# Damped Oscillations and Q factor

We know that in reality, a mass on a spring  or a pendulum won’t oscillate for ever. Frictional forces will diminish the amplitude of oscillation until eventually the system comes to rest.

To remind you, here’s a mass-spring oscillator

We can add frictional forces to a mass and spring oscillator. Imagine that the mass was put in a liquid like engine oil. You won’t like cleaning the weights afterwards, but be that as it may, when the mass is inside the oil,  the amplitudes of successive oscillations will be reduced, and probably it’ll hardly oscillate at all, just bounce once, overshoot a little bit, then come to rest.

On the other hand, as you’ve probably discovered, a mass in air oscillates many times before it comes to rest. To incorporate friction, we can just say that there is an opposing frictional force that’s proportional to the velocity of the mass. This is a pretty good approximation for a body moving at a low velocity in air, or in a liquid. This is known as damping.

The spreadsheet damped-oscillations allows you to play with the amount of damping. Investigate what happens to the time period of the oscillations as the “damping factor” increases.Change the yellow cell to any value between 0 and -2 – this is a damping factor, related to Q (see below) then look at the resultant graph. The graphic above is a screenshot from it, showing moderate damping.

In a car shock absorber system, do we want the damping to be light or heavy?

Look at either of the diagrams. What is happening to amplitude with oscillation number? Have you seen a curve shaped like this before? The amplitude decays exponentially, so if you’ve seen charge decay off a capacitor, or voltage across one during discharge, or radioactive decay, they behave in a similar way mathematically. In this case,

where A= amplitude, k represents damping and n is the number of oscillations. The – sign indicates that the amplitudes decrease with increasing n.

For a single damped mass-spring system, the Q or Quality factor represents the effect of simplified viscous damping or drag where the damping force or drag force is proportional to velocity. The formula for the Q factor is:

where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation

where v is the velocity.

A system with low quality factor (Q < 12) is said to be overdamped. Such a system doesn’t oscillate at all, but when displaced from its equilibrium steady-state output it returns to it by exponential decay, approaching the steady state value asymptotically.  A system with high quality factor (Q > 12) is said to be underdamped Underdamped systems combine oscillation at a specific frequency with a decay of the amplitude. Underdamped systems with a low quality factor (a little above Q12) may oscillate only once or a few times before dying out. As the quality factor increases, the relative amount of damping decreases. A high-quality bell rings with a single pure tone for a very long time after being struck. A purely oscillatory system, such as a bell that rings forever, has an infinite quality factor. A system with an intermediate quality factor (Q = 12) is said to be critically damped Like an overdamped system, the output does not oscillate, and does not overshoot its steady-state output (i.e., it approaches a steady-state asymptote). Critical damping results in the fastest response (approach to the final value) possible without overshoot. Use the spreadsheet to try to duplicate these conditions.