User:Eml4507.s13.team3.steiner/Team Negative Damping (3): Report 4

=Problem 4.1= On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
Spring-damper-body arrangement as shown. Two separate forces applied to masses.

$$ M= \begin{bmatrix} m_{1} & 0\\ 0 & m_{2}\\ \end{bmatrix} $$ $$ d= \begin{bmatrix} d_{1}\\ d_{2}\\ \end{bmatrix} $$ $$ C= \begin{bmatrix} C_{1}+C_{2} & -C_{2}\\ -C_{2} & C_{2}+C_{3}\\ \end{bmatrix}$$ $$ K= \begin{bmatrix} (k_{1}+k_{2}) & -k_{2}\\ -k_{2} & (k_{2}+k_{3})\\ \end{bmatrix}$$

$$ k_1=1, k_2=3, k_3=3 $$

$$

K= \begin{bmatrix} 3 & -2\\ -2 & 5 \end{bmatrix}$$

$$[K-$$γ$$I]x=$$ $$ \begin{bmatrix} 3 & -2\\ -2 & 5 \end{bmatrix}$$ $$-$$γ $$ \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$$ $$)$$ $$ \begin{Bmatrix} x_1\\ x_2 \end{Bmatrix}$$ $$=$$ $$ \begin{Bmatrix} 0\\ 0 \end{Bmatrix}$$ $$det \begin{Bmatrix} 3-\gamma & -2\\ -2 & 5-\gamma \end{Bmatrix} =\gamma^2-8\gamma+11=0$$

Find
Find the eigenvector $$ x_{2} $$ corresponding to the eigenvalue $$\gamma_2$$ for the spring-mass-damper system on p.53-113. Plot and comment on this mode shape. Verify that the eigenvectors are orthogonal to each other

Solution
Eigenvalues are found $$\gamma_1=4+\sqrt{5}>0$$ $$\gamma_2=4-\sqrt{5}>0$$ We find the eigenvectors from $$\gamma_2$$ $$\gamma_2=4+\sqrt{5}$$ $$[K-\gamma_2I]x=$$ $$ \begin{bmatrix} -1-\sqrt{5} & -2\\ -2 & 1-\sqrt{5} \end{bmatrix}$$ $$ \begin{Bmatrix} x_1\\ x_2 \end{Bmatrix}$$ $$=$$ $$ \begin{Bmatrix} 0\\ 0 \end{Bmatrix}$$ Set $$x_2=1$$ $$(-1-\sqrt{5})x_1-(2)x_2=0$$ $$x_1=\frac{2}{-1-\sqrt{5}}$$

Eigenvectors are orthogonal to each other: EDU>> x= [-.8507;-.5257]; EDU>> y= [-.5257;.8507]; EDU>> transpose(y)*x

ans = 0 =Problem 4.2= On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
Use same given values as in problem 4.1

Find
Find the eigenvectors for $$\gamma_1$$ and $$\gamma_2$$ when setting $$x_1=1$$

Solution
We find the eigenvectors from $$\gamma_1$$ $$\gamma_1=4-\sqrt{5}$$ $$[K-\gamma_1I]x=$$ $$ \begin{bmatrix} -1+\sqrt{5} & -2\\ -2 & 1+\sqrt{5} \end{bmatrix}$$ $$ \begin{Bmatrix} x_1\\ x_2 \end{Bmatrix}$$ $$=$$ $$ \begin{Bmatrix} 0\\ 0 \end{Bmatrix}$$ Set $$x_1=1$$ $$(-1-\sqrt{5})x_1-(2)x_2=0$$ $$x_2=\frac{-1+\sqrt{5}}{2}$$ We find the eigenvectors from $$\gamma_2$$ $$\gamma_2=4+\sqrt{5}$$ $$[K-\gamma_2I]x=$$ $$ \begin{bmatrix} -1-\sqrt{5} & -2\\ -2 & 1-\sqrt{5} \end{bmatrix}$$ $$ \begin{Bmatrix} x_1\\ x_2 \end{Bmatrix}$$ $$=$$ $$ \begin{Bmatrix} 0\\ 0 \end{Bmatrix}$$ Set $$x_1=1$$ $$(-1-\sqrt{5})x_1-(2)x_2=0$$ $$x_2=\frac{-1-\sqrt{5}}{2}$$

=Problem 4.3 (fead.f08 p.11-3 (Method 1: Square Root Sum of Squares), p.14-3 (Method 2: Transformation Matrix))= On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Method 1:

 * $$P_1^{(e)} = \sqrt{(f_1^{(e)})^2 + (f_2^{(e)})^2}$$


 * $$P_2^{(e)} = \sqrt{(f_3^{(e)})^2 + (f_4^{(e)})^2}$$

Method 2:

 * $$\mathbf P^{(e)} = \mathbf T^{(e)} \mathbf f^{(e)}$$


 * $$\begin{Bmatrix} P_1^{(e)} \\ P_2^{(e)} \end{Bmatrix} = \begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0 \\0 & 0 & l^{(e)} & m^{(e)} \end{bmatrix} \begin{Bmatrix} f_1^{(e)} \\ f_2^{(e)} \\ f_3^{(e)} \\ f_4^{(e)} \end{Bmatrix}$$

Find

 * Discuss computational efficiency of each method.
 * Reconcile analytically using both algebra and geometry

Given

 * $$P_1^{(e)} = \sqrt{(f_1^{(e)})^2 + (f_2^{(e)})^2}$$


 * $$P_2^{(e)} = \sqrt{(f_3^{(e)})^2 + (f_4^{(e)})^2}$$

Discussion
The first method uses the Pythagorean Theorem, which is also a distance formula, to find the axial member forces from the nodal forces. This method only requires the two nodal forces on a node to find an axial member force. To use this method, the nodal forces are defined, and then put into the distance formula. Additional axial forces are found by defining additional nodal forces. The formula must be repeated each time to give each axial member force.

Given
$$\mathbf P^{(e)} = \mathbf T^{(e)} \mathbf f^{(e)}$$

$$\begin{Bmatrix} P_1^{(e)} \\ P_2^{(e)} \end{Bmatrix} = \begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0 \\0 & 0 & l^{(e)} & m^{(e)} \end{bmatrix} \begin{Bmatrix} f_1^{(e)} \\ f_2^{(e)} \\ f_3^{(e)} \\ f_4^{(e)} \end{Bmatrix}$$

Discussion
The second method uses the transformation matrix to add the projections of the nodal forces along the element. This method requires the two nodal forces and an angle. However, the angle can be applied to both ends of the node. To use this method, the transformation matrix, $$\mathbf T^{(e)}$$ is created, with $$l^{(e)} = \cos{\theta}$$ and $$m^{(e)} = \sin{\theta}$$, as well as the nodal force matrix $$\mathbf f^{(e)}$$.

The matrix multiplication performs the following operations:


 * $$P_1 = l^{(e)}f_1^{(e)} + m^{(e)}f_2^{(e)} + 0 + 0 $$
 * $$P_1 = f_1^{(e)} \cos\theta + f_2^{(e)} \sin\theta$$


 * $$P_2 = 0 + 0 + l^{(e)}f_3^{(e)} + m^{(e)}f_4^{(e)} $$
 * $$P_2 = f_3^{(e)} \cos\theta + f_4^{(e)} \sin\theta$$

To find additional axial member forces, new nodal forces and angles can be defined, and the transformation can be expanded. The matrix multiplication only has to be performed once for each element. The multiplication will give a matrix with all the axial forces.

Reconciliation using algebra and geometry


The Pythagorean Theorem is used for the first method. To reconcile this method with the second method, geometry is used to define the nodal forces in terms of $$P_1$$.


 * $$f_1^{(e)} = P_1^{(e)} \cos\theta$$
 * $$f_2^{(e)} = P_1^{(e)} \sin\theta$$

These nodal force definitions are substituted into the Pythagorean Theorem, and the equation is simplified.


 * $$P_1^{(e)} = \sqrt{(f_1^{(e)})^2 + (f_2^{(e)})^2}$$


 * $$P_1^{(e)} = \sqrt{(P_1^{(e)} \cos\theta)^2 + (P_1^{(e)} \sin\theta)^2}$$


 * $$P_1^{(e)} = \sqrt{(P_1^{(e)})^2(\cos^2 \theta + \sin^2 \theta)}$$


 * $$P_1^{(e)} = \sqrt{(P_1^{(e)})^2(1)}$$


 * $$P_1^{(e)} = \sqrt{(P_1^{(e)})^2}$$


 * $$P_1^{(e)} = P_1^{(e)}$$

= Problem 4.4: Mode Shapes for Project 2.3, p. 97 Kim and Sankar using Matlab and CALFEM =

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: Three Lowest Eigen Pairs
Assemble a lumped mass matrix and find the three lowest eigen pairs $$ \left ( \omega _{j},\phi _{j} \right ) $$ for $$ j = 1,2,3 $$. Then plot and animate the three lowest eigen pairs. Finally verify the results with CALFEM.

Solution
The mass matrix was computed from the local element mass matrices, which had the form: $$\mathbf{M^{e}}=\frac{\rho Al}{2}\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$$

The local element matrices were used to form the global mass matrix using the code shown below:

This is the resulting mass matrix

m = Columns 1 through 10 0.1280   0.0530         0         0         0         0         0         0         0         0      0.0530    0.1280         0         0         0         0         0         0         0         0           0         0    0.0750         0         0         0         0         0         0         0           0         0         0    0.0750         0         0         0         0         0         0           0         0         0         0    0.2030    0.0530         0         0         0         0           0         0         0         0    0.0530    0.1280         0         0         0         0           0         0         0         0         0         0    0.2030    0.0530         0         0           0         0         0         0         0         0    0.0530    0.1280         0         0           0         0         0         0         0         0         0         0    0.2030    0.0530           0         0         0         0         0         0         0         0    0.0530    0.1280           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         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      0.2030    0.0530         0         0         0         0         0         0         0         0      0.0530    0.1280         0         0         0         0         0         0         0         0           0         0    0.2030    0.0530         0         0         0         0         0         0           0         0    0.0530    0.1280         0         0         0         0         0         0           0         0         0         0    0.2030    0.0530         0         0         0         0           0         0         0         0    0.0530    0.1280         0         0         0         0           0         0         0         0         0         0    0.2030    0.0530         0         0           0         0         0         0         0         0    0.0530    0.1280         0         0           0         0         0         0         0         0         0         0    0.2030    0.0530           0         0         0         0         0         0         0         0    0.0530    0.1280           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         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      0.2030    0.0530         0         0         0         0         0         0      0.0530    0.1280         0         0         0         0         0         0           0         0    0.2030    0.0530         0         0         0         0           0         0    0.0530    0.1280         0         0         0         0           0         0         0         0    0.0750         0         0         0           0         0         0         0         0    0.0750         0         0           0         0         0         0         0         0    0.1280    0.0530           0         0         0         0         0         0    0.0530    0.1280

and reduced mass matrix

m_red = Columns 1 through 10 0.1280        0         0         0         0         0         0         0         0         0           0    0.2030    0.0530         0         0         0         0         0         0         0           0    0.0530    0.1280         0         0         0         0         0         0         0           0         0         0    0.2030    0.0530         0         0         0         0         0           0         0         0    0.0530    0.1280         0         0         0         0         0           0         0         0         0         0    0.2030    0.0530         0         0         0           0         0         0         0         0    0.0530    0.1280         0         0         0           0         0         0         0         0         0         0    0.2030    0.0530         0           0         0         0         0         0         0         0    0.0530    0.1280         0           0         0         0         0         0         0         0         0         0    0.2030           0         0         0         0         0         0         0         0         0    0.0530           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         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      0.0530         0         0         0         0         0         0         0         0         0      0.1280         0         0         0         0         0         0         0         0         0           0    0.2030    0.0530         0         0         0         0         0         0         0           0    0.0530    0.1280         0         0         0         0         0         0         0           0         0         0    0.2030    0.0530         0         0         0         0         0           0         0         0    0.0530    0.1280         0         0         0         0         0           0         0         0         0         0    0.2030    0.0530         0         0         0           0         0         0         0         0    0.0530    0.1280         0         0         0           0         0         0         0         0         0         0    0.2030    0.0530         0           0         0         0         0         0         0         0    0.0530    0.1280         0           0         0         0         0         0         0         0         0         0    0.2030           0         0         0         0         0         0         0         0         0    0.0530           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         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      0.0530         0         0         0         0      0.1280         0         0         0         0           0    0.0750         0         0         0           0         0    0.0750         0         0           0         0         0    0.1280    0.0530           0         0         0    0.0530    0.1280

k_red was determined using the code presented in Problem 3.1 from the inputs specified in Problem 3.6. It is given here

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

The eigenpairs were computed using the following code [v,d]=eig(k_red,m_red);

The eigenvalues are

1.0e+08 * 0.0041     0.0693      0.1266      0.3046      0.6578      0.9839      1.1121      1.2959      1.5555      1.9351      2.8228      3.0685      3.5869      4.3452      4.3790      4.8092      5.5736      6.8144      7.0151      7.1125      7.2195      7.5625      8.0215      8.2804      8.7946

and the eigenvectors are

v = Columns 1 through 10 0.0124   0.1245    0.0275   -0.2806   -0.3702   -0.0306    0.4167    0.0952   -0.2672   -0.2172      0.0461    0.0015    0.2846    0.0774    0.2247    0.4536    0.0657   -0.7231   -0.0394    0.3204      0.1141    0.6740   -0.0697   -1.1568   -1.0461   -0.5229    0.6353    0.1969   -0.3137    0.3679     -0.0543   -0.0997    0.1981    0.0456   -0.1082    0.5797    0.3052   -0.0470   -0.0368   -0.9208      0.1018    0.5668   -0.0965   -1.0270   -1.0447   -0.6641    0.7866    0.2775   -0.5347    0.5459      0.0800   -0.0937    0.5240    0.1947    0.2296    0.4494    0.4088   -0.8865   -0.3820    0.7000      0.3022    1.0713   -0.1024   -0.8397    0.4214   -0.0024   -1.1137   -0.2888    0.7612   -0.2194     -0.0961   -0.0802    0.4091   -0.1153   -0.3404    0.8969    0.5032   -0.0663   -0.0140   -0.9801      0.2905    1.0080   -0.1248   -0.9592    0.1959   -0.2087   -1.0306   -0.5434    0.8437    0.1126      0.1024   -0.2325    0.7083    0.1083    0.0033    0.0394    0.3518   -0.8718   -0.1083    0.7065      0.5500    1.0656   -0.1350    0.3516    0.9674   -0.1051    0.3541    0.7815   -0.9445   -0.2754     -0.1258    0.0207    0.6156   -0.2065   -0.2174    0.9770    0.1084    0.0493    0.2314   -0.4689      0.5396    1.0582   -0.1465    0.1509    1.1370   -0.1430    0.0925    0.4191   -1.1044   -0.1378      0.1146   -0.3549    0.8369   -0.1696    0.0888   -0.4493   -0.0542   -0.4832   -0.2493    0.1155      0.8318    0.5992   -0.2122    1.0003   -0.5379   -0.3348    0.5723   -0.1144    1.1249   -0.2918     -0.1444    0.1449    0.7944   -0.0585   -0.0313    0.6309   -0.0560    0.6206    0.0882    0.2530      0.8235    0.6361   -0.2103    0.9548   -0.1698   -0.4943    0.8978   -0.2779    0.9831   -0.2474      0.1192   -0.4199    0.9137   -0.4011    0.4366   -0.9727    0.0356   -0.0097    0.0361   -0.6587      1.1248   -0.1798   -0.3308    0.3177   -0.8924    0.4526   -0.9517   -0.1720   -0.8896   -0.1491     -0.1537    0.2375    0.9199    0.2085   -0.2292    0.0555   -0.2932    0.9125    0.3691    0.6025      1.1195   -0.1317   -0.3193    0.4430   -0.9030    0.1686   -0.7737   -0.6050   -0.8189   -0.3915      0.1193   -0.4265    0.9405   -0.4307    0.5124   -1.2492    0.0475   -0.0136    0.0556   -1.1667      1.4112   -0.9972   -0.4412   -0.9017    0.6631    0.5821    0.6238    0.5108    0.6005    0.4705     -0.1564    0.2726    0.9724    0.3520   -0.4823   -0.2231   -0.7553    0.9289    0.0344    0.4993      1.4099   -0.9816   -0.4286   -0.8400    0.5649    0.4532    0.4677    0.3619    0.3903    0.2657    Columns 11 through 20 -0.5078   1.5302   -1.7136   -0.0841    0.5844    1.0718   -0.3703   -0.1987   -0.2072   -0.0823     -1.0309   -0.6706   -0.2388    0.0643    1.0425    0.1980    0.7164    0.8024    0.0773   -0.0243      0.3245   -0.6684    0.5244   -0.0017   -0.0280    0.2747   -0.3516   -0.6803   -1.2665   -0.3467     -0.6828    0.8394    0.4075    0.3460   -0.0222   -1.0380    0.6555    0.2757   -0.4944    0.1104      0.2959   -0.0822   -0.2903   -0.2709   -0.5208   -0.4585    0.1690    0.4345    1.2803    0.2103     -0.5748   -0.2905   -0.4836   -0.2680   -0.4897   -0.4685   -0.2844   -0.7840    0.4582    0.0313      0.2687    0.5390   -0.2658   -0.2454   -0.5184   -0.4708    0.3183    0.1363   -0.9149   -1.1976     -0.2817   -0.1232    0.7376    0.0198   -0.0210    0.4074   -0.6418   -0.7426    0.0460   -0.3531     -0.0295    0.2803    0.1943    0.6865    0.5081    0.2754   -0.1335    0.2873    0.9697    1.1633      0.4840    0.6042    0.2978    0.6157   -0.5413    0.1157   -0.6245    0.3937   -0.1670    0.4311     -0.0485   -0.0171    0.3529    0.6577    0.5215    0.1659   -0.1003    0.2537    0.5618   -1.5041      0.7567   -0.4634   -0.4261   -0.6058   -0.5043    0.5230   -0.0450    0.8058    0.0872   -0.1858     -0.3133    0.0478    0.1918   -0.8609    0.0241   -0.0993    0.1426   -0.4853   -0.3481    1.4953      0.6323    0.7177    0.7222   -0.5801    0.4901    0.3638    0.7189    0.0142   -0.2883    0.0435     -0.2888   -0.2124   -0.0988   -0.8617    0.0082   -0.0849   -0.1909    0.1455    1.0379   -0.5843      0.8997   -0.1956   -0.5862    0.6300    0.5273   -0.3356    0.6006   -0.1219    0.2136   -0.0304     -0.1814   -0.1782   -0.1631    0.6301   -0.5546    0.2502    0.1612   -0.2257   -0.9995    0.7163     -0.0985   -0.2723   -0.1631    0.0078    0.0010    0.0678    0.3719   -0.5569    0.1012   -0.1177      0.0292   -0.2140   -0.2466    0.6600   -0.5405    0.3576    0.5869   -0.5161    0.4614    0.0168     -0.3227    0.1851    0.1326   -0.2419    0.5362   -0.5323   -0.4101   -0.6507    0.1255    0.0680      0.4449   -0.0594   -0.1079   -0.2174    0.5491   -0.4802   -0.7049    0.7399   -0.6131    0.0882     -0.2700   -0.8795   -0.8455    0.3483    0.0670   -0.8258   -1.4636    1.0445   -0.1750    0.1961      0.1616    0.1654    0.0552   -0.2425    0.5519   -0.4747   -0.4342   -0.3920   -0.0304    0.1236     -1.2684    0.1147    0.4116    0.0241   -1.0601    0.7411    0.0157    0.6942   -0.2519    0.0185      0.0590    0.0512    0.0107   -0.0054    0.0081    0.0390    0.1103    0.2090    0.0176   -0.0742    Columns 21 through 25 -0.1157  -0.0544    0.0130    0.0134   -0.0006      0.2932    0.5471   -0.3595   -0.3521    0.2102     -1.0468   -0.3696    0.5501    0.3305   -0.2133     -0.6858   -0.0262    0.5530    0.2414   -0.2106      1.1504    0.2650   -0.6165   -0.3109    0.2143      0.1306   -0.7619    0.4050    0.5821   -0.4060      0.1850    0.5213   -0.7301   -0.5382    0.4133      0.5279   -0.1468   -0.8363   -0.3546    0.4086     -0.2383   -0.2961    0.8303    0.4858   -0.4146     -0.5791    0.5916   -0.0497   -0.6315    0.5746      0.4421   -0.1236    0.4237    0.5305   -0.5865      0.0628    0.5458    0.7312    0.2517   -0.5818     -0.4252   -0.1644   -0.5693   -0.4317    0.5889      0.4434   -0.3193   -0.5533    0.5341   -0.7038     -1.2562   -0.1247    0.1364   -0.2642    0.7231     -0.4297   -0.8580   -0.3753    0.0780    0.7202      1.1598    0.4436    0.0449    0.1230   -0.7271      0.4008    0.2126    1.0950   -0.3956    0.7838     -0.8362    0.6572   -0.9398    0.2364   -0.8018     -0.0582    0.8446    0.0454   -0.5444   -0.8168      0.8241   -0.7851    0.6051    0.1056    0.8146     -0.6419   -0.3030   -1.3605    0.4583   -0.8008      0.0896    1.3583   -0.2151    1.8897    0.9248      0.1808   -0.2208   -0.1505    1.1893    0.8741     -0.0560   -0.9529    0.1731   -1.6310   -0.9052

Therefore, the 3 lowest eigenpairs are as follows:

$$ \omega_{1}=4.1e5; \phi_{1}=\begin{bmatrix}0.0124\\0.0461\\0.1141\\-0.0543\\0.1018\\0.0800\\0.3022\\-0.0961\\0.2905\\0.1024\\0.5500\\-0.1258\\0.5396\\0.1146\\0.8318\\-0.1444\\0.8235\\0.1192\\1.1248\\-0.1537\\1.1195\\0.1193\\1.4112\\-0.1564\\1.4099\end{bmatrix};

\omega_{2}=6.93e6; \phi_{2}=\begin{bmatrix}0.1245\\0.0015\\0.6740\\-0.0997\\0.5668\\-0.0937\\1.0713\\-0.0802\\1.0080\\-0.2325\\1.0656\\0.0207\\1.0582\\-0.3549\\0.5992\\0.1449\\0.6361\\-0.4199\\0.1798\\ 0.2375\\-0.1317\\-0.4265\\-0.9972\\0.2726\\-0.9816\end{bmatrix}

\omega_{3}=1.266e7; \phi_{3}=\begin{bmatrix}0.0275\\0.2846\\-0.0697\\0.1981\\-0.0965\\0.5240\\-0.1024\\0.4091\\-0.1248\\0.7083\\-0.1350\\0.6156\\-0.1465\\0.8369\\-0.2122\\0.7944\\-0.2103\\0.9137\\-0.3308\\0.9199\\-0.3193\\0.9405\\-0.4412\\0.9724\\-0.4286\end{bmatrix} $$

Plotted Eigen Modes of the Plane Truss
Plotted below are the three eigen modes found with the matlab program. They are plotted with undeformed truss system shown with dotted lines to compare the deformations from the different eigen modes.

CALFEM Verification
First, provide material data - modulus, cross sectional area, and the density.

>> E=100e9; A=.0001; I=0; rho=5000; >> ep=[E A I rho*A];

Construct the Edof  matrix. Column 1 corresponds to the element number; columns 2 though 5 correspond to the degrees of freedom that correspond to the first and second local nodes respectively. Three degrees of freedom as given for each node - translation in the x and y directions and a moment. >> Edof=[ 1    1     2     3     7     8     9 2    7     8     9    13    14    15             3    13    14    15    19    20    21             4    19    20    21    25    26    27             5    25    26    27    31    32    33             6    31    32    33    37    38    39             7     4     5     6    10    11    12             8    10    11    12    16    17    18             9    16    17    18    22    23    24            10    22    23    24    28    29    30            11    28    29    30    34    35    36            12    34    35    36    40    41    42            13     1     2     3     4     5     6            14     7     8     9    10    11    12            15    13    14    15    16    17    18            16    19    20    21    22    23    24            17    25    26    27    28    29    30            18    31    32    33    34    35    36            19    37    38    39    40    41    42            20     1     2     3    10    11    12            21     7     8     9    16    17    18            22    13    14    15    22    23    24            23    19    20    21    28    29    30            24    25    26    27    34    35    36            25    31    32    33    40    41    42];

The construct the coordinate matrix. >> Coord = [    0         0 0   0.3000              0.3000         0              0.3000    0.3000              0.6000         0              0.6000    0.3000              0.9000         0              0.9000    0.3000              1.2000         0              1.2000    0.3000              1.5000         0              1.5000    0.3000              1.8000         0              1.8000    0.3000];

Construct the list of degrees of freedom. This describes the degrees of freedom assigned to each node (which is described by the row number). >>Dof =[ 1    2     3 4    5     6            7     8     9           10    11    12           13    14    15           16    17    18           19    20    21           22    23    24           25    26    27           28    29    30           31    32    33           34    35    36           37    38    39           40    41    42];

Construct the coordinate matrices. >> [Ex, Ey]=coordxtr(Edof,Coord,Dof,2);

Then generate and assemble the stiffness and mass matrices. >> K=zeros(42); M=zeros(42); >> for i=1:25 [k,m]=beam2d(Ex(i,:),Ey(i,:),ep); K=assem(Edof(i,:),K,k); M=assem(Edof(i,:),M,m); end

Denote the degrees of freedom that are known to be 0. >> b =[ 1 3  4  5  6  9 12 15 18 21 24 27 30 33 36 39 42];

Compute the eigenpairs for the stiffness and mass matrices. >> [La,Egv]=eigen(K,M,b);

The three lowest eigenpairs are as follows

$$ \omega_{1}=1.7e5; \phi_{1}=\begin{bmatrix}0.0565\\-0.0565\\0.0565\\-0.0565\\0.0000\\-0.1130\\0.0565\\-0.1130\\0.0000\\-0.1695\\0.0565\\-0.1695\\0.0000\\-0.2260\\0.0565\\-0.2260\\0.0000\\-0.2825\\0.0565\\-0.2825\\0.0000\\-0.3390\\0.0565\\-0.3390\\0.0000\end{bmatrix};

\omega_{2}=2.87e6; \phi_{2}=\begin{bmatrix}-0.0937\\0.1772\\-0.0775\\0.1649\\-0.0144\\02647\\-0.0512\\0.2608\\-0.0434\\0.2245\\-0.0248\\0.2299\\-0.0802\\0.0700\\-0.0071\\0.0813\\-0.1146\\-0.1392\\-0.0010\\-0.1291\\-0.1375\\-0.3262\\-0.0012\\-0.3235\\-0.1462\end{bmatrix}

\omega_{3}=7.08e6; \phi_{3}=\begin{bmatrix}-0.1604\\-0.0298\\-0.1744\\-0.0129\\-0.0743\\-0.0864\\-0.1987\\-00729\\-01282\\-0.1168\\-0.2272\\-0.1097\\-01618\\-0.0892\\-0.2521\\-00886\\-0.1791\\-00099\\-0.2670\\-0.0125\\-0.1866\\0.0834\\-0.2709\\0.0824\\-0.1893\end{bmatrix} $$

It was assumed that the moment of inertia of each beam was 0 because we were given no information regarding that property. As this is changed, the eigenpairs fluctuate. This is likely a reason that the eigenpairs don't match. Additionally, the eigenvector magnitudes depend on the initial value. Our code and CALFEM normalize them differently.



Comparing these plots to the ones found using the matlab code it is apparent that they are very similar. For each of the the three eigen modes the plane truss deforms the same. Therefore these plots confirm that the matlab code was an acceptable approximation in finding the eigen modes.