Civil Engineering Reference
In-Depth Information
where
is circular frequency of oscillation. That the equivalent damping coeffi-
cient depends on frequency is somewhat troublesome, since the implication is
that different coefficients are required for different
freq
uencies. If we consider a
single degree-of-freedom system for which
=
√
k
/
m
,
the equivalent damping
coefficient given by Equation 10.136 becomes
c
eq
=
m
=
√
km
k
=
k
√
k
(10.137)
/
indicating that the damping coefficient is proportional, at least in a general sense,
to both stiffness and mass. We return to this observation shortly.
Next we consider the application of the transformation using the normal-
ized matrix as described in Section 10.7. Applying the transformation to Equa-
tion 10.132 results in
{¨
[
A
]
T
[
C
][
A
]
2
]
[
A
]
T
p
}+
{˙
p
}+
[
{
p
}=
{
F
(
t
)
}
(10.138)
The transformed damping matrix
[
C
]
[
A
]
T
[
C
][
A
]
(10.139)
is easily shown to be a symmetric matrix, but the matrix is
not necessarily diag-
onal.
The transformation does not necessarily result in decoupling the equations
of motion, and the simplification of the mode superposition method is not neces-
sarily available. If, however, the damping matrix is such that
[
C
]
=
=
[
M
]
+
[
K
]
(10.140)
where
and
are constants, then
[
C
]
2
]
(10.141)
is
a diagonal matrix and the differential equations of motion are decoupled. Note
that the assertion of Equation 10.140 leads directly to the diagonalization of the
damping matrix as given by Equation 10.141. Hence, Equation 10.138 becomes
{¨
[
A
]
T
[
M
][
A
]
[
A
]
T
[
K
][
A
]
=
+
=
[
I
]
+
[
2
])
2
]
[
A
]
T
(10.142)
As the differential equations represented by Equation 10.142 are decoupled, let
us now examine the solution of one such equation
p
}+
(
+
[
{˙
p
}+
[
{
p
}=
{
F
(
t
)
}
P
p
i
+
+
i
˙
A
(
i
)
j
2
2
i
¨
p
i
+
p
i
=
F
j
(
t
)
(10.143)
j
=
1
where
P
is the total number of degrees of freedom. Without loss of generality and
for convenience of illustration, we consider Equation 10.143 for only one of the
terms on the right-hand side, assumed to be a harmonic force such that
¨
i
2
2
i
p
i
(10.144)
p
i
+
+
p
i
+
˙
=
F
0
sin
f
t
and assume that the solution is
p
i
(
t
)
(10.145)
=
X
i
sin
f
t
+
Y
i
cos
f
t
