Simultaneous Real Eigenvalue Embedding in Second Order Systems

In the context of a second order system governed by

$\begin{array}{l} M \ddot{q}(t) + D \dot{q}(t) + K q(t) = B u(t) \\ y(t) = C_1 q(t) + C_2 \dot{q}(t), \end{array}$

where

• $M, D$ e $K$ are the mass, damping, and stiffness matrices;
• $B$ is the input matrix;
• $C_1$ e $C_2$ are the feedback matrices;
• $q(t)$, $\dot{q}(t)$ e $\ddot{q}(t)$ 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 $(M,D,K)$ 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 $\tilde{M}, \tilde{D}$ e $\tilde{K}$, such that an specific measured frequency $\mu_1$ is properly inserted (embedded) in a new model $(\tilde{M},\tilde{D},\tilde{K})$, in the place of a real isolated (distinct) real frequency $\lambda_1$.

Theorem: Updating of a real frequency $\mu_1$

Let $(\lambda_1 , x_1)$, $\lambda_1 \neq 0$ be a real eigenvalue - eigenvector isolated pair and let $(M,D,K)$ be a symmetric positive semidefinite model. Suppose we want to change $\lambda_1$ by a real quantity $\mu_1$ in the updated model. We further assume that $\mu_1$ is such that $x_1^T K x_1 - \lambda_1 \mu_1 x_1^T M x_1 \neq 0$.

Then, the updated model $(\tilde{M},\tilde{D},\tilde{K})$ defined by

$\begin{array}{l} \tilde{M} = M - \epsilon_1 \lambda_1 M x_1 x_1^T M \\ \tilde... ... M) \\ \tilde{K} = K - \frac{\epsilon_1}{\lambda_1} K x_1 x_1^T K \end{array}$

where

$$\epsilon_1 = \frac{ \lambda_1 - \mu_1}{x_1^T K x_1 - \lambda_1 \mu_1 x_1^T M x_1}$$

is clearly symmetric and has the following properties:

(i)

The quantity $\mu_1$ is an eigenvalue of $(\tilde{M},\tilde{D},\tilde{K})$ corresponding to $x_1$

(ii)

if $(\lambda_k,x_k),k=2,\dots,2n$ are eigenvalue - eigenvector pairs of $(M,D,K)$ and $\lambda_k \neq \lambda_1$, $k=2,\dots,n$, then these quantities are also eigenvalues of the updated model $(\tilde{M},\tilde{D},\tilde{K})$.

$\Box$

Simultaneous Real Eigenvalue Embedding

Let $\{ \lambda_1,\dots,\lambda_r, \lambda_{r+1} , \dots, \lambda_{2n} \}$ a set of real natural frequencies of a positive semidefinite model $(M,D,K)$ of order $n$, that is, having $n$ state variables, where the frequencies are named $\{ \lambda_1,\dots,\lambda_r \}$ and their corresponding mode vectors $\{x_1,\dots,x_r\}$ are real.

Given a set of measured frequencies $\{ \mu_1,\dots,\mu_r \}$, we want to build and updated representation $(\tilde{M},\tilde{D},\tilde{K})$ whose natural frequencies are $\{\mu_1,\dots,\mu_r, \lambda_{r+1},\dots,\lambda_{2n}\}$, with the same vibration modes $\{x_1 , x_2 , \dots, x_r \}$.

Therefore, given matrices

$$\Lambda_1 = \left[ \begin{array}{cccc} \lambda_1 & & & \\ & \lambda_2 & & \\ & & \dots & \\ & & & \lambda_r \end{array} \right] ,$$

$$\Sigma_1 = \left[ \begin{array}{cccc} \mu_1 & & & \\ & \mu_2 & & \\ & & \dots & \\ & & & \mu_r \end{array} \right] ,$$

$$X_1 = \left[ \begin{array}{cccc} x_1 & x_2 & \dots & x_r \end{array} \right]$$

where (compatibility equation)

$$M X_1 \Lambda_1^2 + D X_1 \Lambda + K X_1 = 0 ,$$

we calculate matrices $W$ and $Z$, together with diagonal matrices $E_m$, $E_d$, e $E_k$, such that the matrices of the symmetric model $(\tilde{M},\tilde{D},\tilde{K})$, given by

$$\begin{array}{l} \tilde{M} = M - W E_m W^T \\ \tilde{D} = D + Z E_d W^T + W E_d Z^T \\ \tilde{K} = K - Z E_k Z^T \end{array}$$

satisfy the updating equation

$$\tilde{M}X_1 \Sigma_1^2 + \tilde{D} X_1 \Sigma_1 + \tilde{K} X_1 = 0$$

and the no spill-over equation

$$\tilde{M}X_2 \Sigma_2^2 + \tilde{D} X_2 \Sigma_2 + \tilde{K} X_2 = 0$$

where $(\Sigma_2, X_2)$ 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 $E_m, E_d , E_k, W, Z$. Once given the input data corresponding to matrices $M, D, K$ , $\Lambda_1$ and $X_1$, the quantity

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

is computed, in order to validate the compatibility equation.