Second Order Systems
1.
Governing equations
All mechanical systems are governed by Newton's laws of motion, the most familiar of
which is
Fm
dx
dt
=
∑
2
2
(
l
)
where
m
is the mass and
x
is the displacement and the
F
components are the forces acting
on the system. Since the forces are usually proportional to
x
and d
x
/d
t
, eq.(1) shows that
such mechanical systems are always second order. These are called linear second order
systems as long as no nonlinear forces proportional to
x
2
, (d
x
/d
t
)
2
,... are disturbing the
system.
This turns out to be of great practical significance, since almost all dynamical mechanical
systems are indeed
linear second order, so that knowledge acquired about one system can
be readily transferred to another problem.
[Brief translation:
Learn this stuff, and you will find it useful for more than just this
lab/report/class/degree.
Really!]
k
m
c
x
The simplest mechanical system is the massspring dashpot
problem shown in the figure. When displaced from its
equilibrium at
x
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 Spring '08
 Spedding
 Radioactive Decay, Second Order Systems, undamped natural frequency

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