In the context of a second order system governed by
In this case, the vibration engineer needs to adjust the theoretical model do guarantee it is valid for future use.
The following result, due to J. Carvalho, B. Datta, W. Lin and C. Wang (2001),
establishes how to define updated matrices
e
,
such that an specific measured frequency
is properly inserted
(embedded) in a new model
, in the
place of a real isolated (distinct) real frequency
.
Theorem:
Updating of a real frequency
Let
,
be a real eigenvalue - eigenvector
isolated pair and let
be a symmetric positive semidefinite model.
Suppose we want to change
by a real quantity
in the updated model. We further assume that
is such that
.
Then, the updated model
defined by
Simultaneous Real Eigenvalue Embedding
Let
a set of real natural frequencies of a positive semidefinite model
of order
, that is, having
state variables, where
the frequencies are named
and their corresponding mode
vectors
are real.
Given a set of measured frequencies
, we want to build and updated
representation
whose
natural frequencies are
,
with the same vibration modes
.
Therefore, given matrices
Here, we provide an computer program to compute the set of
updating matrix parameters
. Once given the
input data corresponding to matrices
,
and
, the quantity