User:Eml4500.f08.RAMROD.A/HW7

Frame Element Definition
A frame element is an element that is made up of a beam element and a truss element. Recall that for a truss element, all deformations were axial while in a beam element, the deformation was transverse to the axis. Knowing this, we can assume that for a frame element, the deformation will be some linear combination of both axial and transverse displacements at each node.

Model Problem for Frame
As with other types of elements, we will do our simple analysis using a model problem. The model here is a two-bar frame with nodes 1 and 3 simply supported and node 2 is a rigid connection between elements 1 and 2, as shown in the diagram below:



Combining the two elemental FBDs to a global FBD is shown in the picture below.



Elemental and Global Stiffness Matrices
The next step is deriving the stiffness matrix for our model problem. Since there are more dof's at each node now, the elemental stiffness matrices will each be a 6x6 matrix. We combine them very similarly as with trusses, with the overlapping dof's being at node two, such that the global stiffness matrix will be a 9x9 and be made up as shown below:



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$$\tilde{K}_{6x6}^{(e)}\tilde{d}_{6x1}^{(e)}=\tilde{F}_{6x1}^{(e)} $$

$$\tilde{d}_{6x1}^{(e)}=\begin{Bmatrix} \tilde{d}_{1}^{(e)}\\ \tilde{d}_{2}^{(e)}\\ \tilde{d}_{3}^{(e)}\\ \tilde{d}_{4}^{(e)}\\ \tilde{d}_{5}^{(e)}\\ \tilde{d}_{6}^{(e)} \end{Bmatrix} $$

$$\tilde{F}_{6x1}^{(e)}=\begin{Bmatrix} \tilde{F}_{1}^{(e)}\\ \tilde{F}_{2}^{(e)}\\ \tilde{F}_{3}^{(e)}\\ \tilde{F}_{4}^{(e)}\\ \tilde{F}_{5}^{(e)}\\ \tilde{F}_{6}^{(e)} \end{Bmatrix}$$

Note: $$\tilde{F}_{3}^{(e)}=F_{3}^{(e)}$$ and $$\tilde{F}_{6}^{(e)}=F_{6}^{(e)}$$ are moments around the z-axis.

$$\tilde{K}_{6x6}^{(e)}=\begin{bmatrix} \frac{EA}{L} & 0 & 0 & \frac{-EA}{L} & 0 & 0\\ & \frac{12EI}{L^3} & \frac{6EI}{L^2} & 0 & \frac{-12EI}{L^3} & \frac{-6EI}{L^2}\\ & & \frac{4EI}{L} & 0 & \frac{-6EI}{L^2} & \frac{2EI}{L}\\ & &  & \frac{EA}{L} & 0 & 0\\ & SYM. & &  & \frac{12EI}{L^3} & \frac{-6EI}{L^2}\\ & &  &  &  & \frac{4EI}{L} \end{bmatrix}$$

Dimensional Analysis
$$\left[\tilde{d}_1 \right]=L=\left[\tilde{d}_i \right] \; \; i=1,2,4,5$$

$$ \left[\tilde{d}_3 \right]=L=\left[\tilde{d}_6 \right]$$

Let's take an arc of a circle of radius r



The arc length $$\bar{AB}=R\theta $$ where $$\bar{AB}$$ is the arc length and $$\theta$$ is the angle measure.

$$ \theta =\frac{\bar{AB}}{R}$$

$$\left[\theta \right]=\frac{\left[\bar{AB} \right]}{\left[R \right]}= \frac{L}{L}=1$$ no dimension

$$ \sigma =E\varepsilon \Rightarrow \left[\sigma \right]=\left[E \right]\left[\varepsilon  \right]$$

Dimensional Analysis
Dimensions of common parameters

$$ [\varepsilon ]=\frac{[du]}{[dx]}=\frac{L}{L}=1 $$

$$ [\sigma ]=[E]=\frac{F}{L^2} $$

$$ [A]=L^2 $$

$$ [I]=L^4 $$

$$ [\frac{EA}{L}]=[K_{11}^{N}]=\frac{(F/L^2)(L^2)}{L}=\frac{F}{L} $$

$$ [\tilde{K}_{11}\tilde{d}_1]=[\tilde{K}_{11}][\tilde{d}_1]=F $$

$$[\tilde{K}_{23}\tilde{d}_3]=[\tilde{K}_{23}][\tilde{d}_3]=F $$

FD relationship and the Transpose
Converting the element FD relationship in global coordinates from element FD in local coordianates

The relationship is presented as follows:

$$ K^{(e)}=\tilde{T}^{(e)T}\tilde{k}^{(e)}\tilde{T}^{(e)} $$

$$ k^{(e)}_{6x6}d^{(e)}_{6x1}=f^{(e)}_{6x1} $$

$$ k^{(e)}_{6x6}=\tilde{T}^{(e)T}_{6x6}\tilde{k}^{(e)}_{6x6}\tilde{T}^{(e)}_{6x6} $$

$$ \tilde{k}^{(e)}_{6x6}\tilde{d}^{(e)}_{6x1}=\tilde{f}^{(e)}_{6x1} $$

where

$$ \tilde{d} = \begin{Bmatrix}\tilde{d}_1\\\tilde{d}_2\\\tilde{\underline{d}}_3\\\tilde{d}_4\\\tilde{d}_5 \\\tilde{\underline{d}}_6\end{Bmatrix}$$

And

$$ \tilde{T} = \begin{bmatrix}\underline{R}_{2x2} & & 0 & 0 & 0 & 0\\ &  & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & \underline{R}_{2x2} & & 0\\0 & 0 & 0 &  &  & 0\\0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} $$

Deformed Shape of Truss Element
The motivation for this section is to find the deformed shape of the truss element through interpolation of the transverse displacement $$\displaystyle v(s) = v(\tilde{x})$$. We will first look at the Principal of Virtual Work for beams;

$$\displaystyle \int_{0}^{L}{w(\tilde{x})\left[-\frac{\partial ^2}{\partial x^2}\left((EI)\frac{\partial^2v}{\partial x^2} \right)+f_t -m\ddot{u} \right]dx} = 0$$ for all possible $$\displaystyle w(x)$$

For the first term we must integrate by parts. This is done below.

$$\displaystyle \alpha := \int_{0}^{L}{w(\tilde{x}) \left[-\frac{\partial ^2}{\partial x^2}\left((EI)\frac{\partial^2v}{\partial x^2} \right)\right]}$$

$$\displaystyle \alpha := \frac{\partial }{\partial x}\left[ \frac{\partial }{\partial x}\left((EI)\frac{\partial^2v}{\partial x^2} \right)\right]$$

What lies inside the square brackets will be designated as $$\displaystyle r(x)$$ and the whole expression is designated as $$\displaystyle r'(x)$$. Performing the integration by parts yields;

$$\displaystyle \alpha := \left[w\frac{\partial }{\partial x} \left( (EI)\frac{\partial^2 v}{\partial x^2} \right)\right]- \int_{0}^{L}{\frac{dw}{dx}\frac{\partial }{\partial x}\left((EI)\frac{\partial ^2v}{\partial x^2} \right)dx}$$

The first term (non integral term) will be denoted by $$\displaystyle \beta_1$$, the first derivative in the interal will be represented by $$\displaystyle s'(x)$$, and as your can see $$\displaystyle r(x)$$ is imbedded in the integral. Performing another integration by parts yields the following;

$$\displaystyle \alpha := \beta_1 - \alpha := \left[ \frac{dw}{dx} (EI)\frac{\partial^2 v}{\partial x^2} \right] + \int_{0}^{L}{\frac{d^2w}{dx^2}\left((EI)\frac{\partial ^2v}{\partial x^2} \right)dx}$$

The term after $$\displaystyle \beta_1 $$ will be denoted $$\displaystyle \beta_2$$ and the whole integral term will be denoted as $$\displaystyle \gamma$$. It is important to note that this new equation is symmetric. Inputting the new variables the new equation of the Principal of Virtual Work for beams is as follows;

$$\displaystyle -\beta_1 + \beta_2 - \gamma +\int_{0}^{L}{w(\tilde{x})f_t dx} - \int_{0}^{L}{Wm\ddot{u}dx} = 0$$ for all possible $$\displaystyle w(x)$$.

Beam Stiffness Matrix
Let us now focus on the stiffness term $$\displaystyle \gamma$$ so we can derive the beam stiffness matrix and identify the beam shape.



We need to plot the following eqaution;

$$\displaystyle v(\tilde{x}) = N_2(\tilde{x})\tilde{d_2} + N_3(\tilde{x})\tilde{d_3} + N_5(\tilde{x})\tilde{d_5} + N_6(\tilde{x})\tilde{d_6}$$

We recall that

$$\displaystyle u(\tilde{x}) = N_1(\tilde{x})\tilde{d_1} +N_4(\tilde{x})\tilde{d_4}$$





The equations for $$\displaystyle N_2(\tilde{x}),N_3(\tilde{x}),N_5(\tilde{x}),N_6(\tilde{x})$$ can now be found from these drawings and are shown below.

$$\displaystyle N_2(\tilde{x}) = 1 - \frac{3\tilde{x}^2}{L^2} + \frac{2\tilde{x}^3}{L^3}$$, $$\displaystyle \tilde{d_2}$$

$$\displaystyle N_3(\tilde{x}) = \tilde{x} - \frac{2\tilde{x}^2}{L} + \frac{\tilde{x}^2}{L^2}$$, $$\displaystyle \tilde{d_3}$$

$$\displaystyle N_5(\tilde{x}) = \frac{3\tilde{x}^2}{L^2} - \frac{2\tilde{x}^3}{L^3}$$, $$\displaystyle \tilde{d_5}$$

$$\displaystyle N_6(\tilde{x}) = -\frac{\tilde{x}^2}{L} + \frac{\tilde{x}^3}{L^2}$$, $$\displaystyle \tilde{d_6}$$



$$\mathbf{\tilde{d}^{(e)}}_{6x1} = \mathbf{\tilde{T}^{(e)}}_{6x6}\underbrace{\mathbf{d^{(e)}}_{6x1}}$$
 * $$\mbox{known after solving FE system}$$

Compute $$U(\tilde{x})\; \mbox{and} \; V(\tilde{x})$$


$$\mathbf{U(\tilde{x})} = U(\tilde{x})\vec{\tilde{i}} + V(\tilde{x})\vec{\tilde{j}}$$

$$\mathbf{U(\tilde{x})} = U_x(\tilde{x})\vec{i} + U_y({x})\vec{j}$$

Compute $$U(\tilde{x})\; \mbox{and}\; V(\tilde{x})$$ using Equations (1) and (2) from above

Compute $$U_x(\tilde{x}) \; \mbox{and} U_y(\tilde{x}) \; \mbox{from} \; U(\tilde{x}) \; \mbox{and} \; V(\tilde{x})$$
$$\begin{Bmatrix} U_x(\tilde{x})\\ V(\tilde{x}) \end{Bmatrix} = \mathbf{R}^T\begin{Bmatrix}U(\tilde{x})\\V(\tilde{x})\end{Bmatrix}$$


 * Note: The RT matrix can be found above.

$$\begin{Bmatrix}U(\tilde{x})\\V(\tilde{x})\end{Bmatrix} = \underbrace{\begin{bmatrix} N_1 & 0 & 0 & N_4 & 0 & 0\\ 0 & N_2 & N_3 & 0 & N_5 & N_6 \end{bmatrix}}_{\mathbb{N}(\tilde{x})}\begin{Bmatrix} \tilde{d}_1^{(e)}\\ \tilde{d}_2^{(e)}\\ \tilde{d}_3^{(e)}\\ \tilde{d}_4^{(e)}\\ \tilde{d}_5^{(e)}\\ \tilde{d}_6^{(e)} \end{Bmatrix}$$

$${\color{Blue}\begin{Bmatrix} U_x(\tilde{x})\\ U_y(\tilde{x}) \end{Bmatrix}= \mathbf{R}^T\mathbf{\mathbb{N}}(\tilde{x})\mathbf{\tilde{T}}^{(e)}\mathbf{d}^{(e)}}$$

Equation One: $$ \begin{Bmatrix}U_x(\tilde{x})\\\mu_y(y)\end{Bmatrix} = R^T \begin{Bmatrix}U(\tilde{x})\\V(\tilde{x}) \end{Bmatrix}$$

Equation Two: $$ \begin{Bmatrix}U(\tilde{x}\\V(\tilde{x})\end{Bmatrix} = \begin{bmatrix}N_1 & 0 & 0 & N_4 & 0 & 0\\ 0 & N_2 & N_3 & 0 & N_5 & N_6\end{bmatrix} \begin{Bmatrix}\tilde{d}_1\\\tilde{d}_2\\\tilde{d}_3\\\tilde{d}_4\\\tilde{d}_5 \\\tilde{d}_6\end{Bmatrix}^{(e)} $$

Equation Three: $$ \begin{Bmatrix}U_x(\tilde{x})\\U_y(\tilde{x})\end{Bmatrix} = R_{2x2}^TN_{2x6}(\tilde{x})\tilde{T}^{(e)}d^{(e)} $$

Once again, we check the dimensional analysis:

$$ [U]=[V]=L $$

$$ [N_1]=[N_4]=1 $$

$$ [N_1][d_1]+[N_4][d_4]=L+L $$

d = displacement or rotation

displacement:

$$ [d_1]=[d_2]=[d_4]=[d_5]=L $$

rotation:

$$ [d_3]=[d_6]=\frac{L}{[\theta] } =\frac{L}{rad}=L $$

Equations of V and V' of Ni
Looking at N5:

$$V(0)=0,V'(L)=-1$$

$$V'(0)=0,V(L)=0 $$

Looking at N6:

$$V(0)=0,V(L)=0 $$

$$V'(0)=0,V'(L)=1 $$

Derived Coefficient in $$ \tilde{k}$$:

$$ \tilde{k}_{22}=\frac{12EI}{L^3}=\int_{0}^{L}{\frac{d^2N_2}{dx^2}(EI)\frac{d^2N_2}{dx^2}dx} $$

$$ \tilde{k}_{23}=\frac{6EI}{L^2}=\int_{0}^{L}{\frac{d^2N_2}{dx^2}(EI)\frac{d^2N_2}{dx^2}dx} $$

Elastodynamics
Studying frames, beams and trusses in 3-D

discrete PVW:

$$w\cdot{(m\ddot{d}+kd-F)}=0$$, for all w

$$d(0)$$ is denoted d0

and $$\dot{d}(0)$$ is denoted V0

Method:
1) Complete ordinary diff. equations (ODE's)

(2nd order in time) and initial conditions are used

Next: Governing the elastodynamics of discrete and continuous problems

For Unforced Vibration:

general equation:

$$ M\ddot{V}+kV=0$$

Assume:

$$ v(t)=\sin(wt)\phi$$

$$-w^2\sin(wt)M\phi+\sin(t)k\phi=0$$

$$ -w^2k\phi =w^2M\phi $$ general Eigenvalue

General Form Eigenvalue:

$$ A_x=\lambda B_x$$

$$I=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$$

And:

$$ \lambda=w^2$$

2) Model Superposition

$$ M\phi=\lambda^{-1}k\phi $$

finally,

$$ M(\varepsilon,\ddot{\xi},\phi)+k(\varepsilon,\dot{\xi},\phi)=F $$

Functions
The functions PlaneTrussElement.m, NodalSoln.m, PlaneTrussResults.m were all called in the code for the 5 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.

MATLAB Code to analyze a varying area and modulus of elasticity
Below is the code for a truss system that contains a varying modulus of elasticity and varying area.

Results from the Varying system
>> k_local =

0.375      -0.375       -0.375        0.375

k =

0.28125     0.16238     -0.28125     -0.16238      0.16238      0.09375     -0.16238     -0.09375     -0.28125     -0.16238      0.28125      0.16238     -0.16238     -0.09375      0.16238      0.09375

k_local =

4.875      -4.875       -4.875        4.875

k =

2.4375     -2.4375      -2.4375       2.4375      -2.4375       2.4375       2.4375      -2.4375      -2.4375       2.4375       2.4375      -2.4375       2.4375      -2.4375      -2.4375       2.4375

K =

Columns 1 through 5

0.28125     0.16238     -0.28125     -0.16238            0      0.16238      0.09375     -0.16238     -0.09375            0     -0.28125     -0.16238       2.7187      -2.2751      -2.4375     -0.16238     -0.09375      -2.2751       2.5313       2.4375            0            0      -2.4375       2.4375       2.4375            0            0       2.4375      -2.4375      -2.4375

Column 6

0           0       2.4375      -2.4375      -2.4375       2.4375

R =

0    0     0     7     0     0

d =

0           0        9.337       11.158            0            0

reactions =

-4.4378     -2.5622       4.4378      -4.4378

results =

3.4162      5.1244       5.1244       0.6437       3.2185        6.276

>>

2 Bar Frame System
Below is the code for solving the 2 bar system as a frame. The displaced results are shown in the plot the system in the section below.

Results from Frame System
>> k_h =

0.09375    -0.09375     -0.09375      0.09375

k_h =

1.2188     -1.2188      -1.2188       1.2188

K_avg =

Columns 1 through 5

0.070313    0.040595    -0.070313    -0.040595            0     0.040595     0.023438    -0.040595    -0.023438            0    -0.070313    -0.040595      0.67969     -0.56878     -0.60938    -0.040595    -0.023438     -0.56878      0.63281      0.60938            0            0     -0.60938      0.60938      0.60938            0            0      0.60938     -0.60937     -0.60938

Column 6

0           0      0.60938     -0.60937     -0.60938      0.60937

R =

0    0     0     7     0     0

d =

0           0       37.348       44.631            0            0

reactions =

-4.4378     -2.5622       4.4378      -4.4378

results =

13.665      20.497       20.497       2.5748       12.874       25.104

>>

Plots Comparing Different 2 Bar Truss Systems
Below is the code to plot the three 2 bar truss systems that were analyzed above. The dotted line is the original 2 bar truss system, the solid blue line is the deformed original system, the red line is the deformed varying 2 bar truss system, and the green is the frame system.

Electric Pylon Frame Analysis
The following code attempts to analyze the electric pylon with the frame technique.

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   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  -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    -0.008012     0.011078    0.0038609    0.0079472   -0.0026624     0.012219    0.0047374    -0.017665   0.00096509   0.00047387    -0.009129   -0.0035153    -0.002223    0.0069672   -0.0068487    0.0092304     0.002932    0.0098031  -0.00061368     0.017688    0.0044436    -0.017665   0.00096509   0.00047387    -0.009129   -0.0032428  -0.00042462    0.0074362   -0.0055158    0.0071134    0.0018676      0.01193   0.00018008     0.020205    0.0034991    -0.017665   0.00096509   0.00047387    -0.009129   -0.0034892    0.0014233    0.0075404   -0.0041829    0.0049963   0.00080324     0.014056   0.00095992     0.018184    0.0017389    -0.017665   0.00096509   0.00047387    -0.009129   -0.0040526    0.0029032    0.0073444   -0.0030439    0.0031872  -0.00010632     0.015874    0.0028869     0.013143  -0.00040913    -0.017665   0.00096509   0.00047387    -0.009129   -0.0045661    0.0043338     0.006219   -0.0018321    0.0012626   -0.0010739     0.017807    0.0067302    0.0039641   -0.0024242    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Columns 8 through 14

0.0066546    0.018818   -0.0072996     0.021862     0.015029    -0.015567    -0.013693    -0.014606     -0.01366     0.014578    -0.018684     -0.01909     0.026004      0.02282    -0.007662     0.017069    0.0086158     0.024413     0.014404     0.016827      0.02152    -0.015646    0.0099807     0.015714     0.023035     0.017577     0.022624     0.023672   0.00057936    0.0052153   -0.0013058  -0.00028167   -0.0042208   -0.0033513   -0.0055633    -0.012112   -0.0078425     0.011319    -0.008877     -0.01045     0.016385     0.013797  -5.195e-006    0.0062641  5.8781e-005    0.0018738   -0.0032159    0.0019347    0.0019017 -0.0068736 -0.00084155    0.0073392    0.0011035  -0.00035545     0.013768     0.014389  -0.00056303    0.0047573    0.0013334 -4.0125e-006   -0.0043484    0.0065872    0.0073062 -0.012905   0.0050606     0.012051     0.012486    0.0096137     0.014271     0.011743   0.00045534   -0.0076039  -0.00051373     -0.01946    -0.017665    0.0013262   -0.0011838    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0.00090541    0.0024708   -0.0008352    0.0048302  -0.00022073   -0.0018631   0.00089764    -0.010716  -0.00061777    0.0049711    0.0010323   0.00016098   -0.0068412   -0.0094633    0.0018121    0.0041723   -0.0010859    0.0074942  -0.00032555   0.00064445    0.0041615    -0.009112    -0.011122    0.0044496   -0.0078457    0.0019855   -0.0086869    -0.012305    0.0015331     0.012465   -0.0007433     0.015077    0.0015499    0.0003494  -0.00090943   -0.0057456       0.0148    0.0016457     0.010274   -0.0026705   -0.0032403    -0.010062    0.0015559      0.01156  -0.00076085     0.013719    0.0014912    0.0004219   0.00010342   -0.0099528    0.0079996     0.003304    0.0089868   -0.0024377   -0.0077263   -0.0097806   0.00056878     0.011285   -0.0011587     0.013286    0.0014097   -0.0009701    0.0021191    -0.010692   -0.0087847    0.0044835   -0.0069132    0.0029417     -0.01184    -0.015201   0.00061577     0.011764   -0.0012315     0.013934    0.0014093  -0.00070067    0.0028758   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-0.00035847   -0.0020919   -0.0047725   -0.0062583     0.011705     0.012888      0.00285    0.0023535     0.013259    -0.014923   -0.0039882    -0.016594    -0.024673     0.027007    0.0014884   -0.0023001   -0.0034625   -0.0062617     0.011893     0.011306    0.0031097    0.0024398     0.014651    -0.010587   -0.0061214   -0.0097526   -0.0033283   -0.0021663    0.0024289   -0.0027026   -0.0019896   -0.0060187     0.011988    0.0083103    0.0046988    0.0094551     0.021195   -0.0062669    -0.012177   -0.0069277    0.0023437    -0.010074    0.0020794   -0.0050287   -0.0019391   -0.0036606     0.011002    0.0058826    0.0043496     0.016292     0.022583   -0.0029114    -0.014072   -0.0031642    0.0091915    -0.014098   0.00083095   -0.0075368  -0.00055325   -0.0018046     0.010501   0.00060967     0.004508     0.021099     0.011645   -0.0038619   -0.0082723  -0.00021821     0.016225    -0.014995   -0.0012132   -0.0076824    0.0018541   -0.0013014    0.0098783    -0.006317    0.0025992  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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      0.01505   0.00089382    0.0032012    0.0026687   -0.0020297      0.01066    0.0095801    0.0057458    0.0031065     0.011512   -0.0048719    0.0031567    -0.024386     0.024395    -0.019224    0.0036904     0.019809   -0.0026503  -0.00082283    0.0024218    -0.011856    0.0067476  -0.00051099    0.0053281   -0.0025344   0.00055909   -0.0092062

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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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





MediaWiki vs. E-Learning
Both systems are completely different and are intended for different uses. Wiki is a great way to share ideas with people around the world and E-learning is a great way to turn in papers, labs, look up class grades, etc. But ultimately we feel like neither of the two systems were intended to be used to teach an advanced engineering course.

Due to Wiki, countless hours were spent trying to figure out how to input some obscure symbol into the editor. Instead of learning the finite element method, we learned how to somewhat use Wiki. Wiki provides great sources of information but it is not intended as a means of teaching a course, especially a course which requires computation and working out practice problems. It can be used as a supplement to the course but not the entire course. Ultimately, we feel like ourlearning experience in the class was hindered do to trying to learn the wiki system. The focus of the class should of been more on the Finite Element Method instead of MediaWiki.

We feel we would have been better served had the course been taught the traditional method of assigning actual HW problems and using E-learning or Wikiversity only as a supplement to the class. That is how most of the other engineering courses are taught in the department. Very few professors rely heavily on E-learning. E-learning has wonderful benefits such as posting announcements, HW assignments, taped lectures, pdf files, well organized course pages, the power of forums and many other helpful aspects. E-learning is a system that has helped in many of our classes.

Wiki on the other hand was given to us as a blank slate where we had to develop and teach ourselves. Since the goal of this course was for us the students to regurgitate every detailed note given in class, wiki was more suitable than e-learning. However, the course webpage was hardly useful, It was impossible to relocate important links to exam solutions or peer evaluation forms and all the notes we wasted hours retyping/creating could have been done as powerpoint files and posted on an e-learning hosted course page. The wiki system could have worked but only if it had been used in a different way.

Contributing Members
EML4500.F08.RAMROD.B 21:12, 9 December 2008 (UTC)

EML4500.f08.RAMROD.E 02:28, 9 December 2008 (UTC)

Eml4500.f08.RAMROD.F 18:11, 9 December 2008 (UTC)

Eml4500.f08.RAMROD.A 20:58, 9 December 2008 (UTC)

Eml4500.f08.ramrod.D 19:31, 9 December 2008 (UTC)

Eml4500.f08.ramrod.c 20:16, 9 December 2008 (UTC)