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2.004 Dynamics and Control II
Spring 2008
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i n
K
9 ± ±
G
J
B
J
9 ± ±
f r i c t i o n a l d r a g p l a t e s
�
Massachusetts Institute of Technology
Department of Mechanical Engineering
2.004 Dynamics and Control II
Spring Term 2008
Lecture 15
1
Reading:
• Class Handout: Modeling Part 1: Energy and Power Flow in Linear Systems
Sec. 3.
• Class Handout: Modeling Part 2: Summary of One-Port Primitive Elements
Nise: Secs. 2.4 and 2.6. •
1
Rotational Systems (continued)
Example 1
The diagram shows a mechanical tachometer that uses frictional drag plates to
create a torque proportional to angular velocity difference. The angular velocity
is indicated by the displacement θ of a torsional spring.
Find the transfer function relating the displacement of the indicator θ to the
input angular velocity Ω
in
θ ( s )
H ( s ) =
Ω
in
( s )
and show that for a constant input angular velocity the steady-state indicated
speed θ
ss
∝ Ω
i
.
Solution: There are two distinct angular velocities, and the system graph is:
1
copyright
c
D.Rowell 2008
15–1

J
B
K
9 = 0
i n
9 ± ±
a c r o s s - v a r i a b l e
s o u r c e
Using impedances, redraw the graph combining the inertia and the spring into a
single impedance Z
2
:
Z
Z
9 = 0
i n
9 ± ±
J
9 ± ±
1
2
T
Z =
1
1
B
1
Z =
2
1
J s
s
K
=
1 / J K
s / K + 1 / J s
Then
Z
2
Ω
J
= Ω
in
( s )
Z
1
+ Z
2
s/ ( Js
2
+ K )
= Ω
in
( s )
1 /B + s/ ( Js
2
+ K )
Bs
= Ω
in
( s ) .
Js
2
+ Bs + K
But the angular displacement θ ( s ) = Ω
J
( s ) /s so that
B
H ( s ) =
Js 2 + Bs + K
If the input velocity is a step function at t = 0, the Ω
in
( s ) = Ω /s and the
steady-state indicated value will be
Ω B
θ
ss
= lim θ ( t ) = lim sθ ( s ) = lim sH ( s ) = Ω
t →∞ s → 0 s → 0
s K
show that