Designing effective speed bumps

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?

Instructor

Juris Steprans