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User:Eml4500.f08.RAMROD.A/HW6

Principle of Virtual Work (the continuous case) of the dynamics of an elastic bar
The partial differential equation that describes this phenomena is below:

 * $$ {\partial \over \partial x}[ (EA) {\partial u \over \partial x} ]+ f = m{\ddot{u}} $$   (equation 1)

The representative equation of motion to the above equation is:

 * $$ \mathbf{-k} \mathbf{d} + \mathbf{k} \mathbf{d} = \mathbf{M} \mathbf{\ddot{d}}$$

 * $$ \mathbf{M} \mathbf{\ddot{d}} + \mathbf{k} \mathbf{d} = \mathbf{F} $$  (equation 2)

The above equation is called the multiple degrees of freedom system which is from vibrations.

Deriving (equation 2) from (equation 1)

 * $$\int_0^L w(x)\ [ { {\partial \over \partial x} [ (EA) {\partial u \over \partial x} ] } +f - m{\ddot{u}} ] = 0$$

This equation works for all possible weighting functions, $$ w(x) $$. This equation is the principal of virtual work for a partial differential equation.

Equation 1 can be transformed into Equation 3 trivially but Equation 3 does not transform into Equation 1 trivially.

Equation 3 can be rewritten as:

 * $$\int w(x) g(x) dx = 0 $$ for all $$ w(x)$$

Since Equation 3 holds true for all $$w(x)$$ then choose $$w(x)$$ equal to $$g(x)$$. Equation 3 becomes:

 * $$\int g^2(x) dx = 0$$

Where $$\ g^2(x) \ge 0$$

Therefore $$\ g(x) = 0$$

Solving the Problem
To solve continuous Principle of Virtual Work problem, we must use integration by parts. Recall from calculus integration by parts shown below.

Applying this principle to the equation for continuous PVW equation:

$$\int_{0}^{L}{\left\{u(x)[EA\frac{\delta w}{\delta x}] + f - m\ddot{u} \right\}dx}$$

Denoting the first term, $$\displaystyle u(x)(EA)\frac{\delta u}{\delta x}$$, as $$\displaystyle r(x)$$ and $$\displaystyle w(x)$$ as $$\displaystyle s(x)$$, we can plug these terms into the above formula and simplify the continuous PVW equation using integration by parts.

$$\int (r(x)s(x))' = \int_{0}^{L}{\left\{u(x)[EA\frac{\delta w}{\delta x}] + f - m\ddot{u} \right\}dx} = \left[w(EA)\frac{\delta u}{\delta x} \right]_{x=0}^{x=L} - \int_{x=L}^{x=0}{\frac{\delta w}{\delta x}(EA)\frac{\delta u}{\delta x}}dx$$

$$= \left[w(L) (EA)_{L}\frac{\delta u} {\delta x}_{L,t} \right] - \left[w(0) (EA)_{0}\frac{\delta u}{\delta x}_{0,t} \right] - \int \frac{\delta w}{\delta x}(EA)\frac{\delta u}{\delta x}dx$$

Where the first term is denoted as $$\displaystyle N(L,t)$$ and the second term is denoted by $$\displaystyle N(0,t)$$.

<\br>Now, return to the model problem of a beam shown below.

The boundary conditions for this model are:

$$\displaystyle u(0,t) = 0$$ and $$\displaystyle N(L,t) = F(t)$$

At x=0, select the weighting function, $$\displaystyle w(x)$$, such that $$\displaystyle w(0) = 0$$. (i.e. kinematically admissible)

Back to Continuous PVW

By selecting $$\displaystyle w(0) = 0$$, unknown reaction, $$\displaystyle N(0,t)$$ term cancels.

So, continuous PVW simplifies down to:

$$\int_{0}^{L}{w(m\ddot{u})dx} + \int_{0}^{L}{\frac{\delta w}{\delta x}(EA)\frac{\delta u}{\delta x}dx} = w(L)F(t) + \int_{0}^{L}{(wf)dx}$$ for all $$\displaystyle w(x)$$ such that $$\displaystyle w(0)=0$$.

Continuous vs Discrete
The following chart compares the continuous and discrete cases. Both cases are formulated using the principal of virtual work.

For the continuous case, the equations are valid for all $$\displaystyle w(x)$$ at $$\displaystyle w(0)=0$$ which is the boundary equation for the elastic bar. For the discrete case, the equations are valid for all $$\displaystyle \mathbf{w}$$. Here the boundary conditions are eliminated here.

Stiffness Term
the below picture depicts an elastic bar with all nodes.

This photo depicts the element displacement more closely.

Here we will assume the displacement $$\displaystyle u(x) $$ for $$\displaystyle x_i\leq x\leq x_{i+1}$$. This means that $$\displaystyle x \epsilon [x_i,x_{i+1}]$$ where the character $$\displaystyle \epsilon $$ means 'belongs to'.

Motivation For Linear Interpolation of $$\displaystyle u(x)$$ For The Two Bar Truss
The function $$\displaystyle u(x) $$ must be interpolated because the deformed shape is a straight line which means there must have been an assumption of displacement between two nodes. This allows a connection between the two cases where there are only axial displacements, or zero transverse displacements. The below picture shows this.

Ethics, Honesty and Imagination
It is always important to take a moment to remember the importance of being creative and imaginative. However, we must focus on our own creativity and not plagiarize the works of others. We cannot grow if we simply copy the imagination of others.

Rube Goldberg Device
Here is an impressive Rube Goldberg Device to inspire the creative mind in all. Rube Goldberg devices are impressive, complicated machines designed to do simple tasks - a show of human imagination.

Conversion of PVW to Discrete PVW
a look at Ni(x) and Ni+1(x)

Ni(x) and Ni+1(x) are linear functions and thus any linear conversion of Ni(x) to Ni+1(x) is also linear.

The following expressions illustrate the conversion:

$$ N_i(x)=\alpha_i+\beta_{i}x $$

$$ N_{i+1}(x)=\alpha _{i+1}+\beta_{i+1}x $$

we will use di to to create the following expression:

$$ N_id_i+N_{i+1}d_{i+1}=(\alpha_i+\beta_ix)d_i+(\alpha_{i+1}+\beta_{i+1}x)d_{i+1}

= (\alpha_id_i+\alpha_{i+1}d_{i+1}) + (\beta_id_i+\beta_{i+1}d_{i+1})x $$

The above is clearly a linear function in x

Images of Ni and Ni+1
images of Ni(x) and right: Ni+1(x)

Element stiffness matrix for element i
The Element stiffness matrix originates from the following equation:

$$\int_{x_{x}}^{x_{i+1}}{\left[N^1_iW_i+N^1_{i+1}W_{i+1} \right](EA)}\left[N^1_id_i+N^1_{i+1}d_{i+1} \right]dx$$

$$N^1_i=\frac{dN_i(x)}{dx}$$ Likewise for N_{i+1}

Note: $$u(x)=\begin{vmatrix} N_i(x) & N_{i+1}(x) \end{vmatrix}\begin{Bmatrix} d_i\\ d_{i+1} \end{Bmatrix}$$ and $$\frac{du(x)}{dx}=\begin{vmatrix} \dot{N}_i(x) & \dot{N}_{i+1}(x) \end{vmatrix}\begin{Bmatrix} d_i\\ d_{i+1} \end{Bmatrix}$$ where $$\mathbf{N}(x)_{1x2}=\begin{vmatrix} N_i(x) & N_{i+1}(x) \end{vmatrix}$$ and $$\mathbf{B}(x)_{1x2}=\begin{vmatrix} \dot{N}_i(x) & \dot{N}_{i+1}(x) \end{vmatrix}$$

Similiarly, the matrix $$\mathbf{W}(x)$$ can be written as:

$$\mathbf{W}(x)= \mathbf{N}(x)\begin{Bmatrix} W_i\\ W_{i+1} \end{Bmatrix}$$

$$\frac{d\mathbf{W}(x)}{dx}= \mathbf{B}(x)\begin{Bmatrix} W_i\\ W_{i+1} \end{Bmatrix}$$

Recalling the Degree of Freedom to establish these two in a matrix conventions:

$$ \mathbf{d}^{(i)}=\begin{Bmatrix} d_i\\ d_{i+1} \end{Bmatrix}\begin{Bmatrix} d^{(1)}_1\\ d^{(1)}_{2} \end{Bmatrix}$$ and $$ \mathbf{W}^{(i)}=\begin{Bmatrix} W_i\\ W_{i+1} \end{Bmatrix}\begin{Bmatrix} W^{(1)}_1\\ W^{(1)}_{2} \end{Bmatrix}$$

$$B=\int_{x_i}^{x_{i+1}}{\left[ (BW^{(i)})(EA)(Bd^{(i)})\right]dx}$$ $$B=W^{(i)}\cdot (K^{(i)}d^{(i)})$$

Using the transpose of matrix B to prove that the global stiffness matrix K will be listed as follows:

$$\mathbf{K}^{(i)}_{2x2}=\int_{x_i}^{x_{i+1}}{\left[ (B^{\tau } (x)(EA)(B(x)\right]dx}$$

which leads to the formation of the matrix K when EA is a constant being:

$$\mathbf{K}^{(i)}_{2x2}=\frac{EA}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$$

Transformation of Variable (coordinate) from $$x$$ to $$\tilde{x}$$
$$\tilde{x}=x-x_i$$ and $$d\tilde{x}=dx$$

so the B matrix in the new coordinate system can be expressed this way:

$$ \mathbf{K}^{(i)}_{2x2}=\int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}}(B^{\tau } (\tilde{x})(EA){B(\tilde{x})d\tilde{x}}$$

which leads to the formation of the matrix K when EA and L is the length of element i is a constant being:

$$\mathbf{K}^{(i)}_{2x2}=\frac{EA}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$$

Here the area matrix and the elasticity matrix are:

$$A(\tilde{x})=N^{(i)}_1(\tilde{x})A_1+N^{(i)}_2(\tilde{x})A_2 $$

$$E(\tilde{x})=N^{(i)}_1(\tilde{x})E_1+N^{(i)}_2(\tilde{x})E_2$$ Giving us the Stiffness matrix K equaling: $$\mathbf{K}^{(i)}(\tilde{x})=\frac{\left[N^{(i)}_1(\tilde{x})E_1+N^{(i)}_2(\tilde{x})E_2 \right]\left[N^{(i)}_1(\tilde{x})A_1+N^{(i)}_2(\tilde{x})A_2 \right]}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1& 1 \end{bmatrix}$$

$$N_1^{(i)}(\tilde{x}) = N_2^{(i)}(\tilde{x}) = \frac{\tilde{x}}{L^{(i)}} = \begin{cases} 0 \; at \; \tilde{x}=0\\ 1 \;at \; \tilde{x}=L^{(i)}\end{cases}$$

Shape Functions N1(i), N2(i)

Comparing Value in Notes with Value in Textbook
Set E1 = E2 = E

Let $$A(\tilde{x})$$ be linear. k(1) will be obtained from the problem above and will be compared to expression given in the textbook, which uses the average area.

From Above: $$\mathbf{K}^{(i)}(\tilde{x})=\frac{\left[N^{(i)}_1(\tilde{x})E_1+N^{(i)}_2(\tilde{x})E_2 \right]\left[N^{(i)}_1(\tilde{x})A_1+N^{(i)}_2(\tilde{x})A_2 \right]}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1& 1 \end{bmatrix}$$ From Book: $$\frac{E}{L^{(i)}}\frac{(A_1+A_2)}{2}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} = \mathbf{k}^{(i)}$$

$$\frac{(A_1+A_2)}{2}$$ = Average Area

Comparing General Value with Average Value
Next, compare the general k(i) from above to the stiffness matrix obtained by using 1/2(A1+A2) and 1/2(E1+E2)
 * Note: $$ E_1 \neq E_2$$

General Value: $$\mathbf{k}^{(i)}=\frac{EA}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$$

Average Value: $$\mathbf{k}_{ave}^{(i)}=\frac{(E_1+E_2)(A_1+A_2)}{4L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$$ Find k(i) - k(i)ave $$\mathbf{k}^{(i)}-\mathbf{k}_{ave}^{(i)} = \frac{EA}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} \; - \; \frac{(E_1+E_2)(A_1+A_2)}{4L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} = \frac{4EA-(E_1+E_2)(A_1+A_2)}{4L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$$

Remark: Mean Value Theorem
Recall the Mean Value Theorem (MVT) and its relation to the centroid.

MVT: $$\int_{x=a}^{x=b}{f(x)dx} = f(\bar{x})[b-a]$$
 * for $$\bar{x} \;in \;interval \;[a,b]$$ (that is $$a \leq \bar{x}\leq b$$)

$$\int_{x=a}^{x=b}{f(x)g(x)dx} = f(\bar{x})g(\bar{x})[b-a]$$

 * $$a\leq\bar{x}\leq b$$

But in general:

 * $$f(\bar{x})\neq \frac{1}{b-a}\int_{a}^{b}{f(x)dx}$$

 * $$\frac{1}{b-a}\int_{a}^{b}{f(x)dx}$$  is the average value of f

 * $$g(\bar{x})\neq \frac{1}{b-a}\int_{a}^{b}{g(x)dx}$$

 * $$\frac{1}{b-a}\int_{a}^{b}{g(x)dx} $$ is the average value of g

Modifying 2-Bar Truss Code
The 2-bar truss code is modified to accommodate for the general stiffness, k(i)

Electric Pylon Code
Below is the code to solve the the electric pylon truss system using the finite element method. The code solves the systems reaction forces, plots the undeformed and deformed system and calculates the systems maximum tensile stress and maximum compressive stress.

MATLAB Code Results
Below are the values for the variable results which has the values of axial stress of all 91 trusses.

stress =

2.1562e+006 -1.3071e+006 1.3468e+006 1.2911e+006 -7.0304e+005 -4.7596e+006 -2.465e+006 1.4236e+006 3.8538e+006 -5.7655e+006 -37882      1755.6  3.8655e+006 2.9395e+006 -3.6988e+006 -5.7661e+006 3.5896e+005 5.242e+006 1.9797e+006 -3.7749e+006 -6.5159e+006 5.5784e+005 6.4388e+005 5.9656e+006 -8.1704e+005 -3.9616e+006 2.32e+006 -8.1307e+005 -5.7556e+006 -28474 -9.9353e+005 -15214 5.9216e+006 55813 -4.8717e+006 1.3915e+006 -3023.3 -5.7588e+006 -4.854e+006 1.3904e+006 -1.8831e+005 2.258e+006 -6.32e+005 -4.0102e+006 0 9.7644e-008 64147 2.7001e+005 2.7149e+005 -2.0778e+006 -4.4271e+006 -5.8777e+006 -7.3549e+006 -8.662e+006 -8.6957e+006 -2.5807e-008 0 -1.0794e-007 4.8819e-008 -1.7704e+005 2.2069e+005 -1.9193e+005 1.9887e+005 2.1464e+006 -2.083e+006 2.2895e+006 -2.2643e+006 2.1035e+006 -2.1464e+006 1.4138e+006 -1.3772e+006 1.5835e+006 -3.6928e+006 -97393      -23690        31208  9.0479e+006 0 9.0317e+006 0 9.0511e+006 -1.3175e+005 -2.9645e+005 1.7966e+006 4.4876e+006 6.6233e+006 7.7755e+006 1.0741e-008 9.6069e-009 1.441e-008 2.1482e-008

Below are the resulting maximum tensile stress and maximum compressive stress with their corresponding element number.

Functions
The functions PlaneTrussElement.m, NodalSoln.m, PlaneTrussResults.m were all called in the code for solving the electric pylon truss system. They are all necessary for the completion of the truss problem.

PlaneTrussElement.m

This function take the Young's Modulus (e), the Area of the element (A) and the coordinates of the element ends (coord) and generates the stiffness matrix for a plane truss element. The function first calculates the lengths of the the elements. Once those are calculated the stiffness matrix is calculated.

NodalSoln.m

The function takes the global coefficient matrix (K), the global right hand side vector (R), the list of degrees of freedom with specified values, and the specified values to determine the displacements and reactions at each node. The dof of the the system is first found by using the  command which finds the longest dimension of R. df then finds the difference between the dofs that have known values (a value of zero) and the dof that were found in the previous line. The displacements and the reactions are then calculated.

PlaneTrussResults.m

This function computes the plane truss element results. It takes in the modulus of elasticity (e), the area of the cross-section (A), the coordinates at the element ends (coord), and the displacements at the elemtent ends (disps) and calculates the axial strain (eps), stress(sigma) and force (force). The results are stored in the variable  and sent back to the main program.

Plots
Below is the plot of the underformed and deformed truss system. The undeformed system is plotted with dotted lines and the deformed system is plotted in solid lines. The elements that contain the maximum tensile and maximum compressive stresses are marked by the lines.

Electric Pylon Eigenvector Problem
In order to construct the lumped mass matrix, a matrix that stores the mass associated with each degree of freedom at each node, the density, 7.8 g/cm^3, is multiplied by the cross sectional area, 4 cm^2, and by half of the length of the associated element. To do this, a length matrix was also created using the x and y coordinates of each node and the Pythagorean theorem. After the lumped mass matrix, M, was created, a simpler mass matrix, m, was made from the diagonals of M. Lastly, assuming K_bar and M_bar are the simplified stiffness and mass matrices using PVW, we can solve for the eigen values and eigen vectors of the generalized eigenvector problem:

$$\displaystyle \bar{K}v=\lambda \bar{M}v$$

In this code, matrix D are the eigenvectors and the diagonal of matrix V are the eigenvalues.

Finally, we isolate only the three smallest eigenvalue/eigenvector pairs and use them to plot the associated three deformations of the structure with respect to the undeformed structure. This is shown in the three figures following the MATLAB code below. This code is intended to be executed following the above techniques for plotting the deformed shape.

The following are the results and plots from the above MATLAB Code:

V =

Columns 1 through 7

-0.0014944    -0.05031     0.038896     0.014198    -0.032696    -0.001106    0.0071974   -0.0094129     0.021545    -0.011552    0.0065082     0.020884    0.0011283   -0.0094644   -0.0026749    0.0029728    -0.063282     0.017282    -0.032582  -0.00085494    0.0066026   -0.0020615     0.009503     -0.01669     0.018202    -0.020147  -0.00055093    0.0079385   -0.0049418    -0.019243   -0.0096861     0.011171    -0.015833  -0.00026065  -0.00036456   -0.0083028      0.01154    0.0040931    0.0074832     0.015165   0.00083623   -0.0067624   -0.0049418    -0.019243   -0.0096861     0.011171    -0.016113  -0.00029973  -0.00025865   -0.0049431    -0.018737      0.05144     0.010434   0.00010155   0.00012578  -0.00032828   -0.0049418    -0.019243   -0.0096861     0.011171    -0.015761  -0.00031107  -0.00042716   -0.0027531    0.0027253  -0.00033855     0.016338    -0.014736  -0.00037307    0.0055664   -0.0073676     -0.01539   -0.0077489    0.0073001  -0.00091268   0.00031983   -0.0065232   -0.0075273     0.010308    0.0034739    0.0087205    0.0092429   0.00055298   -0.0040796   -0.0073676     -0.01539   -0.0077489    0.0073001  -0.00087416   0.00031888   -0.0064171   -0.0055158    0.0071134    0.0018676      0.01193  9.7725e-005   0.00035942  -0.00046686 -0.0073676    -0.01539   -0.0077489    0.0073001  -0.00091143   0.00032318   -0.0065745   -0.0035286     0.003957   0.00028073       0.0151   -0.0088887  -0.00015875    0.0029767   -0.0097911    -0.011541   -0.0058137    0.0034335     0.011117   0.00045368    -0.009729   -0.0067275    0.0090379    0.0028352    0.0099965    0.0040127   0.00040091   -0.0021783   -0.0097911    -0.011541   -0.0058137    0.0034335     0.011098    0.0010523   -0.0096099   -0.0043041    0.0051888       0.0009     0.013863   -0.0038229  8.9657e-005    0.0012635 -0.011003  -0.0096163   -0.0048461    0.0015002     0.014537   0.00035216    -0.011152    -0.007188    0.0097693    0.0032029    0.0092618    0.0040206    0.0005884   -0.0025633    -0.011003   -0.0096163   -0.0048461    0.0015002     0.014342     0.000869    -0.010619   -0.0055158    0.0071134    0.0018676      0.01193  8.9107e-005  -0.00070175  -0.00060388 -0.011003  -0.0096163   -0.0048461    0.0015002     0.014591    0.0014347    -0.011022   -0.0038679    0.0044959   0.00055166     0.014559    -0.003822   0.00025881    0.0017894    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016733    0.0014591    -0.011545   -0.0076242     0.010462    0.0035513    0.0085658    0.0033657    0.0017022   -0.0021194    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016631    0.0014314    -0.011354   -0.0065094    0.0086915    0.0026611     0.010344    0.0030594  -0.00063799   -0.0029699    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016621   0.00044098    -0.011493   -0.0045222    0.0055352    0.0010742     0.013515   -0.0028964   -0.0009389    0.0017509    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016676   0.00043757    -0.011595   -0.0034316    0.0038031   0.00020332     0.015255   -0.0031524    0.0014629    0.0018065    -0.012988   -0.0064638   -0.0032611   -0.0016665       0.0163    0.0036404   -0.0098054   -0.0071395    0.0096923    0.0031642    0.0093392    0.0036554    0.0003512   -0.0032446    -0.012988   -0.0064638   -0.0032611   -0.0016665     0.016242   -0.0019621    -0.010752   -0.0038921    0.0045344   0.00057101      0.01452   -0.0034357   3.967e-005    0.0023552 -0.014616  -0.0038772   -0.0019607   -0.0042649    0.0050086      0.01472    0.0038994   -0.0084967     0.011848     0.004248    0.0071739   -0.0038492     0.007872    0.0048045    -0.014616   -0.0038772   -0.0019607   -0.0042649    0.0048501     -0.01516   -0.0019165   -0.0025592    0.0024174  -0.00049337     0.016647    0.0040403     0.008784   -0.0022205    -0.016623  -0.00069005  -0.00035828   -0.0074664 -1.7925e-005    0.0063344     0.015577 -0.012859    0.018776    0.0077314   0.00021403    -0.026786    -0.046864    -0.055085    -0.016623  -0.00069005  -0.00035828   -0.0074664  -0.00066765    0.0044429     0.011469     -0.01029     0.014696      0.00568    0.0043126    -0.010806   -0.0066796   -0.0028508    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0010076    0.0035273    0.0095149   -0.0092237     0.013003    0.0048285    0.0060139   -0.0064944    0.0032482    0.0044614    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0012259    0.0028835    0.0082753   -0.0084482     0.011771    0.0042093    0.0072512   -0.0039643    0.0092514    0.0047445    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0015742    0.0018451    0.0073598   -0.0074788     0.010231    0.0034352    0.0087978   -0.0013382     0.014987    0.0044037    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0019923   0.00059961    0.0069186   -0.0062913    0.0083451    0.0024869     0.010692  -0.00020806     0.019347    0.0042055    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0019997   -0.0011384    0.0065969   -0.0047645    0.0059201    0.0012677     0.013128   0.00055104     0.019693    0.0025555    -0.016623  -0.00069005  -0.00035828   -0.0074664    -0.001584    -0.002481    0.0065811   -0.0035528    0.0039955   0.00030008     0.015062    0.0016607      0.01567   0.00083891    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0012435   -0.0036022    0.0071142   -0.0026077    0.0024944  -0.00045466      0.01657    0.0041473      0.01023   -0.0015388    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0010142   -0.0043633     0.008214   -0.0018321    0.0012626   -0.0010739     0.017807     0.006589    0.0042216   -0.0034572    -0.016623  -0.00069005  -0.00035828   -0.0074664  -0.00064793    -0.005506     0.010106  -0.00074159   -0.0004695   -0.0019448     0.019547     0.010882   -0.0063904     0.001122    -0.016623  -0.00069005  -0.00035828   -0.0074664   0.00015962   -0.0080805     0.014385     0.001803   -0.0045111   -0.0039768     0.023607     0.027895    -0.053301     0.047945     -0.01718   0.00019526  8.6827e-005   -0.0083557   -0.0032142    -0.001658    0.0057069 -0.01092    0.015697    0.0061832    0.0033073    -0.013746     -0.01368   -0.0095212     -0.01718   0.00019526  8.6827e-005   -0.0083557   -0.0032288      0.00119    0.0056899 -0.00013572  -0.0014318   -0.0024286     0.020514     0.013702    -0.013691     0.005942    -0.017483   0.00067641   0.00032873    -0.008839   -0.0042235   -0.0039517    0.0046073   -0.0098538     0.014003    0.0053317    0.0050086   -0.0089283   -0.0023228   0.00075412    -0.017483   0.00067641   0.00032873    -0.008839   -0.0041914    0.0034874    0.0054104   -0.0011778   0.00022335   -0.0015965     0.018851    0.0090606   -0.0018008   -0.0012768    -0.017665   0.00096509   0.00047387    -0.009129   -0.0046243   -0.0048842    0.0049671   -0.0092237     0.013003    0.0048285    0.0060139   -0.0066272    0.0030477    0.0034716    -0.017665   0.00096509   0.00047387    -0.009129   -0.0041038   -0.0036227    0.0063496    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Columns 8 through 14

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   0.001671    -0.004461   -0.0041803   -0.0053147      0.01218    0.0096731    0.0095258    0.0047436     0.011054     -0.01493   -0.0030084    -0.017299    -0.032257     0.039949   -0.0019723   -0.0042239    0.0066862   -0.0033804    0.0086251    -0.010771   -0.0065418    0.0020221   -0.0088225    -0.013041 -9.4872e-005     0.015658    -0.018801  -0.00041176 0.00051585  -0.0044888   -0.0058896   -0.0041654    0.0088317    0.0020712     0.016636    0.0018162     0.013824    -0.013417   -0.0048381    -0.013663    -0.014768     0.013904  -0.00050477   -0.0045787    0.0074429   -0.0023817    0.0067334   -0.0078129   -0.0056683    0.0014353    -0.011726    -0.013411    0.0015688     0.015089    -0.015569   0.00012618   -0.0013241   -0.0050527   -0.0062288   -0.0031643      0.00773  -0.00058405     0.017498    0.0039112     0.015896    -0.012608   -0.0064073    -0.011502   -0.0013877   -0.0039055   -0.0021876   -0.0033004   -0.0048132   -0.0037569    0.0072832   0.00021689     0.010294     0.013102     0.023381   -0.0028371    -0.015183   -0.0038303    0.0037028   -0.0094951   -0.0016637  -0.00019033   -0.0015339   -0.0057123    0.0076202    -0.000718    0.0053884     0.019296     0.017713   -0.0038519    -0.011726   -0.0014782     0.014865    -0.015231  -0.00046877    0.0014309    0.0016713   -0.0066267    0.0074881  -0.00063158  -0.00050724     0.022383    0.0018164   -0.0060926   -0.0025434    0.0016601     0.016725   -0.0088003   0.00090305 -4.2758e-005    0.0043743   -0.0052761    0.0067932   -0.0014754   -0.0036199 0.020483   -0.014426   -0.0078766    0.0072917    0.0042673    0.0064927    0.0028295    0.0017978   -0.0031447    0.0065795   -0.0027328     0.005644   -0.0039043   -0.0038288      0.01475    -0.020824   -0.0081054     0.011844    0.0057379   -0.0068911    0.0067412    0.0013652   -0.0050503    0.0073897   -0.0016611    0.0059395   -0.0059134   -0.0042477    0.0044209    -0.013918    -0.013985    0.0027983     0.014728    -0.010425    0.0029632    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Columns 15 through 21

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  0.010414    0.0001695    0.0029418     0.010775   -0.0041438    0.0023338    -0.018613   0.00083312   -0.0067808  3.7708e-005    0.0016876   -0.0011102   0.00089045   -0.0021173 0.0082002   0.0082228    0.0017345    0.0017853    0.0031477   -0.0051086     0.017713   -0.0016483    0.0017548   0.00031409    0.0027261  -0.00059936  -0.00070476    0.0036851   -0.0027068     0.010359  -0.00097276    -0.013468    0.0071251   -0.0085515     0.031612   -0.0068619     0.010398  -0.00013534 -1.3711e-005   0.00024221   -0.0016681    0.0065751 -0.011715   0.0073271   -0.0024546     -0.01419     0.003744   -0.0030331    0.0088153    -0.011638     0.015843  -0.00065777   -0.0019653 -9.9703e-005  -0.00025788   0.00051247 -0.0065645 -0.00061978  -0.00068125    0.0024727   -0.0017063    0.0011157     -0.01085    -0.016384     0.021402   -0.0012811    -0.005145   5.911e-005    0.0014507   -0.0074758 -0.00053815  -0.0018033    0.0012826     0.013839   -0.0033199  -0.00024533   -0.0038127     0.004437   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Columns 22 through 28

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Columns 29 through 35

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Columns 36 through 42

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Columns 43 through 49

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0.011545   -0.0087859   -0.0037034  -0.00069165    0.0039811   0.00031606   -0.0013449    -0.046464     0.032506    0.0057607     0.012025    -0.029002   -0.0038893     0.004829  -0.00094844    0.0009777    -0.005068   -0.0016749   0.00056231   0.00047929  7.0358e-005 -0.0054843   0.0040512    -0.028855   -0.0071942   -0.0014312    0.0041773   -0.0010221

Columns 50 through 56

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    0.015113     0.009726    0.0039381    0.0024891   -0.0013181     0.015064     0.011426    -0.012761    -0.015871    0.0020118    0.0098753     0.010041    -0.016817    0.0040206    -0.012812   -0.0055371    0.0069382    0.0088158    0.0042083     0.007583    0.0094597    -0.013058    -0.017194   -0.0032357    0.0015587    -0.018009     0.027801   -0.0013722     0.012565    0.0089768    -0.015572    -0.016531     0.013123    0.0027056    0.0052886   -0.0094852    -0.006471     0.010941     0.018354    -0.037765    -0.018675    0.0015459   -0.0099522   -0.0037733     0.012373     0.011433  6.6585e-006 -0.001715   0.0057923    -0.010016    -0.020309    0.0034609    0.0081577    -0.032959   -0.0024337    0.0032152   -0.0072448    -0.012821    0.0027169    -0.022813    0.0014779     0.010942    0.0035025  -0.00066599    -0.026014    0.0056328    0.0062643   -0.0079112    0.0024113   -0.0044425    0.0077403     0.031044   -0.0050946     0.035508    -0.014243   -0.0025078   -0.0022409    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Columns 57 through 63

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Columns 64 through 70

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Columns 71 through 77

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Columns 78 through 84

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Column 92

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D =

Columns 1 through 7

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Columns 8 through 14

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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            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Columns 22 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            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Columns 29 through 35

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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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Columns 36 through 42

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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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Columns 43 through 49

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Columns 50 through 56

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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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Columns 57 through 63

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Columns 64 through 70

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Columns 71 through 77

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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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Columns 78 through 84

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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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Columns 85 through 91

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1.5193e+006            0            0 0           0            0            0            0  1.8084e+006            0 0           0            0            0            0            0  2.1778e+006 0           0            0            0            0            0            0

Column 92

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min_evec1 =

-0.0014944  -0.0094129   -0.0026749   -0.0020615   -0.0049418   -0.0083028   -0.0049418   -0.0049431   -0.0049418   -0.0027531   -0.0073676   -0.0075273   -0.0073676   -0.0055158   -0.0073676   -0.0035286   -0.0097911   -0.0067275   -0.0097911   -0.0043041    -0.011003    -0.007188    -0.011003   -0.0055158    -0.011003   -0.0038679    -0.012215   -0.0076242    -0.012215   -0.0065094    -0.012215   -0.0045222    -0.012215   -0.0034316    -0.012988   -0.0071395    -0.012988   -0.0038921    -0.014616   -0.0084967    -0.014616   -0.0025592    -0.016623    -0.012859    -0.016623     -0.01029    -0.016623   -0.0092237    -0.016623   -0.0084482    -0.016623   -0.0074788    -0.016623   -0.0062913    -0.016623   -0.0047645    -0.016623   -0.0035528    -0.016623   -0.0026077    -0.016623   -0.0018321    -0.016623  -0.00074159    -0.016623     0.001803     -0.01718     -0.01092     -0.01718  -0.00013572    -0.017483   -0.0098538    -0.017483   -0.0011778    -0.017665   -0.0092237    -0.017665    -0.008012    -0.017665   -0.0068487    -0.017665   -0.0055158    -0.017665   -0.0041829    -0.017665   -0.0030439    -0.017665   -0.0018321    -0.020088   -0.0092237    -0.020088   -0.0018321

min_evec2 =

-0.05031    0.021545    0.0029728     0.009503    -0.019243      0.01154    -0.019243    -0.018737    -0.019243    0.0027253     -0.01539     0.010308     -0.01539    0.0071134     -0.01539     0.003957    -0.011541    0.0090379    -0.011541    0.0051888   -0.0096163    0.0097693   -0.0096163    0.0071134   -0.0096163    0.0044959   -0.0076917     0.010462   -0.0076917    0.0086915   -0.0076917    0.0055352   -0.0076917    0.0038031   -0.0064638    0.0096923   -0.0064638    0.0045344   -0.0038772     0.011848   -0.0038772    0.0024174  -0.00069005     0.018776  -0.00069005     0.014696  -0.00069005     0.013003  -0.00069005     0.011771  -0.00069005     0.010231  -0.00069005    0.0083451  -0.00069005    0.0059201  -0.00069005    0.0039955  -0.00069005    0.0024944  -0.00069005    0.0012626  -0.00069005   -0.0004695  -0.00069005   -0.0045111   0.00019526     0.015697   0.00019526   -0.0014318   0.00067641     0.014003   0.00067641   0.00022335   0.00096509     0.013003   0.00096509     0.011078   0.00096509    0.0092304   0.00096509    0.0071134   0.00096509    0.0049963   0.00096509    0.0031872   0.00096509    0.0012626    0.0048143     0.013003    0.0048143    0.0012626

min_evec3 =

0.038896   -0.011552    -0.063282     -0.01669   -0.0096861    0.0040931   -0.0096861      0.05144   -0.0096861  -0.00033855   -0.0077489    0.0034739   -0.0077489    0.0018676   -0.0077489   0.00028073   -0.0058137    0.0028352   -0.0058137       0.0009   -0.0048461    0.0032029   -0.0048461    0.0018676   -0.0048461   0.00055166   -0.0038785    0.0035513   -0.0038785    0.0026611   -0.0038785    0.0010742   -0.0038785   0.00020332   -0.0032611    0.0031642   -0.0032611   0.00057101   -0.0019607     0.004248   -0.0019607  -0.00049337  -0.00035828    0.0077314  -0.00035828      0.00568  -0.00035828    0.0048285  -0.00035828    0.0042093  -0.00035828    0.0034352  -0.00035828    0.0024869  -0.00035828    0.0012677  -0.00035828   0.00030008  -0.00035828  -0.00045466  -0.00035828   -0.0010739  -0.00035828   -0.0019448  -0.00035828   -0.0039768  8.6827e-005 0.0061832 8.6827e-005 -0.0024286  0.00032873    0.0053317   0.00032873   -0.0015965   0.00047387    0.0048285   0.00047387    0.0038609   0.00047387     0.002932   0.00047387    0.0018676   0.00047387   0.00080324   0.00047387  -0.00010632   0.00047387   -0.0010739    0.0024091    0.0048285    0.0024091   -0.0010739

Eigenvector Plots

Contributing Team Members
Eml4500.f08.RAMROD.A 19:11, 17 November 2008 (UTC)

EML4500.F08.RAMROD.B 19:07, 21 November 2008 (UTC)

Eml4500.f08.RAMROD.F 19:37, 18 November 2008 (UTC)

Eml4500.f08.ramrod.c 18:54, 19 November 2008 (UTC)

Eml4500.f08.ramrod.D 01:56, 21 November 2008 (UTC)

EML4500.f08.RAMROD.E 17:27, 21 November 2008 (UTC)

title: User:Eml4500.f08.RAMROD.A/HW6