In the context of a second order system governed by

- 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;

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

- (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

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

Joao Carvalho 2005-08-25