Civil Engineering Reference
In-Depth Information
In most FE codes, the 'mass and stiffness proportional' damping is used as an effi cient technique
of assembling a damping matrix without reference to the element contribution. If two modes only
are involved, this is termed 'Rayleigh damping' and is given by the following expression:
C
=
α
MK
(4.16.1)
The parameters
α
and
β
can be evaluated if the damping ratio
ξ
i
is known for any two modes.
Using the following relationship:
2
αβω
+
=
2
ωξ
(4.16.2)
i
i
i
two simultaneous equations in
α
and
β
are derived for two known values of
ξ
i
. Consequently,
the damping ratio
ξ
i
in any mode can be calculated as below:
2
αβω
ω
+
j
ξ
=
(4.16.3)
i
2
j
The above assumption is essential to retain the option of solving decoupled equations of motion.
Since the mode shapes are orthogonal to
M
and
K
, they are also orthogonal to the Rayleigh
damping matrix.
(e) Formulate the equations of motion in terms of normal (or generalized) coordinates Y
i
:
2
Y
+
2
ξω
Y
+
ω
Y
= −
Γ
x
g
(4.17.1)
i
i
i
i
i
i
i
where the angular frequency
ω
i
for the
i
th mode is:
ˆ
ˆ
K
M
i
ω
i
=
(4.17.2)
i
in which
M
ˆ
i
is the generalized mass given as follows:
M
i
i
T
(4.17.3)
= FF
M
i
and
K
ˆ
i
represents the generalized stiffness expressed by:
K
i
= FF
i
T
K
(4.17.4)
i
The factor
i
is called the ' modal participation factor ' and provides a measure of the degree to
which the
i
th mode participates to the global dynamic response. This factor is as below:
Γ
L
M
i
Γ
i
=
(4.17.5)
ˆ
i
where:
i
T
L
i
= F
M
I
(4.17.6)
(f)
Compute the solutions of the system of
N
uncoupled equations in normal coordinates given in
equation (4.17.1). The response of the
i
th mode of vibration at any time
t
can be expressed by
the convolution (Duhamel) integral in the form:
