Problem 5.2 -- General overview of types of solution shapes to expect
This system has no damping, and the forces are coupled to the rotation.
You expect solutions of the form:
Where 'omega' is the rate of rotation of system described in the homework problem.
You may already notice that the paths that can described by these
equations are all:
Elliptical
With the major axes aligned with the x and y axes
At the extremes:
If Ay=0, you just have motion in the x direction (a straight line side to side), only.
If Ax=0, you have only y-direction motion (a straight line, up and down)
When Ax and Ay are BOTH non-zero:
Ax and Ay of the SAME SIGN will describe an ellipse which orbits in the SAME direction as the rotation (omega) of the system.
Ax and Ay of OPPOSITE SIGN will orbit in the opposite direction.
Either Ax=Ay or Ax=-Ay will describe a perfect circle (orbiting with or against the omega direction).
Use MATLAB to play around with different values of Ax and Ay, if you don't have intuition about this:
t=2*pi*[0:.01:.9];
Ax=2;
Ay=1;
x=Ax*cos(t);
y=Ay*sin(t);
plot(x,y)
axis image
alim=1.5*max(abs(Ax),abs(Ay));
axis([-alim alim -alim alim])
grid on
(You can copy and paste the lines above.)
Then you can repeat these commands for other values for Ax and/or Ay.
(I intentionally left the ellipse not-quite-closed, so it would be obvious in which 'direction' the ellipse was drawn...)
