Up to this point we have only examined the special case in which the net force on an oscillating particle is always proportional to the displacement of the particle. Oftentimes, however, there are other forces in addition to this restoring force, which create more complex oscillations. Though much of the study of this motion lies in the realm of differential equations, we will give at least an introductory treatment to the topic.

Damped Harmonic Motion

In most real physical situations, an oscillation cannot go on indefinitely. Forces such as friction and air resistance eventually dissipate energy and decrease both the speed and amplitude of oscillation until the system is at rest at its equilibrium point. The most common dissipative force encountered is a damping force, which is proportional to the velocity of the object, and always acts in a direction opposite the velocity. In the case of the pendulum, air resistance always works against the motion of the pendulum, counteracting the gravitational force, shown below.

Figure %: A pendulum subject to air resistance of magnitude bv, where b is a positive constant.

We denote the force as Fd, and relate it to the velocity of the object: Fd = - bv, where b is a positive constant of proportionality, dependent on the system. Recall that we generated the differential equation for simple harmonic motion using Newton's Second Law:

- kx = m

We must add our damping force to the left side of this equation:

- kx - b = m    

Unfortunately generating a solution to this equation requires more advanced mathematics than just calculus. We will simply state the final solution and discuss its implications. The position of the damped oscillating particle is given by:

x = xme-bt/2mcos(σâ≤t)    


σâ≤ =    

Clearly this equation is a complicated one, so let's take it apart piece by piece. The most notable change from our simple harmonic equation is the presence of the exponential function, e-bt/2m. This function gradually decreases the amplitude of the oscillation until it reaches zero. We still have our cosine function, though we must calculate a new angular frequency. As we can tell by our equation for σâ≤, this frequency is smaller than with simple harmonic motion--the damping causes the particle to slow down, decreasing the frequency and increasing the period. Shown below is a graph of typical damped harmonic motion:
Figure %: The graph of damped harmonic motion, with position plotted versus time, denoted by F(t). Also shown is the exponential function which "frames" the sinusoidal function.
We can see from the graph that the motion is a superposition of an exponential function and a sinusoidal function. The exponential function, on both the positive and negative sides, acts as a limit for the amplitude of the sinusoidal function, resulting in a gradual decrease of oscillation. Another important concept from the graph is that the period of the oscillation does not change, even though the amplitude is constantly decreasing. This property allows grandfather clocks to work: the pendulum of the clock is subject to frictional forces, gradually decreasing the amplitude of the oscillation but, since the period remains the same, it can still accurately measure the passage of time.

The study of damped harmonic motion could be a chapter in and of itself; we have simply given an overview of the concepts that give rise to this complex motion.


The second example of complex harmonic motion we will examine is that of forced oscillations and resonance. Up to this point we have only looked at natural oscillations: cases in which a body is displaced and then released, subject only to natural restoring and frictional forces. In many cases, however, an independent force acts on the system to drive the oscillation. Consider a mass spring system in which the mass oscillates on the spring (as usual) but the wall to which the spring is attached oscillates at a different frequency, as shown below:

Figure %: A mass-spring system experiencing forced oscillation by the oscillating wall. The wall oscillates over a distance a, while the spring oscillates over a distance b.

Usually the frequency of the external force (in this case the wall) differs from the frequency of the natural oscillation of the system. As such, the motion is quite complex, and can sometimes be chaotic. Considering the complexity, we will omit the equations governing this motion, and simply examine the special case of resonance in forced oscillations.