. parms, s] sol = OutputResponse[tf, DiracDelta[t], t]; F 0 m 2ω √ ξ 2 −1 eλ 2ωt Plot[sol, {t, 0, 60}, PlotRange -> All, Frame -> True, FrameLabel -> {{z[t], None}, {Row[{t, " (sec)"}], eq}},GridLines -> Automatic] 26
5.5 Impulse sin function Input is given by F (t) = F 0 sin (ϖt) where ϖ = 2π 2t 1 = π t 1 ζ = 0 ζ < 1 ζ = 1 ⎧ ⎨ −iω roots ⎩ +iω ⎧ ⎧ ⎪⎨ ϖ = ω → ( ⎪⎨ ⎪⎩ −u (0) + F 0 π k 2 ⎧ ( u (t) ⎪⎨ u (0) cos ωt + ϖ ≠ ω → ⎪⎩ ⎪⎩ ⎧ √ ⎨ −ξω + iω n 1 − ξ 2 roots ⎩ √ −ξω − iω n 1 − ξ 2 ⎧ ⎪⎨ u (t) = ⎪⎩ u (0) cos ωt + u′ (0) ω sin ωt − F 0 k ωt ) cos ωt + −u ′ (0)+ F 0 k u ′ (0) ϖ ( − (F π/t1 0/k) ω π/t1 1−( ω 2 cos (ωt) 0 ≤ t ≤ t 1 π 2t 1 )) ) 2 ω sin ωt t > t 1 sin ωt + F 0/k 1−( π/t1 ω ( ) ) 2 sin πt t 1 0 ≤ t ≤ t 1 u (t 1 ) cos ωt + u′ (t 1 ) ω sin ωt t > t 1 time constant τ = 1 ζω n e −ξωt (A cos ω d t + B sin ω d t) + F 0 √ 1 k ( e −ξωt u (t 1 ) cos ω d t + u′ (t 1 )+u(t 1 )ξω ω d sin ω d t A = u (0) + F 0 k 1 √(1−r 2 ) 2 +(2ξr) 2 sin θ B = u′ (0) ω d ⎧ ⎨ roots + u(0)ξω ω d + F 0 −ω √ 1 k ω d (1−r 2 ) 2 +(2ξr) ( sin (1−r 2 ) 2 +(2ξr) ) 2 2 (ξω sin θ − ϖ cos θ) ) t 1 − θ ⎩ −ω ⎧ ( ) ⎪⎨ (A + Bt) e −ωt + F 0 √ 1 sin k π t u (t) = (1−r 2 ) 2 +(2r) 2 t 1 − θ 0 ≤ t ≤ t 1 ⎪⎩ u (t) = (u (t 1 ) + (u ′ (t 1 ) + u (t 1 ) ω) t) e −ωt t > t 1 π t 0 ≤ t ≤ t 1 t > t 1 A = u (0) + F 0 k 1 √(1−r 2 ) 2 +(2r) 2 sin θ B = u ′ (0) + u (0) ω + F 0 ( √ 1 k (1−r 2 ) 2 +(2r) 2 ) ω sin θ − π t 1 cos θ 27
Page 1 and 2: Dynamics cheat sheet Nasser M. Abba
Page 3 and 4: 18.7.2 Bi-Elliptic transfer orbit .
Page 5 and 6: are the natural frequencies of the
Page 7 and 8: Similarly, µ 2 is found ⎧ ⎫T
Page 9 and 10: 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: 5.3 no loading (free) mu ′′ + c
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 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 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
Page 89 and 90:
89
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
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http://en.wikipedia.org/wiki/Flight
Page 103 and 104:
http://www.grc.nasa.gov/WWW/k-12/UE
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http://en.wikipedia.org/wiki/Lift_c
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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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