User:Eml4507.s13.team3.chen/R6

= Problem 6.1: Solving a general eigenvalue problem by transforming into a SEP = On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
Consider the following numerical values for the MDOF system:
 * $$m_{1}=3, m_{2}=2$$
 * $$k_{1}=10, k_{2}=20, k_{3}=15$$

Find: The eigenvalues and eigenvectors for the GEP and SEP
The general eigenvalue problem $$\mathbf {K x}=\lambda \mathbf {M x}$$

The standard eigenvalue problem $$\mathbf {K^\star x^\star}=\lambda \mathbf x^\star$$


 * $$\mathbf K=\begin{bmatrix} (k_1 + k_2) & -k_2 \\ -k_2 & (k_2 + k_3) \end{bmatrix}=\begin{bmatrix} 30 & -20 \\ -20 & 35 \end{bmatrix}$$
 * $$\mathbf M=\begin{bmatrix} m_1 & 0 \\ 0 & m_2 \end{bmatrix}=\begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix}$$

Solution
Find $$\mathbf K^\star$$, where:
 * $$\mathbf K^\star = \mathbf {M^{-1/2}KM^{-1/2}}$$
 * $$\mathbf M^{-1/2}=\text{Diag}[(m_1)^{-1/2} \text{...} (m_n)^{-1/2}]$$


 * $$\mathbf K^\star = \begin{bmatrix} \frac{1}{\sqrt{3}} & 0 \\ 0 & \frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} 30 & -20 \\ -20 & 35 \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{3}} & 0 \\ 0 & \frac{1}{\sqrt{2}} \end{bmatrix}=\begin{bmatrix} 10.0000 & -8.1650 \\ -8.1650 & 17.5000 \end{bmatrix}$$

Use the eig command in MATLAB to calculate the eigenvectors for $$\mathbf K^\star$$, where V is the matrix of eigenvectors, and D is the vector of eigenvalues. EDU>> [V,D]=eig(Kstar) V = -0.8418  -0.5397   -0.5397    0.8418 D = 4.7651        0         0   22.7349

Define a new matrix $$\mathbf x^\star$$, which is equal to the eigenvector of $$\mathbf K^\star$$:
 * $$\mathbf x^\star := \mathbf {M^{1/2} x} \rightarrow \mathbf x := \mathbf {M^{-1/2} x^\star}$$

Recover the eigenvectors $$\mathbf x$$ by pre-multiplying $$\mathbf x^\star$$ by $$\mathbf M^{-1/2}$$ EDU>> M^(-0.5)*V ans = -0.4860  -0.3116   -0.3817    0.5953

Confirm eigenvector by solving general eigenvalue problem: EDU>> [V,D]=eig(K,M) L = 4.7651 22.7349 X = -0.4860  -0.3116 -0.3817   0.5953