Damped oscillations in one dimensions
- Simple spring equation + linear resistance, $F_r=-b\dot{x}$.
- Solving the (differential) equation of motion.
- Under, over, and critical damping.
Simulation (Mathematica .cdf file can be played with the free CDF Player.)
Let's add a viscous damping force to the spring equation... $$m \ddot{x} = -kx -b \dot{x}.$$
In this 1-d equation of motion, $-kx$ is the spring's restoring force, and the new piece is the velocity dependent damping force $-b \dot{x}$ acting opposite the velocity: this is viscous damping.
Let's re-write this (dividing everything by $m$) as: $$\ddot{x} + (b/m) \dot{x} + (k/m)x = 0$$
We'll find a damping constant to be useful: $$\beta \equiv \frac{b}{2m}$$
And we can define the natural frequency: $$\omega_0 \equiv \sqrt{\frac{k}{m}}$$
The equation of motion, using these constants, that we'll need to solve is:
$$\ddot{x} + 2\beta \dot{x} + \omega_0^2x = 0$$
How to solve?
Any guesses?
Indeed we will appeal to guessing. There is no obvious integration technique or variable separation, let alone any 'cookbook' way of finding a solution to this one.
$e^{kt}$ "been berry berry good to me"[*]...
[*] "Chico Escuela"'s catchphrase "Beisbol been berry berry..." on Saturday Night Live, 1980'ish
Let's guess $x(t)=e^{kt}$, $\Rightarrow \dot{x} = ke^{kt}$; $\ddot{x} = k^2e^{kt}$
Substituting these into our equation of motion gives: $$k^2e^{kt} + 2\beta ke^{kt} + \omega_0^2e^{kt}=0$$
Dividing both sides by $e^{kt}$ gives us a quadratic equation for $k$: $$k^2 + 2\beta k + \omega_0^2=0$$
The two solutions are $$ k_1 = -\beta + \sqrt{\beta^2 - \omega_0^2}$$ $$ k_2 = -\beta - \sqrt{\beta^2 - \omega_0^2}$$
The general solution is then a linear combination of the two solutions $$x(t)=C_1e^{k_1 t}+C_2e^{k_2 t}$$ which I'll factor just a bit like this:
$$x(t)=e^{-\beta t}\left(C_1 e^{\sqrt{\beta^2-\omega_0^2}t} + C_2 e^{-\sqrt{\beta^2-\omega_0^2}t}\right).$$
Sanity check: If we set $\beta=0$ (no damping) we ought to get good ol' simple harmonic oscillations...
Let's consider 2 different parts of parameter space:
$\beta < \omega_0$: underdamping
When $\beta < \omega_0$ then the argument of the square roots is negative. We can re-write those pieces in terms of a new positive real number $\omega_1$: $$\sqrt{\beta^2-\omega_0^2} = i\sqrt{\omega_0^2-\beta^2} \equiv i \omega_1$$
where the new angular frequency $\omega_1 < \omega_0$ is smaller than the undamped oscillator frequency.
Notice that $\beta$ and $\omega_0$ and $\omega_1$ are all frequencies (1/sec). Is that obvious from the equation of motion? The solution is now $$x(t)=e^{-\beta t}\left(C_1 e^{i\omega_1 t}+C_2 e^{-i \omega_1t}\right) = Ae^{-\beta t}\cos (\omega_1 t + \delta). $$
This is the product of exponential decay and simple harmonic motion.
blue: $Ae^{-\beta t}(cos(\omega_1 t + \delta))$
pink: $\pm Ae^{-\beta t}$
Oscillations are dieing out with a characteristic time of $1/\beta$
$\beta > \omega_0$: overdamping
When $\beta > \omega_0$ then the argument of the square roots is positive, and thus real. The general solution is a sum of two exponential decays $$x(t)=C_1 e^{-\left(\beta- \sqrt{\beta^2-\omega_0^2}\right) t} + C_2 e^{-\left(\beta+\sqrt{\beta^2-\omega_0^2}\right) t}$$
The $t \gg 0$ behavior approaches the one that dies off slower, $$x(t \rightarrow \infty) = C_1 e^{-\left(\beta- \sqrt{\beta^2-\omega_0^2}\right)t}$$
The characteristic time for oscillations to die out is $1/\left(\beta - \sqrt{\beta^2-\omega_0^2}\right)$.
The blue function is a sum of the two gray exponentials.
$\beta = \omega_0$: critical damping
When $\beta = \omega_0$, the square root in our two quadratic solutions is 0, and so instead of two solutions, we have one: $A_1e^{-\beta t}$. Hmmm, we really *ought* to have one more linearly independent solution. Maybe something with a whiff of exponential-ness...
You can verify that this is another solution: $$x(t)=te^{kt}$$ The general solution is now: $$x(t)=C_1 e^{-\beta t} + C_2 t e^{-\beta t}.$$
And the characteristic time for oscillations to die away is now $1/\beta = 1/\omega_0$
Characteristic times
It is interesting to look at that 'characteristic time' for oscillations to die away across all the regimes.
This gives you a ready design criteria for things like automobile shock absorbers: $\beta \approx \omega_0$ to damp out oscillations as fast as possible.
Homework
Chapter 5: 20, 21, 28, 31