390	Week 9: Oscillations

• From the velocity equation above, we can easily find the kinetic energy as a function of time:

	K(t) =	1	mv2	=	1	mA2	ω2 sin2	(ωt + φ) =		1	kA2 sin2	(ωt + φ)	(772)
		2			2				2

• The potential energy of an oscillator is found by integrating:

U (x) = −Z0	−kx′dx′ =	2 kx2 =		2 kA2 cos2(ωt + φ)		(773)
	x	1		1

if we use the (usual but not necessary) convention that U (0) = 0 when the mass is at the equilibrium displacement, x = 0.

• The total mechanical energy is therefore obviously a constant:

E(t) =	1	mv2 +	1	kx2 =	1		kA2 sin2(ωt + φ) +	1		kA2 cos2(ωt + φ) =	1		kA2	(774)
	2		2			2			2			2

•As usual, the relation between the angular frequency, the regular frequency, and the period of the oscillator is given by:

	ω =	2π	= 2πf	(775)
		T

(where f = 1/T ). SI units of frequency are Hertz – cycles per second. Angular frequency has units of radians per second. Since both cycles and radians are dimensionless, the units themselves are dimensionally inverse seconds but they are (obviously) related by 2π radians per cycle.

•A (non-ideal) harmonic oscillator in nature is almost always damped by friction and drag forces. If we assume damping by viscous drag in (low Reynolds number) laminar flow – not unreasonable for smooth objects moving in a damping fluid, if somewhat itself idealized – the

equation of motion becomes:

d2x		b dx				k
	+ +				+		x = 0
dt2		m		dt		m

•The solution to this equation of motion is:

x (t) = A e −b t cos(ω′t + φ)

±± 2m

where r

ω′ = ω0 1 − b2 4km

(776)

(777)

(778)

9.1: The Simple Harmonic Oscillator

Oscillations occur whenever a force exists that pushes an object back towards a stable equilibrium position whenever it is displaced from it. Such forces abound in nature – things are held together in structured form because they are in stable equilibrium positions and when they are disturbed in certain ways, they oscillate.

When the displacement from equilibrium is small, the restoring force is often linearly related to the displacement, at least to a good approximation. In that case the oscillations take on a special character – they are called harmonic oscillations as they are described by harmonic functions (sines and cosines) known from trigonometery.

In this course we will study simple harmonic oscillators, both with and without damping (and harmonic driving) forces. The principle examples we will study will be masses on springs and various penduli.

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	equilibrium	Fx = − kx

k

m

x

Figure 117: A mass on a spring is displaced by a distance x from its equilibrium position. The spring exerts a force Fx = −kx on the mass (Hooke’s Law).

9.1.1: The Archetypical Simple Harmonic Oscillator: A Mass on a Spring

Consider a mass m attached to a perfect spring (which in turn is a xed to a wall as shown in figure 117. The mass rests on a surface so that gravitational forces are cancelled by the normal force and are hence irrelevant. The mass will be displaced only in one direction (call it x) and otherwise constrained so that motion in or out of the plane is impossible and no drag or frictionless forces are (yet) considered to be relevant.

We know that the force exerted by a perfect spring on a mass stretched a distance x from its equilibrium position is given by Hooke’s Law:

Fx = −kx

(779)

where k is the spring constant of the spring. This is a linear restoring force, and (as we shall see) is highly characteristic of the restoring forces exerted by any system around a point of stable equilibrium.

Although thus far we have avoided trying to determine the most general motion of the mass, it is time for us to tackle that chore. As we shall see, the motion of an undamped simple harmonic oscillator is very easy to understand in the ideal case, and easy enough to understand qualitatively or semi-quantitatively that it serves as an excellent springboard to understanding many of the properties of bulk materials, such as compressibility, stress, and strain.

We begin, as one might expect, with Newton’s Second Law, used to obtain the (second order, linear, homogeneous, di erential) equation of motion for the system. Note well that although this sounds all complicated and everything – like “real math” – we’ve been solving second order di erential equations from day one, so they shouldn’t be intimidating. Solving the equation of motion for the simple harmonic oscillator isn’t quite as simple as just integrating twice, but as we will see neither is it all that di cult.

Hooke’s Law combined with Newton’s Second Law can thus be written and massaged algebraically as follows:

d2x

max = m

= Fx = −kx

dt2

m

d2x

+ kx

=

0

dt2

d2x

k

+

x

=

0

dt2

m

d2x

+ ω2x

=

0

(780)

dt2

where we have defined the angular frequency of the oscillator,

ω2 = k/m

(781)

392	Week 9: Oscillations

This must have units of inverse time squared (why?). We will momentarily justify this identification, but it won’t hurt to start learning it now.

Equation 780 (with the ω2) is the standard harmonic oscillator di erential equation of motion (SHO ODE). As we’ll soon see with quite a few examples and an algebraic argument, we can put the equation of motion for many systems into this form for at least small displacements from a stable equilibrium point. If we can properly solve it once and for all now, whenever we can put an equation of motion into this form in the future we can just read o the solution by identifying similar quantities in the equation.

To solve it171, we note that it basically tells us that x(t) must be a function that has a second derivative proportional to the function itself.

We know at least three functions whose second derivatives are proportional to the functions themselves: cosine, sine and exponential. In this particular case, we could guess cosine or sine and we would get a perfectly reasonable solution. The bad thing about doing this is that the solution methodology would not generalize at all – it wouldn’t work for first order, third order, or even general second order ODEs. It would give us a solution to the SHO problem (for example) but not allow us to solve the damped SHO problem or damped, driven SHO problems we investigate later this week. For this reason, although it is a bit more work now, we’ll search for a solution assuming that it has an exponential form.

Note Well!

If you are completely panicked by the following solution, if thinking about trying to understand it makes you feel sick to your stomach, you can probably skip ahead to the next section (or rather, begin reading again at the end of this chapter after the real solultion is obtained).

There is a price you will pay if you do. You will never understand where the solution comes from or how to solve the slightly more di cult damped SHO problem, and will therefore have to memorize the solutions, unable to rederive them if you forget (say) the formula for the damped oscillator frequency or the criterion for critical damping.

As has been our general rule above, I think that it is better to try to make it through the derivation to where you understand it, even if only a single time and for a moment of understanding, even if you do then move on and just learn and work to retain the result. I think it helps you remember the result with less e ort and for longer once the course is over, and to bring it back into mind and understand it more easily if you should ever need to in the future. But I also realize that mastering a chunk of math like this doesn’t come easily to some of you and that investing the time to do given a limited amount of time to invest might actually reduce your eventual understanding of the general content of this chapter. You’ll have to decide for yourself if this is true, ideally after at least giving the math below a look and a try. It’s not really as di cult as it looks at first.

The exponential assumption:

x(t) = Aeαt

(782)

makes solutions to general linear homogeneous ODEs simple.

171Not only it, but any homogeneous linear N th order ordinary di erential equation – the method can be applied to first, third, fourth, fifth... order linear ODEs as well.

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Let’s look at the pattern when we take repeated derivatives of this equation:

x(t) = Aeαt

	dx			=	αAeαt
		dt
	d2x			= α2Aeαt
	dt2
	d3x			= α3Aeαt
	dt3
	...				(783)

where α is an unknown parameter and A is an arbitrary (possibly complex) constant (with the units of x(t), in this case, length) called the amplitude. Indeed, this is a general rule:

	dnx	= αnAeαt	(784)
	dtn

for any n = 0, 1, 2....

Substituting this assumed solution and its rule for the second derivative into the SHO ODE, we get:

	d2x			+ ω2x	=	0
		dt2
d2Aeαt		+ ω2Aeαt			=	0
dt2
α2Aeαt + ω2Aeαt					=	0
¡α2 + ω2¢ Aeαt					=	0	(785)

There are two ways this equation could be true. First, we could have A = 0, in which case x(t) = 0 for any value of α. This indeed does solve the ODE, but the solution is boring – nothing moves! Mathematicians call this the trivial solution to a homogeneous linear ODE, and we will reject it out of hand by insisting that we have a nontrivial solution with A =6 0.

In that case it is necessary for

¡α2 + ω2¢ = 0

(786)

This is called the characteristic equation for the linear homogeneous ordinary di erential equation. If we can find an α such that this equation is satisfied, then our assumed answer will indeed solve the ODE for nontrivial (nonzero) x(t).

Clearly:	α = ±iω
			(787)
where	√
	i = + −1		(788)

We now have a solution to our second order ODE – indeed, we have two solutions – but those solutions are complex exponentials172 and contain the imaginary unit173 , i.

In principle, if you have satisfied the prerequisites for this course you have almost certainly studied imaginary numbers and complex numbers174 in a high school algebra class and perhaps again in college level calculus. Unfortunately, because high school math is often indi erently well taught, you may have thought they would never be good for anything and hence didn’t pay much attention to them, or (however well they were or were not covered) at this point you’ve now forgotten them.

172Wikipedia: http://www.wikipedia.org/wiki/Euler Formula.

173Wikipedia: http://www.wikipedia.org/wiki/Imaginary unit.

174Wikipedia: http://www.wikipedia.org/wiki/Complex numbers.

394	Week 9: Oscillations

In any of these cases, now might be a really good time to click on over to my online Mathematics for Introductory Physics book175 and review at least some of the properties of i and complex numbers and how they relate to trig functions. This book is still (as of this moment) less detailed here than I would like, but it does review all of their most important properties that are used below. Don’t hesitate to follow the wikipedia links as well.

If you are a life science student (perhaps a bio major or premed) then perhaps (as noted above) you won’t ever need to know even this much and can get away with just memorizing the real solutions below. If you are a physics or math major or an engineering student, the mathematics of this solution is just a starting point to an entire, amazing world of complex numbers, quaternions, Cli ord (geometric division) algebras, that are not only useful, but seem to be essential in the development of electromagnetic and other field theories, theories of oscillations and waves, and above all in quantum theory (bearing in mind that everything we are learning this year is technically incorrect, because the Universe turns out not to be microscopically classical). Complex numbers also form the basis for one of the most powerful methods of doing certain classes of otherwise enormously di cult integrals in mathematics. So you’ll have to decide for yourself just how far you want to pursue the discovery of this beautiful mathematics at this time – we will be presenting only the bare minimum necessary to obtain the desired, general, real solutions to the SHO ODE below.

Here are the two linearly independent solutions:

x+(t)	=	A+e+iωt	(789)
x−(t)	=	A−e−iωt	(790)

that follow, one for each possible value of α. Note that we will always get n independent solutions for an nth order linear ODE, because we will always have solve for the roots of an nth order characteristic equation, and there are n of them! A± are the complex constants of integration – since the solution is complex already we might as well construct a general complex solution instead of a less general one where the A± are real.

Given these two independent solutions, an arbitrary, completely general solution can be made up of a sum of these two independent solutions:

x(t) = A+e+iωt + A−e−iωt

(791)

We now use a pair of True Facts (that you can read about and see proven in the wikipedia articles linked above or in the online math review). First, let us note the Euler Equation:

eiθ = cos(θ) + i sin(θ)

(792)

This can be proven a number of ways – probably the easiest way to verify it is by noting the equality of the taylor series expansions of both sides – but we can just take it as “given” from here on in this class. Next let us note that a completely general complex number z can always be written as:

z= x + iy

=|z|cos(θ) + i|z|sin(θ)

=|z|(cos(θ) + i sin(θ))

= |z|eiθ

(793)

(where we used the Euler equation in the last step so that (for example) we can quite generally write:

	A+	=	|A+|e+iφ+	(794)
	A−	=	|A−|e−iφ−	(795)
175http://www.phy.duke.edu/˜rgb/Class/math for intro physics.php There is an entire chapter on this:				Complex

Numbers and Harmonic Trigonometric Functions, well worth a look.

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395

for some real amplitude |A±| and real phase angles ±φ±176.

If we substitute this into the two independent solutions above, we note that they can be written

as:

x+(t)	=	A+e+iωt	=	|A+|eiφ+ e+iωt = |A+|e+i(ωt+φ+)	(796)
x−(t)	=	A−e−iωt	=	|A−|e−iφ− e−iωt = |A−|e−i(ωt+φ−)	(797)

Finally, we wake up from our mathematical daze, hypnotized by the strange beauty of all of these equations, smack ourselves on the forehead and say “But what am I thinking! I need x(t) to be real, because the physical mass m cannot possibly be found at an imaginary (or general complex) position!”. So we take the real part of either of these solutions:

x+(t) = |A+|e+i(ωt+φ+)
= |A+| (cos(ωt + φ+) + i sin(ωt + φ+))
= |A+|cos(ωt + φ+)	(798)
and
x−(t) = |A−|e−i(ωt+φ−)
= |A−| (cos(ωt + φ−) − i sin(ωt + φ−))
= |A−|cos(ωt + φ−)	(799)

These two solutions are the same. They di er in the (sign of the) imaginary part, but have exactly the same form for the real part. We have to figure out the amplitude and phase of the solution in any event (see below) and we won’t get a di erent solution if we use x+(t), x−(t), or any linear combination of the two! We can finally get rid of the ± notation and with it, the last vestige of the complex solutions we used as an intermediary to get this lovely real solution to the position of (e.g.) the mass m as it oscillates connected to the perfect spring.

If you skipped ahead above, resume reading/studying here!

Thus:
x(t) = A cos(ωt + φ)	(800)

is the completely general, real solution to the SHO ODE of motion, equation 780 above, valid in any context, including ones with a di erent context (and even a di erent variable) leading to a di erent algebraic form for ω2.

A few final notes before we go on to try to understand this solution. There are two unknown real numbers in this solution, A and φ. These are the constants of integration! Although we didn’t exactly “integrate” in the normal sense, we are still picking out a particular solution from an infinity of two-parameter solutions with di erent initial conditions, just as we did for constant acceleration problems eight or nine weeks ago! If you like, this solution has to be able to describe the answer for any permissible value of the initial position and velocity of the mass at time t = 0. Since we can independently vary x(0) and v(0), we must have at least a two parameter family of solutions to be able to describe a general solution177.

176It doesn’t matter if we define A− with a negative phase angle, since φ− might be a positive or negative number anyway. It can also always be reduced via modulus 2π into the interval [0, 2π), because eIφ is periodic.

177In future courses, math or physics majors might have to cope with situations where you are given two pieces of

data about the solution, not necessarily initial conditions. For example, you might be given x(t1) and x(t2) for two specified times t1 and t2 and be required to find the particular solution that satisfied these as a constraint. However, this problem is much more di cult and can easily be insu cient data to fully specify the solution to the problem. We will avoid it here and stick with initial value problems.