Page 53 - Anatomy of a Robot

02_200256_CH02/Bergren 4/17/03 11:23 AM Page 38
38 CHAPTER TWO
shows the two roots. Notice that the two roots are shown with a - notation in the fol-
lowing sections:
2
a x b x c 0
x ( b (b 4 a c) 2 )/2 a
1 ⁄
2
We are going to use the quadratic equation to solve our characteristic equation. First,
we are going to cheat a little, because we already know the answer. We’re going to
change some of the constants in the characteristic equation before solving for the roots.
This allows us to easily see the final result. Here are the three changes we make:
Divide by K so

2
m s B s K 0
changes to

2
m>K s B>K s 1 0

2
Substitute 1/v for m/K. Take a look at the second and third behaviors (Figures 2-
10 and 2-11) of the bouncing weight we showed above, and you’ll start to appre-
ciate this substitution.
Substitute 2 d/v for B/K. The damping coefficient d, integral to slowing down
the system over time, is directly related to the coefficient of friction, as we might
expect.

The equation changes with the substitution from

2
m>K s B>K s 1 0
to

2
2
11>v 2 s 12 d>v2 s 1 0
Using the quadratic equation, the two roots are

2
2
2
s 1 12 d>v2 112 d>v2 4 11>v 2 12 1>2 2>2 11>v 2
Take out the factors of 2:

2
2
2
s 1 1d>v2 11d>v2 11>v 22 1>2 2>11>v 2