# Damped Oscillations, Forced Oscillations and Resonance

"The bible tells you how to go to heaven, not how the heavens go"
Galileo Galilei - at his trial

DAMPED OSCILLATIONS
• To date our discussion of SHM has assumed that the motion is frictionless, the total energy (kinetic plus potential) remains constant and the motion will continue forever.  Of course in real world situations this is not the case, frictional forces are always present such that, without external intervention, oscillating systems will always come to rest.  The frictional (damping) force is often proportional (but opposite in direction) to the velocity of the oscillating body such that

where b is the damping constant.
• This differential equation has solutions

where when the damping is small (small b).
Notice that this solution represents oscillatory motion with an exponentially decreasing amplitude

See damped oscillation applet courtesy, Davidson College, North Carolina.

FORCED OSCILLATIONS AND RESONANCE

• Suppose now that instead of allowing our system to oscillate in isolation we apply a "driving force".  For example, in the case of the (vertical) mass on a spring the driving force might be applied by having an external force (F) move the support of the spring up and down.  In this case the equation of motion of the mass is given by,

One common situation occurs when the driving force itself oscillates, in which case we may write

where is the (angular) frequency of the driving force.

• This equation has solutions of the form

where the amplitude of these oscillations, B, depends on the parameters of the motion,

The amplitude, B, has a maximum value when .  This is called the resonance condition.  Note that at resonance, B, can become extremely large if b is small.  (In the diagram at right is the natural frequency of the oscillations, , in the above analysis). In designing physical systems it is very important to identify the system's natural frequencies of vibration and provide sufficient damping in case of resonance.  This clearly did not happen in the design of the Tacoma Narrows Bridge (Tacoma Narrows Newsreel) in 1940.

• There are many physical and engineering systems where resonance is very important e.g. shock absorbers, earthquakes, loudspeakers, NMR, microwave ovens etc. etc.  A very important subject which, unfortunately, we do not have time to discuss in any more detail.
• A good example of "coupled" oscillations, where forcing the oscillators causes their oscillations to synchronize may be found here.

One day a researcher is doing an experiment with frogs. He cuts one of the frog's legs off. He ordered the frog to jump, and it did. Then he cut off one of the frog's arms off. He then ordered the frog to jump again, and it did as it was told. He continued to do this until he had cut all of the frog's appendages off. He then ordered the frog to jump, and the frog couldn't. The rescearcher then wrote in his notes: when I cut all of a frog's legs off it becomes deaf.

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu