User:Eml4507.s13.team3.steiner/Team Negative Damping (3): Report 6

= 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) V = 4.7651 22.7349 D = -0.4860  -0.3116 -0.3817   0.5953 = Problem 6.2 = On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given: Plane Truss
Consider the plane truss in the figure above. The horizontal and vertical members have length L, while inclined members have length 1.414*L. Assume the Young's modulus E = 100 GPa, cross-sectional area A = 1.0 cm2, L = 0.3 m, and the density if 5000 kg/m3.

Find: Transform GEP into SEP.
Transform the generalized eigenvalue problem into a standard eigenvalue problem.

Solution
The mass matrix was computed in the same way as it was in Report 5. The following code was used to find it.

%Mass Matrix Construction m = zeros(3*G,3*G); for i=1:G for j=1:E if(l1(j)==i || l2(j)==i) m(3*i-2,3*i-2)=(Ro(j)*Area(j)*L(j))/2 + m(3*i-2,3*i-2); m(3*i-1,3*i-1)=(Ro(j)*Area(j)*L(j))/2 + m(3*i-1,3*i-1); m(3*i,3*i)=(Ro(j)*Area(j)*L(j))/2 + m(3*i,3*i); end end end

This is the resulting mass matrix

m = Columns 1 through 10 0.2561        0         0         0         0         0         0         0         0         0           0    0.2561         0         0         0         0         0         0         0         0           0         0    0.1500         0         0         0         0         0         0         0           0         0         0    0.1500         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      0.3311         0         0         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0         0         0           0         0         0    0.3311         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 21 through 28 0        0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0      0.3311         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0           0         0         0    0.3311         0         0         0         0           0         0         0         0    0.1500         0         0         0           0         0         0         0         0    0.1500         0         0           0         0         0         0         0         0    0.2561         0           0         0         0         0         0         0         0    0.2561 The reduced mass matrix is as follows.

mm = Columns 1 through 10 0.2561        0         0         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0         0         0           0         0         0    0.3311         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      0.3311         0         0         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0         0         0           0         0         0    0.3311         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 21 through 25 0        0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0      0.3311         0         0         0         0           0    0.1500         0         0         0           0         0    0.1500         0         0           0         0         0    0.2561         0           0         0         0         0    0.2561

k_red was calculated using the same code as in Report 3. The resulting K matrix is as follows.

k_red = 1.0e+07 * Columns 1 through 10 4.5118        0         0   -1.1785   -1.1785         0         0         0         0         0           0    7.8452    1.1785         0         0   -3.3333         0   -1.1785   -1.1785         0           0    1.1785    4.5118         0   -3.3333         0         0   -1.1785   -1.1785         0     -1.1785         0         0    7.8452    1.1785         0         0   -3.3333         0         0     -1.1785         0   -3.3333    1.1785    4.5118         0         0         0         0         0           0   -3.3333         0         0         0    7.8452    1.1785         0         0   -3.3333           0         0         0         0         0    1.1785    4.5118         0   -3.3333         0           0   -1.1785   -1.1785   -3.3333         0         0         0    7.8452    1.1785         0           0   -1.1785   -1.1785         0         0         0   -3.3333    1.1785    4.5118         0           0         0         0         0         0   -3.3333         0         0         0    7.8452           0         0         0         0         0         0         0         0         0    1.1785           0         0         0         0         0   -1.1785   -1.1785   -3.3333         0         0           0         0         0         0         0   -1.1785   -1.1785         0         0         0           0         0         0         0         0         0         0         0         0   -3.3333           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0   -1.1785           0         0         0         0         0         0         0         0         0   -1.1785           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0   -1.1785   -1.1785         0         0         0         0         0         0         0           0   -1.1785   -1.1785         0         0         0         0         0         0         0           0   -3.3333         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      1.1785         0         0   -3.3333         0   -1.1785   -1.1785         0         0         0      4.5118         0   -3.3333         0         0   -1.1785   -1.1785         0         0         0           0    7.8452    1.1785         0         0   -3.3333         0         0         0         0     -3.3333    1.1785    4.5118         0         0         0         0         0         0         0           0         0         0    7.8452    1.1785         0         0   -3.3333         0   -1.1785           0         0         0    1.1785    4.5118         0   -3.3333         0         0   -1.1785     -1.1785   -3.3333         0         0         0    7.8452    1.1785         0         0   -3.3333     -1.1785         0         0         0   -3.3333    1.1785    4.5118         0         0         0           0         0         0   -3.3333         0         0         0    7.8452    1.1785         0           0         0         0         0         0         0         0    1.1785    4.5118         0           0         0         0   -1.1785   -1.1785   -3.3333         0         0         0    7.8452           0         0         0   -1.1785   -1.1785         0         0         0   -3.3333    1.1785           0         0         0         0         0         0         0   -3.3333         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0   -1.1785   -1.1785   -3.3333           0         0         0         0         0         0         0   -1.1785   -1.1785         0    Columns 21 through 25 0        0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0     -1.1785         0         0         0         0     -1.1785         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0   -3.3333         0   -1.1785   -1.1785     -3.3333         0         0   -1.1785   -1.1785      1.1785         0         0   -3.3333         0      4.5118         0         0         0         0           0    3.3333         0         0         0           0         0    3.3333         0   -3.3333           0         0         0    4.5118    1.1785           0         0   -3.3333    1.1785    4.5118

Then we decompose the mass matrix:

$$M=M^{\frac{1}{2}}M^{\frac{1}{2}}$$

$$ M^{\frac{1}{2}}= $$

Columns 1 through 10 0.5061        0         0         0         0         0         0         0         0         0           0    0.5754         0         0         0         0         0         0         0         0           0         0    0.5754         0         0         0         0         0         0         0           0         0         0    0.5754         0         0         0         0         0         0           0         0         0         0    0.5754         0         0         0         0         0           0         0         0         0         0    0.5754         0         0         0         0           0         0         0         0         0         0    0.5754         0         0         0           0         0         0         0         0         0         0    0.5754         0         0           0         0         0         0         0         0         0         0    0.5754         0           0         0         0         0         0         0         0         0         0    0.5754            0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0       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  0           0         0         0         0    0.5754         0         0         0         0         0           0         0         0         0         0    0.5754         0         0         0         0           0         0         0         0         0         0    0.5754         0         0         0           0         0         0         0         0         0         0    0.5754         0         0           0         0         0         0         0         0         0         0    0.5754         0           0         0         0         0         0         0         0         0         0    0.5754            0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 21 through 25 0        0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0      0.5754         0         0         0         0           0    0.3873         0         0         0           0         0    0.3873         0         0           0         0         0    0.5061         0           0         0         0         0    0.5061

$$ M^{\frac{-1}{2}}= $$

Columns 1 through 10 1.9760        0         0         0         0         0         0         0         0         0           0    1.7379         0         0         0         0         0         0         0         0           0         0    1.7379         0         0         0         0         0         0         0           0         0         0    1.7379         0         0         0         0         0         0           0         0         0         0    1.7379         0         0         0         0         0           0         0         0         0         0    1.7379         0         0         0         0           0         0         0         0         0         0    1.7379         0         0         0           0         0         0         0         0         0         0    1.7379         0         0           0         0         0         0         0         0         0         0    1.7379         0           0         0         0         0         0         0         0         0         0    1.7379            0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      1.7379         0         0         0         0         0         0         0         0         0           0    1.7379         0         0         0         0         0         0         0         0           0         0    1.7379         0         0         0         0         0         0         0           0         0         0    1.7379         0         0         0         0         0         0           0         0         0         0    1.7379         0         0         0         0         0           0         0         0         0         0    1.7379         0         0         0         0           0         0         0         0         0         0    1.7379         0         0         0           0         0         0         0         0         0         0    1.7379         0         0           0         0         0         0         0         0         0         0    1.7379         0           0         0         0         0         0         0         0         0         0    1.7379            0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 21 through 25 0        0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0      1.7379         0         0         0         0           0    2.5819         0         0         0           0         0    2.5819         0         0           0         0         0    1.9760         0           0         0         0         0    1.9760 The following steps are taken to transform the generalized eigenvalue problem,  $$ Kx=\lambda Mx $$ into the standard eigenvalue problem, $$ K^{*}x^{*}=\lambda x^{*} $$  :

$$ M^{\frac{-1}{2}}Kx=\lambda M^{\frac{-1}{2}} Mx $$

$$ M^{\frac{-1}{2}}Kx=\lambda M^{\frac{-1}{2}} Mx=\lambda M^{\frac{1}{2}} $$

$$ x^{*}:=M^{\frac{1}{2}}x\Leftrightarrow x=M^{\frac{-1}{2}}x^{*} $$

$$ \left (M ^{\frac{-1}{2}}K M ^{\frac{1}{2}} \right )x^{*}=Kx^{*} $$ $$ K^{*}x^{*}=\lambda x^{*} $$

The matrix $$\mathbf K^\star$$ is as follows: kstar= 1.0e+008 * Columns 1 through 7 1.7618        0         0   -0.4047   -0.4047         0         0         0    2.3694    0.3559         0         0   -1.0067         0         0    0.3559    1.3627         0   -1.0067         0         0   -0.4047         0         0    2.3694    0.3559         0         0   -0.4047         0   -1.0067    0.3559    1.3627         0         0         0   -1.0067         0         0         0    2.3694    0.3559         0         0         0         0         0    0.3559    1.3627         0   -0.3559   -0.3559   -1.0067         0         0         0         0   -0.3559   -0.3559         0         0         0   -1.0067         0         0         0         0         0   -1.0067         0         0         0         0         0         0         0         0         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 8 through 14 0        0         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0   -1.0067         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -1.0067         0   -0.3559   -0.3559         0         0   -1.0067         0         0   -0.3559   -0.3559         0    2.3694    0.3559         0         0   -1.0067         0         0    0.3559    1.3627         0         0         0         0         0         0         0    2.3694    0.3559         0         0   -1.0067         0         0    0.3559    1.3627         0   -1.0067         0   -1.0067         0         0         0    2.3694    0.3559         0         0         0         0   -1.0067    0.3559    1.3627         0         0         0   -1.0067         0         0         0    2.3694         0         0         0         0         0         0    0.3559         0         0   -0.3559   -0.3559   -1.0067         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0         0         0         0         0   -1.0067         0         0         0         0         0         0         0         0         0         0         0         0         0   -0.3559         0         0         0         0         0         0   -0.3559         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 15 through 21 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0   -1.0067         0         0         0         0         0         0         0         0         0         0         0         0    0.3559         0         0   -1.0067         0   -0.3559   -0.3559    1.3627         0   -1.0067         0         0   -0.3559   -0.3559         0    2.3694    0.3559         0         0   -1.0067         0   -1.0067    0.3559    1.3627         0         0         0         0         0         0         0    2.3694    0.3559         0         0         0         0         0    0.3559    1.3627         0   -1.0067   -0.3559   -1.0067         0         0         0    2.3694    0.3559   -0.3559         0         0         0   -1.0067    0.3559    1.3627         0         0         0   -1.4957         0         0         0         0         0         0         0         0         0         0         0         0         0   -0.4047   -0.4047   -1.1447         0         0         0         0   -0.4047   -0.4047         0         0  Columns 22 through 25 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -1.4957         0   -0.4047   -0.4047         0         0   -0.4047   -0.4047         0         0   -1.1447         0         0         0         0         0    2.2222         0         0         0         0    2.2222         0   -1.7007         0         0    1.7618    0.4602         0   -1.7007    0.4602    1.7618 The eig command in MATLAB is used to calculate the eigenvectors for $$\mathbf K^\star$$.The eigenvectors are represented by the matrix V, and D is the vector of eigenvalues.

EDU>> [V,D]=eig(kstar)

V = Columns 1 through 7 -0.0110  -0.0315   -0.0474    0.0455   -0.0792   -0.0889   -0.1931    0.1557    0.2277   -0.1435   -0.3004   -0.1617    0.2041   -0.4005    0.0395    0.0784    0.0091   -0.1050    0.0065    0.1288   -0.2570    0.0821    0.1971    0.2085   -0.2278    0.3017    0.2841    0.3472   -0.0018    0.0014    0.0341    0.0026    0.0593    0.0043    0.2715   -0.2742   -0.2610    0.3190    0.2376    0.3316    0.0538    0.1634   -0.0563   -0.0673    0.0268    0.0048    0.0601    0.2775    0.0064   -0.1877   -0.3668   -0.2722    0.2983   -0.3174   -0.1619   -0.0642   -0.0237   -0.0583   -0.0308    0.1022   -0.0527   -0.2660    0.1207    0.3496    0.0886   -0.3889    0.0026   -0.2603    0.0434    0.3646    0.0647    0.0170   -0.0334    0.0985   -0.0458   -0.0028    0.2341    0.2782    0.3812    0.2241   -0.0062    0.1242   -0.3824   -0.0281    0.0452    0.0799   -0.0039   -0.0781   -0.0219   -0.1467   -0.1805   -0.3809    0.1944    0.2865   -0.0505   -0.0603   -0.2520   -0.2453   -0.0628    0.0417    0.0091   -0.0879   -0.0446   -0.2807    0.1110   -0.3281   -0.2238   -0.1774   -0.4031    0.0188    0.2610   -0.0404   -0.0599   -0.0540    0.0316   -0.0379    0.0716    0.2108   -0.2000    0.3820   -0.4494   -0.0719   -0.1934    0.3121   -0.1435   -0.0836    0.0661   -0.1100    0.1125   -0.0884   -0.0389   -0.1323   -0.0628    0.3192   -0.0062    0.1944    0.5157   -0.0137    0.2596   -0.2266    0.0611    0.0083   -0.0604    0.1340    0.0319    0.2425    0.0179   -0.2290    0.3219    0.0668    0.1875   -0.3376    0.2516    0.1494    0.1054   -0.2025    0.4186   -0.2325   -0.4546    0.0415    0.1507   -0.2091    0.1365   -0.2034   -0.1894    0.0408   -0.1486    0.2193   -0.1545    0.2486   -0.3967    0.2109    0.3697   -0.0208   -0.0742  Columns 8 through 14 -0.1082   0.2590    0.1593    0.0995    0.0920   -0.5045    0.4512   -0.1368   -0.1865   -0.2806   -0.0426    0.0795   -0.1556   -0.4220   -0.2536    0.2527    0.1979    0.2892    0.0125    0.3888   -0.1230    0.1028   -0.2806    0.0152    0.2436    0.0029    0.2378    0.1403    0.2126   -0.3367   -0.2557   -0.3286   -0.0702   -0.2821   -0.0295   -0.1487    0.2270    0.1129   -0.0085   -0.0606   -0.0487   -0.2568   -0.3531    0.0392   -0.0611   -0.3971   -0.2713    0.0113    0.2019    0.0602   -0.1237   -0.1545   -0.0928   -0.0609    0.2550   -0.0933    0.3267   -0.0786    0.0356    0.3746    0.3146   -0.0098   -0.1418   -0.0781    0.0143    0.1930   -0.0162   -0.0455    0.1400    0.3221   -0.3730   -0.0322   -0.1677    0.0755    0.4672   -0.0899   -0.0142    0.1165    0.2732    0.0272   -0.2226    0.0569   -0.1608   -0.0633    0.3772    0.0256    0.1585   -0.0146   -0.4776    0.1282    0.0376   -0.1190   -0.2665   -0.2509    0.0779    0.0728    0.1526    0.3975   -0.2858   -0.1814    0.0304    0.2665   -0.3923   -0.1843   -0.1236    0.1844   -0.0581    0.1921    0.1731    0.0270   -0.2992    0.0815    0.3254    0.1235    0.0285   -0.3059    0.3661    0.1684    0.1462   -0.0360   -0.0120   -0.0337   -0.0057   -0.0012   -0.0598   -0.1177   -0.1124   -0.2951    0.4610   -0.2556    0.1775    0.0683   -0.0818    0.0533   -0.2026   -0.0266    0.1385   -0.0682    0.0114    0.0633    0.1749    0.2819   -0.3911    0.2471   -0.1284   -0.1470    0.0246    0.0748    0.0357    0.3347   -0.0738   -0.0106   -0.1803   -0.3144    0.0467    0.1988   -0.1670    0.0211   -0.0367    0.0616    0.0180   -0.0085    0.3412   -0.2219   -0.1508    0.0171    0.2337   -0.1101   -0.0198   -0.0589    0.0148    0.0014   -0.0036    0.0180    0.0059  Columns 15 through 21 -0.5574   0.1148    0.0041    0.0377    0.0178   -0.0802   -0.1034   -0.2613   -0.1480    0.0161   -0.0131    0.1253    0.0463    0.0756    0.1005    0.0286    0.0436    0.2403    0.0239   -0.4054   -0.3811   -0.3591    0.3670   -0.0209   -0.0421    0.0693    0.0377   -0.0098   -0.0288   -0.1465    0.0389    0.2037    0.0051   -0.3640   -0.3777   -0.1200   -0.2985    0.0276   -0.0560    0.2248    0.0823    0.0434    0.0332    0.1448    0.1141    0.3747    0.0589   -0.2894    0.1347   -0.1033    0.4069   -0.0371   -0.0411    0.1528   -0.0121   -0.1127   -0.1098   -0.1100    0.1097    0.3538    0.0449   -0.3326    0.0600    0.1036   -0.2702    0.0348   -0.1084    0.2987    0.0324   -0.0414    0.1442    0.1126    0.2060    0.3598    0.0738    0.1333    0.3411    0.2859    0.1959   -0.0487   -0.0088    0.2429   -0.0554   -0.0987   -0.1063   -0.0294    0.2022    0.3583    0.0688    0.0643    0.3955    0.1364   -0.0095    0.0385   -0.1498    0.3518   -0.0749    0.0214    0.0349    0.0324    0.3098    0.1805    0.0432    0.3491   -0.1906    0.3490   -0.0934   -0.0560    0.0347    0.3259   -0.0159   -0.0187    0.0075    0.0259    0.3068    0.1938    0.0465    0.3336   -0.0703   -0.0074    0.2857    0.0397   -0.1692    0.3867   -0.1490    0.1529   -0.1021   -0.0206    0.4168   -0.1050   -0.0146    0.0868   -0.3058   -0.0625   -0.2595   -0.0597    0.0691    0.3861    0.0790   -0.0716    0.1662    0.2111    0.4150   -0.0894   -0.0089    0.1258   -0.3066   -0.0149    0.3453    0.0267   -0.1153    0.2688   -0.1058    0.1157   -0.0761   -0.1647    0.3505   -0.2686   -0.0376   -0.2421    0.1823   -0.3571   -0.1733   -0.0536    0.0736    0.3633    0.1235   -0.1592   -0.0332   -0.1199    0.4577   -0.3468   -0.0476   -0.2999    0.2119  Columns 22 through 25 -0.0882   0.0483    0.0971   -0.0110    0.0980    0.3005    0.0433   -0.1621   -0.2775    0.0483    0.1312    0.0617    0.0451    0.0476    0.1733   -0.1872   -0.3426    0.0782    0.1281    0.1545   -0.0169    0.4537    0.1277   -0.1073    0.3627   -0.0396   -0.2270   -0.2080    0.0028    0.0868    0.3271   -0.2778    0.3107    0.0815   -0.3302   -0.1818   -0.0344    0.3814   -0.0693   -0.0822   -0.1177   -0.0489    0.3473    0.2131    0.1336    0.0013    0.2536   -0.3414   -0.0444    0.1491    0.2615    0.2781   -0.0114    0.2047   -0.1428    0.1717   -0.2388   -0.1493   -0.2096   -0.2993    0.1211   -0.1785    0.2699   -0.1266   -0.3494    0.0481   -0.2430   -0.2606   -0.1885   -0.0232   -0.2832    0.2031    0.3830   -0.0360    0.1279    0.2418    0.0340   -0.3964    0.0618   -0.0248    0.2943    0.1901   -0.0218    0.2570   -0.1544   -0.0229   -0.2468    0.1826   -0.0974   -0.1963    0.1089   -0.1756    0.1215   -0.3768   -0.0340    0.1917   -0.1046   -0.1748    0.1099   -0.1718

D = 1.0e+008 * Columns 1 through 7 4.7170        0         0         0         0         0         0         0    4.3103         0         0         0         0         0         0         0    3.8339         0         0         0         0         0         0         0    3.7647         0         0         0         0         0         0         0    3.6051         0         0         0         0         0         0         0    3.0752         0         0         0         0         0         0         0    3.0589         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 8 through 14 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    2.9417         0         0         0         0         0         0         0    2.7262         0         0         0         0         0         0         0    2.3728         0         0         0         0         0         0         0    2.1074         0         0         0         0         0         0         0    2.0577         0         0         0         0         0         0         0    1.7262         0         0         0         0         0         0         0    1.6623         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 15 through 21 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    1.4801         0         0         0         0         0         0         0    0.9847         0         0         0         0         0         0         0    0.0017         0         0         0         0         0         0         0    0.0267         0         0         0         0         0         0         0    0.0702         0         0         0         0         0         0         0    0.1154         0         0         0         0         0         0         0    0.2452         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 22 through 25 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    0.3962         0         0         0         0    0.7073         0         0         0         0    0.5060         0         0         0         0    0.5581 Next, the answers to the SEP need to be checked for accuracy.

The eigenvectors are recovered. EDU>> X=mm^(-1/2)*V X = Columns 1 through 7 -0.0217  -0.0623   -0.0936    0.0899   -0.1566   -0.1757   -0.3815    0.2705    0.3957   -0.2493   -0.5221   -0.2809    0.3547   -0.6960    0.0686    0.1362    0.0158   -0.1824    0.0113    0.2238   -0.4467    0.1427    0.3425    0.3624   -0.3959    0.5242    0.4938    0.6035   -0.0032    0.0024    0.0592    0.0045    0.1030    0.0075    0.4718   -0.4766   -0.4536    0.5544    0.4130    0.5762    0.0934    0.2840   -0.0978   -0.1170    0.0465    0.0084    0.1044    0.4823    0.0112   -0.3262   -0.6375   -0.4731    0.5185   -0.5517   -0.2814   -0.1116   -0.0412   -0.1013   -0.0535    0.1777   -0.0916   -0.4622    0.2098    0.6076    0.1540   -0.6759    0.0046   -0.4523    0.0754    0.6337    0.1125    0.0295   -0.0581    0.1712   -0.0797   -0.0049    0.4068    0.4834    0.6625    0.3894   -0.0107    0.2158   -0.6645   -0.0489    0.0785    0.1388   -0.0068   -0.1358   -0.0380   -0.2549   -0.3137   -0.6620    0.3378    0.4979   -0.0878   -0.1048   -0.4379   -0.4264   -0.1091    0.0724    0.0158   -0.1527   -0.0775   -0.4877    0.1929   -0.5702   -0.3890   -0.3082   -0.7006    0.0327    0.4535   -0.0702   -0.1042   -0.0939    0.0549   -0.0658    0.1244    0.3663   -0.3476    0.6639   -0.7810   -0.1250   -0.3361    0.5424   -0.2494   -0.1453    0.1148   -0.1911    0.1956   -0.1537   -0.0676   -0.2300   -0.1092    0.5547   -0.0107    0.3379    0.8963   -0.0239    0.4512   -0.3938    0.1062    0.0145   -0.1050    0.2329    0.0555    0.4214    0.0311   -0.5913    0.8312    0.1724    0.4842   -0.8716    0.6497    0.3858    0.2720   -0.5228    1.0808   -0.6003   -1.1739    0.1072    0.3892   -0.4131    0.2697   -0.4020   -0.3742    0.0807   -0.2936    0.4334   -0.3054    0.4912   -0.7838    0.4167    0.7305   -0.0412   -0.1465  Columns 8 through 14 -0.2138   0.5118    0.3148    0.1967    0.1818   -0.9970    0.8917   -0.2377   -0.3242   -0.4876   -0.0740    0.1382   -0.2705   -0.7333   -0.4407    0.4392    0.3439    0.5027    0.0218    0.6756   -0.2138    0.1787   -0.4876    0.0264    0.4234    0.0051    0.4133    0.2439    0.3694   -0.5852   -0.4444   -0.5711   -0.1221   -0.4903   -0.0512   -0.2584    0.3945    0.1962   -0.0147   -0.1054   -0.0846   -0.4462   -0.6136    0.0682   -0.1062   -0.6901   -0.4715    0.0196    0.3509    0.1046   -0.2151   -0.2685   -0.1613   -0.1059    0.4432   -0.1621    0.5677   -0.1365    0.0619    0.6510    0.5468   -0.0170   -0.2464   -0.1357    0.0249    0.3353   -0.0281   -0.0790    0.2433    0.5598   -0.6482   -0.0560   -0.2915    0.1312    0.8120   -0.1562   -0.0246    0.2024    0.4748    0.0472   -0.3868    0.0988   -0.2794   -0.1100    0.6555    0.0445    0.2754   -0.0254   -0.8301    0.2228    0.0653   -0.2069   -0.4632   -0.4360    0.1354    0.1265    0.2652    0.6907   -0.4966   -0.3153    0.0528    0.4631   -0.6818   -0.3203   -0.2148    0.3204   -0.1011    0.3339    0.3008    0.0470   -0.5199    0.1417    0.5656    0.2146    0.0496   -0.5316    0.6362    0.2927    0.2541   -0.0625   -0.0209   -0.0585   -0.0099   -0.0020   -0.1039   -0.2046   -0.1953   -0.5128    0.8011   -0.4442    0.3085    0.1188   -0.1421    0.0926   -0.3521   -0.0463    0.2406   -0.1185    0.0199    0.1101    0.3040    0.4899   -0.6797    0.4294   -0.2232   -0.2554    0.0427    0.1931    0.0921    0.8641   -0.1907   -0.0275   -0.4654   -0.8119    0.1206    0.5134   -0.4313    0.0544   -0.0948    0.1590    0.0465   -0.0168    0.6743   -0.4386   -0.2979    0.0338    0.4618   -0.2175   -0.0391   -0.1164    0.0292    0.0028   -0.0070    0.0355    0.0117  Columns 15 through 21 -1.1014   0.2269    0.0082    0.0745    0.0352   -0.1585   -0.2043   -0.4542   -0.2573    0.0279   -0.0227    0.2178    0.0805    0.1314    0.1746    0.0497    0.0757    0.4176    0.0416   -0.7045   -0.6624   -0.6241    0.6379   -0.0364   -0.0732    0.1204    0.0655   -0.0171   -0.0500   -0.2547    0.0677    0.3540    0.0089   -0.6326   -0.6564   -0.2085   -0.5187    0.0479   -0.0973    0.3907    0.1430    0.0754    0.0576    0.2516    0.1982    0.6511    0.1023   -0.5029    0.2342   -0.1796    0.7071   -0.0645   -0.0714    0.2656   -0.0210   -0.1958   -0.1908   -0.1912    0.1906    0.6149    0.0780   -0.5780    0.1042    0.1800   -0.4695    0.0605   -0.1884    0.5191    0.0564   -0.0719    0.2506    0.1956    0.3581    0.6252    0.1283    0.2317    0.5929    0.4969    0.3405   -0.0847   -0.0153    0.4221   -0.0962   -0.1715   -0.1847   -0.0511    0.3514    0.6227    0.1195    0.1118    0.6874    0.2370   -0.0165    0.0670   -0.2603    0.6115   -0.1301    0.0372    0.0607    0.0564    0.5384    0.3137    0.0751    0.6067   -0.3313    0.6066   -0.1624   -0.0974    0.0603    0.5663   -0.0277   -0.0325    0.0130    0.0451    0.5332    0.3367    0.0809    0.5798   -0.1222   -0.0128    0.4965    0.0690   -0.2941    0.6721   -0.2590    0.2657   -0.1774   -0.0358    0.7243   -0.1825   -0.0253    0.1508   -0.5315   -0.1087   -0.4511   -0.1038    0.1201    0.6709    0.1374   -0.1245    0.2889    0.3669    0.7213   -0.1553   -0.0154    0.2186   -0.5329   -0.0384    0.8916    0.0690   -0.2976    0.6941   -0.2732    0.2986   -0.1965   -0.4254    0.9051   -0.6936   -0.0971   -0.6251    0.4707   -0.7057   -0.3424   -0.1058    0.1455    0.7178    0.2440   -0.3145   -0.0656   -0.2369    0.9044   -0.6853   -0.0940   -0.5926    0.4187  Columns 22 through 25 -0.1742   0.0954    0.1920   -0.0217    0.1703    0.5222    0.0753   -0.2817   -0.4822    0.0840    0.2280    0.1072    0.0784    0.0828    0.3012   -0.3254   -0.5953    0.1359    0.2226    0.2685   -0.0293    0.7884    0.2220   -0.1865    0.6304   -0.0688   -0.3946   -0.3615    0.0048    0.1509    0.5684   -0.4828    0.5400    0.1416   -0.5738   -0.3160   -0.0597    0.6628   -0.1205   -0.1428   -0.2045   -0.0850    0.6036    0.3703    0.2322    0.0022    0.4407   -0.5933   -0.0772    0.2591    0.4545    0.4833   -0.0198    0.3558   -0.2481    0.2984   -0.4151   -0.2595   -0.3643   -0.5201    0.2105   -0.3102    0.4690   -0.2200   -0.6072    0.0837   -0.4223   -0.4529   -0.3275   -0.0403   -0.4921    0.3530    0.6657   -0.0626    0.2223    0.4202    0.0592   -0.6889    0.1074   -0.0430    0.5115    0.3304   -0.0379    0.4467   -0.3986   -0.0591   -0.6372    0.4714   -0.2516   -0.5067    0.2811   -0.4534    0.2400   -0.7445   -0.0672    0.3789   -0.2067   -0.3454    0.2171   -0.3395 Then, the GEP is solved and the resulting eigenvectors are found. If the two sets match up, then the results from the SEP are correct.

EDU>> [VV,DD]=eig(k_red,mm) VV = Columns 1 through 7 -0.0082  -0.0745   -0.0352   -0.1585   -0.2043    0.1742    0.1920   -0.0279    0.0227   -0.2178    0.0805    0.1314   -0.1703    0.0753   -0.0757   -0.4176   -0.0416   -0.7045   -0.6624    0.4822    0.2280    0.0364    0.0732   -0.1204    0.0655   -0.0171   -0.0784    0.3012   -0.0677   -0.3540   -0.0089   -0.6326   -0.6564    0.5953    0.2226   -0.0479    0.0973   -0.3907    0.1430    0.0754    0.0293    0.2220   -0.1982   -0.6511   -0.1023   -0.5029    0.2342   -0.6304   -0.3946    0.0645    0.0714   -0.2656   -0.0210   -0.1958   -0.0048    0.5684   -0.1906   -0.6149   -0.0780   -0.5780    0.1042   -0.5400   -0.5738   -0.0605    0.1884   -0.5191    0.0564   -0.0719    0.0597   -0.1205   -0.3581   -0.6252   -0.1283    0.2317    0.5929    0.2045    0.6036    0.0847    0.0153   -0.4221   -0.0962   -0.1715   -0.2322    0.4407   -0.3514   -0.6227   -0.1195    0.1118    0.6874    0.0772    0.4545   -0.0670    0.2603   -0.6115   -0.1301    0.0372    0.0198   -0.2481   -0.5384   -0.3137   -0.0751    0.6067   -0.3313    0.4151   -0.3643    0.0974   -0.0603   -0.5663   -0.0277   -0.0325   -0.2105    0.4690   -0.5332   -0.3367   -0.0809    0.5798   -0.1222    0.6072   -0.4223   -0.0690    0.2941   -0.6721   -0.2590    0.2657    0.3275   -0.4921   -0.7243    0.1825    0.0253    0.1508   -0.5315   -0.6657    0.2223    0.1038   -0.1201   -0.6709    0.1374   -0.1245   -0.0592    0.1074   -0.7213    0.1553    0.0154    0.2186   -0.5329   -0.5115   -0.0379   -0.0690    0.2976   -0.6941   -0.2732    0.2986    0.3986   -0.6372   -0.9051    0.6936    0.0971   -0.6251    0.4707    0.2516    0.2811    0.1058   -0.1455   -0.7178    0.2440   -0.3145   -0.2400   -0.0672   -0.9044    0.6853    0.0940   -0.5926    0.4187    0.2067    0.2171  Columns 8 through 14 0.0217  -0.0954   -0.2269    1.1014   -0.8917   -0.9970   -0.1818    0.2817   -0.5222    0.2573    0.4542    0.7333   -0.2705   -0.1382   -0.1072   -0.0840   -0.0497   -0.1746    0.2138    0.6756   -0.0218    0.3254   -0.0828   -0.6379    0.6241   -0.2439    0.4133   -0.0051   -0.2685   -0.1359    0.2547    0.0500    0.0512   -0.4903    0.1221    0.1865   -0.7884    0.5187    0.2085    0.4462   -0.0846    0.1054    0.3615    0.0688   -0.2516   -0.0576   -0.3509    0.0196    0.4715    0.4828   -0.1509   -0.7071    0.1796    0.1621    0.4432    0.1059    0.3160   -0.1416    0.1912    0.1908    0.2464   -0.0170   -0.5468    0.1428   -0.6628    0.4695   -0.1800   -0.5598    0.2433    0.0790   -0.3703    0.0850   -0.1956   -0.2506    0.0246   -0.1562   -0.8120    0.5933   -0.0022   -0.3405   -0.4969    0.1100   -0.2794   -0.0988   -0.4833   -0.2591    0.0511    0.1847   -0.0653    0.2228    0.8301   -0.2984   -0.3558    0.0165   -0.2370   -0.6907    0.2652   -0.1265    0.5201    0.2595   -0.0564   -0.0607    0.2148   -0.3203    0.6818    0.2200    0.3102    0.1624   -0.6066   -0.1417   -0.5199   -0.0470    0.4529   -0.0837   -0.0451   -0.0130   -0.2541    0.2927   -0.6362   -0.3530    0.0403   -0.4965    0.0128    0.2046   -0.1039    0.0020   -0.4202    0.0626    0.0358    0.1774    0.1421    0.1188   -0.3085    0.0430    0.6889    0.4511    0.1087   -0.1101    0.0199    0.1185   -0.4467   -0.3304   -0.3669   -0.2889   -0.0427   -0.2554    0.2232   -0.4714    0.0591   -0.8916    0.0384    0.8119   -0.4654    0.0275    0.4534    0.5067    0.4254    0.1965   -0.0465    0.1590    0.0948   -0.3789    0.7445    0.3424    0.7057    0.2175    0.4618   -0.0338    0.3395    0.3454    0.2369    0.0656   -0.0117    0.0355    0.0070  Columns 15 through 21 0.1967   0.3148   -0.5118    0.2138    0.3815    0.1757   -0.1566   -0.0740   -0.4876    0.3242    0.2377    0.6960   -0.3547   -0.2809    0.5027    0.3439   -0.4392    0.4407    0.4467   -0.2238    0.0113    0.4234    0.0264    0.4876   -0.1787   -0.6035   -0.4938    0.5242   -0.5711   -0.4444    0.5852   -0.3694   -0.4718   -0.0075    0.1030   -0.0147    0.1962   -0.3945    0.2584   -0.2840   -0.0934    0.5762   -0.6901   -0.1062   -0.0682    0.6136   -0.0112   -0.4823    0.1044   -0.1613   -0.2685    0.2151   -0.1046    0.1116    0.2814   -0.5517    0.6510    0.0619    0.1365   -0.5677   -0.2098    0.4622   -0.0916   -0.0281    0.3353   -0.0249    0.1357   -0.6337   -0.0754   -0.4523    0.1312   -0.2915    0.0560    0.6482   -0.4068    0.0049   -0.0797   -0.3868    0.0472   -0.4748   -0.2024    0.0489    0.6645    0.2158   -0.0254    0.2754   -0.0445   -0.6555    0.3137    0.2549   -0.0380    0.1354   -0.4360    0.4632    0.2069    0.4264    0.4379   -0.1048    0.4631    0.0528    0.3153    0.4966   -0.1929    0.4877   -0.0775    0.3008    0.3339    0.1011   -0.3204    0.0702   -0.4535    0.0327   -0.5316    0.0496   -0.2146   -0.5656    0.3476   -0.3663    0.1244   -0.0099   -0.0585    0.0209    0.0625    0.1453    0.2494    0.5424   -0.4442    0.8011    0.5128    0.1953    0.1092    0.2300   -0.0676    0.2406   -0.0463    0.3521   -0.0926    0.3938   -0.4512   -0.0239    0.4294   -0.6797   -0.4899   -0.3040   -0.0311   -0.4214    0.0555   -0.1907    0.8641   -0.0921   -0.1931   -0.3858   -0.6497   -0.8716    0.0544   -0.4313   -0.5134   -0.1206   -0.3892   -0.1072   -1.1739   -0.2979   -0.4386   -0.6743    0.0168   -0.4334    0.2936    0.0807    0.0028    0.0292    0.1164    0.0391    0.1465    0.0412    0.7305  Columns 22 through 25 0.0899   0.0936    0.0623    0.0217   -0.5221    0.2493   -0.3957   -0.2705   -0.1824   -0.0158   -0.1362   -0.0686   -0.3959   -0.3624   -0.3425   -0.1427    0.0045   -0.0592   -0.0024    0.0032    0.4130   -0.5544    0.4536    0.4766    0.0084   -0.0465    0.1170    0.0978    0.5185    0.4731    0.6375    0.3262    0.1777    0.0535    0.1013    0.0412    0.0046    0.6759   -0.1540   -0.6076    0.1712    0.0581   -0.0295   -0.1125   -0.0107   -0.3894   -0.6625   -0.4834   -0.1358    0.0068   -0.1388   -0.0785   -0.0878   -0.4979   -0.3378    0.6620   -0.1527   -0.0158   -0.0724    0.1091   -0.7006    0.3082    0.3890    0.5702   -0.0658   -0.0549    0.0939    0.1042   -0.3361    0.1250    0.7810   -0.6639   -0.1537   -0.1956    0.1911   -0.1148    0.8963   -0.3379    0.0107   -0.5547    0.2329    0.1050   -0.0145   -0.1062    0.4842   -0.1724   -0.8312    0.5913   -0.6003   -1.0808    0.5228   -0.2720   -0.3742    0.4020   -0.2697    0.4131    0.4167    0.7838   -0.4912    0.3054

DD = 1.0e+008 * Columns 1 through 7 0.0017        0         0         0         0         0         0         0    0.0267         0         0         0         0         0         0         0    0.0702         0         0         0         0         0         0         0    0.1154         0         0         0         0         0         0         0    0.2452         0         0         0         0         0         0         0    0.3962         0         0         0         0         0         0         0    0.5060         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 8 through 14 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    0.5581         0         0         0         0         0         0         0    0.7073         0         0         0         0         0         0         0    0.9847         0         0         0         0         0         0         0    1.4801         0         0         0         0         0         0         0    1.6623         0         0         0         0         0         0         0    1.7262         0         0         0         0         0         0         0    2.0577         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 15 through 21 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    2.1074         0         0         0         0         0         0         0    2.3728         0         0         0         0         0         0         0    2.7262         0         0         0         0         0         0         0    2.9417         0         0         0         0         0         0         0    3.0589         0         0         0         0         0         0         0    3.0752         0         0         0         0         0         0         0    3.6051         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 22 through 25 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    3.7647         0         0         0         0    3.8339         0         0         0         0    4.3103         0         0         0         0    4.7170

=Problem 6.3: Redoing 3.1 with zero displacement degree of freedoms.= On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
Plane truss as shown. A force, F, of 20000 N is applied as shown. The length of each element is 1 meter. Element 2 is vertical and the elements are spaced evenly around node 1. Each element has the same physical properties: Young's modulus of 206 GPa and cross sectional area of .0001 m. Additionally, node 2 experiences an initial displacement of 2 cm in the x direction and -1 cm in the y direction; node 3 experiences an initial displacement of -3 cm in the x direction and 5 cm in the y direction.

Find
Find the unknown displacements and forces at each global node as well as the member forces of each element using Matlab and verify using CALFEM. Plot the deformed and undeformed shape on the same plot.Perform the same tasks once again with zero prescribed displacements, and come compare the outcomes.

Using Matlab
Enter the number of global nodes. >> G=4;

Enter the x, y, and z coordinates (this program was designed solve 1D, 2D, and 3D problems, so the z coordinates must also be entered. As long as all nodes have the same z coordinate, the solution will match the 2D problem). >> x=[0 0 sqrt(3)/2 -sqrt(3)/2]; >> y=[0 1 -.5 -.5]; >> z=[0 0 0 0];

Enter the input force matrix. The first element of each row corresponds to the global degree of freedom; the second element corresponds to the force associated with that degree of freedom. Only the known forces are entered into this matrix. >> F_in=[1 20000/sqrt(2); 2 20000/sqrt(2)];

Enter the input displacement matrix. The first element of each row corresponds to the global degree of freedom; the second element corresponds to the displacement associated with that degree of freedom. Only the known displacements are entered into this matrix. If the force for a particular degree of freedom is unknown, then its displacement is known. >> Q_in=[3 0; 4 .02; 5 -.01; 6 0; 7 -.03; 8 .05; 9 0; 10 0; 11 0; 12 0];

Enter the number of elements. >> E=3;

Enter the global node corresponding to the first local node of each element. The column number corresponds to the element number of the given node. >> l1=[1 1 1];

Enter the global node corresponding to the second local node of each element. >> l2=[3 2 4];

Enter the Young's Modulus associated with each element. The column number corresponds to the element number of the given node. >> Modulus=10^9*[206 206 206];

Enter the Young's Modulus associated with each element. The column number corresponds to the element number of the given node. >> Area=10^-4*[1 1 1];

Run the displacement program. >> [k Q F Q_bar F_bar]=displacement(G, x, y, z, F_in, Q_in, E, l1, l2, Modulus, Area);

The stiffness matrix k: k = 1.0e+07 * 3.0900        0         0         0   -1.5450    0.8920   -1.5450   -0.8920            0    3.0900         0   -2.0600    0.8920   -0.5150   -0.8920   -0.5150            0         0         0         0         0         0         0         0            0   -2.0600         0    2.0600         0         0         0         0      -1.5450    0.8920         0         0    1.5450   -0.8920         0         0       0.8920   -0.5150         0         0   -0.8920    0.5150         0         0      -1.5450   -0.8920         0         0         0         0    1.5450    0.8920      -0.8920   -0.5150         0         0         0         0    0.8920    0.5150

The image to the right shows a plot of the deformed and undeformed truss. The displacement matrix Q: Q = 1.0000  -0.0290       2.0000    0.0108       3.0000         0       4.0000    0.0200       5.0000   -0.0100       6.0000         0       7.0000   -0.0300       8.0000    0.0500       9.0000         0      10.0000         0      11.0000         0      12.0000         0

This shows that the first and second degrees of freedom (x and y directions) for global node 1 experience a displacement of -2.9 cm and 1.08 cm respectively. The remaining displacements were given in the problem.

The Force matrix, F F = 1.0e+05 * 0.1414      0.1414            0            0      -4.2816            0      -3.6562       2.1109            0       3.5148       2.0293            0

This shows the forces at each node. Every 3 rows correspond to 1 node. Global node 1 (rows 1-3) shows the force that was given in the problem.

The internal spring force matrix F_bar F_bar = 1.0e+05 * 4.2219   4.2816    4.0586      -4.2219   -4.2816   -4.0586

Row 1 gives the member forces with respect to the Global nodes and row 2 gives the member forces with respect to the members.

Matlab Code
This code is equivalent to that used in R2.3 except that it uses the given spring constants rather than than the modulus and area to determine the stiffness matrix.

CALFEM Verification
First begin by constructing the Edof  matrix. Column 1 corresponds to the element number; columns 2 and 3 correspond to the global node that corresponds to the first and second local nodes respectively. >> Edof=[1 1 2 5 6 2 1 2 3 4           3 1 2 7 8];

Construct an empty stiffness matrix K that has side lengths equal to the number of degrees of freedom times the number of global nodes. For this problem, there are 4 global nodes and motion is constrained to 2 degrees of freedom. >> K=zeros(8,8);

Construct the force matrix F. The forces corresponding to global node 1 are known; the forces corresponding to global nodes 2, 3, and 4 are unknown. >> F=zeros(8,1); >> F(1)=20000/sqrt(2); >> F(2)=20000/sqrt(2);

Construct the displacement matrix Q. Global nodes 2, 3, and 4 and known displacements as shown. >> Q=[3 .02 4 -.01        5 -.03         6 .05         7 0         8 0];

The function bar2e generates the element stiffness matrices for a bar with 2 degrees of freedom. Its parameters are the x and y coordinates of the element and a vector containing the Young's modulus and cross sectional area. This information is given in the problem statement.

The Vector containing the Young's modulus and cross sectional area. >> E=206e9; >> A=1e-4; >> ep=[E A];

The x and y coordinates for each element. >> ex1=[0 sqrt(3)/2]; >> ey1=[0 -.5]; >> ex2=[0 0]; >> ey2=[0 1]; >> ex3=[0 -sqrt(3)/2]; >> ey3=[0 -.5];

The commands: >> Ke1=bar2e(ex1, ey1, ep); >> Ke2=bar2e(ex2, ey2, ep); >> Ke3=bar2e(ex3, ey3, ep);

Yield the following stiffness matrices: >> Ke1 = 1.0e+07 * 1.5450  -0.8920   -1.5450    0.8920          -0.8920    0.5150    0.8920   -0.5150          -1.5450    0.8920    1.5450   -0.8920           0.8920   -0.5150   -0.8920    0.5150      Ke2 = 0          0           0           0                0    20600000           0   -20600000                0           0           0           0                0   -20600000           0    20600000   >> Ke3 = 1.0e+07 * 1.5450   0.8920   -1.5450   -0.8920          0.8920    0.5150   -0.8920   -0.5150         -1.5450   -0.8920    1.5450    0.8920         -0.8920   -0.5150    0.8920    0.5150

The global stiffness matrix can be assembled with the element stiffness matrices using the following command. >> K=assem(Edof(1,:),K,Ke1); >> K=assem(Edof(2,:),K,Ke2); >> K=assem(Edof(3,:),K,Ke3);

This results in the following stiffness matrix. >> K = 1.0e+07 * 3.0900        0         0         0   -1.5450    0.8920   -1.5450   -0.8920               0    3.0900         0   -2.0600    0.8920   -0.5150   -0.8920   -0.5150               0         0         0         0         0         0         0         0               0   -2.0600         0    2.0600         0         0         0         0         -1.5450    0.8920         0         0    1.5450   -0.8920         0         0          0.8920   -0.5150         0         0   -0.8920    0.5150         0         0         -1.5450   -0.8920         0         0         0         0    1.5450    0.8920         -0.8920   -0.5150         0         0         0         0    0.8920    0.5150

This stiffness matrix is in accord with that computed in Matlab in the previous section.

The function solveq takes the stiffness matrix, known force matrix, and displacement matrix and returns the complete force and displacement matrices. >> [q r]=solveq(K,F,Q) q = -0.0290      0.0108       0.0200      -0.0100      -0.0300       0.0500            0            0   r = 1.0e+05 * 0.0000      0.0000            0      -4.2816      -3.6562       2.1109       3.5148       2.0293

The displacement and force matrices are in accord with those computed in Matlab in the previous section.

The function extract is used to evaluate the displacements of the local nodes for each beam, which is needed to determine the force in each beam. >> ed1=extract(Edof(1,:),q); >> ed2=extract(Edof(2,:),q); >> ed3=extract(Edof(3,:),q);

This results in the following displacements. The number following 'ed' corresponds to the element number. Each column corresponds to the respective degree of freedom for the beam. ed1 = -0.0290   0.0108   -0.0300    0.0500 ed2 = -0.0290   0.0108    0.0200   -0.0100 ed3 = -0.0290   0.0108         0         0

The function bar2s can be used to determine the force in each beam. Its input parameters are the x and y coordinates for the beam (ex1 and ey1), the physical properties of the beam (Young's modulus and cross sectional area - ep) and the displacement (ed). >> es1=bar2s(ex1, ey1,ep,ed1) >> es1=bar2s(ex2, ey2,ep,ed2) >> es1=bar2s(ex3, ey3,ep,ed3)

This results in the following spring forces. es1 = -4.2219e+05 es2 = -4.2816e+05 es3 = -4.0586e+05

This is in accord with the forces computed in Matlab in the previous section.

Using Matlab: With zero disp dof
The matlab is going to be the same exact format, except we will be using zeros for the prescribed displacement degree of freedoms. Therefore the only thing changed in the displacement inputs.

Enter the input displacement matrix. The first element of each row corresponds to the global degree of freedom; the second element corresponds to the displacement associated with that degree of freedom. Only the known displacements are entered into this matrix. If the force for a particular degree of freedom is unknown, then its displacement is known. >> Q_in=[3 0; 4 0; 5 0; 6 0; 7 0; 8 0; 9 0; 10 0; 11 0; 12 0]; We get the stiffnes matrix, k, which turns out to be the same. k = 1.0e+07 * 3.0900        0         0         0   -1.5450    0.8920   -1.5450   -0.8920         0    3.0900         0   -2.0600    0.8920   -0.5150   -0.8920   -0.5150         0         0         0         0         0         0         0         0         0   -2.0600         0    2.0600         0         0         0         0   -1.5450    0.8920         0         0    1.5450   -0.8920         0         0    0.8920   -0.5150         0         0   -0.8920    0.5150         0         0   -1.5450   -0.8920         0         0         0         0    1.5450    0.8920   -0.8920   -0.5150         0         0         0         0    0.8920    0.5150

We get the displacement matrix, Q

Q = 1.0000   0.0005    2.0000    0.0005    3.0000         0    4.0000         0    5.0000         0    6.0000         0    7.0000         0    8.0000         0    9.0000         0   10.0000         0   11.0000         0   12.0000         0

We get the force matrix

F = 1.0e+04 *

1.4142   1.4142         0         0   -0.9428         0   -0.2989    0.1725         0   -1.1154   -0.6440         0

We get the internal springs force, F_bar matrix

F_bar = 1.0e+04 * 0.3451   0.9428   -1.2879   -0.3451   -0.9428    1.2879

= Problem 6.4: Project 2.1 p.94 sec.2.6 KS 2008 using Matlab and CALFEM = On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
Space truss as shown that consists of 25 elements. Each element has a circular cross-section that initially has a diameter $$d=2\;in$$. At the nodes 1 and 2 a constant force $$F=60,000\;lb$$ is applied in the y-direction. The nodes 7, 8, 9 and 10 are fixed. The elements are made of steel with the following properties, Young's modulus $$E=3\times 10^7\; psi$$, Poisson's ratio $$\nu=0.3$$, yield stress $$\sigma_y=37,000\; psi$$ and mass density $$\rho=7.3\times10^-4\;\frac{lb*sec^2}{in^4}$$. A safety factor $$N=1.5$$ is used in the design of the truss. The diameter of the truss elements should be between 0.1 in and 2.5 in.

Find
1. Solve the initial truss structure problem using matlab. Provide plots of the deformed truss and a table of the stresses in the beams.

2. Minimize the structural weight by changing the cross-sectional diameter for each truss element. Identify the zero force members and provide a plot of the deformed shape under optimal design. Calculate the weights of the original truss and the optimal truss.

3. Verify the problem with CALFEM.

1. Original Truss Deformation Using Matlab
Enter the number of global nodes. >> G=10;

Enter the x, y, and z coordinates. >> x=[-1.5 1.5 -1.5 1.5 1.5 -1.5 -4 4 4 -4]; >> y=[0 0 1.5 1.5 -1.5 -1.5 4 4 -4 -4]; >> z=[8 8 4 4 4 4 0 0 0 0];

Enter the input force matrix. The first element of each row corresponds to the global degree of freedom; the second element corresponds to the force associated with that degree of freedom. Only the known forces are entered into this matrix. >> F_in=[1 0; 2 60000; 3 0; 4 0; 5 60000; 6 0; 7 0; 8 0; 9 0; 10 0; 11 0; 12 0; 13 0; 14 0; 15 0; 16 0; 17 0; 18 0];

Enter the input displacement matrix. The first element of each row corresponds to the global degree of freedom; the second element corresponds to the displacement associated with that degree of freedom. >> Q_in=[19 0; 20 0; 21 0; 22 0; 23 0; 24 0; 25 0; 26 0; 27 0; 28 0; 29 0; 30 0];

Enter the number of elements. >> E=25;

Enter the global node corresponding to the first local node of each element. The column number corresponds to the element number of the given node. >> l1=[1 1 2 1 2 2 2 1 1 3 4 3 5 3 6 4 5 4 3 5 6 6 3 4 5];

Enter the global node corresponding to the second local node of each element. >> l2=[2 4 3 5 6 4 5 3 6 6 5 4 6 10 7 9 8 7 8 10 9 10 7 8 9];

Enter the Young's Modulus associated with each element. The column number corresponds to the element number of the given node. >> Modulus=3e7*[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1];

Enter the Area associated with each element. The column number corresponds to the element number of the given node. >> Area=(pi/4)*2^2*[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1];

Enter the Mass Density associated with each element. The column number corresponds to the element number of the given node. >> Ro=7.3e-4*[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]; Run the truss program. >> [F_bar] = truss(G, x, y, z, F_in, Q_in, E, l1, l2, Modulus, Area, Ro)

The internal spring force matrix F_bar F_bar = 1.0e+04 * Columns 1 through 9 0.0000   3.5997    3.5997   -3.5997   -3.5997    5.5981   -5.5981    5.5981   -5.5981      -0.0000   -3.5997   -3.5997    3.5997    3.5997   -5.5981    5.5981   -5.5981    5.5981     Columns 10 through 18 -0.0000  -0.0000   -0.9053    0.9053    1.8114   -1.8114    1.8114   -1.8114    3.4748       0.0000    0.0000    0.9053   -0.9053   -1.8114    1.8114   -1.8114    1.8114   -3.4748     Columns 19 through 25 3.4748  -3.4748   -3.4748   -6.7822    6.7822    6.7822   -6.7822      -3.4748    3.4748    3.4748    6.7822   -6.7822   -6.7822    6.7822

Row 1 gives the member forces with respect to the Global nodes and row 2 gives the member forces with respect to the members.

The axial stresses are given by dividing the axial forces by the area. stress = 1.0e+04 * Columns 1 through 9 -0.0000  -1.1458   -1.1458    1.1458    1.1458   -1.7819    1.7819   -1.7819    1.7819     Columns 10 through 18 0.0000   0.0000    0.2882   -0.2882   -0.5766    0.5766   -0.5766    0.5766   -1.1061     Columns 19 through 25 -1.1061   1.1061    1.1061    2.1588   -2.1588   -2.1588    2.1588



Deformed (red) and undeformed (black) plot of the Truss. The shown deformation is scaled 10 times the actual deformation.

The weight, which is found by summing up the mass matrix and multiplying by the acceleration due to gravity and dividing by 3 is 117.17 lb.

Matlab Code
This code is equivalent to that used in R2.3 except that it uses the given spring constants rather than than the modulus and area to determine the stiffness matrix.

2. Optimal Truss Deformation Using Matlab
In order to minimize the diameter of the elements the stress in each element was used with the yield stress, a factor of safety, and Poisson's ratio to construct a formula for the new diameter of the beams.

New Diameter Calculations Stress in a beam. $$\sigma =\frac{P}{A}$$ Yield stress scaled with a factor of safety. $$\frac{\sigma_{y}}{1.5} =\frac{P}{A}$$ Solving for deformed diameter. $$D=\sqrt{\frac{4}{\pi}\frac{P}{\sigma_{y}/N}}$$ Poisson's ratio. $$\nu = -\frac{d\varepsilon_\mathrm{trans}}{d\varepsilon_\mathrm{axial}} = -\frac{d\varepsilon_\mathrm{y}}{d\varepsilon_\mathrm{x}}= -\frac{d\varepsilon_\mathrm{z}}{d\varepsilon_\mathrm{x}}$$ Poisson's ratio solved for a circular beam. The variable "d" is the new undeformed diameter, "l" is the deformed length and "L" is the undeformed length. $$\nu=-\frac{\frac{D-d}{d}}{\frac{l-L}{L}}$$ Poisson's ratio solved for a the new diameter. $$d=\frac{D}{\frac{-0.3(l-L)}{L}+1}$$ Matlab Code that calculates the new diameters. Table of the stress in each member with the new diameters. Conclusion The zero force members are 1, 10 and 11. The new diameters allow for the an approximate stress of 24666 psi under the load given in the problem statement. From the figure of the deformed truss it can be seen that this truss deforms much more than the original because of the smaller areas. However, this truss weighs only 54.35 pounds while the original weighs 117.17 pounds.

3. CALFEM Verification
First begin by constructing the Edof matrix. Column 1 corresponds to the element number. It is also noted that for this problem, code formatting from our teams previous reports were used.

>> Edof=[1 1 2 3 4 2 1 2 5 6          3 3 4 7 8           4 1 2 5 6           5 3 4 9 10           6 3 4 5 6           7 3 4 11 12           8 1 2 5 6           9 1 2 9 10           10 9 10 11 12           11 13 14 15 16           12 5 6 3 4           13 5 6 9 10           14 17 18 7 8           15 9 10 5 6             16 19 20 5 6           17 17 18 7 8           18 9 10 3 4           19 17 18 7 8           20 3 4 15 16           21 15 16 3 4           22 17 18 7 8           23 9 10 3 4           24 17 18 7 8           25 1 2 11 12];

Construct an empty stiffness matrix K that has side lengths equal to the number of degrees of freedom times the number of global nodes. For this problem, there are 10 global nodes and motion has 3 degrees of freedom.

>> K=zeros(30,30);

Constructing the force matrix and constructing the displacement matrix Q. Global nodes and known displacements as shown.

>> Q=[1 .3 2 .5       3 .02        4 -.01        5 .3        6 .57        7 0        8 0        9 0        10 0        11 -.03        12 .57        13 0        14 1        15 .86        16 -.01        17 -.03        18 .86        19 0        20 1        21 .64        22 -.16        23 -.31        24 .1        25 0];

The function bar2e generates the element stiffness matrices for a bar with 3 degrees of freedom.

The Vector containing the Young's modulus and cross sectional area are found and and the x and y coordinates for each element are shown below:

>> ex1=[0 0]; >> ey1=[0 3]; >> ez1=[0 0]; >> ex2=[0 3]; >> ey2=[0 1]; >> ez2=[3 1]; >> ex3=[0 8]; >> ey3=[6 3; >> ez3=[.5 1]; >> ex4=[6 0]; >> ey4=[7 3]; >> ez4=[.9 .4]; >> ex5=[8 3]; >> ey5=[4 1]; >> ez5=[0 4]; >> ex6=[4 8]; >> ey6=[3 3; >> ez6=[.5 4]; >> ex7=[4 0]; >> ey7=[5 3]; >> ez7=[.3 1]; >> ex8=[6 3]; >> ey8=[0 1]; >> ez8=[5 .5]; >> ex9=[9 8]; >> ey9=[9 3]; >> ez9=[4 .5]; >> ex10=[0 0]; >> ey10=[6 3]; >> ez10=[0 8]; >> ex11=[6 3]; >> ey11=[8 1]; >> ez11=[8 1]; >> ex12=[9 8]; >> ey12=[8 9]; >> ez12=[8 1]; >> ex13=[0 0]; >> ey13=[5 3]; >> ez13=[05 1]; >> ex14=[9 3]; >> ey14=[8 1]; >> ez14=[4 5]; >> ex15=[0 9]; >> ey15=[8 3; >> ez15=[8 4]; >> ex16=[6 0]; >> ey16=[8 3]; >> ez16=[4 4]; >> ex17=[0 8]; >> ey17=[0 1]; >> ez17=[.5 1]; >> ex18=[6 8]; >> ey18=[7 3]; >> ez18=[0 .3]; >> ex19=[6 8]; >> ey19=[8 3]; >> ez19=[5 8]; >> ex20=[3 3]; >> ey20=[5 1]; >> ez20=[0 0]; >> ex21=[0 8]; >> ey21=[0 3; >> ez21=[0 0]; >> ex22=[8 4]; >> ey22=[0 3]; >> ez22=[0 0]; >> ex23=[5 3]; >> ey23=[8 1]; >> ez23=[0 4]; >> ex24=[0 8]; >> ey24=[8 8]; >> ez24=[0 1]; >> ex25=[0 0]; >> ey25=[0 8]; >> ez25=[8 8];

The following yields 25 stiffness matrices.:

>> Ke1=bar2e(ex1, ey1, ez1, ep); >> Ke2=bar2e(ex2, ey2, ez2 ep); >> Ke3=bar2e(ex3, ey3, ez3 ep); >> Ke4=bar2e(ex4, ey4, ez4, ep); >> Ke5=bar2e(ex5, ey5, ez5, ep); >> Ke6=bar2e(ex6, ey6, ez6, ep); >> Ke7=bar2e(ex7, ey7, ez7, ep); >> Ke8=bar2e(ex8, ey8, ez8, ep); >> Ke9=bar2e(ex9, ey9, ez9, ep); >> Ke10=bar2e(ex10, ey10, ez10, ep); >> Ke11=bar2e(ex11, ey11, ez11, ep); >> Ke12=bar2e(ex12, ey12, ez12, ep); >> Ke13=bar2e(ex13, ey13, ez13, ep); >> Ke14=bar2e(ex14, ey14, ez14 ep); >> Ke15=bar2e(ex15, ey15, ez15 ep); >> Ke16=bar2e(ex16, ey16, ez16, ep); >> Ke17=bar2e(ex17, ey17, ez17, ep); >> Ke18=bar2e(ex18, ey18, ez18, ep); >> Ke19=bar2e(ex19, ey19, ez19, ep); >> Ke20=bar2e(ex20, ey20, ez20, ep); >> Ke21=bar2e(ex21, ey21, ez21, ep); >> Ke22=bar2e(ex22, ey22, ez22, ep); >> Ke23=bar2e(ex23, ey23, ez23, ep); >> Ke24=bar2e(ex24, ey24, ez24, ep); >> Ke25=bar2e(ex25, ey25, ez25, ep);

The global stiffness matrix can be assembled with the element stiffness matrices. This results in a stiffness matrix that is in accord with that computed in Matlab in the previous section. The function solveq takes the stiffness matrix, known force matrix, and displacement matrix and returns the complete force and displacement matrices.

>> [q r]=solveq(K,F,Q);

The displacement and force matrices are in accord with those computed in Matlab in the previous section. The function extract is used to evaluate the displacements of the local nodes for each beam, which is needed to determine the force in each beam.

>> ed1=extract(Edof(1,:),q); >> ed2=extract(Edof(2,:),q); >> ed3=extract(Edof(3,:),q); >> ed4=extract(Edof(4,:),q); >> ed5=extract(Edof(5,:),q); >> ed6=extract(Edof(6,:),q); >> ed7=extract(Edof(7,:),q); >> ed8=extract(Edof(8,:),q); >> ed9=extract(Edof(9,:),q); >> ed10=extract(Edof(10,:),q);

This results in the displacements. The function bar2s can be used to determine the force in each beam. Its input parameters are the x and y coordinates for the beam (ex1 and ey1), the physical properties of the beam (Young's modulus and cross sectional area - ep) and the displacement (ed).

>> es1=bar2s(ex1, ey1,ez1 ep,ed1) >> es1=bar2s(ex2, ey2,ez2 ep,ed2) >> es1=bar2s(ex3, ey3,ez3 ep,ed3) >> es1=bar2s(ex1, ey1,ez4 ep,ed1) >> es1=bar2s(ex2, ey2,ez5 ep,ed2) >> es1=bar2s(ex3, ey3,ez6 ep,ed3) >> es1=bar2s(ex1, ey1,ez7 ep,ed1) >> es1=bar2s(ex2, ey2,ez8 ep,ed2) >> es1=bar2s(ex3, ey3,ez9 ep,ed3) >> es1=bar2s(ex3, ey3,ez10 ep,ed3)

This results in the following spring forces.

F_bar = 1.0e+04 * Columns 1 through 9 0.0000   3.5997    3.5997   -3.5997   -3.5997    5.5981   -5.5981    5.5981   -5.5981     -0.0000   -3.5997   -3.5997    3.5997    3.5997   -5.5981    5.5981   -5.5981    5.5981    Columns 10 through 18 -0.0000  -0.0000   -0.9053    0.9053    1.8114   -1.8114    1.8114   -1.8114    3.4748      0.0000    0.0000    0.9053   -0.9053   -1.8114    1.8114   -1.8114    1.8114   -3.4748    Columns 19 through 25 3.4748  -3.4748   -3.4748   -6.7822    6.7822    6.7822   -6.7822     -3.4748    3.4748    3.4748    6.7822   -6.7822   -6.7822    6.7822

This is in accord with the forces computed in Matlab in the previous section. The axial stresses are given by dividing the axial forces by the area.

stress = 1.0e+04 * Columns 1 through 9 -0.0000  -1.1458   -1.1458    1.1458    1.1458   -1.7819    1.7819   -1.7819    1.7819    Columns 10 through 18 0.0000   0.0000    0.2882   -0.2882   -0.5766    0.5766   -0.5766    0.5766   -1.1061    Columns 19 through 25 -1.1061   1.1061    1.1061    2.1588   -2.1588   -2.1588    2.1588

= Problem 6.5: Vibration Analysis of Space Truss from 6.4 = On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
Create animations of the lowest 3 modes. Given are the animations from three orthogonal views as well as an isometric view.