At resonance the amount of energy lost due to damping is equal to the rate of energy supply from the driver. The driver is the source of external energy that keeps the oscillations going – for example, the person pushing a kid on a swing. Increasing the damping will reduce the size (amplitude) of the oscillations at resonance, but the amount of damping has next to no effect at all on the frequency of resonance.
Damping also has an effect on the sharpness of a resonance. By sharpness we mean how sensitively the resonance is tuned, and is sometimes called the Q-factor by engineers. If damping is very small, a system will only oscillate very close to the resonant frequency. And if the driver hits the resonant frequency bang-on suddenly the oscillations can get very large. Conversely, if damping is large, the amplitude of oscillations at resonance is less, but the driver will get some response from the system at frequencies some way from resonance. This means that resonance will be weaker, but will have an effect over a larger range of frequencies.
The video below shows how the suspension of a car can be used to demonstrate resonance and damping.
The sharpness of a resonance is measured by its Q-Factor. Although this does have a precise mathematical definition, it is most easily understood as roughly the number of free oscillations the oscillator will complete before decaying to zero. At the M.O.T. garage, the mechanic tests the damping of your shock absorbers by bouncing the wings of the car…they probably don’t know they’re measuring Q-factor, but they do know that more than a bounce-and-a-half and it fails!
So, a small amount of damping equates to a large Q, and a large amount of damping equates to a small Q.
When the millennium bridge opened in London, it was found to have too big a Q and too little damping. They cured the wobbling bridge by adding damping to reduce the Q.