I Different Definitions of The Quality Factor

[flyusx]

TL;DR

Taylor and Marion/Thornton give two different definitions for the quality factor that differ when $\beta<\omega$ but $\beta$ is not tiny.

I was reviewing some undergraduate mechanics when I found something I hadn't realised before.

Consider a damped driven oscillator governed by the differential equation $$\ddot{x}+2\beta\dot{x}+\omega^{2}x=F_{d}\cos\left(\omega_{d}t\right)$$ where $\omega$ is the system's natural frequency and $\omega_{d}$ is the driving frequency. Marion and Thornton (5th ed, pg121) states that the system's quality factor is $$Q=\frac{\sqrt{\omega^{2}-2\beta^{2}}}{2\beta}$$ In comparison, Taylor (pg191) defines the quality factor as $$Q=\frac{\omega}{2\beta}$$ The definitions in Taylor and Marion/Thornton are almost equivalent when the system is lightly damped and hence $\frac{\beta}{\omega}\ll1$. When this ratio reaches or exceeds one, the system is no longer underdamped and does not exhibit oscillatory motion; $Q$ does not need to be worried about here. However, this leaves the realm where $\frac{\beta}{\omega}<1$ but is not tiny where the two definitions diverge in agreement. In this case, which definition should be used?

 

[Dale]

In this case, which definition should be used?

What is the intended purpose for using the Q factor?

 

[DaveE]

If you're an EE then it's definitely the 2nd version.
The canonical quadratic form would be ${(\frac{s}{\omega_o})}^2 +\frac{1}{Q} (\frac{s}{\omega_o}) +1$. This would be in the Laplace transform of the second order DE for example.
https://authors.library.caltech.edu/records/cf0mt-rwk16

 

[kuruman]

In this case, which definition should be used?

The second expression is the leading term of a series expansion of the first term. If you use the first term, you can't go wrong because it is exact. The second term gives an estimate for $Q$ which becomes rougher as the ratio $~\beta/\omega~$ gets closer to $1$. Is that what you wanted to know?

 

[Cthugha]

The second expression is the leading term of a series expansion of the first term. If you use the first term, you can't go wrong because it is exact. The second term gives an estimate for $Q$ which becomes rougher as the ratio $~\beta/\omega~$ gets closer to $1$. Is that what you wanted to know?

That is a false friend because it looks mathematically correct, but is interestingly not related to the physics at hand here. Usually, the second expression is considered the correct Q factor.

There are several surprisingly nontrivial factors involved here. First, for a sinusoidally driven harmonic oscillator, the frequency where one gets the resonance in the amplitude and the frequency where one gets the resonance in the velocity (and therefore also in the dissipated power) is not the same. The former arises at $\sqrt{\omega_0^2 -2\beta^2}$, the latter arises at $\omega_0$. As one is usually interested in the dissipation of power in a continuously driven harmonic oscillator, the second expression is the typical definition of the quality factor.

However, things may look different in slightly different situations. For example, one will get a slightly different behavior in the power loss, when the oscillator is just displaced and returns to its lowest energy state without continuous driving because then the oscillation frequency must necessarily be the former one and the definition of the quality factor becomes a bit more complicated. I do not know the book discussed above, but my best guess is that the physical situations described are slightly different.