DAMPING RATIOS

This applet explores the differential equation modeling an unforced spring-mass-damper system. The differential equation is \[x'' + 2\zeta\omega x' + \omega^2 x = 0\] This models a damped harmonic oscillator where, \(\omega\) is the natural frequency of the oscillator and \(\zeta\) (greek zeta) is called the damping ratio. You should check that \(\zeta\) is a dimensionless constant!

The applet shows the solution for three different values of \(\zeta\), labeled \(\zeta_1,\, \zeta_2,\, \zeta_3\). All three solutions use the same value of \(\omega\), and the same initial conditions \(x(0) = b_0, \, x'(0) = b_1\). The values of \(\zeta_1,\, \zeta_2,\, \zeta_3,\, \omega,\, b_0,\, b_1\) are all settable using their respective sliders.

The solution graphs are color-coded to indicate if the system is underdamped, overdamped or critically damped. Just because we can, the applet allows the damping ratio to be set to 0 (undamped) or negative (unstable). In these cases, the plots are drawn in gray.

The time \(t\) can be changed by sliding the small vertical line on the \(t\) axis of the time graphing window (left hand window). The values of time and each of the solutions are displayed below the graphing window. They are given in the same color as the corresponding graph.

The Show roots checkbox determines if the roots (also known as poles) are shown. When it is checked, the (possibly complex) roots for each of the three systems are shown in the root graphing window (right hand window). They are also color coordinated with their corresponding solutions. This window also shows the root locus, which is the curve the roots lie as \(\omega\) is held fixed and \(\zeta\) changes.

Next to each \(\zeta\) slider is a checkbox which determines whether the corresponding solution is displayed.

The time graphing window can be zoomed in both directions using the t-zoom and x-zoom sliders. It can also be shifted left or right by holding the mouse down and dragging inside the graph window.

If your mouse is in either graphing window the coordinates of the mouse position are displayed above the graph.