Supposing that the mass matrix M = I , and the stiffness matrix is given by K = [3 -1 -1 -1 3 -1 -1 -1 3] Use the Jacobi method to find the eigenvalues and eigenvectors for such a system.

Question

Supposing that the mass matrix M = I, and the stiffness matrix is given by

[latex]K=\left[\begin{array}{ccc}3 & -1 & -1 \-1 & 3 & -1 \-1 & -1 & 3\end{array} ight][/latex]

Use the Jacobi method to find the eigenvalues and eigenvectors for such a system.

Accepted Answer

For i = 1, j = 2, from equation (9.248) we get

[latex] heta=\frac{1}{2} an ^{-1} \frac{2 K_{k \ell}^{(j)}}{K_{k k}^{(j)}-K_{\ell \ell}^{(j)}} \quad[/latex] for [latex]\quad K_{k k}^{(j)} eq K_{\ell \ell}^{(j)}[/latex] (9.248)

[latex]\cos heta=0.7571 \sin heta=0.7571[/latex]

and

[latex]P_1=\left[\begin{array}{ccc}0.7571 & -0.7571 & 0 \0.7571 & 0.7571 & 0 \0 & 0 & 1\end{array} ight] (9.250)[/latex]

Thus

[latex]P_1^T K_1 P_1=\left[\begin{array}{ccc}1.999 & 0 & -1.4142 \0 & 3.9999 & 0 \-1.4142 & 0 & 3\end{array} ight] (9.251)[/latex]

For i = 1, j = 3 we get

[latex]\begin{aligned}\cos heta &=0.8165 \sin heta=0.5773 \P_2 &=\left[\begin{array}{ccc}0.8165 & 0 & -0.5773 \0 & 1 & 0 \0.57732 & 0 & 0.8165\end{array} ight] (9.252) \P_2^T P_1^T K P_1 P_2 &=\left[\begin{array}{ccc}0.9999 & 0 & 0 \0 & 3.9999 & 0 \0 & 0 & 3.9999\end{array} ight]=\Lambda (9.253)\end{aligned}[/latex]

Equation (9.253) yields a diagonalization of K after two transformations, the product of which is

[latex]P_1 P_2=\left[\begin{array}{ccc}0.5773 & -0.7571 & -0.4082 \0.5773 & 0.7571 & -0.4082 \0.5773 & 0 & 0.8165\end{array} ight]=\Phi (9.254)[/latex]

The eigenpairs are:

For [latex]\quad \lambda_1=1: x_1=\left[\begin{array}{l}1 \ 1 \ 1\end{array} ight][/latex]

For [latex]\quad \lambda_2=4: x_2=\left[\begin{array}{c}-1 \ 1 \ 0\end{array} ight][/latex]

For [latex]\quad \lambda_3=4: x_3=\left[\begin{array}{c}-1 \ -1 \ 2\end{array} ight][/latex]

As we mentioned earlier, we usually need many steps to diagonalize matrix K. As to the case when [latex]M \pm I[/latex], we can use the generalized Jacobi method following procedures described in reference [5].

Question 9.8.1: Supposing that the mass matrix M = I , and the stiffness mat...

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