MA 36600 LECTURE NOTES: FRIDAY, MARCH 6
Mechanical and Electrical Vibrations
Hooke’s Law.
Say that we have a mass
m
which is attached to a spring. Consider four forces on the mass
m
:
•
Gravity:
Newton’s Law of Gravity states that
F
g
=
mg
.
•
Restoring Force:
Hooke’s Law states that
F
s
=

k
(
L
+
u
) for some positive constant
k
.
•
Damping:
Say that the mass is in a viscous ﬂuid, one which gives a type of damping. For example,
the mass may be aﬀected by air resistance, or the mass may be in a liquid. Then the force of damping
is proportional to the velocity i.e.,
F
d
=

γ u
0
for some positive constant
γ
.
•
External:
Say that the mass is acted upon by an outside mechanical force. We simply write this as
F
e
=
F
(
t
) for some function.
Newton’s Second Law of Motion states that
F
=
ma
. That is,
mu
00
+
γ u
0
+
k u
=
F
(
t
)
.
Hence the position
u
=
u
(
t
) satisﬁes a linear second order diﬀerential equation. Note that in general, if we
have an equation in the form
ay
00
+
by
0
+
cy
=
f
(
t
)
where
a
,
b
, and
c
are constants, we identify
m
=
a
as the “mass,”
γ
=
b
as the “damping constant,”
k
=
c
as the “spring constant,” and
F
(
t
) =
f
(
t
) as the “external force.”
Analysis of Solutions.
We focus on solutions to the diﬀerential equation
mu
00
+
γ u
0
+
k u
=
F
(
t
)
.
We know that the general solution is in the form
u
(
t
) =
Au
1
(
t
) +
B u
2
(
t
) +
U
(
t
)
for some constants
A
and
B
, where we have denoted the functions
u
1
(
t
) =
(
e
λt
cosh
μt
if
γ
2

4
mk >
0,
e
λt
cos
μt
if
γ
2

4
mk <
0.
u
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 Spring '09
 MA

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