Dynamics cheat sheet

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$my Latex and Tex4ht cheat sheet$

ζ > 1 ⎧ √ ⎨ −ω n ξ + ω n ξ 2 − 1 roots ⎩ √ −ω n ξ − ω n ξ 2 − 1 ⎧ ( ⎪⎨ −ξ+ √ ( ξ Ae 2 −1 )ωt + Be u (t) = ( ⎪⎩ −ξ+ √ ) ξ 2 −1 ω u (t) = A 1 e nt + B1 e A = u′ (0)+u(0)ωξ+u(0)ω √ (( ξ 2 −1+ F k β ξ+ √ ) ξ 2 −1 2ω √ ξ 2 −1 B = − u′ (0)+u(0)ωξ−u(0)ω √ ξ 2 −1+ F k β (( ξ− √ ξ 2 −1 1 β = √ (1−r 2 ) 2 +(2ξr) 2 ( A 1 = − ˙u(t 1)+u(t 1 )ω n ξ− √ ) ξ 2 −1 √ 2ω n ( ξ 2 −1 B 1 = ˙u(t 1)+u(t 1 )ξω n ξ+ √ ) ξ 2 −1 2ω n √ ξ 2 −1 2ω √ ξ 2 −1 −ξ− √ ) ξ 2 −1 ωt + F π t ( −ξ− √ ) ξ 2 −1 ω nt k β sin ( ) ω sin θ− π cos θ t 1 ) ) ω sin θ− π cos θ t 1 ) t 1 − θ 0 ≤ t ≤ t 1 t > t 1 6 Tree view look at the different cases This tree illustrates the different cases that needs to be considered for the solution of single degree of freedom system with harmonic loading. There are 12 cases to consider. Resonance needs to be handled as special case when damping is absent due to the singularity in the standard solution when the forcing frequency is the same as the natural frequency. When damping is present, there is no resonance, however, there is what is called practical response which occur when the forcing frequency is almost the same as the natural frequency. 28
Solution to single degree of freedom system, second order, linear, time invariant By Nasser M. Abbasi December 11, 2012 Single_degree_system_tree.vsd undamped damped r c c cr free mü ku 0 forced mü ku Fsint free forced mü cu ku 0 mü cu ku Fsint 1 1 1 r 1 r 1 r 1 r 1 1 1 1 1 1 1 r 1 resonance 1 underdamped, roots a ib with a real and negative 1 critical damping roots a,a 1 overdamped roots both real and negative a,b x x x xx The following is another diagram made sometime ago which contains more useful information and is kept here for reference. 29
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 and 26: 5.3 no loading (free) mu ′′ + c
Page 27: 5.5 Impulse sin function Input is g
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
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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
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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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Dynamics cheat sheet

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