Question

When an oscillator is underdamped, there are two principle effects that we observe as a result (compared to the undamped oscillator). Describe them and explain why they occur?

Answer 1

When an oscillator is underdamped, there are two principle effects that we observe as a result.

Underdamped : Finally, when 0 < < 1,

where, = complex

and the system is 'under-damped'.

In this situation, the system will oscillate at the natural damped frequency _d, which is a function of the natural

frequency and the damping ratio.

In this case, the solution can be generally written as -

x (t) = e^-₀t (A cos _d t + B sin _d t)

where, _d = damped frequency = ₀1 - ²

Undamped :

A simple harmonic follows an equation for sinusoidal oscillations which is given as -

x_undamped = A cos (t + )

For a mass-spring system, the angular frequency is given by -

= k / m

Explanation, why they occur : consider, a door that uses a spring to close the door once open. This can lead to any

of the above types of damping which depending on the strength of damping. If the door is undamped, then it will

swing back and forth forever at a particular resonant frequency.

If it is underdamped, then it will swing back and forth with decreasing size of the swing until it comes to a stop.