Sinusoidally Driven Damped Harmonic Motion

U-drive: choose the F and w:

U-choose the damping and initial v:

 F0 =  Note: keep |F|<6 

 b =  Note: keep b positive 

 w =  Note: keep w<0 

v0 =  Note: keep |v|<12

Check this box to superimpose the force on the position graph

Once the box is selected/deselected you must reinitialize your choice.


1. Driving at the resonant frequency (w = 1.414) yields motion exactly 90 degrees out of phase with drive.  With this phase relationship, the driving force always does positive work on the system.

2. Generally, the motion is seen to be a superposition of oscillations at the natural frequency (the transient) and oscillations at the drive frequency (the steady-state solution).  The magnitude of the damping determines how long the transient persists.

3. The steady-state motion is in phase with the drive when w is less than the natural frequency (for example: b=0.1 and w=0.5) and 180 degrees out of phase when w is greater than the natural frequency (say w=4).

Reference: Jerry B. Marion and Stephen T. Thornton, Classical Dynamics of Particles and Systems (4th edition, 1995) p. 125-131.

* This demonstration was written by Tim Gfroerer and scripted by Mario Belloni.