Math245 Computer Lab set #9, Spring 2009
Hard spring oscillator (nonlinear 2ndorder ODE)
In this lab, we will just glance at the strange eﬀect of nonlinearity by using a hard spring oscillator model
as an example.
Nonlinear spring model
Consider a mechanical spring. When stretching the spring, the force needed to stretch (
F
s
) is linearly
proportional to the stretch (
x
), if the stretch is suﬃciently small. In this case, the spring force is modeled
by the Hooke’s law of
linear spring
.
F
s
=
kx
where
k
is called
spring constant
. If the stretch is larger and larger, this Hooke’s law loses its validity,
because some nonlinearity of the spring material comes into play. In some occasions, the spring force is
modeled by the following nonlinear expression.
F
s
=
kx
+
sx
3
where
s
is a constant parameter. If
s >
0, the spring called
hard spring
, otherwise (
s <
0) the spring is
called
soft spring
.
F
s
x
k x
k x + s x
( s > 0)
3
k x + s x
( s < 0)
3
linear spring
hard spring
soft spring
F
s
x
Spring  mass  damper model
Consider a forced, springmassdamper model as shown below, where we use a hard spring.
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 Spring '07
 Alexander
 Math, Robert Hooke, Fundamental physics concepts, Linear system, Nonlinear system, hard spring

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