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.