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6
CHAPTER 3 SDOF VIBRATION: WITH DAMPING
4. For a massspringdamper system in free vibration: (a) Derive the equation of motion and
solve for the transient response
x
(
t
)
that is driven by the initial conditions
x
(0) =
x
0
and
_
x
(0) =
v
0
. (b) Assume parameter values:
k
= 1
lb/in, weight
W
= 100
lb,
x
0
= 0
,
v
0
= 10
in/s.
Vary the damping constant
c
so that both underdamped and overdamped responses can be
demonstrated. Try values of
c
such that the cases
= 0
:
1
and
= 0
:
9
are obtained. (c) For the
case above with
= 0
:
1
, vary initial velocity
v
0
(zero displacement) as a function of initial velocity.
Solution:
(a) The equation of motion for a free massspringdamper system is given by
m
x
+
c
_
x
+
kx
= 0
or
x
+ 2
n
_
x
+
!
2
n
x
= 0
:
We assume a solution of the form
x
(
t
) =
A
exp(
rt
)
or
x
(
t
) =
A
cos
rt
+
B
sin
rt:
_
x
and
x;
A
exp(
rt
)[
mr
2
+
cr
+
k
] = 0
:
Since neither
A
nor the exponential equal zero, the polynomial in the brackets must equal zero,
mr
2
+
cr
+
k
= 0
:
This second order polynomial is the characteristic equation for the system and it can be solved for
the two values of
r
using the quadratic equation,
r
1
;
2
=
c
±
p
c
2
4
mk
2
m
:
The form of the solution changes depending on the value of
c
2
4
mk:
Overdamped Response:
For
c
2
4
mk >
0
(equivalently
1)
;
the system is overdamped, and the
solution takes the form
x
(
t
) =
Ae
+
p
2
1
±
!
n
t
+
Be
p
2
1
±
!
n
t
:
Upon substituting the initial conditions, the constant coe¢ cients of integration are found to be
A
=
1
2
x
o
+
v
o
+
o
!
n
!
n
p
2
1
!
and
B
=
1
2
x
o
v
o
+
o
!
n
!
n
p
2
1
!
:
Critically Damped Response:
For
c
2
4
mk
= 0
(equivalently
= 1)
;
the system is critically damped,
and the solution takes the form
x
(
t
) = (
A
+
Bt
) exp (
!
n
t
)
;
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where the constant coe¢ cients of integration are found to be
A
=
x
0
and
B
=
v
0
+
x
0
!
n
:
Underdamped Response:
For
c
2
4
mk <
0
(equivalently
0
1)
, the system is underdamped,
and the solution takes the form
x
(
t
) = exp (
n
t
) (
A
cos
!
d
t
+
B
sin
!
d
t
)
;
where
!
d
=
!
n
p
1
2
. Upon substituting the initial conditions, the constant coe¢ cients of inte
gration are found to be
A
=
x
0
and
B
=
v
0
+
x
0
n
!
d
:
An alternate form of the solution for the underdamped system is
x
(
t
) = exp (
n
t
) [
C
cos (
!
d
t
±
)]
:
x
0
=
C
cos
±
and
v
0
=
C
(
n
cos
±
+
!
d
sin
±
)
:
The amplitude and the phase,
C
and
±;
are found to be
C
=
s
x
2
0
+
v
0
+
&!
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 Fall '11
 Benaroya

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