3.6.1.4 Forced vibrations and resonance
Qualitative treatment of free and forced vibrations.
Resonance and the effects of damping on the sharpness of resonance.
Examples of these effects in mechanical systems and situations involving stationary waves.
Resonance
So far only free oscillations have been considered. If an external agent drives the oscillation it becomes a forced oscillation.
How the object responds to being driven depends on how it is being driven. If the object driving the oscillator forces the oscillations at the same frequency as the oscillator’s own natural frequency then resonance occurs.
This can easily be shown by attempting to force a pendulum, first quickly enough so that the change in direction of the displacement stops the oscillator from responding and the pendulum no longer moves, and then slowly so that it barely effects the oscillation. It is easy enough to find the resonant frequency of the pendulum.
It should be noticed that the amplitude of the oscillation increases dramatically. The system is said to be resonating.
At resonance the driver is continuously supplying energy to the system, and without damping the amplitude would increase until the system destroyed itself. This is what happens when an opera singer manages to shatter a glass.
A system is in resonance when the applied frequency is equal to the applied frequency.
The phase difference between the displacement and the periodic force is $\frac{π}{2}$. The force is exactly in phase with the velocity of the system.
If the applied frequency increases, and becomes higher than the natural frequency of the system then the amplitude decreases more and more and the phase difference increases from $\frac{π}{2}$ to $π$ so that the applied force is out of phase with the displacement.
This can be demonstrated very well by Barton’s pendulums.