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

In the context of a model $(M,D,K)$ which represents a system vibrating according to

$$M \ddot{q}(t) + D \dot{q}(t) + K q(t) = 0$$,

where $M, D$ e $K$ are the mass, damping, and stiffness matrices, very frequently some natural frequencies and their mode shape vectors (eigenvalues and eigenvectors) do not match with those obtained in a vibration test of the real structure being modeled. In this case, the representation must be updated, to guarantee its validity for future use.

The following result, due to J. Carvalho (2002), establishes, in the undamped case, how to define an updated matrix $\tilde{K}$ such that part of the spectra remains unchanged.

Theorem 1 Consider the positive semidefinite model $(M,D,K)$ with no damping, that is, $D = 0$. Let matrices $\Lambda \in {\mathbb{R}}^{n \times n}$ and $X \in {\mathbb{R}}^{n \times n}$ , which represent the modal structure of the model, satisfy

$$M X \Lambda^2 + K X = 0$$ (1)

and be partitioned as in

$$\Lambda = \left[ \begin{array}{cc} \Lambda_1 & \\ & \Lambda... ... X = \left[ \begin{array}{cc} X_1 & X_2 \end{array} \right].$$ (2)

Suppose that the diagonal submatrices $\Lambda_1$ and $\Lambda_2$ do not have a common nonzero entry. Then, for every symmetric matrix $\Phi \in {\mathbb{R}}^{m \times m}$, the updated symmetric matrix $\tilde{K}$ defined by

$$\tilde{K} = K - M X_1 \Phi X_1^T M$$ (3)

assures

$$M X_2 \Lambda_2^2 + \tilde{K} X_2 = 0.$$ (4)

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

$\Box$

Suppose now that an incomplete modal data set is available, meaning that a set of $m$ natural frequencies and $m$ corresponding incomplete mode shapes (only first $m$ components) are known from measurement. Assume this information is contained in matrices $\Sigma_1^2 \in {\mathbb{R}}^{m \times m}$ and $Y_{11} \in {\mathbb{R}}^{m \times m}$; the first for the frequencies, the last for the incomplete mode shapes. The next result, due to J. Carvalho (2002), show how to compute $\Phi$ such that the matrix $\tilde{K}$ satisfies

$$M Y_1 \Sigma_1^2 + \tilde{K} Y_1 = 0,$$ (5)

which means that the measured data is incorporated to the updated model, where

$$Y_1 = \left[ \begin{array}{c} Y_{11} \\ Y_{12} \end{array} \right].$$ (6)

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

Theorem 2: Suppose that $MX_1$ has full rank and

$$M X_1 = \left[ \begin{array}{cc} U_1 & U_2 \end{array} \right] \left[ \begin{array}{c} Z \\ 0 \end{array} \right].$$

$$M = \left[ \begin{array}{cc} M_1 & M_2 \end{array} \right]$$

$$K = \left[ \begin{array}{cc} K_1 & K_2 \end{array} \right]$$

where $M_1, K_1 \in {\mathbb{R}}^{n \times m}$ and $M_2, K_2 \in {\mathbb{R}}^{n \times (n-m)}$.

Then a matrix $\Phi \in {\mathbb{R}}^{m \times m}$ implying (5) exists only if $Y_{12}$ is such that

$$U_2^T M_2 Y_{12} \Sigma_1^2 + U_2^T K_2 Y_{12} = -U_2^T( K_1 Y_{11} + M_1 Y_{11} \Sigma_1^2 ).$$ (7)

$\Box$

Theorem 3: Once $Y_{12}$ is computed in order to make equation (7) true, if we form the matrix $Y_1$ using (6), and post-multiply $Y_1$ to a new $Y_1$ such that $Y_1^T M Y_1$ is a diagonal matrix , and compute $\Phi \in {\mathbb{R}}^{m \times m}$ from

$$Y_1^T M Y_1 \Sigma_1^2 + Y_1^T K Y_1 = ( Y_1^T M X_1 ) \Phi (Y_1^T M X_1)^T$$ (8)

then equation (5) holds.

$\Box$

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 $M, K \in {\mathbb{R}}^{n \times n}$; the set of $m$ analytical frequencies and mode shapes to be updated; the complete set of $m$ measured frequencies and mode shapes from the vibration test.

Output: Updated stiffness matrix $\tilde{K}$.

Assumptions: $M = M^T \geq 0$ , $K = K^T \geq 0$ and $MX_1$ has full rank.

Step 1: Form the matrices $\Sigma_1^2 \in {\mathbb{R}}^{m \times m}$ and $Y_{11} \in {\mathbb{R}}^{m \times m}$ from the available data. Form the corresponding matrices $\Lambda_1^2 \in {\mathbb{R}}^{m \times m}$ and $X_1 \in {\mathbb{R}}^{n \times m}$.

Step 2: Compute the matrices $U_1 \in {\mathbb{R}}^{n \times m}$, $U_2 \in {\mathbb{R}}^{n \times (n-m)}$, and $Z \in {\mathbb{R}}^{m \times m}$ from the QR factorization:

$$M X_1 = \left[ \begin{array}{cc} U_1 & U_2 \end{array} \right] \left[ \begin{array}{c} Z \\ 0 \end{array} \right]$$ .

Step 3: Partition $M = \left[ \begin{array}{cc} M_1 & M_2 \end{array} \right] , K = \left[ \begin{array}{cc} K_1 & K_2 \end{array} \right]$ where $M_1, K_1 \in {\mathbb{R}}^{n \times m}$.

Step 4: Solve the following matrix equation to obtain $Y_{12} \in {\mathbb{R}}^{(n-m)\times m}$:

$$U_2^T M_2 Y_{12} \Sigma_1^2 + U_2^T K_2 Y_{12} = -U_2^T [ K_1 Y_{11} + M_1 Y_{11} \Sigma_1^2 ]$$

and form the matrix

$$Y_1 = \left[ \begin{array}{c} Y_{11} \\ Y_{12} \end{array} \right]$$.

Step 5: Compute the matrix $U \in {\mathbb{R}}^{n \times m}$ comming from the SVD decomposition of $Y_1^T M Y_1$. Update the matrix $Y_1$ by $$Y_1 \leftarrow Y_1 U$$.

Step 6: Compute $\Phi \in {\mathbb{R}}^{m \times m}$ by solving the following system of equations:

$$(Y_1^T M X_1) \Phi (Y_1^T M X_1)^T = Y_1^T M Y_1 (\Sigma_1)^2 + Y_1^T K Y_1$$ .