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
; 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
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
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: