Model Updating in Symmetric Positive Semi-definite Models using Incomplete Measured Data

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Suppose that symmetric matrices $M \in {\mathbb{R}}^{n \times n}$, $K \in {\mathbb{R}}^{n \times n}$, are given such that the undamped model $(M,D,K)$, where $D = 0$, has eigenstructure sets $(\Lambda_1,X_1)$ and $(\Lambda_2,X_2)$ (in compact representation) such that

• the first, satisfying $M \Lambda_1^2 X_1 + K X_1 = 0$, where $\Lambda_1^2 \in {\mathbb{R}}^{m \times m}$ is a diagonal matrix with non-positive entries (eigenvalue matrix), $X_1 \in {\mathbb{R}}^{n \times m}$ is the corresponding eigenvector matrix, represent part of the eigenstructure which we want to update, to which the vibration test gives measured data matrices $\Sigma_1^2 \in {\mathbb{R}}^{m \times m}$ and $Y_{11} \in {\mathbb{R}}^{m \times m}$ (only first $m$ entries of corresponding mode shapes are measured);
• the last, satisfying $M \Lambda_2^2 X_2 + K X_2 = 0$, where $\Lambda_2 \in {\mathbb{R}}^{(n-m) \times (n-m)}$ , $X_2 \in {\mathbb{R}}^{n \times (n-m)}$, the eigenvector matrix, represent an unknown part of the eigenstructure which we want to not get changed.

Our purpose is to compute a symmetric matrix $\Phi \in {\mathbb{R}}^{m \times m}$ such that the symmetric matrix

$$\tilde{K} = K - M X_1 \Phi X_1^T M$$

is such that the new symmetric undamped model $(M,D,\tilde{K})$, where $D = 0$, incorporates the information from the vibration test, without changing the unknown frequencies and mode shapes represented by $(\Lambda_2,X_2)$, that is, we do not allow spill-over to occur.

Here, we provide a computer program to compute this matrix $\Phi$, following an algorithm given in J. Carvalho, State Estimation and Finite-Element Model Updating in Vibrating Systems, PhD. Dissert, NIU, 2002.

Although the algorithm itself does not require direct information on $\Lambda_1^2$ to compute $\Phi$, the eigenvalue matrix must still be provided in order to compute the quantity

$$\Vert M X_1 \Lambda_1^2 + K X_1 \Vert _F$$

corresponding to a compatibility requirement.

Here a computer program is provided. To run this program, acess here.