Simultaneous Real Eigenvalue Embedding in Second Order Systems

In the context of a second order system governed by

where
• e are the mass, damping, and stiffness matrices;
• is the input matrix;
• e are the feedback matrices;
• , e are called, respectively, displacement, velocity and acelleration vectors;
very frequently some natural frequencies and their mode shape vectors (eigenvalues and eigenvectors) of a finite element model do not match with those obtained in a vibration test of a real structure.

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

where
is clearly symmetric and has the following properties:
(i)
The quantity is an eigenvalue of corresponding to
(ii)
if are eigenvalue - eigenvector pairs of and , , then these quantities are also eigenvalues of the updated model .

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

where (compatibility equation)
we calculate matrices and , together with diagonal matrices , , e , such that the matrices of the symmetric model , given by
satisfy the updating equation
and the no spill-over equation
where stands for any set (generally unknown) of eigenvalues - eigenvectors, which are not included in the updating.

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

is computed, in order to validate the compatibility equation.

Joao Carvalho 2005-08-25