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MEEN 364
Notes from Session on September 29, 2004
304 Fermier Hall, 6:00~7:00 pm
1) Following figure shows a system consisting of a gear of radius r and moment of inertia J, a
rack of mass m, a linear spring of stiffness k and a torsional spring with stiffness K.
(a) Obtain the governing differential equation of motion for the given system.
(b) Represent the system in statespace form.
Fig.4
Kinematics stage
Notice that, the rack and gear move in synchronization and hence
x
and
θ
are related by the
expression
r
x
=
. Thus, the system has just onedegree of freedom, represented by the
displacement of rack,
x
(or
the rotation of the gear
). This completes the kinematics stage.
Kinetics stage
(a) Taking
as the chosen coordinate,
The total kinetic energy of the system is
( )
2
2
2
2
2
2
1
2
1
2
1
2
1
&
&
&
&
eq
J
mr
J
x
m
J
T
=
+
=
+
=
(1)
since, from geometry of motion,
r
x
=
.
Similarly the total potential energy of the system is
( )
2
2
2
2
2
2
1
2
1
2
1
2
1
eq
K
kr
K
kx
K
V
=
+
=
+
=
(
2
)
Thus, the resulting system become equivalent to the one shown in the fig. 4(b) which is a simple
rotational springinertia system. The governing equation of motion of this system is given by
0
=
+
eq
eq
K
J
&
&
(
3
)
where
2
2
K
and
kr
K
mr
J
J
eq
eq
+
=
+
=
1
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View Full DocumentMEEN 364
Notes from Session on September 29, 2004
304 Fermier Hall, 6:00~7:00 pm
(b) Alternatively, taking
x
as the chosen coordinate,
The total kinetic energy of the system is
2
2
2
2
2
2
1
2
1
2
1
2
1
x
m
x
m
r
J
x
m
J
T
eq
&
&
&
&
=
⎟
⎠
⎞
⎜
⎝
⎛
+
=
+
=
θ
(4)
Similarly the total potential energy of the system is
2
2
2
2
2
2
2
1
2
1
2
1
2
1
x
k
x
k
r
K
kx
K
V
eq
=
⎟
⎠
⎞
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 Spring '08
 Staff
 Controls, Moment Of Inertia, Torsion

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