Fall 2003 Math 308/501–502
4 Linear SecondOrder Equations
4.8 Free Mechanical Vibrations
Fri, 03/Oct
c
±
2003, Art Belmonte
Summary
Background material on springmassdashpot systems may be
found in Section 4.1 of your textbook. For electric circuits, we
refer you to the component laws and Kirchhoff’s laws on pages
119–120. Familiarize yourselves with the following terms.
•
spring equilibrium
•
springmass equilibrium
•
mass
m
[inertia]
•
damping constant
b
[due to a dashpot, friction, etc.]
•
spring constant
k
[stiffness]
Motion of a vibrating spring
The governing DE of motion is
my
00
+
by
0
+
ky
=
F
(
t
)
,where
the righthand side is a function which represents an external
force. In this section,
F
(
t
)
=
0, so that the motion is
unforced
.
Electric circuits
Remarkably, the mathematical treatment of certain circuits
directly parallels that of massspring systems. For example,
LQ
00
+
RQ
0
+
1
C
Q
=
E
(
t
)
or
LI
00
+
RI
0
+
1
C
I
=
E
0
(
t
)
.
Harmonic motion
Putting the DEs into SLF, we make the following identiﬁcations.
c
=
b
2
m
,ω
0
=
r
k
m
,
f
(
t
)
=
F
(
t
)
m
,
x
=
y
c
=
R
2
L
0
=
r
1
LC
,
f
(
t
)
=
E
(
t
)
L
,
x
=
Q
c
=
R
2
L
0
=
r
1
,
f
(
t
)
=
E
0
(
t
)
L
,
x
=
I
This gives the equation of harmonic motion, where
c
and
ω
0
are
nonnegative:
x
00
+
2
cx
0
+
ω
2
0
x
=
f
(
t
)
.
Simple harmonic motion
When
c
=
0and
f
(
t
)
=
0, we have
x
00
+
ω
2
0
x
=
0, whence
x
=
c
1
cos
ω
0
t
+
c
2
sin
ω
0
t
via §4
.
3 techniques. The natural
frequency of the motion is
ω
0
=
√
k
/
m
and its period is
T
=
2
π/ω
0
=
2
π
√
m
/
k
.
Damped harmonic motion
Here
c
>
0. There are three cases according to the nature of the
roots of the characteristic equation
r
2
+
2
cr
+
ω
2
0
=
0, namely
r
=
c
∓
q
c
2

ω
2
0
, which we’ll label
r
1
and
r
2
, respectively.
•
When
c
<ω
0
, the motion is
underdamped
.Wehave
x
(
t
)
=
e

ct
(
c
1
cos
ω
t
+
c
2
sin
ω
t
)
, with
ω
=
q
ω
2
0

c
2
.
•
When
c
>ω
0
, the motion is
overdamped
x
(
t
)
=
c
1
e
r
1
t
+
c
2
e
r
2
t
;here
r
1
<
r
2
<
0.
•
When
c
=
ω
0
, the motion is
critically damped
x
(
t
)
=
c
1
e

ct
+
c
2
te

ct
.
Note that in the overdamped and critially damped cases, there is
no
oscillation.
Hand Examples
Example A
Plot
y
√
3cos2
t
+
sin 2
t
over the interval 0
≤
t
≤
2
π
.Then
place this function in the form
y
=
A
cos
(ω
t

φ)
and graph it.