In the context of a model which represents a system vibrating according to

,

where e are the 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 that the diagonal submatrices and do not have a common nonzero entry. Then, for every symmetric matrix , the updated symmetric matrix defined by

assures

which means that the rest of the spectrum is not changed.

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

The following result, due to J. Carvalho (2002), establishes how to define matrix such that the measured data is incorporated to the updated model.

**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:

.

Joao Carvalho 2006-03-17