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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 Fuid, one which gives a type of damping. ±or example,
the mass may be a²ected 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
=
−
γu
°
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
°°
+
γu
°
+
ku
=
F
(
t
)
.
Hence the position
u
=
u
(
t
) satis³es a linear second order di²erential equation. Note that in general, if we
have an equation in the form
ay
°°
+
by
°
+
cy
=
f
(
t
)
where
a
,
b
, and
c
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue UniversityWest Lafayette.
 Spring '09
 EdrayGoins

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