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are important for the analysis.
Distribution of kinetic and potential energy in the main mode shapes is the crucial for
the modification. Therefore, it is necessary to know potential energy of the system
,
1 { } [ ]{ }
2
T
p r
r
r
E
Q K Q
=
(2)
and also kinetic energy
,
1 { } [ ]{ }
2
T
k r
r
r
r
E
Q M Q
λ
=
(3)
for the
r
th
shape of oscillation.
Analysis is directed to the elements of structure which have the best energy percentage
at some shapes, in order to be monitored. These elements are the most sensitive to the
change of design parameters (especially the eigenvalues). In order to accurately diagnose
this sensitivity for all elements, arbitrarily modified structural model is formed or the
model of just one particular, the most sensitive element. The crucial condition of the
statements is that these changes must be very small. These small changes purport the
change of some structural variable in the respective elements, being most sensitive.
A series of structural variables depend on the kind of considered construction. This
might be: thickness of the plate or beam, density of the material, length of finite element.
Then, the expression for eigenvalue increment can be used with a sufficient confidence:
{ }
{ }
{ }
{ }
{ } { }
1
1
[ ]
[ ]
2
2
1
[ ]
2
T
T
i
i
i
i
i
i
T
i
i
i
i
Q K Q
Q M Q
Q M Q
λ
λ
λ
λ
Δ
−
Δ
Δ
=
(4)
Denominator in above expression remains constant, insomuch that increment of
eigenvalue depends only on numerator that is the difference
,
,
p r
k r
E E
Δ − Δ
. From this
stage of the procedure next values can be determined:
- eigenvalues of the modified structure or respective element,
'
i
λ
; and
- eigenvectors of the modified construction
{ }
'
i
Q
; as well as
increments of the potential and kinetic energy,
,
,
p r
k r
E E
Δ − Δ
Dynamic analysis and model diagnostics include analysis and model behaviour
explanation and its modification, if necessary. Analysing the effect of energy distribution
on the main mode shapes of construction elements, few specific cases can be recognized.
Using these cases, the algorithm for reanalysis of similar structure can be made /12/.
The specific cases can be classified in following groups:
(i) The elements with kinetic and potential energy, which values are negligible
comparing to the other elements. These elements require a detailed analysis. It is
important to say that these elements are not convenient for modification, since they are
not sensitive to changes. The elements close to supports of a simple beam or thin plate
supported in a simple way are typical example for this group.
(ii) Elements with the kinetic energy greater than the potential energy. These elements
have large displacements, for example elements at the end of cantilever beam. It is
recommended to place the new support, if it is possible, or decrease existing mass or
select a new material of lower density.