# Physicopoeia

###### φυσικοποιΐα

# Resonance

Many textbooks will give you a small selection of graphs showing the decay of damped SHM, and similar for resonance. Sketching these accurately by hand can be tricky (and I've seen textbooks do them quite sloppily as well), so here are two Excel simulations I did to plot them automatically. You can change the value in any cell highlighted in yellow and the graph will re-plot itself to show you the change.

The second Excel file also has this graph to display or print out, showing resonance with different degrees of damping:

[Graph screen shot]

Notice that the resonant frequency actually decreases slightly as the amount of damping increases (red dashed line). This shows that the usual description of resonance as occurring when driver frequency *exactly *equals natural frequency is, at best, a bit of a simplification.

Also notice that the graph above is *not *symmetrical about the peak. At very low frequencies, the driven object will simply move in step with the driving object so will have a non-zero amplitude. At very high frequencies, the object transferring the energy from the driving source to the driven object (*i.e.* usually a spring) will absorb so much of the vibration that almost none will be transferred, hence why the amplitude above tails off at high frequencies.

Finally, it's also worth pointing out that this analysis is based on a frictional damping force which varies linearly with velocity (*F* = -*bv, *where *b* is the damping coefficient in the above simulation - see the legend to the right of the graph). Damping forces which vary differently to this would produce different results.

PHASE LAG BETWEEN THE DRIVER AND THE DRIVEN OBJECT'S MOTION

Also in the second Excel file are graphs which might be less familiar, showing the phase difference between the driving force and the motion of the oscillator (what used to be called the 'master' and 'slave' in the days when such vocabulary didn't make people feel quite so uncomfortable). If you set up a mass on a spring being driven by a vibration generator and vary the generator frequency, not only will the response quite obviously follow the amplitude graph above, but you might also notice a phase difference between the two motions. At resonance, the driver should be 90° ahead of the object being driven whereas far below and far above the resonant frequency they will be exactly in phase and 180° out of phase respectively. The higher the damping, the more gradually the phase response transitions between the two extremes.

Well, that's the general idea. This graph from the Excel file (using the same values as the graph above) summarises it so much more elegantly: