ME3015 PROBLEM SET #3 (25 points).
Solutions
1.
(6 points) Assume that the system is initially in equilibrium. The gravitational force of mass
causes a static deflection of the spring. The external force
u
applies as input and the
displacement
y
is the output of the system.
Find the statespace representation of the springmasspulley shown below. Then convert the
statespace representation to the corresponding transfer function from
u
to
y
.
Solution
Define the tension acting in the wire as
T
’=
T
+
k
δ
where
k
=
mg.
Since the displacement y is defined
from the equilibrium position, the effect of gravity is not included in the dynamic equations.
The equations of motion for the mass and the pulley are given as
2
R
k
TR
J
u
T
y
m
θ
−
=
+
−
=
Since
R
y
=
, the equation of motion for the pulley can be written as
y
kR
TR
y
J
2
2
−
=
J
=
TR
2
R
k
m
k
δ
y
(Origin: Equilibrium position)
J
Moment of Inertia
θ
u
mg
R
Neutral length
without MASS
m
y
T
=
m
u
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 Fall '08
 UEDA
 Trigraph, R1 R2 R3

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