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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
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where t represents time, x(t) the position of the mass at time t and tex2html_wrap_inline31 is a constant which depends on the system in question. (In the case of a spring, tex2html_wrap_inline31 would be related to the stiffness of the spring.) Alternatively, a system of two equations can describe the system,
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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
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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
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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.




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Juris Steprans
Thu Mar 27 17:09:04 EST 1997