MAE 171A
Introduction to Feedback and Control Systems
Fall 2010 HW#1 Solutions
Exercises from
Feedback Control of Dynamic S ystems
(Sixth Edition)
by Gene F. Franklin, J.David Powell and Abbas EmamiNaeini
2.3 Write the equations of motion for the doublependulum system shown in the figure below. Assume
that the displacement angles of the pendulums are small enough to ensure that the spring is always
horizontal. The pendulum rods are taken to be massless, of length
l
, and the springs are attached 3/4
of the way down.
Double pendulum
From the FBD we proceed by summing the moments about the pivot point of the left pendulum.
(
29
1
1
2
1
1
3
3
sin
sin
sin
cos
4
4
M
I
mgl
k l
l
θ
=
= 
+

1
3
cos
4
l
(
29
2
1
3
sin
sin
4
k l

(
29
1
2
3
sin
sin
4
k l

mg
mg
We begin by developing the FBD.
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View Full DocumentModify the equation of motion for the cruise control in derived in Example 2.1,
so that it has a control law; that is, let
where
(
),
r
u
K
ν
=

,
b
u
m
m
+
=
This is a “proportional” control law in which the difference between
v
r
and the actual speed is used
as a signal to speed the engine up or slow it down. Revise the equations of motion with
v
r
as the
input and
v
as the output and find the transfer function. Assume that
m
= 1000 kg and
b
= 500 N
·
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 Winter '09
 IDAN
 FBD, mgl sin

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