Dynamics cheat sheet

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$my Latex and Tex4ht cheat sheet$

One degree freedom linear system unforced response y t c m y t k m yt 0 y t 2 n y t n2 yt 0 Some defintions k n m rad/sec damping ratio c c c r c 2 km 2 n m Solution roots s 1,2 n n 2 1 3 cases d n 1 2 30 Defined only for under damped case(else zero) 1 1 1 Underdamped 2 roots, complex conjugate Critical damped (one real root, double multiplicity) Over damped, 2 real roots, distinct yt e nt Acos d t Bsin d t A y0, B y 0A n d yt A Bte nt A y0, B y 0 A n T d 2 d 1 n Nasser M. Abbasi May 25, 2011 One_DOF_system.vsd Damped period of oscillation Time constant yt Ae 2 1 n t Be 2 1 n t A y 0y0 n 2 n B y 0y0 n 2 n 2 1 2 1 2 1 2 1
7 Some common formulas These definitions are used throughout the derivations below. ξ = c c r = c 2 √ km = c 2ω n m u st = F static deflection √ k k ω n = m ω D = ω n √ 1 − ξ 2 note: not defined for ξ > 1 since becomes complex r = ϖ ω n T d = 2π damped period of oscillation ω { d −1 τ = , −1 } time constants where λ i are roots of characteristic equation λ 1 λ2 1 β = magnification factor √(1 − r 2 ) 2 + (2rξ) 2 β max when r = √ 1 − 2ξ 2 1 β max = 2ξ √ 1 − ξ 2 y n = e ζωn2π ω D y ( n+1 ) yn ln = ζ2π y n+1 ( ) 1 M ln yn = ζ2π y n+M small damping ⇒ √ ζ2π 1−ζ 2 e ⇒ e ζ2π This table shows many cycles it takes for the peak to decay by half its original value as a function of the damping ζ. For example, we see that when ζ = 2.7% then it takes 4 cycles for the peak (i.e. displacement) to reduce to half its value. data = Table[{i, (1/i Log[2]/(2*Pi)*100)}, {i, 1, 20}]; TableForm[N@data, TableHeadings -> {None, {Column[{"number of cycles", "needed for peak", "to decay by half"}], "\[Zeta] (%)"}}] 31
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 and 28: 5.5 Impulse sin function Input is g
Page 29: Solution to single degree of freedo
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
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2. Find speed of second planet V 2
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1 mu = 324859; 2 alt = 1475.776; 3
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18.11.3 Two satellite, separate orb
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1 TOF:=hohmann_rendezvous_2(theta=0
Page 89 and 90:
89
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18.14 Orbit changing by low contiuo
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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
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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
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From http://www.solar-city.net/2010
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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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Dynamics cheat sheet

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