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CONTROL SYSTEMS 37
The force in a closed loop of mechanical elements adds up to zero. From this, we get
the “characteristic” differential equation of this mechanical system:
2
2
m d x>dt B dx>dt K x 0
This says the spring force acts trying to accelerate the mass and overcome friction.
In calculus, many ways exist for solving a differential equation like this. The mathe-
matics get a bit difficult, but French mathematician Laplace provided a shortcut in the
form of his Laplace transforms. They basically eliminate the requirement for integral
calculus and reduce the problem to algebra and searching some tables. We will perform
a Laplace transform on our differential equation, do some algebra, and then use the
tables to perform an invervse Laplace transform to get back our real-world answer (see
Figure 2-15).
First, we transform our differential equation using the methods of Laplace. Substitute
the variable s to stand for a single differentiation. As such, the differential equation
becomes

2
M s B s K 0
We’re going to use algebra to find the roots of this quadratic equation. Remember the
old formula for finding the roots of the quadratic equation? I bet you thought you’d
never use it! Stay awake in school! The following restates the quadratic equation and

FIGURE 2-15 Pierre-Simon Laplace