Dynamics cheat sheet

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⎡ ⎤ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫ ⎣ m 1 0 ⎨x ⎦ ′′ ⎬ 1 0 m ⎩ 2 x ′′ ⎭ + ⎣ k 1 + k 2 −k 2 ⎨x ⎦ 1 ⎬ ⎨ 2 −k 2 k ⎩ 2 x ⎭ = f 1 (t) ⎬ ⎩ 2 f 2 (t) ⎭ ⎡ ⎤ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫ ⎣ 1 0 ⎨x ⎦ ′′ ⎬ 1 0 3 ⎩ x ′′ ⎭ + ⎣ 3 −2 ⎨x ⎦ 1 ⎬ ⎨ −2 2 ⎩ x ⎭ = 0 ⎬ ⎩ 2 sin (5t) ⎭ 2 In this example m 11 = 1, m 12 = 0, m 21 = 0, m 22 = 3 and k 11 = 3, k 12 = −2, k 21 = −2, k 22 = 2 and f 1 (t) = 0 and f 2 (t) = sin (5t) step 2 is now applied which solves the eigenvalue problem in order to find the two natural frequencies Let ω 2 = λ hence The solution is λ 1 = 3. 475 and λ 2 = 0.192, therefore And det ( [K] − ω 2 [M] ) = 0 ⎛⎡ ⎤ ⎡ ⎤⎞ det ⎝⎣ 3 −2 ⎦ − ω 2 ⎣ 1 0 ⎦⎠ = 0 −2 2 0 3 ⎡ ⎤ det ⎣ 3 − ω2 −2 ⎦ = 0 −2 2 − 3ω 2 ( 3 − ω 2 ) ( 2 − 3ω 2) − (−2) (−2) = 0 3ω 4 − 11ω 2 + 2 = 0 3λ 2 − 11λ + 2 = 0 ω 1 = √ 3.475 = 1.864 ω 2 = √ 0.192 = 0.438 step 3 is now applied which finds the non-normalized eigenvectors. For each natural frequency ω 1 and ω 2 the corresponding shape function is found by solving the following two sets of equations for the eigen vectors ϕ 1 , ϕ 2 ⎛⎡ ⎤ ⎡ ⎤⎞ ⎧ ⎫ ⎧ ⎫ ⎝⎣ 3 −2 ⎦ − ω1 2 ⎣ 1 0 ⎨ϕ ⎦⎠ 11 ⎬ ⎨ −2 2 0 3 ⎩ ϕ ⎭ = 0⎬ ⎩ 21 0 ⎭ For ω 1 = 1. 864 ⎛⎡ ⎤ ⎡ ⎤⎞ ⎧ ⎫ ⎧ ⎫ ⎝⎣ 3 −2 ⎦ − 1.864 2 ⎣ 1 0 ⎨ϕ ⎦⎠ 11 ⎬ ⎨ −2 2 0 3 ⎩ ϕ ⎭ = 0⎬ ⎩ 21 0 ⎭ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫ ⎣ −0.475 −2 ⎨ 1 ⎬ ⎨ ⎦ −2 −8.424 ⎩ ϕ ⎭ = 0⎬ ⎩ 21 0 ⎭ This gives one equation to solve for ϕ 21 (the first row equation is only used) −0.475 − 2ϕ 21 = 0 10
Hence The first eigen vector is Similarly for ω 2 = 0.438 ϕ 21 = 0.475 −2 = −0.237 ⎧ ⎫ ⎧ ⎫ ⎨ϕ 11 ⎬ ⎨ ϕ 1 = ⎩ ϕ ⎭ = 1 ⎬ ⎩ 21 −0.237 ⎭ ⎛⎡ ⎤ ⎡ ⎤⎞ ⎧ ⎫ ⎧ ⎫ ⎝⎣ 3 −2 ⎦ − 0.438 2 ⎣ 1 0 ⎨ϕ ⎦⎠ 12 ⎬ ⎨ −2 2 0 3 ⎩ ϕ ⎭ = 0⎬ ⎩ 22 0 ⎭ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫ ⎣ 2.808 −2 ⎨ 1 ⎬ ⎨ ⎦ −2 1.425 ⎩ ϕ ⎭ = 0⎬ ⎩ 22 0 ⎭ This gives one equation to solve for ϕ 22 (the first row equation is only used) 2.808 − 2ϕ 22 = 0 Hence The second eigen vector is ϕ 22 = −2.808 −2 = 1.404 ⎧ ⎫ ⎧ ⎫ ⎨ϕ 12 ⎬ ⎨ ϕ 2 = ⎩ ϕ ⎭ = 1 ⎬ ⎩ 22 1.404 ⎭ Now step 4 is applied, which is mass normalization of the shape vectors (or the eigenvectors) µ 1 = ϕ T 1 [M] ϕ 1 Hence Similarly, µ 2 is found Hence ⎧ ⎫ ⎨ 1 ⎬ µ 1 = ⎩ −0.237 ⎭ = 1. 169 ⎧ ⎫ ⎨ 1 ⎬ µ 2 = ⎩ 1.404 ⎭ = 6.914 ⎤ ⎧ ⎫ ⎣ 1 0 ⎨ 1 ⎬ ⎦ 0 3 ⎩ −0.237 ⎭ T ⎡ µ 2 = ϕ T 2 [M] ϕ 2 ⎤ ⎧ ⎫ ⎣ 1 0 ⎨ 1 ⎬ ⎦ 0 3 ⎩ 1.404 ⎭ T ⎡ 11
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: Or in full form and ⎧ ⎫ ⎡ ⎨
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 and 62:
14.2.2 Derivation for τ = Iω in 3
Page 63 and 64:
15 Wheel spinning precession x Mg
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18 Astrodynamics 18.1 Ellipse main
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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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73
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18.7.3 semi-tangential elliptical t
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18.8.1 Rocket equation with plane c
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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
Page 87 and 88:
1 TOF:=hohmann_rendezvous_2(theta=0
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89
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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
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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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