User:Eml4500.f08.team.guan/fixed1013

 HW Redo: The eigenvectors and values were determined once again (correctly), more explanation and derivation was added to the math, and the eigenvectors were plotted and explained.

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Eml4500.f08.team.foskey.ckf 00:46, 3 November 2008 (UTC)

Eigenvalues and Eigenvectors
Recall the eigenvalue problem of the global stiffness matrix:$$\displaystyle KV=\lambda V$$ Let $$\displaystyle $$ be the pure eigenvectors corresponding to 4 non zero eigenvalues. Zero Eigenvalues correspond to rigid body motion and zero stored elastic energy.

$$ \mathbf{k}\mathbf{u}\mathbf{i} = \mathbf{0}\mathbf{u}\mathbf{i} $$

The linear combination of $$ \left\{\mathbf{u}_{i}, i=1,...,4 \right\}$$ The equation which defines $$ \mathbf{w} $$ is $$ \sum_{i=1}^{4}{\alpha i\mathbf{u}_{i}}=:\mathbf{w} $$ w is also an eigenvector corresponding a zero eigenvalue. $$Kw=K(\sum_{i=1}^4{\alpha_i u_i})=\sum_{i=1}^4\alpha_i(Ku_i)=0=0w$$ w describes the linear combination of modal (eigenvectors) shapes. In the expression above, Ku_i equals zero and is a 6 by 1 matrix of zeros.