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,
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.
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 .
