Damped Harmonic Oscillator
In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases proportional to the acting frictional force. Whereas Simple harmonic motion oscillates with only the restoring force acting on the system, Damped Harmonic motion experiences friction. In many vibrating systems the frictional force Ff can be modeled as being proportional to the velocity v of the object: Ff = −cv, where c is called the viscous damping coefficient.
Balance of forces (Newton's second law) for damped harmonic oscillators is then
This is rewritten into the form
where
- is called the 'undamped angular frequency of the oscillator' and
- is called the 'damping ratio'.
The value of the damping ratio ζ critically determines the behavior of the system. A damped harmonic oscillator can be:
- Overdamped (ζ > 1): The system returns (exponentially decays) to steady state without oscillating. Larger values of the damping ratio ζ return to equilibrium slower.
- Critically damped (ζ = 1): The system returns to steady state as quickly as possible without oscillating. This is often desired for the damping of systems such as doors.
- Underdamped (ζ < 1): The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. The angular frequency of the underdamped harmonic oscillator is given by
The Q factor of a damped oscillator is defined as
Q is related to the damping ratio by the equation
Read more about this topic: Harmonic Oscillator
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