Quality Factor

The total energy of a damped oscillator is the sum of its kinetic and potential energies: i.e.,

$$E = \frac{1}{2}\,m\left(\frac{dx}{dt}\right)^2 + \frac{1}{2}\,m\,\omega_0^{\,2}\,x^2.$$ (94)

Differentiating the above expression with respect to time, we obtain

$$\frac{dE}{dt} = m\,\frac{dx}{dt}\,\frac{d^2 x}{dt^2} + m\,\o... ...ac{dx}{dt}\left(\frac{d^2 x}{dt^2} + \omega_0^{\,2}\,x\right).$$ (95)

It follows from Equation (83) that

$$\frac{dE}{dt} = -2\,m\,\nu\left(\frac{dx}{dt}\right)^2.$$ (96)

We conclude that the presence of damping causes the oscillator energy to decrease monotonically in time, and, hence, causes the amplitude of the oscillation to eventually become negligibly small [see Equation (81)].

The energy loss rate of a weakly damped (i.e., $\nu\ll\omega_0$) oscillator is conveniently characterized in terms of a parameter, $Q$, which is known as the quality factor. This parameter is defined to be $2\pi$ times the energy stored in the oscillator, divided by the energy lost in a single oscillation period. If the oscillator is weakly damped then the energy lost per period is relatively small, and $Q$ is therefore much larger than unity. Roughly speaking, $Q$ is the number of oscillations that the oscillator typically completes, after being set in motion, before its amplitude decays to a negligible value. Let us find an expression for $Q$.

Now, the most general solution for a weakly damped oscillator can be written in the form [cf., Equation (91)]

$$x = x_0\,{\rm e}^{-\nu\,t}\,\cos(\omega_r\,t-\phi_0),$$ (97)

where $x_0$ and $\phi_0$ are constants, and $\omega_r = \sqrt{\omega_0^{\,2}-\nu^2}$. It follows that

$$\frac{dx}{dt} =- x_0\,\nu\,{\rm e}^{-\nu\,t}\,\cos(\omega_r\... ...)- x_0\,\omega_r\,{\rm e}^{-\nu\,t}\,\sin(\omega_r\,t-\phi_0).$$ (98)

Thus, making use of Equation (96), the energy lost during a single oscillation period is

$$\Delta E = -\int_0^{T_r} \frac{dE}{dt}\,dt$$ (99)

$$= 2\,m\,\nu\,x_0^{\,2}\int_0^{T_r}{\rm e}^{-2\,\nu\,t}\left[\nu\,\cos(\omega_r\,t-\phi_0) + \omega_r\,\sin(\omega_r\,t-\phi_0)\right]^2 dt,$$

where $T_r=2\pi/\omega_r$. In the weakly damped limit, $\nu\ll \omega_r$, the exponential factor is approximately unity in the interval $t=0$ to $t=T_r$, so that

$$\Delta E \simeq \frac{2\,m\,\nu\,x_0^{\,2}}{\omega_r}\int_0^... ...heta\,\sin\theta + \omega_r^{\,2}\,\sin^2\theta\right)d\theta.$$ (100)

Thus,

$$\Delta E \simeq \frac{2\pi\,m\,\nu\,x_0^{\,2}}{\omega_r}\,(\... ...m\,\omega_0^{\,2}\,x_0^{\,2}\left(\frac{\nu}{\omega_r}\right),$$ (101)

since $\cos^2\theta$ and $\sin^2\theta$ both have the average values $1/2$ in the interval $0$ to $2\pi$, whereas $\cos\theta\,\sin\theta$ has the average value $0$. According to Equation (81), the energy stored in the oscillator (at $t=0$) is

$$E = \frac{1}{2}\,m\,\omega_0^{\,2}\,x_0^{\,2}.$$ (102)

It follows that

$$Q = 2\pi\,\frac{E}{\Delta E} = \frac{\omega_r}{2\,\nu}\simeq \frac{\omega_0}{2\,\nu}.$$ (103)