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Phasor Analysis of Linear Mechanical Systems
and Linear Differential Equations
ME104, Prof. B. Paden
In this set of notes, we aim to imitate for linear mechanical systems and linear differential
equations, the phasor analysis we learned for electric circuits.
Recall how we derived the complex impedance for an inductor. Starting with the
differential equation for the VI characteristic for an inductor
I
dt
d
L
V
=
(
1
)
we substitute complex sinusoids
t
j
e
V
V
ω
ˆ
→
(
2
)
t
j
e
I
I
ˆ
→
(
3
)
So that equation (1) becomes
(
t
j
t
j
e
I
dt
d
L
e
V
ˆ
ˆ
=
)
(
4
)
Differentiating and solving yields
I
L
j
V
ˆ
ˆ
=
(
5
)
and the impedance of the inductor is defined by
ˆ
ˆ
V
Z
jL
I
=
±
(
6
)
where “
” denotes “defined equal to”.
Having done this calculation once, we see that
we can jump directly from (1) to (5) by making the substitution
±
j
dt
d
→
(
7
)
Phasor Analysis of Linear Mechanical Systems
Consider a mechanical damper (a.k.a. shock absorber) which produces a velocity
dependent force according to the linear differential equation
x
dt
d
b
f
=
(
8
)
Making the substitution
j
dt
d
→
, we get
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b
j
f
ˆ
ˆ
ω
=
(
9
)
And defining the mechanical impedance to be the ratio of force to displacement, we have
ˆ
ˆ
f
Z
jb
x
=
±
(Newtons/meter)
(10)
Note that the damper has a low stiffness at low frequencies, and a high stiffness at high
frequencies. The units of impedance are Newtons/meter in mechanical systems and
volts/amp = Ohms in electrical systems.
For a mass,
m
, we have
2
2
dd
d
f
mx
m
dt
dt
dt
⎛⎞
==
⎜⎟
⎝⎠
x
(
1
1
)
Substituting
j
dt
d
→
yields
()
2
2
ˆ
ˆ
f
j
m
x
ωω
−
(
1
2
)
The impedance of a mass increases very rapidly with frequency. This explains why anvils
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 Fall '08
 Staff

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