In the context of a model which represents a system
vibrating according to
The following result, due to J. Carvalho (2002),
establishes, in the undamped case, how to define an updated matrix
such that part of the spectra remains unchanged.
Theorem 1
Consider the positive semidefinite model with no damping,
that is,
.
Let matrices
and
, which represent the modal structure of the model, satisfy
Suppose now that an incomplete modal data set is available, meaning that
a set of natural frequencies and
corresponding incomplete mode shapes
(only first
components) are known from measurement.
Assume this information is contained in matrices
and
; the first for the frequencies, the last
for the incomplete mode shapes. The next result, due to J. Carvalho (2002),
show how to compute
such that the matrix
satisfies
Theorem 2: Suppose that has full rank and
Then a matrix
implying (5)
exists only if
is such that
Theorem 3: Once is computed in order to make
equation (7) true, if we
form the matrix
using (6), and post-multiply
to
a new
such that
is a diagonal matrix , and
compute
from
An algorithm for solving the Model Updating problem with Incomplete Measured Data is proposed:
Algorithm 1:
: Model Updating of an Undamped Symmetric Positive Semidefinite Model
Using Incomplete Measured Data
Input: The symmetric matrices
;
the set of
analytical frequencies and mode shapes to be updated;
the complete set of
measured frequencies and mode shapes from the
vibration test.
Output: Updated stiffness matrix .
Assumptions:
,
and
has full rank.
Step 1: Form the matrices
and
from the available data. Form the
corresponding matrices
and
.
Step 2: Compute the matrices
,
, and
from the QR
factorization:
Step 3: Partition
where
.
Step 4: Solve the following matrix equation to
obtain
:
Step 5:
Compute the matrix
comming from the SVD decomposition of
.
Update the matrix
by
.
Step 6: Compute
by solving
the following system of equations: