Dynamics cheat sheet

18.7.2 Bi-Elliptic transfer orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
18.7.3 semi-tangential elliptical transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
18.8 Rocket engines, Hohmann transfer, plane change at equator . . . . . . . . . . . . . . . . . . . . . 75
18.8.1 Rocket equation with plane change not at equator . . . . . . . . . . . . . . . . . . . . . . 77
18.9 Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
18.10interplanetary transfer orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
18.10.1 interplanetary hohmann transfer orbit, case one . . . . . . . . . . . . . . . . . . . . . . . . 79
18.11rendezvous orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
18.11.1 Two satellite, walking rendezvous using Hohmann transfer . . . . . . . . . . . . . . . . . . 81
18.11.2 Two satellite, separate orbits, rendezvous using Hohmann transfer, coplaner . . . . . . . . 83
18.11.3 Two satellite, separate orbits, rendezvous using bi-elliptic transfer, coplaner . . . . . . . . 85
18.12Semi-tangential transfers, elliptical, parabolic and hyperbolic . . . . . . . . . . . . . . . . . . . . 88
18.13Lagrange points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
18.14Orbit changing by low contiuous thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
18.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
19 Dynamic of flights 92
19.1 Wing geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
19.2 Summary of main equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
19.2.1 Writing the equations in linear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
19.3 definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
19.4 images and plots collected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
19.5 Some strange shaped airplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
19.6 links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
19.7 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
1 Modal analysis for two degrees of freedom system
Detailed steps to perform modal analysis are given below for a standard undamped two degrees of freedom
system. The main advantage of solving a multidegree system using modal analysis is that it decouples the
equations of motion (assuming they are coupled) making solving them much simpler.
In addition it shows the fundamental shapes that the system can vibrate in, which gives more insight into
the system. Starting with standard 2 degrees of freedom system
x 1
x 2
k 1 k 2
m 1 m 2
f 1 t
f 2 t
In the above the generalized coordinates are x 1 and x 2 . Hence the system requires two equations of motion
(EOM’s).
1.1 Step one, finding the equations of motion in normal coordinates space
The two EOM’s are found using any method such as Newton’s method or Lagrangian method. Using Newton’s
method, free body diagram is made of each mass and then F = ma is written for each mass resulting in the
equations of motion. In the following it is assumed that both masses are moving in the positive direction and
that x 2 is larger than x 1 when these equations of equilibrium are written
3