Optimal Estimation of Dynamic Systems, 2nd Edition

640 Optimal Estimation of Dynamic Systems

kx-

FF

m

k

cx-



m

x

c

Figure A.15: Simple Spring-Mass-Damper System

Since Ae

st

is never zero, Equation (A.290) holds true if and only if

ms

2

+ cs+ k = 0 (A.291)

Equation (A.291) is called the characteristic equation of the system. The same equa-

tion can also be derived by taking the Laplace transform of Equation (A.289), with

F = 0 again. The roots of this equation are clearly given by

s

1,2

=

−c ±

√

c

2

−4mk

2m

(A.292)

Three possibilities for s

1,2

exist: 1) the roots are real and unequal for c

2

−4mk > 0;

2) the roots are real and repeated for c

2

−4mk = 0; and 3) the roots are complex

conjugates for c

2

−4mk < 0. The solution for