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The Harmonic Oscillator

The harmonic oscillator is a canonical system discussed in every freshman course of physics. Thus, you might skip this lecture if you are familiar with it.

1. Basic equations of motion and solutions

The linearized equation of motion of an undamped and undriven pendulum is called a harmonic oscillator:

(1)	d2/dt2 + 02 = 0,

Nearly all oscillators and oscillations in physics are modeled by this equation of motion, at least in a first approximation, because it can be solved analytically. It is a linear second-order differential equation with constant coefficients. Such a differential equation can be solved easily by the ansatz = exp(t), where is some constant which has to be calculated. Each d/dt puts a factor before the exponential function. Thus, a linear ordinary differential equation like (1) turns into a polynomial because the common exponential function (a nonzero factor) can be dropped. Thus, ² + ₀² = 0. The roots of this so-called characteristic polynomial are = ±i₀. Hence, we get two solutions. Because the equation of motion is linear, one gets new solutions by multiplying a known solution by an arbitrary factor or by adding a solution to another one. This so-called superposition principle of linear differential equations is very powerful. It says that any solution can be built up by only a few different types of "building blocks" called fundamental solutions. The solutions exp(-i₀t) and exp(+i₀t) are a set of two independent fundamental solutions of the harmonic oscillator. Thus, the general solution of (1) reads

(2)	(t) = C exp(i0t) + C* exp(-i0t),

where C is a complex number and C^* denotes its conjugated complex. Because of the superposition principle, the set of fundamental solutions can be arbitrary, except that a fundamental solution has to be independent from the other ones. That is, each fundamental solution can not be written as a weighted sum of the other fundamental solutions. Instead of the complex valued exponential functions, one often chooses sin(₀t)=(exp(i₀t)-exp(-i₀t))/2i and cos(₀t)=(exp(i₀t)+exp(-i₀t))/2 as another set of fundamental solutions. Therefore,

(3)	(t) = A sin(0t) + B cos(0t),

is a general solution, too. The constants A and B can be expressed in term of C and vice versa.

Using the porperties of trigonometric functions, the general solution can also be expressed in a physically more relevant way: (4) (t) = max sin(0t-), where max and are the amplitude and the phase of oscillation, respectively. Again they can be expressed in terms of the integration constants of (2) or (3). Note, that these constants are determined by the initial conditions (0) and d/dt|0. The period of oscillation is T0=2/0. These three quantities are independent from each other. This isn't a triviality as nonlinear oscillations show. In fact, the independence of the frequency from the amplitude is a more subtile definition of a harmonic oscillator.

2. Damping

What happens if the harmonic oscilliator is damped? In this case, the equation of motions has an additional term which comes from the damping force:

(5)	d2/dt2 + d/dt + 02 = 0.

Again, with the ansatz = exp(t), one gets a characteristic polynomial: ² + + ₀² = 0. It has the solutions

(6)	± = -/2 ± (2/4 - 02)1/2.

Depending on wether 2/4 is less than, equal, or greater than 02, one has to distinguish three cases: The underdamped case: /2 < 0 The eigenvalues ± are a conjugated complex pair. The imaginary part (02 - 2/4 )1/2 is just the frequency of a damped oscillation. The real part determines the rate of decay. The general solution reads (7) (t) = max exp(-t/2) sin(t+), The exponentially decaying amplitude of the oscillation is called the envelope. The critically damped case: /2 = 0 Both eigenvalues are identically. In this so-called degenerated case, the ansatz = exp(t) is not generally enough because it leads only to one solution. But for a fundamental system two independent solutions are needed. Where do we get a second solution? In such a case the ansatz = t exp(t) helps. Thus the general solution reads (8) (t) = (At+B) exp(-t/2). The overdamped case: /2 > 0 Both eigenvalues are negative and real. The general solution reads (8) (t) = A+ exp(+t) + A- exp(-t). Any initial condition leads to an exponentially decay. There is only one extrema (minimum or maximum) contrary to the underdamped case with infinitely many maxima and minima. The decay is dominated by the eigenvalue most closest to zero.

3. Resonance

Oscillations in a damped harmonic oscillator decay to zero. When it is driven by a periodic force, one oscillation survives. The equation of motion of a damped and driven harmonic oscillator reads

(9)	d2/dt2 + d/dt+ 02 = a cos 2f t.

This is still a linear differential equation, but the sum of two solutions are no longer a solution. The reason for that is the inhomogenous term on the right-hand side. Nevertheless, the superposition principle holds in a modified version. A solution of (9) plus an arbitray solution of (5) (a so-called homogeneous solution) is also a solution of (9). One needs only one solution of (9) (the so-called particular solution) to generate any solution of (9) with the help of the homogeneous solution. Here, a particular solution can be found by the ansatz

(10)	(t) = A sin(2f t) + B cos(2f t).

It turns (9) into a sum with terms proportionally to cos(2f t) and sin(2f t). Because the equation should hold for any value of t, the sum of all cosinus terms and the sum of all sinus terms have to be zero independently. Hence, [₀²-(2f)²] B + 2f A = a and [₀²-(2f)²] A - 2f B = 0, respectively. The solution of these equations reads

(11)

A = 2fa / ([02-(2f)2]2 + (2f)2),

B = [02-(2f)2] a / ([02-(2f)2]2 + (2f)2).

In the long-time limit any solution of (9) approaches the particular solution (10) since any solution of (5) decays to zero. The harmonic oscillator, therefore, oscillates not with its eigenfrequency ₀ but with the frequency of the periodic force. The amplitude of oscillations depend on the driving frequency. It has its maximum when the driving frequency matches the eigenfrequency. This phenomenon is called resonance. To see this, rewrite equation (10) in the form (t) = _max cos(2ft-), with

(12)	max = a / ([02-(2f)2]2 + (2f)2)1/2

and

(13)	= arctan (2f/[02-(2f)2]).

In the underdamped case, the amplitude as a function of the driving frequency has a maximum near the eigenfrequency f₀ = ₀/2 of the oscillator. The deviation is of quadratic order in . The maximum is a/(₀) in leading order of . This is by a factor ₀/ larger than the amplitude for f0. That is, the amplitude of oscillation is larger than the amplitude of driving. This can be easily observed in the lab at the horizontally driven pendulum. The width of the so-called resonance line is proportional to . In the critically damped and overdamped case the resonance line disappears.

In the limit f0 the phase of oscillation is identical with the driving phase. In resonance, the phase is just 90° behind the driving phase. At high frequencies the phase approaches 180°. Thus, the pendulum moves just opposite to the drive. Again, this can be easily observed in the lab at the horizontally driven pendulum.

QUESTIONS worth to think about:

What happens with an undamped harmonic oscillator driven exactly by its eigenfrequency? Another look at the dynamics of the damped and driven harmonic oscillator is the following one: Instead of discussing the solution as a function of time, discuss it as a function of the initial conditions. For example, calculate the angle and the angular velocity at a given time as a function of the angle and the angluar velocity for the previous period of driving. That is, given are the initial conditions (0) and d/dt|0. Calculate (1/f) and d/dt|1/f as a function of the initial conditions. This calculation leads to a Poincaré map or stroboscopic map.

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© 1998 Franz-Josef Elmer,  elmer@ubaclu.unibas.ch last modified Sunday, July 19, 1998.