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18.03 Class 38
, May 10, 2010
Linearization: The nonlinear pendulum and phugoid oscillation
[1]
Nonlinear pendulum
[2]
Phugoid oscillation
[3]
Buckling bridge
[1] The bob of a pendulum is attached to a rod, so it can swing clear
around the pivot. This system is determined by three parameters:
L
length of pendulum
m
mass of bob
g
acceleration of gravity
We will assume that the motion is restricted to a plane.
To describe it we need a dynamical variable.
We could use a horizontal
displacement, but it turns out to be easier to write down the equation
controlling it if you use the angle of displacement from straight down.
Write
theta
for that angle, measured to the right.
Here is a force diagram:
*
\
theta

\

\
mg 
/\
(this is supposed to be a right angle!)

\/

/

/ mg sin(theta)
 /
/
*
Write
s
for arc length along the circle, with
s = 0
straight down.
Of course,
s
=
L theta
Newton's law says
ms"
=
F
The force has the
 mg sin(theta)
component of the force of gravity
(and notice the sign!),
and also a frictional force which depends upon
s'
=
L theta'
Make the simplest model for friction,
 cs' = cL theta'.
So:
m L theta"
=
 mg sin(theta)  cL theta'
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View Full DocumentDivide through by
mL
and we get
theta" +
b theta' + k sin(theta)
=
0
where
k = g/L
and
b = c/m .
This is a nonlinear second order equation. It still has a "companion
first order system," obtained by setting
x
=
theta
,
y
=
x'
so
y' = theta" =  k sin(theta)  b theta'
or
x'
=
y
y'
=
 k sin(x)  by
This is an autonmous system. Let's study its phase portrait.
Equilibria:
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 Spring '09
 vogan

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