# $Q$-factor for damped oscillator (not driven)?

How would this be defined?

Some of the Q-factor definitions I have encountered include:

$$Q=2\pi\frac{Energy \space stored}{Mean \space power \space per \space cycle}\\Q=2\pi\frac{Energy \space stored}{Energy \space lost \space per \space period \space of \space oscillation}\\Q=2\pi\frac{1}{Fractional \space power \space lost \space per \space cycle}$$

However, none of these seem to work for a non-driven, damped oscillator. The first two won't work because energy stored is not a constant, and unless fractional power lost per cycle is a constant (is it, and if it how then how do you show that?) the third won't work either.

The Q-factor tells you something about the frequency response of a driven system to a constant amplitude driver when a steady state (constant amplitude of driven system) has been reached.
The driver supplies energy to the driven system which at steady state results in the energy stored in the driven system staying constant (constant amplitude) and there is also a constant rate of energy (power) dissipation from the driven system..

With a non-driven system there is no input of energy into the system and so the energy (amplitude) of the oscillating system just decreases with time.
The Q-value is ratio of the total energy stored in the oscillating system (at some time) divided by the energy lost in the following single cycle.
For small amounts of damping (large values of Q) the Q-value is the number of oscillation such that the amplitude drops off to approximately $$\frac {1}{25}^{\rm th}$$ of its original value.
The Q-value is also equal to $$\frac{\pi f_0}{\alpha}$$ where $$f_0$$ is the natural frequency of the undamped oscillator and $$\alpha$$ appears in the term $${\rm e}^{-\alpha t}$$, where $$t$$ is the time, which controls the rate at which the amplitude of the oscillations decay.

I have just found out that this is a duplicate question - Definition of the Q-factor?

• So then how would one work out the Q factor? – Pancake_Senpai Jan 2 '19 at 15:35

A practical way to measure the Q factor for a non-driven oscillator is to measure the logarithmic decrement of the amplitude as the response decays after an impulse, and use that to find the damping ratio and hence Q.

Note that the value of Q is only a constant for linear systems. For a nonlinear oscillator, in general it is amplitude dependent, and might not be a very useful concept anyway. For a nonlinear oscillator the resonant frequency may also be amplitude-dependent which makes it even more non-intuitive what (if anything) the Q value means in practice.

The Q factor is $$2\pi$$ divided by the fraction of energy lost per cycle. If you drive the oscillator externally and keep the stored energy constant then you have to supply this fraction at every cycle.

Note: this definition does not require Q to be constant.

• How do we know if the fraction of energy lost per cycle is constant in non-driven systems? – Pancake_Senpai Jan 2 '19 at 15:36
• This is incorrect. It's the fraction of energy lost per radian of oscillation. – DanielSank Jan 2 '19 at 17:31
• Really? I haven't read that anywhere. Do you have a source? – Pancake_Senpai Jan 2 '19 at 19:27
• @Pancake_Senpai It isn't constant, in general, "Q" is only a useful concept when it is constant - for example, if the system response is linear. – alephzero Jan 2 '19 at 20:07
• @Pancake_Senpai well, for starters the very question at the top of this post :-) – DanielSank Jan 2 '19 at 21:40