6.4 Damped Oscillations

An object moving with simple harmonic motion will continue to do so indefinitely. A more realistic scenario is to include the effects of other forces, such as friction and air resistance, that slow down the oscillations until they eventally decay away to nothing. These effects are collectively known as damping.

Depending on the strength of the damping forces, they can result one of of three scenarios:

•

Underdamping — The damping effects aren’t that strong, there are still oscillations but they decay away over time.
•

Overdamping — Damping is very strong, there are no oscillations and the system smoothly decays to equilibrium.
•

Critial damping — The system returns to equilibrium in the fastest possible way without overshooting.

Damping forces can be proportional to velocity (as with viscous drag), proportional to the square of velocity (quadratic drag e.g. air resistance), or something else (friction etc.). In this chapter we will focus on linear drag as it provides an instructive example of damping effects which we can solve analytically. We write the damping force as $F=-bv$ , where $b$ is some constant with units of $\mathrm{kg}\text{\,}{\mathrm{s}}^{-1}$ . The minus sign indicates that the damping force acts in the opposite direction of the velocity. Using Newton II with the two forces we have now, the equation of motion becomes

m\frac{\operatorname{\mathrm{d}}\!^{2}x}{\operatorname{\mathrm{d}}\!t^{2}}+b% \frac{\operatorname{\mathrm{d}}\!x}{\operatorname{\mathrm{d}}\!t}+kx=0.

(6.54)

To solve this we use an ansatz of the form $x(t)=Ae^{\xi t}$ . Differentiating twice and substituting into the above, we get an equation for $\xi$ :

\xi^{2}+\frac{b}{m}\xi+\omega_{0}^{2}=0,

(6.55)

where we have written the resonant frequency as $\omega_{0}^{2}=\frac{k}{m}$ . The subscript 0 is important to distinguish the resonant frequency from other frequencies we will define later. If we also define the damping coefficient $\gamma$ as

\gamma=\frac{b}{2m},

(6.56)

then we can write $\xi$ as

\xi=-\gamma\pm\sqrt{\gamma^{2}-\omega_{0}^{2}},

(6.57)

and the general solution is

x(t)=A_{1}e^{(-\gamma+\sqrt{\gamma^{2}-\omega_{0}^{2}})t}+A_{2}e^{(-\gamma-% \sqrt{\gamma^{2}-\omega_{0}^{2}})t}.

(6.58)

6.4.1 Different Damping Cases

Now, we have four different cases that can arise depending on the strength of the damping force i.e. the size of $b$ compared to $\omega_{0}$ .

Case 1:

$b=0$ (no damping). Then $\gamma=0$ and the general solution becomes

$\displaystyle x(t)$ $\displaystyle=A_{1}e^{i\omega_{0}^{2}t}+A_{2}e^{-i\omega_{0}^{2}t}$ (6.59)

$\displaystyle=A\cos(\omega_{0}t+\phi),$ (6.60)

which is exactly the SHM that we found before. $\xi$ is purely imaginary.
Case 2:

$\gamma^{2}-\omega_{0}^{2}<0$ . In this case we define the damped frequency $\omega_{d}=\sqrt{\omega_{0}^{2}-\gamma^{2}}$ , so the solution becomes

$\displaystyle x(t)$ $\displaystyle=A_{1}e^{(-\gamma+i\omega_{d})t}+A_{2}e^{(-\gamma-i\omega_{d})t}$ (6.61)

$\displaystyle=Ae^{-\gamma t}\cos(\omega_{d}t+\phi).$ (6.62)

So there are still oscillations, but the amplitude decays over time ( $x_{m}(t)=Ae^{-\gamma t}$ ). This is an underdamped system. $\xi$ is a complex number.
Case 3:

$\gamma^{2}-\omega_{0}^{2}>0$ . Then $\xi$ is a real number and the solution just becomes the sum of two decaying exponentials.

$x(t)=A_{1}e^{(-\gamma+\sqrt{\gamma^{2}-\omega_{0}^{2}})t}+A_{2}e^{(-\gamma-% \sqrt{\gamma^{2}-\omega_{0}^{2}})t}.$ (6.63)

There are two slightly different decay rates, the second term decays faster since its decay constant is larger. This is an overdamped system.
Case 4:

$\gamma^{2}=\omega_{0}^{2}$ . Here, $\xi$ simplifies to just $-\gamma$ (a real number) and the general solution becomes

$x(t)=(A_{1}+A_{2}t)e^{-\gamma t}.$ (6.64)

This is a critically damped system.

6.4.2 $Q$ -Factor

In the case of an underdamped system, notice that the total mechanical energy, given by equation 6.37, will decay exponentially.

E_{\text{mech}}(t)=E_{\text{mech}}(0)e^{-\frac{b}{m}t}.

(6.65)

To quantify the level of underdamping, it is common to define the $Q$ -factor as the ratio of the initial energy in the oscillator to the energy dissipated in one radian of the oscillation, or

Q=2\pi\frac{\text{initial energy}}{\text{energy dissipated in one cycle}}.

(6.66)

It can be shown that for our linear damping the $Q$ -factor is equal to

Q=\frac{m}{b}\omega_{d}=\tau\omega_{d},

(6.67)

where $\tau=\frac{m}{b}$ is the time constant for the decay.

	$\displaystyle x(t)$	$\displaystyle=A_{1}e^{i\omega_{0}^{2}t}+A_{2}e^{-i\omega_{0}^{2}t}$		(6.59)
		$\displaystyle=A\cos(\omega_{0}t+\phi),$		(6.60)

	$\displaystyle x(t)$	$\displaystyle=A_{1}e^{(-\gamma+i\omega_{d})t}+A_{2}e^{(-\gamma-i\omega_{d})t}$		(6.61)
		$\displaystyle=Ae^{-\gamma t}\cos(\omega_{d}t+\phi).$		(6.62)

6.4 Damped Oscillations

6.4.1 Different Damping Cases

6.4.2 Q-Factor

6.4.2 $Q$ -Factor