DAMPING RATIO

The graphing window at the top right displays a solution of the differential equation $$\ddot{x} + 2 \zeta \omega_n \dot{x} + \omega_n^2 x = 0$$. Positions on the graph are set using a time slider under the window. The [>>] key starts an animation. The [<<] key resets $$t$$ to $$t = 0$$. Rolling the cursor over this window creates crosshairs and a readout of the values of $$t$$ and $$x$$.

A yellow diamond in the left window represents the initial condition in the ($$\dot{x}, x)$$ phase plane. The trajectory of $$(\dot{x}, x)$$ in the phase plane is also displayed (limited to ten loops of the spiral ahead and behind $$t = 0$$ in the underdamped case). The yellow diamond moves along the trajectory when time is varied.

The initial condition is set by dragging the yellow diamond or the sliders at bottom and left side of the phase plane window.

The [Relate diagrams] key toggles a faint yellow line connecting the point on the $$(\dot{x}, x)$$ plane with the corresponding point on the graph. A slider below this key controls a zoom feature in $$x$$ and $$\dot{x}$$ and the maximal displayed values of $$x$$ and $$\dot{x}$$ is read out to the right of the slider.

Adjust the values of the natural circular frequency $$\omega_n$$ and the damping ratio $$\zeta$$; using sliders at bottom right.

The damping ratio slider actually displays $$-\zeta$$. The graphing window above that slider displays the complex number $$-\zeta$$ in green, and, in yellow, the two roots of the characteristic polynomial. These roots are also read out to the right.