7.5 Damping in MDF Systems
281
Now, using
[K ]
φ
(r )
= p
2
r [M]
φ
(r )
and performing several algebraic operations, we can show that the damping coefficient associated with any mode r may be
written as
¯
c r =
i
a i p
2i
r ¯
m r = 2ζ r p r ¯
m r
(7.38)
from which
ζ r =
1
2 p r
i
a i p
2i
e
(7.39)
Equation (7.39) may be used to determine constants a i for any desired value of
modal damping ratios corresponding to any specified number of modes. For example,
to evaluate these constants specifying the first four modal damping ratios ζ 1 , ζ 2 , ζ 3
and ζ 4 , we may choose i = 1, 2, 3, 4. In this case, Eq. (7.39) gives the following
system of equations
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
ζ 1
ζ 2
ζ 3
ζ 4
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
=
1
2
⎡
⎢
⎢
⎣
p 1 p
3
1 p
5
1 p
7
1
p 2 p
3
2 p
5
2 p
7
2
p 3 p
3
3 p
5
3 p
7
3
p 4 p
3
4 p
5
4 p
7
4
⎤
⎥
⎥
⎦
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
a 1
a 2
a 3
a 4
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(7.40)
In general, Eq. (7.40) may be written symbolically as
{ζ } =
1
2
[Q]{a}
(7.41)
where [Q] is a square matrix having different powers of the natural frequencies> On
solving Eq. (7.41), yields
{a} = 2[Q]
−1
{ζ }
(7.42)
Finally, the damping matrix is obtained after the substitution of Eq. (7.42) into
Eq. (7.35)
Figure 7.7 illustrates the relationship between damping ratio and frequency that
would result from this matrix. To simplify the figure, it has been assumed that the same
damping ratio ¯
ζ was specified for all four frequencies; however, each of the damping
ratios could have been specified arbitrarily. p 1 is the fundamental frequency and p 4 is
intended to approximate the frequency of the highest mode that contributes significantly to the response, while p 2 and p 3 are spaced about equally within the frequency
range. It is evident in Fig. 7.7 that damping ratio remains close to the desired value ¯
ζ
throughout the frequency range, being exact at four specified frequencies and ranging
slightly above or below at other frequencies. It is important to note, however, that the
damping increases monotonically with frequency for frequencies above p 4 . This has
281
Now, using
[K ]
φ
(r )
= p
2
r [M]
φ
(r )
and performing several algebraic operations, we can show that the damping coefficient associated with any mode r may be
written as
¯
c r =
i
a i p
2i
r ¯
m r = 2ζ r p r ¯
m r
(7.38)
from which
ζ r =
1
2 p r
i
a i p
2i
e
(7.39)
Equation (7.39) may be used to determine constants a i for any desired value of
modal damping ratios corresponding to any specified number of modes. For example,
to evaluate these constants specifying the first four modal damping ratios ζ 1 , ζ 2 , ζ 3
and ζ 4 , we may choose i = 1, 2, 3, 4. In this case, Eq. (7.39) gives the following
system of equations
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
ζ 1
ζ 2
ζ 3
ζ 4
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
=
1
2
⎡
⎢
⎢
⎣
p 1 p
3
1 p
5
1 p
7
1
p 2 p
3
2 p
5
2 p
7
2
p 3 p
3
3 p
5
3 p
7
3
p 4 p
3
4 p
5
4 p
7
4
⎤
⎥
⎥
⎦
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
a 1
a 2
a 3
a 4
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(7.40)
In general, Eq. (7.40) may be written symbolically as
{ζ } =
1
2
[Q]{a}
(7.41)
where [Q] is a square matrix having different powers of the natural frequencies> On
solving Eq. (7.41), yields
{a} = 2[Q]
−1
{ζ }
(7.42)
Finally, the damping matrix is obtained after the substitution of Eq. (7.42) into
Eq. (7.35)
Figure 7.7 illustrates the relationship between damping ratio and frequency that
would result from this matrix. To simplify the figure, it has been assumed that the same
damping ratio ¯
ζ was specified for all four frequencies; however, each of the damping
ratios could have been specified arbitrarily. p 1 is the fundamental frequency and p 4 is
intended to approximate the frequency of the highest mode that contributes significantly to the response, while p 2 and p 3 are spaced about equally within the frequency
range. It is evident in Fig. 7.7 that damping ratio remains close to the desired value ¯
ζ
throughout the frequency range, being exact at four specified frequencies and ranging
slightly above or below at other frequencies. It is important to note, however, that the
damping increases monotonically with frequency for frequencies above p 4 . This has