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Combined
ξ
n
ξ
m
Stiffness proportional
a
0
= 0
Mass proportional
a
1
= 0
ω
m
ω
n
Figure 17.3
Relationship between damping ratio and frequency for Rayleigh damping.
Modal analysis is the most popular and efficient method for solving engi-
neering dynamic problems. In order to apply modal analysis of damped
systems, it is common to assume proportional damping. Mathematically
the most common and easy way is to use Rayleigh damping method, with a
linear combination of the mass and the stiffness matrices as
c
=
a m a k
−
1
(17.3)
0
where:
c
,
m
, and
k
are the damping, the mass, and the stiffness matrix,
respectively
a
0
and
a
1
are proportional constants
The relationship between damping ratio and frequency for Rayleigh damp-
ing is shown in Figure
17.3. By simplification, this relationship leads to the
next equation:
1
ω
n
ξ
ξ
a
a
ω
1
2
n
n
0
(17.4)
=
1
m
ω
1
m
ω
m
where ω
n
and ω
m
are the damping ratios (ξ
n
and ξ
m
) associated with two spe-
cific angular frequencies (ω
n
and ω
m
in radian/second) are known, the two
Rayleigh damping factors (
a
0
and
a
1
) can be calculated by Equation 17.4.
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