User:Eml4500.f08.gravy.bje/Notes4

HW Proof
$$ \underline{\tilde{f}}^{(e)} = \underline{\tilde{T}}^{(e)} \underline{f}^{(e)} $$

$$ \underline{\tilde{T}}^{(e)} = \begin{bmatrix} l^{(e)} & -m^{(e)} & 0 & 0 \\ m^{(e)} & l^{(e)} & 0 & 0 \\ 0 & 0 & l^{(e)} & -m^{(e)} \\ 0 & 0 & m^{(e)} & l^{(e)} \end{bmatrix}_{4x4}\,$$

$$\ l^{(e)}= \vec \tilde{i} \cdot \vec i= cos\theta^{(e)}\,$$

$$\ m^{(e)}= \vec i \cdot \vec j= cos\left (\frac{\pi}{2}- \theta^{(e)} \right )= sin\left (\theta^{(e)} \right )\,$$

$$\ \tilde{f}_{1}^{(e)} = \begin{bmatrix} l^{(e)} & m^{(e)} \end{bmatrix} \begin{Bmatrix} f_{1}^{(e)} \\ f_{2}^{(e)} \end{Bmatrix} $$

$$\ \tilde{f}_{2}^{(e)} = \begin{bmatrix} -m^{(e)} & l^{(e)} \end{bmatrix} \begin{Bmatrix} f_{1}^{(e)} \\ f_{2}^{(e)} \end{Bmatrix} $$

$$\begin{Bmatrix}\ \tilde{f}_{1}^{(e)} \\ \tilde{f}_{2}^{(e)} \end{Bmatrix} = \begin{bmatrix} l^{(e)} & m^{(e)} \\ -m^{(e)} & l^{(e)} \end{bmatrix} \begin{Bmatrix} f_{1}^{(e)} \\ f_{2}^{(e)} \end{Bmatrix} $$

$$\begin{Bmatrix}\ \tilde{f}_{1}^{(e)} \\ \tilde{f}_{2}^{(e)} \\ \tilde{f}_{3}^{(e)} \\ \tilde{f}_{4}^{(e)} \end{Bmatrix} = \begin{bmatrix} l^{(e)} & -m^{(e)} & 0 & 0 \\ m^{(e)} & l^{(e)} & 0 & 0 \\ 0 & 0 & l^{(e)} & -m^{(e)} \\ 0 & 0 & m^{(e)} & l^{(e)} \end{bmatrix} \begin{Bmatrix} f_{1}^{(e)} \\ f_{2}^{(e)} \\ f_{3}^{(e)} \\ f_{4}^{(e)} \end{Bmatrix} $$

For θ(e) = 0 we have

$$\ l^{(e)} = cos\theta^{(e)} = 1\,$$

$$\ m^{(e)} = sin\theta^{(e)} = 0\,$$

$$\begin{Bmatrix}\ \tilde{f}_{1}^{(e)} \\ \tilde{f}_{2}^{(e)} \\ \tilde{f}_{3}^{(e)} \\ \tilde{f}_{4}^{(e)} \end{Bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{Bmatrix} f_{1}^{(e)} \\ f_{2}^{(e)} \\ f_{3}^{(e)} \\ f_{4}^{(e)} \end{Bmatrix} $$

$$\tilde{f}_{1}^{(e)} = f_{1}^{(e)}$$

$$\tilde{f}_{2}^{(e)} = f_{2}^{(e)}$$

$$\tilde{f}_{3}^{(e)} = f_{3}^{(e)}$$

$$\tilde{f}_{4}^{(e)} = f_{4}^{(e)}$$

Since $$x = \tilde{x}$$ at θ(e) = 0

Eigenvector Homework
$$\ \underline{k} = \begin{bmatrix} K_{11}^{(1)} & K_{12}^{(1)} & K_{13}^{(1)} & K_{14}^{(1)} & 0 & 0  \\ K_{21}^{(1)}& K_{22}^{(1)} & K_{23}^{(1)} & K_{24}^{(1)} & 0 & 0 \\ K_{31}^{(1)} & K_{32}^{(1)} & (K_{33}^{(1)} + K_{11}^{(2)}) & (K_{34}^{(1)} + K_{12}^{(2)}) & K_{13}^{(2)} & K_{14}^{(2)} \\ K_{41}^{(1)}& K_{42}^{(1)} & (K_{43}^{(1)} + K_{21}^{(2)}) & (K_{44}^{(1)} + K_{22}^{(2)}) & K_{23}^{(2)} & K_{24}^{(2)} \\ 0 & 0 & K_{31}^{(2)} & K_{32}^{(2)} & K_{33}^{(2)} & K_{34}^{(2)} \\ 0 & 0 & K_{41}^{(2)} & K_{42}^{(2)} & K_{43}^{(2)} & K_{44}^{(2)}  \end{bmatrix}_{6x6}\,$$

$$\ \ = \begin{bmatrix} \frac{9}{16}& \frac{3\sqrt{3}}{16} & -\frac{9}{16} & -\frac{3\sqrt{3}}{16} & 0 & 0 \\  \frac{3\sqrt{3}}{16} & \frac{3}{16} & -\frac{3\sqrt{3}}{16} & -\frac{3}{16} & 0 & 0 \\ -\frac{9}{16} & -\frac{3\sqrt{3}}{16} & \frac{49}{16} & \frac{3\sqrt{3} - 40}{16} & -\frac{5}{2} & \frac{5}{2} \\ -\frac{3\sqrt{3}}{16} & -\frac{3}{16} & \frac{3\sqrt{3} - 40}{16} & \frac{43}{16} & \frac{5}{2} & -\frac{5}{2} \\  0 & 0 & -\frac{5}{2} & \frac{5}{2} & \frac{5}{2} & -\frac{5}{2} \\ 0 & 0 & \frac{5}{2} & -\frac{5}{2} & -\frac{5}{2} & \frac{5}{2}  \end{bmatrix}_{6X6}\,$$

MATLAB Code

Values of the stiffness matrix are assigned to matrix K >> K= [9/16 (3*sqrt(3))/16 -9/16 -(3*sqrt(3))/16 0 0; (3*sqrt(3))/16 3/16 -(3*sqrt(3))/16 -3/16 0 0; -9/16 -(3*sqrt(3))/16 49/16 ((3*sqrt(3)-40)/16) -5/2 5/2; -(3*sqrt(3))/16 -3/16 ((3*sqrt(3)-40)/16) 43/16 5/2 -5/2; 0 0 -5/2 5/2 5/2 -5/2; 0 0 5/2 -5/2 -5/2 5/2]

Values of K in decimal form K = 0.5625   0.3248   -0.5625   -0.3248         0         0    0.3248    0.1875   -0.3248   -0.1875         0         0   -0.5625   -0.3248    3.0625   -2.1752   -2.5000    2.5000   -0.3248   -0.1875   -2.1752    2.6875    2.5000   -2.5000         0         0   -2.5000    2.5000    2.5000   -2.5000         0         0    2.5000   -2.5000   -2.5000    2.5000

Matlab function to calculate eigenvectors and eigenvalues >> [V,D] = eigs(K)

Eigenvectors of the matrix V = -0.0139   0.6174    0.0117    0.0155   -0.7776   -0.1162   -0.0080    0.3565    0.0016    0.1013    0.1481    0.9169    0.5123   -0.5409   -0.0835   -0.2936   -0.4947    0.3273   -0.4904   -0.4329    0.1665    0.6366   -0.3420    0.1486   -0.4984   -0.0765    0.5583   -0.6459   -0.0723    0.1075    0.4984    0.0765    0.8083    0.2843    0.0803   -0.0712

Eigenvalues of the matrix shown in the diagonal D = 10.0295        0         0         0         0         0         0    1.4705         0         0         0         0         0         0    0.0000         0         0         0         0         0         0   -0.0000         0         0         0         0         0         0    0.0000         0         0         0         0         0         0   -0.0000