#### MDOF Systems with Proportional Damping - Saeed Ziaei-Rad

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MDOF Systems with Proportional Damping - Saeed Ziaei-Rad

MDOF Systems with
Proportional Damping
Saeed Ziaei Rad
General Concepts
Proportional damping has the advantage of being
easy to include in the analysis so far.
The modes of a structure with proportional damping
are almost identical to those of the undamped
version of the model.
It is possible to derive the modal properties of a
proportionally damped system by analyzing the
undamped version in full and then making a
correction for the damping.
Saeed Ziaei Rad
Proportional damping –
special case
Consider the general equation of motion for a MDOF with a
viscous damping:
[ M ]{ x} [ C ]{ x } [ K ]{ x} { f }
Assume that the damping matrix is directly proportional to the
Stiffness matrix:
[C ] [ K ]
Then:
[ ] [ C ][ ] [ ] [ K ][ ] [ k r ] [ c r ]
T
T
Saeed Ziaei Rad
Proportional damping –
Let’s define {p} as:
special case
The undamped modal matrix
{ x } [ ]{ p }
Then the equation of motion becomes (f=0):
[ m r ]{ p} [ c r ]{ p } [ k r ]{ p } { 0}
From which:
m r p r c r p r k r p r 0
This is a SDOF system and therefore:
kr
cr
2
2
r
, d r 1 r ,
r
0 . 5 r
mr
2 krmr
Saeed Ziaei Rad
Proportional damping –
The receptane matrix can be defined as:
[ H ( )] ([ K ] i [ C ] [ M ])
2
or
jr kr
N
H
jk
( )
(k
r 1
m r ) i c r
2
r
Saeed Ziaei Rad
1
special case
Proportional damping –
general case
The general form of proportional damping is:
[C ] [ K ] [ M ]
Again assume:
{ x } [ ]{ p }
And then:
[ ] [ C ][ ] [ k r ] [ m r ]
T
In this case, the damped system has eigenvalues and
eigenvectors as follow:
d r 1 r ,
2
r 0 . 5 r 0 . 5 / r
Saeed Ziaei Rad
Proportional Hysteretic damping
Consider the general equation of motion for a MDOF with a
hysteretic damping:
[ M ]{ x} ([ K ] i[ D ]){ x } { f }
Assume the hysteretic damping as:
[D ] [K ] [M ]
And again
2
r
kr
mr
,
r r (1 i r ) ,
2
2
r / r
2
Exercise: Extract the above relations from the equation of motion.
Saeed Ziaei Rad