MDOF Systems with
Proportional Damping
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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.
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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
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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
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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
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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
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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.
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