The term *harmonic oscillator* is applied to any system (moving
in one dimension) which experiences a force towards the origin which is
proportional to the distance from the origin. A mass attached to a
spring is a good example. The further away the mass is from the
equilibrium point the greater the forced exerted on it by the
spring. Notice that the spring always acts in s ucha way as to pull
the mass towards the centre.
The differential equation describing such a system is given by

where *t* represents time,
*x*(*t*) the position of the mass at time *t* and is a constant
which depends on the system in question. (In the case of a spring,
would be related to the stiffness of the spring.)
Alternatively, a system of two equations can describe the system,

where *v*(*t*) represents the velocity of themass at time *t*.

A driven harmonic oscillator arises when some external force is
applied to the moving mass -- for example, if the mass attached to a
spring is given an occasional push. The system of differential equations
describing this system is

where *F*(*t*,*x*(*t*),*v*(*t*) is the external force exerted on the system. Of
course, if *F*(*t*,*x*(*t*),*v*(*t*) has negative values then it acts as a
damping force rather than a driving one. Typically a damping force is
due to friction and is proportional to the velocity of the mass.
An harmonic oscillator satisfying the differential equations

where *K* is only slightly greater than 0 is known as an underdamped
harmonic oscillator. It smotion tends to oscillate with decreasing
amplitude.

Assume that the vertical motion of a car body on it suspension is a damped harmonic oscillator. Design speed bumps which will cause a car moving at excessive speed to bounce up and down increasingly more violently as it passes over the speed bumps. Obviously you will have to make some assumptions.

- What is excessive speed?
- What are the values of
*K*and is the equations of a damped harmonic oscillator for a typical car? You should be able to make an intelligent guess about the mass of a car. Then consider what happens to a typical car if you press down hard on its hood and release it. How many times does it bob up and down before coing to a rest? This provides atest of your assumptions about*K*and . - How shape is a speed bump?
- For what duration does it apply an upwards force?
- Is there a downwards force exerted?

Thu Mar 27 17:09:04 EST 1997