# Oscillations I: Heavily Damped Oscillator

*Michael Fowler *

### Introduction

In the next three lectures, we'll look at a wide variety of oscillatory phenomena. After a brief recap of undamped simple harmonic motion, we analyze the motion of a heavily damped oscillator. Why start with that? Well, the math is very straightforward (no complex numbers) and there is some nontrivial physics. Following that, we'll cover the lightly damped oscillator, where we do need complex numbers---avoiding them leads to very cumbersome trigonometry. We'll also discuss the important intermediate case called critical damping. Finally, in the third lecture, we examine the physically very relevant case of a driven damped oscillator---and when that leads to disaster.

### Brief Review of Undamped Simple Harmonic Motion

Our basic model simple harmonic oscillator is a mass $m$ moving back and forth along a line on a smooth
horizontal surface, connected to an inline horizontal spring, having spring
constant $k,$ the other end of the string being attached to
a wall. The spring exerts a restoring force equal to $-kx,$ on the mass when it is a distance
$x$ from the equilibrium point. By “equilibrium point” we mean the point
corresponding to the spring resting at its natural length, and therefore
exerting no force on the mass. The in-class realization of this model was an
aircar, with a light spring above the track (actually, we used *two* light
springs, going in opposite directions$\u2014$we found
if we just one it tended to sag on to the track when it was slack, but two in
opposite directions could be kept taut.
The two springs together act like a single spring having spring constant
the sum of the two).

Newton’s Law gives:

$F=ma,\text{or}m\frac{{d}^{2}x}{d{t}^{2}}=-kx.$

Solving this differential equation gives the position of the mass (the aircar) relative to the rest position as a function of time:

$x(t)=A\mathrm{cos}({\omega}_{\text{\hspace{0.17em}}0}\text{\hspace{0.05em}}t+\phi ).$

Here $A$ is the maximum displacement, and is called the
**amplitude** of the motion. ${\omega}_{0}t+\phi $ is called the **phase**. $\phi $ is called the **phase constant**: it
depends on where in the cycle you start, that is, where is the oscillator at
time zero.

The velocity and acceleration are given by differentiating $x\left(t\right)$ once and twice:

$v(t)=\frac{dx}{dt}=-A{\omega}_{\text{\hspace{0.17em}}0}\mathrm{sin}({\omega}_{\text{\hspace{0.05em}}0}\text{\hspace{0.05em}}t+\phi )$

and

$a(t)=\frac{{d}^{2}x(t)}{d{t}^{2}}=-A{\omega}_{\text{\hspace{0.05em}}0}{}^{2}\mathrm{cos}({\omega}_{\text{\hspace{0.05em}}0}\text{\hspace{0.05em}}t+\phi ).$_{}

We see immediately that this $x\left(t\right)$ does indeed satisfy Newton’s Law provided ${\omega}_{0}$ is given by

${\omega}_{\text{\hspace{0.05em}}0}=\sqrt{k/m}.$

*Exercise*: Verify that,
apart from a possible overall constant, this expression for ${\omega}_{0}$ could have been figured out using
dimensions.

### Energy Conservation in Undamped Simple Harmonic Motion

The spring stores *potential energy*: if you push one
end of the spring from some positive extension $x$ to $x+dx$ (with the other end of the spring fixed, of
course) the force $-kx$ opposes the motion, so you must push with
force $+kx,$ and
therefore do work $kxdx.$ To find the *total* potential energy
stored by the spring when the end is ${x}_{0}$ away from the equilibrium point (the natural spring
length) we must find the total work required to stretch the spring from that resting length to an extension ${x}_{0}.$ This
means adding up all the little bits of work $kxdx$ needed to get the spring from no extension at
all to an extension of ${x}_{0}.$ In other words, we need to do an integral to
find the potential energy $U\left({x}_{0}\right):$

_{$U({x}_{0})={\displaystyle \underset{0}{\overset{{x}_{0}}{\int}}kxdx=}{\scriptscriptstyle \frac{1}{2}}k{x}_{0}{}^{2}.$}_{}

So the potential energy plotted as a function of distance
from equilibrium is *parabolic*:

The
oscillator has *total* energy equal to kinetic energy + potential energy,

$E={\scriptscriptstyle \frac{1}{2}}m{v}^{2}+{\scriptscriptstyle \frac{1}{2}}k{x}^{2}$

when the mass is at position $x.$ Putting in the values of $x\left(t\right),v\left(t\right)$ from the equations above, it is easy to check that $E$ is independent of time and equal to ${\scriptscriptstyle \frac{1}{2}}k\text{\hspace{0.05em}}{A}^{2},\text{\hspace{0.17em}}A$ being the amplitude of the motion, the maximum displacement. Of course, when the oscillator is at $x=\pm A,$ it is momentarily at rest, so has no kinetic energy.

### Heavily Damped Oscillator: Pendulum in Molasses

Once we allow damping, it turns out that heavy damping is a little easier to deal with than light damping, so we'll look at this first. (When, exactly, does damping count as heavy? We'll see shortly.)

Suppose, then, the oscillator is subject to a drag force *proportional to velocity*.

The equation of motion is:

$m\frac{{d}^{2}x}{d{t}^{2}}=-kx-b\frac{dx}{dt}.$

Although this equation looks much more difficult than the no-drag case, it really isn’t!

The
important point is that the terms are just *derivatives* of $x$ with respect to time, multiplied by *constants*.
The math would be a lot more difficult if we had a drag force proportional to
the square of the velocity, or if the force exerted by the spring were not a
constant times $x$ (this means we can’t stretch the string too
far$\u2014$they all
go nonlinear for a big enough stretch).
Anyway, it is easy to find *exponential
functions *that are solutions to this equation. Let us guess a solution:

$x={x}_{0}{e}^{-\alpha t}.$

We're guessing that, thanks to the molasses it won't actually oscillate. Can that be right? We'll find out. Inserting this proposed solution in the equation of motion, using

$\frac{dx}{dt}=-\alpha {x}_{0}{e}^{-\alpha t},\text{}\frac{{d}^{2}x}{d{t}^{2}}={\alpha}^{2}{x}_{0}{e}^{-\alpha t}$

we find that it *is* a valid solution provided that $\alpha $ satisfies:

$m{\alpha}^{2}-\alpha b+k=0$

from which

$\alpha =\frac{b\pm \sqrt{{b}^{2}-4mk}}{2m}.$

Staring at this expression for $\alpha $, we notice that for $\alpha $ to be real, we need to have

${b}^{2}>4mk.$

What can that mean?
Remember $b$ is the damping parameter$\u2014$we’re
finding that our proposed exponential solution only works for *large* damping! (We know physically, of course, that if the damping is small, the pendulum will swing, a motion not described by our exponential decay function.) We'll analyze the large damping case now,
then after that go on to see how to extend the solution to small damping.

### Interpreting the Two Different Exponential Solutions

It’s worth looking at the case of *very* large damping
(imagine our mass and spring being in a tub of molasses), where the two
exponential solutions turn out to decay at very different rates. For ${b}^{2}$ *much* greater than $4mk,$ we can write

$\alpha =\frac{b\pm b\sqrt{1-\frac{4mk}{{b}^{2}}}}{2m}={\alpha}_{1},{\alpha}_{2}$

and then expand the square root using

${(1-x)}^{1/2}\cong 1-{\scriptscriptstyle \frac{1}{2}}x,$

valid for small $x,$ to find that approximately$\u2014$for large $b$ $\u2014$the two possible values of $\alpha $ are:

$\text{}{\alpha}_{1}=\frac{b}{m}\text{and}{\alpha}_{2}=\frac{k}{b}\text{}\text{.}$

That is to say, there are two possible highly damped decay modes,

$x={A}_{1}{e}^{-{\alpha}_{1}t}\text{and}x={A}_{2}{e}^{-{\alpha}_{2}t}.$

Note that since the damping $b$ is large, ${\alpha}_{1}$ is *large*, meaning *fast* decay,
and ${\alpha}_{2}$ is *small*, meaning *slow* decay.

*Question*: what,
physically, is going on in these two different highly damped exponential
decays? Can you construct a plausible
scenario of a mass on a spring, all in molasses, to see why two very different
rates of change of speed are possible?

*Hint*: look again at
the equation of motion of this damped oscillator. Notice that in each of these highly damped
decays, one term doesn’t play any part$\u2014$but the
irrelevant term is a *different* term for the two decays!

*Answer 1*: for $\alpha =k/b$,
evidently the *mass* doesn’t play a role. This decay is what you get if you pull the
mass to one side, let go, then, after it gets moving, it will very slowly
settle towards the equilibrium point.
Its rate of approach is determined by balancing the spring’s force
against the speed-dependent damping force, to give the speed. The rate of change of speed$\u2014$the
acceleration$\u2014$is so tiny
that the *inertial* term$\u2014$the mass$\u2014$is
negligible.

*Answer 2*: for $\alpha =b/m$,
the *spring* is negligible. And, this is very *fast* motion ( $b/m\gg k/b,$ since we said ${b}^{2}\gg 4mk.$ ) The
way to get this motion is to pull the mass to one side, then give it a very
strong kick towards the equilibrium point. If you give it just the right (high) speed,
all the momentum you imparted will be spent overcoming the damping force as the
mass moves to the center$\u2014$the force
of the spring will be negligible.

### *The Most General Solution for the Highly Damped Oscillator

The damped oscillator equation

$m\frac{{d}^{2}x}{d{t}^{2}}=-kx-b\frac{dx}{dt}$

is a linear equation. This means that if ${x}_{1}\left(t\right)$ is a solution, and ${x}_{2}\left(t\right)$ is another solution, that is,

$\begin{array}{l}m\frac{{d}^{2}{x}_{1}(t)}{d{t}^{2}}=-k{x}_{1}(t)-b\frac{d{x}_{1}(t)}{dt}\\ m\frac{{d}^{2}{x}_{2}(t)}{d{t}^{2}}=-k{x}_{2}(t)-b\frac{d{x}_{2}(t)}{dt}\end{array}$

then just adding the two equations we get:

_{$m\frac{{d}^{2}({x}_{1}(t)+{x}_{2}(t))}{d{t}^{2}}=-k({x}_{1}(t)+{x}_{2}(t))-b\frac{d({x}_{1}(t)+{x}_{2}(t))}{dt}.$}_{}

It is also clear that *multiplying a solution by a
constant produces another solution*: if $x\left(t\right)$ satisfies the equation, so does $3x\left(t\right).$

This means, then, that given two solutions ${x}_{1}\left(t\right)$ and ${x}_{2}\left(t\right),$ and two arbitrary constants ${A}_{1}$ and ${A}_{2},$ the function

$${A}_{1}{x}_{1}\left(t\right)+{A}_{2}{x}_{2}\left(t\right)$$

is also a solution of the differential equation.

In fact, all possible motions of the highly damped
oscillator have this form. The way to understand this is to realize that the
oscillator’s motion is *completely determined* if we specify at an initial
instant of time *both* the position *and* the velocity of the
oscillator. The equation of motion gives
the acceleration as a function of position and velocity, so, at least in
principle, we can work out step by step
how the mass must move; technically, we are integrating the equation of motion,
either mathematically, or numerically such as by using a spreadsheet. So, by
suitably adjusting the two arbitrary constants ${A}_{1}$ and ${A}_{2},$ we can match our sum of solutions to any given
initial position and velocity.

To summarize, for the highly damped oscillator any solution is of the form:

$x(t)={A}_{1}{e}^{-{\alpha}_{1}t}+{A}_{2}{e}^{-{\alpha}_{2}t}={A}_{1}{e}^{-\frac{b+b\sqrt{1-\frac{4mk}{{b}^{2}}}}{2m}t}+{A}_{2}{e}^{-\frac{b-b\sqrt{1-\frac{4mk}{{b}^{2}}}}{2m}t}.$

*Exercises on highly damped
oscillations*

1. If the oscillator
is pulled aside a distance ${x}_{0},$ and released from rest at $t=0,$ what are ${A}_{1},{A}_{2}$? Describe the subsequent motion, especially
the very beginning: what is the initial acceleration? (*Hint*:
think carefully about how important the damping term is immediately after
release from rest$\u2014$you should
be able to *guess* the initial
acceleration.)

2. If the oscillator is initially at the equilibrium position ${x}_{0}=0,$ but is given a kick to a velocity ${v}_{0},$ find ${A}_{1}$ and ${A}_{2}$ and describe the subsequent motion.

### *The Principle of Superposition for Linear Differential Equations

The equation for the highly damped oscillator is a *linear* differential equation, that is,
an equation of the form (in more usual notation):

${c}_{0}f(x)+{c}_{1}\frac{df(x)}{dx}+{c}_{2}\frac{{d}^{2}f(x)}{d{x}^{2}}=0$

where ${c}_{0},{c}_{1}$ and ${c}_{2}$ are constants, that is, independent of $x.$

For such a linear differential equation, if ${f}_{1}\left(x\right)$ and ${f}_{2}\left(x\right)$ are solutions, so is ${A}_{1}{f}_{1}\left(x\right)+{A}_{2}{f}_{2}\left(x\right)$ for any constants ${A}_{1},{A}_{2}.$

This is called the ** Principle of Superposition**,
and is proved in general

*exactly*as we proved it for the highly damped oscillator in the preceding section.

Even more important, this Principle of Superposition is
valid, using analogous arguments, for linear differential equations in *more
than one variable*, such as the wave equations we'll be considering
shortly. In that case, it gives insight
into how waves can pass through each other and emerge unchanged.