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User:Eas4200c.f08.gator.edwards/Chapter One Notes

Emphasis of the course includes: the understanding of mechanics, the ability to formulate problems, to be able to judge correctness of solutions, and to avoid the old ad hoc methods of structural analysis.

There are two basic aircraft design goals: we want the aircraft to be both light and strong.

The desired aircraft material properties include: high stiffness, high strength, and lightweight.

Stiffness is the Young's Modulus  $$\displaystyle E$$ (slope) in linear relation between stress  $$\displaystyle \sigma$$ and strain  $$\displaystyle \epsilon$$, i.e.,   $$\displaystyle \sigma$$ =  $$\displaystyle E$$  $$\displaystyle \epsilon$$

Strength includes yield stess  $$\displaystyle \sigma$$ Y, and ultimate stress (rupture stress)  $$\displaystyle \sigma$$ u.

Toughness is the material's ability to resist fracture and damage. This is also called fracture toughness.

Some high stiffness and high strength materials include: steel alloys, titanium alloys (which have lower stiffness and strength than steel alloys), and aluminum alloys (which have lower stiffness and strength than titanium alloys).

A high stiffness and low toughness material is glass.

Some low stiffness and high toughness materials include: plastic and nylon. An example of a material with good fracture toughness is aluminum (which has lower stiffness, strength, and fracture toughness than steel alloys) and it is used in aircraft skin.

Some high stiffness and high toughness materials include: composite materials.
 * composite materials- fiber reinforced composites where the material used as the base material (called the matrix) is reinforced with many fibers (which increases strength and stiffness).

There are two aspects of an aircraft structure we will consider: its geometry and the material that it is made out of. Two styles of the geometry of the aircraft is monocoque and semi-monocoque.

Some other considerations that need to be made for aircraft construction include the fact that the geometry is limited by aerodynamic considerations such as lift and drag of the airfoil. Also the fact that we want the aircraft to be used to its full potential and for there not to be any wasted weight. Optimizing the aircraft materials used is also important. This can be done by replacing aluminum and titanium with fiber reinforced composites, which saves up to 30 to 40 percent in weight.

 References

Boeing 787 in Wikipedia

Metal Matrix Composite

Carbon Fiber Reinforced Plastic

Monocoque


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User:Eml4500.f08.gravy.psc

Verify the dimensions of all the terms in the stiffness matrix multiplied by the displacement matrix
Note: The square brackets, in dimensional analysis, indicate that the dimension of the enclosed are being found. Note: The unit 'F' stands for force, such as newtons or pounds-force. Note: The unit 'L' stands for length, such as inches, meters, or something similar.

To find the dimensionals, first we need the equations for the values, which is given by $$\tilde{k_{ij}}$$ and $$\tilde{d_{j}}$$.

$$\tilde{k_{ij}}\tilde{d_{j}}=\begin{bmatrix} \frac{EA}{L} & 0 & 0 & -\frac{EA}{L} & 0 & 0 \\ 0 & \frac{12EI}{L^{3}} & \frac{6EI}{L^{2}} & 0 & -\frac{12EI}{L^{3}} & -\frac{6EI}{L^{2}} \\ 0 & \frac{6EI}{L^{2}} & \frac{4EI}{L} & 0 & -\frac{6EI}{L^{2}} & \frac{2EI}{L} \\ -\frac{EA}{L} & 0 & 0 & \frac{EA}{L} & 0 & 0 \\ 0 & -\frac{12EI}{L^{3}} & \frac{6EI}{L^{2}} & 0 & \frac{12EI}{L^{3}} & \frac{6EI}{L^{2}} \\ 0 & -\frac{6EI}{L^{2}} & \frac{2EI}{L} & 0 & \frac{6EI}{L^{2}} & -\frac{4EI}{L} \end{bmatrix} \begin{Bmatrix} \tilde{d_{1}} \\ \tilde{d_{2}} \\ \tilde{d_{3}} \\ \tilde{d_{4}} \\ \tilde{d_{5}} \\ \tilde{d_{6}} \end{Bmatrix}$$

The dimensions of the displacements $$\tilde{d_{j}}$$ are split into two different groups. The first group, where j = 1, 2, 4 and 5, are transverse displacements. The dimensional analysis is shown below:

$$[\tilde{d_{1}}] = [\tilde{d_{2}}] = [\tilde{d_{4}}] = [\tilde{d_{5}}] = L$$

The displacements $$\tilde{d_{j}}$$ where j = 3 and 6 are rotational displacements. The dimensions of rotational displacements are unit less, which will be shown as '1'. Shown below is the dimensional analysis.

$$[\tilde{d_{3}}] = [\tilde{d_{6}}] = 1$$

Other important units are:

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

$$[A] = L^{2}$$

$$[I] = L^{4}$$

Using the unit analysis listed above, and applying it to the different values used in the stiffness matrix multiplied by the displacement matrix are:

$$[\frac{EA}{L}] = \frac{F}{L}$$

$$[\frac{EI}{L}] = FL$$

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

$$[\frac{EI}{L^{3}}] = \frac{F}{L}$$

The only other value used by the the stiffness matrix multiplied by the displacement matrix are numbers, including 'zero' and negitives. All numbers, like the rotational displacements, are dimensionless, therefore their units are equal to '1'.

Listed below are all of the non-zero values and their associated units from the top half of the resultant matrix.

$$ [\tilde{k_{11}}\tilde{d_{1}}] = [\frac{EA}{L} * L] = F $$ $$ [\tilde{k_{14}}\tilde{d_{4}}] = [\frac{EA}{L} * L] = F$$ $$ [\tilde{k_{22}}\tilde{d_{2}}] = [\frac{EI}{L^2} * 1] = FL$$ $$ [\tilde{k_{23}}\tilde{d_{3}}] = [\frac{EI}{L^3} * L] = F$$ $$ [\tilde{k_{25}}\tilde{d_{5}}] = [\frac{EI}{L^3} * L] = F$$ $$ [\tilde{k_{26}}\tilde{d_{6}}] = [\frac{EI}{L^2} * 1] = FL$$ $$ [\tilde{k_{32}}\tilde{d_{2}}] = [\frac{EI}{L^2} * L] = FL$$ $$ [\tilde{k_{33}}\tilde{d_{3}}] = [\frac{EI}{L} * 1] = FL$$ $$ [\tilde{k_{35}}\tilde{d_{5}}] = [\frac{EI}{L^2} * L] = FL$$ $$ [\tilde{k_{36}}\tilde{d_{6}}] = [\frac{EI}{L} * 1] = FL$$ $$ [\tilde{k_{41}}\tilde{d_{1}}] = [\frac{EA}{L} * L] = F$$ $$ [\tilde{k_{44}}\tilde{d_{4}}] = [\frac{EA}{L} * L] = F$$ $$ [\tilde{k_{52}}\tilde{d_{2}}] = [\frac{EI}{L^3} * L] = F$$ $$ [\tilde{k_{53}}\tilde{d_{3}}] = [\frac{EI}{L^3} * L] = F$$ $$ [\tilde{k_{55}}\tilde{d_{5}}] = [\frac{EI}{L^3} * L] = F$$ $$ [\tilde{k_{56}}\tilde{d_{6}}] = [\frac{EI}{L^2} * 1] = FL$$ $$ [\tilde{k_{62}}\tilde{d_{2}}] = [\frac{EI}{L^2} * L] = FL$$ $$ [\tilde{k_{63}}\tilde{d_{3}}] = [\frac{EI}{L} * 1] = FL$$ $$ [\tilde{k_{65}}\tilde{d_{5}}] = [\frac{EI}{L^2} * L] = FL$$ $$ [\tilde{k_{66}}\tilde{d_{6}}] = [\frac{EI}{L} * 1] = FL$$

N(i)
In order to solve the system, we need to express $$ u\bigl(\begin{matrix}x\end{matrix}\bigr)$$ in terms of $$d_{i}=u\bigl(\begin{matrix}x_{i}\end{matrix}\bigr)$$ and $$d_{i+1}=u\bigl(\begin{matrix}x_{i+1}\end{matrix}\bigr)$$ as a linear function in x (i.e. linear interpolation)

$$u\bigl(\begin{matrix}x_{i}\end{matrix}\bigr)=N_{i}(x)d_{i}+N_{i+1}(x)d_{i+1}$$

where $$N_{i}\bigl(\begin{matrix}x\end{matrix}\bigr)$$ and $$N_{i+1}\bigl(\begin{matrix}x\end{matrix}\bigr)$$ are linear functions of x.

From this plot it can be easily observed that:

$$ N_{i}(x)=\frac{x-x_{i}}{x_{i+1}-x_{i}} $$

and

$$ N_{i+1}(x)=\frac{x-x_{i+1}}{x_{i}-x_{i+1}} $$

Consider EA = Const
HW 6: Consider EA = constant, then the equation for k simplifies to:

$$ k^{(i)} {=} \frac{EA}{L^{(i)}} \begin{vmatrix} 1 & -1 \\ -1 & 1 \end{vmatrix} $$

27.1 - $$(AB)^T = B^T * A^T$$
Prove: $$ (A_{i x j} B_{j x k})^T = B^T A^T$$

Although this will work for any i, j or k selected, for ease of proving, a low value for each was selected and filled in randomly.

The results were $$ A_{2x3} $$ and $$ B_{3x3} $$with values of:

$$ A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{bmatrix} $$

$$ B = \begin{bmatrix} 7 & 8 & 9\\ 1 & 2 & 3\\  4 & 5 & 6 \end{bmatrix} $$

Step 1: Multiply A and B which gives:

$$ AB = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{bmatrix} \begin{bmatrix} 7 & 8 & 9\\ 1 & 2 & 3\\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} 21 & 27 & 33\\ 57 & 72 & 87 \end{bmatrix} $$

Step 2: Take the Transpose of the resultant matrix:

$$ (AB)^T = \begin{bmatrix} 21 & 57\\ 27 & 72\\ 33 & 87 \end{bmatrix} $$

Step 3: Solve the right side of the equation:

Step 3.a: Solve $$B^T$$

$$ B^T = \begin{bmatrix} 7 & 1 & 4\\ 8 & 2 & 5\\ 9 & 3 & 6 \end{bmatrix}$$

Step 3.b: Solve $$A^T$$

$$ A^T = \begin{bmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{bmatrix}$$

Step 4: Multiply the two resultant matrices together:

$$ B^T A^T = \begin{bmatrix} 7 & 1 & 4\\ 8 & 2 & 5\\ 9 & 3 & 6 \end{bmatrix}\begin{bmatrix}1 & 4\\ 2 & 5\\ 3 & 6 \end{bmatrix} = \begin{bmatrix} 21 & 57\\ 27 & 72\\ 33 & 87 \end{bmatrix}$$

Since both sides give the same answer, the equation $$ (A_{i x j} B_{j x k})^T = B^T A^T$$ is valid for all i, j and k.

27.4
Show that the following equation is true:

$$( T^{(e)}W)\cdot [\hat{k}^{(e)}q^{(e)} - p^{(e)}] = 0 $$

$$( T^{(e)}W)\cdot [\hat{k}^{(e)}(T^{(e)}d^{(e)}) - p^{(e)}]=0 $$

$$(T^{(e)}W)^T [\hat{k}^{(e)}(T^{(e)}d^{(e)})-p^{(e)}] = 0 $$

By applying the relation shown above in Homework 27.1, the equation above can be simplified to:

$$ W^{T}T^{(e)T}[\hat{k}^{(e)}(T^{(e)}d^{(e)}) - p^{(e)}] = 0 $$

$$ W^{T}[T^{(e)T}\hat{k}^{(e)}(T^{(e)}d^{(e)}) - T^{(e)T}p^{(e)}] = 0 $$

$$ W^{T}[(T^{(e)T}\hat{k}^{(e)}T^{(e)})d^{(e)} - T^{(e)T}p^{(e)}] = 0 $$

The following proof is then used: []

$$\ W^{T}[ k^{(e)}d^{(e)} - T^{(e)T}p^{(e)}] = 0 \,$$

$$\ W^{T}[k^{(e)}d^{(e)} - f^{(e)}] = 0 \,$$

Then, apply the dot product property shown above in reverse, and the follow equation comes forth:

$$ W \cdot [k^{(e)}d^{(e)} - f^{(e)}] = 0 $$

Which was proved to be true earlier in this report.

HW 4 -20.4
Verify that the equation: $$ \mathbf k^{(e)} = \tilde \mathbf T^{(e)^{T}}\tilde \mathbf k^{(e)}\tilde \mathbf T^{(e)}$$ is true.

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

$$\mathbf k^{(e)} = k^{(e)}\begin{bmatrix} l^{(e)} & -m^{(e)} & 0 & 0 \\ m^{(e)} & l^{(e)} & 0 & 0 \\ 0 & 0 & l^{(e)} & -m^{(e)} \\ 0 & 0 & m^{(e)} & l^{(e)} \end{bmatrix}\begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}\begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0 \\ -m^{(e)} & l^{(e)} & 0 & 0 \\ 0 & 0 & l^{(e)} & m^{(e)} \\ 0 & 0 & -m^{(e)} & l^{(e)} \end{bmatrix}$$

$$\mathbf k^{(e)} = \underset{}{k^{(e)}} \begin{bmatrix} (l^{(e)})^{2} & l^{(e)}m^{(e)}  &  -(l^{(e)})^{2}  &  -l^{(e)}m^{(e)}   \\ l^{(e)}m^{(e)} &  (m^{(e)})^{2}  &  -l^{(e)}m^{(e)}  &  -(m^{(e)})^{2}   \\ -(l^{(e)})^{2} & -l^{(e)}m^{(e)}  &  (l^{(e)})^{2}  &  l^{(e)}m^{(e)} \\ -l^{(e)}m^{(e)} &  -(m^{(e)})^{2}  &  l^{(e)}m^{(e)}  &  (m^{(e)})^{2} \\ \end{bmatrix} $$

The above is the definition of k(e), therefore the equation is verified and true.

HW 4 - 23.2
Derive Equation 2 :

$$f_4^{(1)}+f_2^{(2)}=P $$  (2)

Writing the forces in terms of displacements and element stiffness:

$$f_4^{(1)}=\begin{bmatrix} k_{41}^{(1)}d_{1}^{(1)} + k_{42}^{(1)}d_{2}^{(1)} + k_{43}^{(1)}d_{3}^{(1)} + k_{44}^{(1)}d_{4}^{(1)} \end{bmatrix}$$

$$ f_2^{(2)}=\begin{bmatrix} k_{21}^{(2)}d_{1}^{(2)} + k_{22}^{(2)}d_{2}^{(2)} + k_{23}^{(2)}d_{3}^{(2)} + k_{24}^{(2)}d_{4}^{(2)} \end{bmatrix}$$

Substituting the previous two equations into Equation 2: $$ \begin{bmatrix} k_{41}^{(1)}d_{1}^{(1)} + k_{42}^{(1)}d_{2}^{(1)} + k_{43}^{(1)}d_{3}^{(1)} + k_{44}^{(1)}d_{4}^{(1)} \end{bmatrix}+\begin{bmatrix} k_{21}^{(2)}d_{1}^{(2)} + k_{22}^{(2)}d_{2}^{(2)} + k_{23}^{(2)}d_{3}^{(2)} + k_{24}^{(2)}d_{4}^{(2)} \end{bmatrix}= P $$

Then, replace the elemental d with the global equivalent:

$$ \begin{bmatrix} k_{41}^{(1)}d_1 + k_{42}^{(1)}d_2 + k_{43}^{(1)}d_3 + k_{44}^{(1)}d_4\end{bmatrix} + \begin{bmatrix} k_{21}^{(2)}d_3 + k_{22}^{(2)}d_4 + k_{23}^{(2)}d_5 + k_{24}^{(2)}d_6 \end{bmatrix}= P$$

Combing terms of the same d value and then turn that equation into a multiplication of matrices to form:

$$\ \begin{bmatrix} K_{41}^{(1)}& K_{42}^{(1)} & (K_{43}^{(1)} + K_{21}^{(2)}) & (K_{44}^{(1)} + K_{22}^{(2)}) & K_{23}^{(2)} & K_{24}^{(2)}  \end{bmatrix} \begin{Bmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5 \\ d_6 \end{Bmatrix} = P \,$$

Which is the fourth row of the $$\ \underline{K} \underline{d} = \underline{F} \,$$ equation.

Combining the Results of Eq 1 and Eq 2, and plugging them into a single equation gives the following:

$$\ \begin{bmatrix} K_{31}^{(1)} & K_{32}^{(1)} & (K_{33}^{(1)} + K_{11}^{(2)}) & (K_{34}^{(1)} + K_{12}^{(2)}) & K_{13}^{(2)} & K_{14}^{(2)} \\ K_{41}^{(1)}& K_{42}^{(1)} & (K_{43}^{(1)} + K_{21}^{(2)}) & (K_{44}^{(1)} + K_{22}^{(2)}) & K_{23}^{(2)} & K_{24}^{(2)}  \end{bmatrix} \begin{Bmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5 \\ d_6 \end{Bmatrix} = \begin{Bmatrix} 0 \\ P \end{Bmatrix}\,$$

HW2 MATLab Redo HW problem
To solve the system above using an automated FEM process, we've turned to MATLab. Below is the code that was used:

In order to run this code, there are a few functions that are needed. All these functions were downloaded from the book's Website. They are listed below.

After running the preceding code, this was the output results: (Note: code was changed from MATLab laungue to a generic matrix in order to be easier to read.)

14-3 HW problem
Verify the following equation:

$$\ \underline{k}^{(e)} = \underline{T}^{(e)T} * \underline{k}^{carrot(e)} * \underline{T}^{(e)}$$

Where: $$\ \underline{T}^{(e)} = \begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0\\ 0 & 0 & l^{(e)} & m^{(e)} \end{bmatrix}$$ and $$\ \underline{k}^{carrot(e)} = k^{(e)} * \begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix}$$

Since $$k^{(e)}$$ is a scaler, we can move it within the equation without changing the problem. So after moving the scaler and inserting in the matrices, the following equation is found:

$$\ \underline{k}^{(e)} = k^{(e)} * \begin{bmatrix} l^{(e)} & 0\\ m^{(e)} & 0\\ 0 & l^{(e)}\\ 0 & m^{(e)}\end{bmatrix} * \begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix} * \begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0\\ 0 & 0 & l^{(e)} & m^{(e)} \end{bmatrix}$$

Solving the first set of matrices gives:

$$\ \underline{k}^{(e)} = k^{(e)} * \begin{bmatrix} l^{(e)} & -l^{(e)}\\ m^{(e)} & -m^{(e)}\\ -l^{(e)} & l^{(e)}\\ -m^{(e)} & m^{(e)}\end{bmatrix} * \begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0\\ 0 & 0 & l^{(e)} & m^{(e)} \end{bmatrix}$$

Solving the second set of matrices gives:

$$\ \underline{k}^{(e)} = k^{(e)}\begin{bmatrix} (l^{(e)})^2 & l^{(e)}m^{(e)} & -(l^{(e)})^2 & -(l^{(e)}m^{(e)})  \\  l^{(e)}m^{(e)} & (m^{(e)})^2 & -(l^{(e)}m^{(e)}) & -(m^{(e)})^2  \\ -(l^{(e)})^2 & -(l^{(e)}m^{(e)}) & (l^{(e)})^2 & l^{(e)}m^{(e)}\\ -(l^{(e)}m^{(e)}) & -(m^{(e)})^2 & l^{(e)}m^{(e)} & (m^{(e)})^2 \end{bmatrix} $$

Which is the definition of the stiffness matrix, therefore the equation is true.

MATLAB code 15-3

The above picture is the output graph from MATLab after the above code was run. The solid line represents the deformed truss, while the dashed lines show the initial, or un-deformed, truss.

HW#1 - M-Files
M-files M-files are are stored files usually ending with the “.m” extension. They allow MATLAB to run an entire preprogrammed series of commands. They effectivly make up the backbone of MATLAB, and most work is usually done to create or edit these files. There are two main types of m-files: script files and function files.

Script Files Script files are a type of m-file that contains a sequence or normal MATLAB commands. They are normally used for the creation and mataince of large variables, such as matrices. The varibable created in a script file are global. To call a script file, simply enter a command into MATLAB that is the name of the script file with the “.m” extension left off. For example, if the file is named “example1.m” than typing in the command “example1” will cause the script file to run.

Function Files Function files are the true brains behind MATLAB. They allow for the creation of any type of function programmable, which greatly extends the cabalities of MATLAB. Function files, unlike script files, default to creating local variables. However they can be declared global if the need arises. A function files starts with the line “function variable=execute(a,b)”. The word “function” tells MATLAb that the following is a function. “variable” is the output argument, and the name can be changed to be more descriptive. “execute(a,b)” is the name of the function (execute) and the variables that need to be passed into the function (a,b). The list of variables (a,b), called the input arguments, can be expanded or deleted to fit the need of the function. Function files, like all m-files, are able to use and call other files within them, allowing for a greater range of programming options.

An example of a function file is shown below. Remember that the “%” symbol allows for comments without affecting any code. MATLAB ignores all statements after the % symbol.

function a=randint(m,n) %RANDINT Randomly generated integral matrix. %       randint(m,n) returns an m-by-n matrix with entries %      between 0 and 9 a=floor(10*rand(m,n));

Putting the command “z=randint(3,6)” into MATLAB will return a 3-by-6 matrix variable named “z”. The great advantage to function files is the ability to expand and create huge programs. This allows for the creation of many smaller function, similar to the randint(a,b) created before, to be combined into a large file that can be programmed to do thousands of different things. Also, there is the ability to combine self-made functions with some of MATLAB’s built-in functions, much like the built in function “rand(x,y)” was used above.

A useful command to find out a list of all the functions that MATLAB has access to, a combination of built-in and user programmed functions, can be seen by typing the command “type functionname”.

HW#2 - Lectures 8 and 10
Lecture 8

$$\underline{Global F-D Relationship}$$

Noting the fact that the equations for the elemental $$\underline{d^{(e)}}$$ and global $$\underline{d}$$ overlap, as seen in the below picture, the following equations can be derived.

$$d_1 = d_1^{(1)}$$ $$d_2 = d_2^{(1)}$$ $$d_3 = d_3^{(1)} = d_1^{(2)}$$ $$d_4 = d_4^{(1)} = d_2^{(2)}$$ $$d_5 = d_3^{(2)}$$ $$d_6 = d_4^{(2)}$$

Below is a representation of the global $$\underline{K}$$ matrix, which is the stiffness matrix for the system. The two colored boxes represent the contributions of the elemental $$\underline{k^{(e)}}$$. The $$\underline{k^{(1)}}$$ matrix is shown in blue, while the red box represents $$\underline{k^{(2)}}$$.

Lecture 10 $$\underline{Elimination of Known Degrees of Freedom}$$

Knowing that global nodes 1 and 3 are actually fixed points, it is seen that there is no possibility of them moving. Therefore, the displacement at those points is zero.

$$ d_1 = d_2 = d_5 = d_5 = d_6 = 0 $$

$$\ \underline{d}=\begin{Bmatrix} d_1 \\ d_2  \\  d_3 \\ d_4 \\ d_5 \\ d_6 \end{Bmatrix}= \begin{Bmatrix}  0 \\ 0 \\  d_3 \\ d_4 \\ 0 \\ 0 \end{Bmatrix}$$

Since 4 of the values for $$\underline{d}$$ are zero, we can reduce the system. Turning $$\underline{d}$$ into a 2x2 matrix, would mean that you would be able to reduce $$\underline{K}$$ to a 6x2 matrix. This is down by removing the columns that are associated with the rows within $$\underline{d}$$ that are zero. The new $$\underline{K}$$ is shown below.

$$\ \underline{K}=\begin{bmatrix} K_{13} & K_{14} \\ K_{23} & K_{24} \\  K_{33} & K_{34} \\ K_{43} & K_{44} \\ K_{53} & K_{54} \\ K_{63} & K_{64} \end{bmatrix}$$

This makes the F-D equation: $$\ \underline{K}_{6x2} \underline{d}_{2x1} = \underline{F}_{6x1} \,$$

The next step is to see that since global nodes 1 and 3 are fixed, we can able the concept of "Virtual Work" which will allow us to ignore their effects for the moment and let us turn $$\underline{F}$$ into a 2x1 by removing the components from those fixed nodes. However, by doing this, we need to adjust $$\underline{K}$$ to keep the equation true. We can further reduce $$\underline{K}$$ down to a 2x2, and the following equation results:

$$\ \underline{K}_{2x2} * \underline{d}_{2x1} = \underline{F}_{2x1} \,$$

Where $$\underline{K}$$ is equal to:

$$\ \underline{K}=\begin{bmatrix} K_{33} & K_{34} \\ K_{43} & K_{44} \end{bmatrix}$$ Since $$\underline{F}$$ is now only made up of the forces from global node 2, we now know the value of $$\underline{F}$$. $$\ \underline{F} = \begin{Bmatrix} F_3 \\ F_4 \end{Bmatrix}= \begin{Bmatrix}  0 \\ P \end{Bmatrix}$$ We also can solve for $$\underline{K}$$: $$\ \underline{K}=\begin{bmatrix} 3.0625 & 2.1752 \\ 2.1752 & 2.6875 \end{bmatrix}$$

At this point, the only term not known in the global F-D equation is $$\underline{d}$$. In order to solve the system, we must first solve for $$\underline{d}$$, which gives the following equation: $$\ \underline{d}_{2x1} = \underline{K}^{-1}_{2x2} * \underline{F}_{2x1} \,$$

Where $$\underline{K^{-1}}$$ is equal to: $$\ \underline{K^{-1}} = \frac{1}{det(\underline{K})}*\begin{bmatrix} K_{44} & -K_{43} \\ -K_{34} & K_{33} \end{bmatrix}$$

$$\ \underline{d} = \begin{Bmatrix} 4.352 \\ 6.1271 \end{Bmatrix}$$

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User:Eml4500.f08.bottle.brockmiller

Homework
HW1

HW2

HW3

HW4

HW5

HW6

HW7

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User talk:Eml4500.f08.group.cady

 Hello Eml4500.f08.group.cady, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 23:29, 10 September 2008 (UTC)

title: User talk:Eml4500.f08.group.cady


text:
User:Eml4500.f08.gator.mueller

Matlab Tutorial

title: User:Eml4500.f08.gator.mueller


text:
User talk:EML4500.f08.wiki.Lopez

 Hello EML4500.f08.wiki.Lopez, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
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 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
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 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 23:29, 10 September 2008 (UTC)

title: User talk:EML4500.f08.wiki.Lopez


text:
User talk:Eas4200c.f08.vqcrew.a

 Hello Eas4200c.f08.vqcrew.a, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
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 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 23:30, 10 September 2008 (UTC)

title: User talk:Eas4200c.f08.vqcrew.a


text:
User talk:Eml4500.f08.gator.mueller

 Hello Eml4500.f08.gator.mueller, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 23:31, 10 September 2008 (UTC)

title: User talk:Eml4500.f08.gator.mueller


text:
User:Eas4200c.f08.blue.a/hw1lecnotes

Contact Information
Instructor: Dr. L. Vu-Quoc

 * Telephone: 392-6227
 * Email: vu-quoc@ufl.edu
 * Course Website: EGM4200c Aerospace Structures
 * Office: NEB 135 (New Engineering Building)
 * Classroom: MAE-A 303
 * Class Time: MWF, period 4 (10:40 am - 11:30 am)
 * Office Hours: MWF, period 9 (4:05 pm - 4:55 p)

Brief Summary of Course Objectives
The objective of the course "Aerospace Structures" is to establish an understanding of the mechanics of aerospace structures. With this, the student will develop the ability to formulate problems pertaining to aerospace structures and judge the correctness of the solutions formulated. In developing these methods, old methods will be avoided and partial differential equations will be solved by new methods like the finite element method.

Informative Links
| Aerospace Engineering

| Aerospace Structures

| MAE at The University of Florida

The use of Wikipedia for Homework Reports
Setting up wikipedia accounts: Format: Username: EAS4200c.f08.team_name.last_name Password: Up to user

Special Note: Class wikipedia accounts should NOT be used for editing pages other than the class assigned pages. Failure to follow these instructions could lead to punishment.

Wikipedia will be used for the submission of homework reports. The reports will consist of 2 parts. The first being lecture note re-writes and the second part being homework problems.

Students are expected to attend all lectures and take good, accurate notes. The notes should then be compiled with the other group members and posted on the group's wikipedia page. On the page, the notes should be organized according to subject, not lecture number. It is important not to simply re-write the notes but also add to them. The instructor will make suggestions on where to contribute to the notes. Examples of contributions include, explanations of phenomena and derivations of equations.

The Vision for EAS 4200c
The Big Picture

Wikipedia is part of the Mediawiki software. The Mediawiki software is used in many different programs and websites and allows for collaboration from all users from any location in the world. Learning the use of this software will not only allow students to succeed in this class, but also make other contributions on other Mediawiki projects.

Confidentiality

Because some students have expressed concern over having their last names in the wikipedia usernames, groups have the option of using letters (i.e. a,b,c,d,e) to identify their individual userpages.

Method of Work

The new method of work is as follows: 31% for group homewoek reports, and 23% for each of three exams. Also during each exam, students will fill out group evaluations about their fellow group members. These will factor into the fonal homework grades.

Old Approach

The old approach which didn't utilize the Mediawiki software had a different grading scale. The old system had homework worth 10%, and each of the exams are worth 30%. For now, we will try the new method, where homework is worth 31%, and each exam is worth 23%. If at any point during the year, a majority of students feel like the new method doesn't work, we can revert to the old method. We will give it, "the old college try."

E-Learning

Dr. Vu-Quoc has decided not to use the e-learning system provided by UF for this class. He is not familiar with the software and feels that the Mediawiki software is perfect for the course plan.

Collaboration

This class involves collaboration among several groups. The first is collaboration among the students in the group. Next, is the collaboration among the class groups. Lastly is collaboration among all of the groups and the professor. By making all the class notes availiable to everyone, it will enhance the learning experience.

Introduction to Aerospace structures
The goals of aerospace structures is to design aircraft that are very light yet also strong. this is accomplished through the use of materials that have a high stiffness, high strength, and lightweight.

The stiffness of a material is determined by $$E$$ The young's modulus of the material which is relates the stress and strain on the material through hooke's law $$\sigma=E*\epsilon$$ hooke's law is a linear equation and the Young's modulus is the slope of the equation. This means for the higher the Young's modulus the faster the stress will increase meaning little deformation will be able to occur in the material before the yield stress is reached.

The strength of a material is determined by $$\sigma_y $$, $$ \sigma_u$$ which are the yield stress and the ultimate stress respectively.

The higher the stresses that the material is able to handle before yielding the stronger the material.

Toughness is also considered in materials because it is the ability of the material to resist fracturing. Which means that a material with a high toughness would be able to resist fracturing very well. An example of a very stiff material but has a very low toughness would be glass.

Two aspects of aircraft design are the geometry and materials used. The geometry for the aircraft body is constructed by a thin wall shell design instead of an internal truss design. Aerodynamics also play a larger role in the geometry of the aircraft. Materials that are used need to be used to their full capacity to avoid more weight than necessary in the design. Also when able replace aluminum and titanium with fiber reinforced composites which will reduce weight by approximately thirty to forty percent.

Book pg 1-3

Problem 1.1

The problem is to find the optimal $$b/a$$ in a thin walled beam to maximize the load-bearing capacity of a beam while assuming that the moment (M) is equal to the torsion (T).This is done by optimizing the cross section that carries the maximum bending moment and the maximum torsion.

Finding the Solution begins with the following assumptions: 1. The wall thickness is much less than the width and height of the beam (t<<a, t<<b)  2. The Moment on the beam is equal to the Torsion on the beam (M=T)  3. The allowable stress is equal to twice the allowable shear ($$\sigma _{allowable}= 2 \tau _{allowable}$$)  4. The beam is rectangular in shape, ie. the perimeter of the beam face (L) is equal to twice the sum of length a and b (L=2(a+b)).

Next, the shear stress ($$\tau$$) due to Torsion (T) is found. First, the total Torsion is found by separating the beam into four different elements, finding the Torsion in each, and adding them together.

$$T=T_{AB}+T_{BC}+T_{CD}+T_{DA}$$

Knowing that $$\tau =\frac{\nu }{t}$$, we are able to solve for Torsion in each element of the area of the beam.

$$T_{AB}=\frac{b}{2}\times (\nu \times a)=\frac{1}{2}\tau abt$$

$$T_{BC}=\frac{a}{2}\times (\nu \times a)=\frac{1}{2}\tau abt$$

$$T_{CD}=T_{BC}$$

$$T_{DA}=T_{CD}$$

$$T=2 \tau abt$$

$$\tau =\frac{T}{2abt}$$

We consider two cases for the solution.

Case1

In Case 1 we assume that the bending normal stress ($$\sigma $$) reaches the allowable stress ($$\sigma _{allowable}$$). The goal is to verify that the shear stress in the beam ($$\tau $$) is less than or equal to the allowable shear stress ($$\tau _{allowable}$$). The first step in this case is to define the bending moment as follows:

$$\sigma =\frac{Mz}{I}$$

Where M is the bending moment, z is the ordinate of a point adn I is the second area moment of inertia defined as: $$I=\int \int_{A}^{}{z^{2}dydz}$$

Because the shear stress is greatest at the top and bottom of the beam section (where z = b/2), the equation of shear stress can be rewritten as follows: $$\sigma = \frac{Mz}{I}  \Rightarrow   M=\frac{2I\sigma\ _{allowable}}{b} $$

Thus, it is necessary to find the value of I in terms of b (recalling that a can be written as a function of b: a=L/2-b). From the problem parameters we know that for this instance, Mmax = Tmax. Because Mmax has been determined, it must only be put into the relation:

$$\tau _{max}=\frac{T^{(1)}_{max}}{2a^{(1)}b^{(1)}t}=\frac{M^{(1)}_{max}}{2a^{(1)}b^{(1)}t}$$

Now check the values of $$\tau _{max}$$ with the value of $$\tau _{allowed}$$.If $$\tau _{max}$$ is larger than $$\tau _{allowable}$$ the optimal ratio of $$\frac{b^{(1)}}{a^{(1)}}$$would be acceptable. But $$\tau _{max}$$ is smaller than $$\tau _{allowable}$$, then the optimal ratio of $$\frac{b^{(1)}}{a^{(1)}}$$ would be unacceptable

Case 2

For this case we assume that the shear stress is eqaul to the maximum allowable shear stress in the section. Then we will test to see if the normal stress falls within acceptable tolerances (notably, less than $$\sigma _{max}$$)

By the equation given earlier in the problem the value of T becomes:

$$T=(2t\tau _{allowable})(ab)$$

Where the dimensions of the rectangle ab are equal to one another, hence a square. Therefore if the total perimeter of the square is L and all sides are equal, both a and b are equal to L/4. Plugging this quantity into the previous equation and using the given relation that M=T, we have:

$$M^{(2)}_{max}=\frac{1}{8}tL^{2}\tau _{allowable}$$

Using this relation and the previously defined equation for normal stress:

$$\sigma ^{(2)}_{max}=\frac{M^{(2)}_{max}b^{(2)}}{2I^{(2)}}$$

We can find $$\sigma ^{(2)}_{max}$$ and compare it with the allowable normal stress. If the maximum normal stress is less than the allowable shear stress, then the ratio of b to a for case 2 is acceptable. But, if it is greater, then that ratio is not correct.

Solving Case 1

First the moment of inertia, I, must be computed. This is done by summing the moments of inertia of each of the four parts of the section. In order to do this, the Parallel axis theorem must be applied. Solving for the horizontal axis yields:

$$I=\frac{tb^{2}}{6}(3a+b)$$

And with the relation a=L/2-b, the plot of f(b)=I/b is shown to the right.

Taking the derivative of the function f(b) yields the maximum value of the curve to be b(1)=3L/8. Thus the ratio of $$\frac{b^{(1)}}{a^{(1)}}$$ is equal to 3 and the value of a(1) is found to be L/8. Solving for $$\tau _{max}$$ with the values of a(1) and b(1) yield:

$$\tau _{max}=\frac{M^{(1)}_{max}}{2a^{(1)}b^{(1)}t}=\frac{32M^{(1)}_{max}}{3tL^{2}}$$

but because this larger than half of $$\tau _{allowable}$$ this case is not possible

Team Members That Contributed to This Report
Mark Barry Eas4200c.f08.blue.a 18:20, 18 September 2008 (UTC)

Clay Robertson Eas4200c.f08.blue.f 05:11, 19 September 2008 (UTC)

Christian Garabaya Eas4200c.f08.blue.d 05:24, 19 September 2008 (UTC)

Jeffrey Shea Eas4200c.f08.blue.e 12:13, 19 September 2008 (UTC)

Marc Pelletier Eas4200c.f08.blue.b EAS4200c.f08.blue.b 15:24, 19 September 2008 (UTC)

Thomas Sexton EAS4200c.f08.blue.c 16:16, 19 September 2008 (UTC)

title: User:Eas4200c.f08.blue.a/hw1lecnotes


text:
User:Eml4500.f08.gator.mueller/Matlabtutorial

Relations

Many of the relational operators used in MATLAB are used very similar to the operators used in C++ to compare variables. The advantage to using MATLAB for relational operators is when matrices are being used. In a programming language you would need to write a for loop when comparing matrices instead of just a single line of code.

The relational operators are:
 * < (less than)
 * > (greater than)
 * <= (equal or less than)
 * >= (equal or greater than)
 * == (equal to)
 * ~= (not equal to)
 * & (and)
 * | (or)
 * ~ (not)

When using these statements, the line of code will return a 0 or 1 representing false and true respectively.


text:
User:Eml4500.f08.jamama.mueller

Matlab Tutorial

title: User:Eml4500.f08.jamama.mueller


text:
User:Eml4500.f08.jamama.mueller/Matlabtutorial

Relations

Many of the relational operators used in MATLAB are used very similar to the operators used in C++ to compare variables. The advantage to using MATLAB for relational operators is when matrices are being used. In a programming language you would need to write a for loop when comparing matrices instead of just a single line of code.

When using these statements, the line of code will return a 0 or 1 representing false and true respectively.

Scalar Functions

Many functions in MATLAB use matrices but some operate on each element in the matrix called scalar functions.

Vector Functions

Vector functions in MATLAB operate on each column of the matrix. So if A is a 3X3 matrix min(A) would return a 1X3 matrix with each value the minimum of their column. If the matrix is only one row but multiple columns such as the return from the min(A) function, the minimum of the row is found and returned by the function. So if the user wanted to find the minimum of a matrix, min(min(a)) should be used so the matrix can be of any size and the desired effect is achieved.

M-files

Script Files

Script files consist of normal MATLAB commands in an m-file. To run a script in the working directory in MATLAB, simply type the name of the m-file and all of the lines in the script file will be run. Script files are very helpful when tedious data entry is required.

Function Files

This allows users to create their own custom functions in MATLAB if the existing functions are not enough or if the user would like to modularize their MATLAB program. All variables are local by default (only seen within the function) but can be made global (able to be seen by all functions) by using the global command.

Text string, error messages, and input

In MATLAB variables can be set to string values using single quotes. Also, display messages and error messages for MATLAB programs use the same way of defining string values. Input from users can also be taken in string format and saved in a string type variable.

Managing M-files

Comparing the efficiency of algorithms in MATLAB

When using MATLAB there are many different ways to get to the same result in your program. The program can be made more efficient by changing the way a certain operation is completed for the faster and more efficient way. This can be discovered using flops and the tic toc feature.

Output Format

All of the calculations in MATLAB are done in double precision, but sometimes for certain programs the data needs to be formatted. For instance, if the value returned be a function is an amount of money it only makes sense to show it as 2 decimal places.

17. Hardcopy

The list of commands can be written to a file using the diary command. Diary can be turned off by typing diary off and then typing diary on when done.


title: User:Eml4500.f08.ateam.shah/teammembers


text: File:Truss FBD 1.jpg

title: File:Truss FBD 1.jpg


text:
User talk:Eml4500.f08.jamama.mueller

 Hello Eml4500.f08.jamama.mueller, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 00:23, 11 September 2008 (UTC)

title: User talk:Eml4500.f08.jamama.mueller


text: File:Truss FBD 2.jpg

title: File:Truss FBD 2.jpg


text: File:FD Relation.jpg

title: File:FD Relation.jpg


text: File:FD Relation 2.jpg

title: File:FD Relation 2.jpg


text: File:FD Relation Matrix.jpg

title: File:FD Relation Matrix.jpg


text:
User talk:Eml4500.f08.bike.sweeting

 Hello Eml4500.f08.bike.sweeting, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 01:08, 11 September 2008 (UTC)

title: User talk:Eml4500.f08.bike.sweeting


text:
Favourite presentations

The idea for this "pedagogy" learning project is create and discuss a list of favourite presentations.

These presentations can be personal favourites - not everyone has like to like them. But the point is really to explore and explain why each of the suggested presentations "works" and/or why they don't "work".

The purpose of this project is develop a collaborative, inspiring resource for teachers and presenters, with lots of examples, about ways to "present" a topic to an "audience".

Favourite presentations
This section is for suggesting favourite presentations and anyone can add notes analysing the presentation.

Hardt (2005) - Identity 2.0
Identity 2.0. See also Lecture 2.0
 * What works
 * Relatively small amount of content per slide makes for a faster-moving, visually-alive presentation
 * What doesn't

What works and what doesn't
This section is for synthesising a generic set of observations or tips about what works and what doesn't work for presenting.

What works?

 * 1) Relatively small amount of content per slide makes for a faster-moving, visually-alive presentation

Michael De Percy and Amanda Burells idea of lectures as a performance, in my opinion certainly works. After watching Michael present yesterday I would happily go to any "performance" he gave on a topic. As a student it is refreshing to see that kind of commitment and personality from a lecturer rather than the monotonous reading off a slide.

James Neills audio streaming also is an idea to consider as it provides a service for those that cannot make it to the lecture however I don't think it allows for the feedback and communicaton between students and lecturer as well as a lecture does. I also think the lecture provides an arena to meet other students and to make social connections, it allows conversation during breaks or after the lecture and allows an interaction between students about the topic.

I know James also has an assessment piece that is an e-portfolio on wikiversity. If students would fully participate it is an excellent learning tool, the problem is getting everyone to do that.

What doesn't?

 * 1) Reading directly off the slides with no extra information or expansion in a monotone. Students WILL go to sleep!
 * 2) Too many presentations, one after the other.

Some places to search for presentations

 * http://slideshare.net - most favourited
 * http://teacher.tv
 * http://ted.com
 * http://www.wepapers.com

title: Favourite presentations


text:
User:Eml4500.f08.bottle.brockmiller/HW1

Chapter 4 of the textbook Finite Element Analysis With Mathematica and Matlab Computations visits the Finite Element Method approach to understanding trusses, beams, and frame systems. Often, engineers and other problem solvers are faced with statically indeterminate systems. In other words, using standard knowledge of statics and mechanics of materials will not be able to understand these systems. The Finite Element Method is a different way of understanding these systems with the use of matrix algebra. Basically, the method rather than analyzing the system as a whole, parts of the system are analyzed separately. This chapter first treats plane truss systems. Unlike beams and frames, trusses are only subjected to axial loads. It is assumed that they are elastic bars that are deformable.

When faced with a problem dealing with truss systems, it is important to clearly understand the background of the problem and what is being asked. Diagrams are especially useful in visualizing the problem when they are properly labeled. Starting with the entire system itself, a clearly and properly marked global free body diagram (FBD) is necessary. Included in this diagram would be the global coordinate system, loads, reactions, degrees of freedom, and labels for each node and member of the system. Examples of degrees of freedom are the known, fixed degrees of freedom and constraints, or the unknown, often displacements. It is standard to label the global nodes using numbers with circles encircling the numbers. The members of the system should be labeled with numbers encircled by triangles. From this, one can determine whether or not the system is statically indeterminate.

Once the system has been determined statically indeterminate, one may proceed with Finite Element Analysis. Since the members of the system will be analyzed separately, element free body diagrams should be created for each of the members of the system. This process also has its own standard labeling method. Although the nodes should already have been labeled encircled by circles, labeling the local nodes encircled with a square helps clarify the diagram. The reactions, loads, and degrees of freedom should be written the same except for the global nodes that experience the loads, once should choose one member of the system that undergoes the load while the other member doesn't. The local coordinate system may be the same as the global coordinate system, or one may be chosen to further simplify the problem.

Recalling back to fundamental physics, the equation that describes the force (F) of a spring with a spring constant (k) where one end is fixed and the other goes through a displacement (x) is F=kx. However, this single equation can not describe the motion of a spring where both ends are not fixed. This is a statically indeterminate system, therefore the Finite Element Method must be used. The motion of this scenario can be described

$$\begin{Bmatrix} F_1 \\ F_2 \end{Bmatrix} = \begin{bmatrix} k & -k \\ -k & k \end{bmatrix}\begin{Bmatrix} d_1 \\ d_2 \end{Bmatrix}$$

where Fi is the force of node i=1,2, k is the stiffness matrix, and di is the displacement of node i=1,2. This can also be applied to truss systems in a similar manner. However many degrees of freedom determines the size of the matrix. In the spring scenario, there were two degrees of freedom creating a 2X1 force matrix, a 2X1 displacement matrix and a 2X2 stiffness matrix. In truss systems that are unconstrained the stiffness matrix will be of dimensions mXn, where m is the number of unknown displacement degrees of freedom and n is the number of both known and unknown degrees of freedom. Once the known degrees of freedom are eliminated the force and displacement matrix will be mX1, and the stiffness matrix will be mXm. For consistency purposes, the element stiffness and force matrices will be situated in the global coordinate system. Once the stiffness and displacement matrices are established, one may calculate the reactions (the unknown forces).

title: User:Eml4500.f08.bottle.brockmiller/HW1


text:
User:Eml4500.f08.team.guan/2bar

In order to analyze a truss, first a truss must be defined. A truss is a structural object in which its members or bars carry only axial loading. Members that carry loads normal to their longitudinal axis which are loads that cause normal bending stresses are called beams. The global free body diagram of an unconstrained system is shown below. The term global means pertaining to the entire structure. The unconstrained condition means that the system is a "free,free" system in which the determinant of the global sfiffness matrix $$\displaystyle K$$is zero.

Before any analysis can be done, a labeling convention must be established. The global nodes of the system are labeled with numbers circumscribed in circles, the elements are labeled with numbers circumscribed by triangles, and local nodes (elemental nodes) are labeled with numbers circumscribed by squares. Upon drawing the global, unconstrained free body diagram of the truss system, the global displacement degrees of freedom and the global forces must be indicated. In the two-bar truss system above, there are six degrees of freedom, two at each node. Actually, the displacement dofs are partitioned into 1) a known part(i.e. fixed dofs, constraints) 2) an unknown part (these are the dofs solved by the finite element method). Similarly with the global forces, they are partitioned into a 1) known part (applied forces) and 2) an unknown part (reaction forces). Here, all the forces are drawn in accordance with the global coordinate system chosen to be a convenient system in solving the problem. The global coordinate system may or may not be different than local coordinate systems. It all depends on the convenience of calculation. Draw the local elements. Label the elemental displacement dofs and the local forces. The directions of these can be in either local or global coordinate systems. Again, the choice of the coordinate system is for convenience of calculation. A picture of element 1 of the previous two-bar truss is shown below. Now, the global force-displacement relation can be established. The element stiffness matrices, and the element force matrices must be assembled in global coordinates. $$\displaystyle F=Kd$$ Elimination of known dofs to reduce the global force-displacement relation will transition the system into a constrained system where the determinant of the stiffness matrix will not be zero. The stiffness matrix will be non singular and the system can be solved with a system of linear equations. Also, the k matrix is invertible. "n" designates the total number of displace dofs (unknown and known) where "m" designates the number of unknown displacement dofs. Solve for the d matrix by multiplying the F matrix with the inverted sitffness matrix. Finally, compute element forces form the now known d matrix and info about the stresses can be found. Compute the reaction (unknown) forces.

title: User:Eml4500.f08.team.guan/2bar


title: File:2bartruss.JPG


text:
Lecture as performance

"Lecture as performance" refers to the bringing together of dramaturgy and pedagogy in the context of lectures. Thus, lectures can be approached as a drama, and dramatic theories and techniques can be utilised to facilitate affective, social, and intellectual engagement with the topic.

title: Lecture as performance


text:
User talk:Petm1

 Hello Petm1, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 02:38, 11 September 2008 (UTC)

Fundamental entity
How would you describe the "Fundamental entity" page? Is this physics, philosophy, creative writing? --JWSchmidt 02:44, 11 September 2008 (UTC)

I think the page is a question needing an answer. My essay is philosophy in so far as it is more a feeling than a vision, physics because it is about motion and I guess creative writing just so people could read about it. jem

Wiki links
Hi, I have seen you editing this page:. Note that tha wikilinks are beter. So instead of dog i is better to have the wikilink dog. Note that "w" is the abreviation for Wikipedia project, ":" characterises the main namespace and "dog" the page, so "w:dog" is the adress, where the link should go. Than you just place "|" and everything in front of it is just visible on the page (i.e. cat will link me also to the Wikipedia article about the dog, even it will look like this cat).--Juandev 09:11, 25 January 2009 (UTC)

Thank you, it does make it appear cleaner jem

title: User talk:Petm1


text:
User:Eas4200c.f08.wiki.d

Aerospace Structures - EAS 4200C

Team Wiki

/HW1

/HW2

/HW3

/HW4

/HW5

/HW6

/HW7

title: User:Eas4200c.f08.wiki.d


text:
User talk:Eas4200c.f08.wiki.d

 Hello Eas4200c.f08.wiki.d, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 02:45, 11 September 2008 (UTC)

title: User talk:Eas4200c.f08.wiki.d


text:
User:Eas4200c.f08.blue.a/Lecture 5

 * Hey Mark, I hope this is ok. I'm having trouble uploading the images but will add them asap *************

Yea dude, this looks fine. If you can shoot me an e-mail when you get the picts up. I'll check it later 2nite anyway. Thanks Mark

 Problem 1.1

The problem is to find the optimal $$b/a$$ in a thin walled beam to maximize the load-bearing capacity of a beam while assuming that the moment (M) is equal to the torsion (T).This is done by optimizing the cross section that carries the maximum bending moment and the maximum torsion.

Finding the Solution begins with the following assumptions: 1. The wall thickness is much less than the width and height of the beam (t<<a, t<<b)  2. The Moment on the beam is equal to the Torsion on the beam (M=T)  3. The allowable stress is equal to twice the allowable shear ($$\sigma _{allowable}= 2 \tau _{allowable}$$)  4. The beam is rectangular in shape, ie. the perimeter of the beam face (L) is equal to twice the sum of length a and b (L=2(a+b)).

Next, the shear stress ($$\tau$$) due to Torsion (T) is found. First, the total Torsion is found by separating the beam into four different elements, finding the Torsion in each, and adding them together.

$$T=T_{AB}+T_{BC}+T_{CD}+T_{DA}$$

Knowing that $$\tau =\frac{\nu }{t}$$, we are able to solve for Torsion in each element of the area of the beam.

$$T_{AB}=\frac{b}{2}\times (\nu \times a)=\frac{1}{2}\tau abt$$

$$T_{BC}=\frac{a}{2}\times (\nu \times a)=\frac{1}{2}\tau abt$$

$$T_{CD}=T_{BC}$$

$$T_{DA}=T_{CD}$$

$$T=2 \tau abt$$

$$\tau =\frac{T}{2abt}$$

We consider two cases for the solution.

Case1

In Case 1 we assume that the bending normal stress ($$\sigma $$) reaches the allowable stress ($$\sigma _{allowable}$$). The goal is to verify that the shear stress in the beam ($$\tau $$) is less than or equal to the allowable shear stress ($$\tau _{allowable}$$). The first step in this case is to define the bending moment as follows:

$$\sigma =\frac{Mz}{I}$$

Where M is the bending moment, z is the ordinate of a point adn I is the second area moment of inertia defined as: $$I=\int \int_{A}^{}{z^{2}dydz}$$

title: User:Eas4200c.f08.blue.a/Lecture 5


text:
User:Eml4500.f08.delta 6.lopez/Week 1: Lecture Notes, Matlab (Contribution)

Section 6: For, While, and If Statements
The following MATLAB flow control statements operate similarly to most computer languages.

For a given n, the statement below will produce a certain n-vector, which in this case is given a value of 10. EDU>> x=[];

EDU>> for i = 1:10, x=[x,i^2],end

x =

1

x =

1    4

x =

1    4     9

x =

1    4     9    16

x =

1    4     9    16    25

x =

1    4     9    16    25    36

x =

1    4     9    16    25    36    49

x =

1    4     9    16    25    36    49    64

x =

1    4     9    16    25    36    49    64    81

x =

1    4     9    16    25    36    49    64    81   100

The next code produces another n-vector using different parameters.

EDU>> x=[];

EDU>> for i = 1:5

x=[x,i^2]

end

x =

1

x =

1    4

x =

1    4     9

x =

1    4     9    16

x =

1    4     9    16    25

This next code produces the result as above, but in reverse order.

EDU>> x=[];for i = 5:-1:1, x=[x,i^2], end

x =

25

x =

25   16

x =

25   16     9

x =

25   16     9     4

x =

25   16     9     4     1

The statement below will produce the mxn hilbert matrix. For this example, m and n are given a value of 5. This allows for any matrix to be formed, unlike the 1:n statement.

EDU>> for i=1:5

for j= 1:5

H(i,j)=1/(i*j-1);

end

end

EDU>> H

H =

Inf   1.0000    0.5000    0.3333    0.2500

1.0000   0.3333    0.2000    0.1429    0.1111

0.5000   0.2000    0.1250    0.0909    0.0714

0.3333   0.1429    0.0909    0.0667    0.0526

0.2500   0.1111    0.0714    0.0526    0.0417

Once a matrix is defined, the following statements compute the sum of matrix A by adding all of the column sums. The only difference between the two statements are the amount of time taken to complete these computations.

EDU>> A=[1 2 3 4;5 6 7 8]; EDU>> s=0; EDU>> for c=A s=s+sum(c); end EDU>> s

s =

36

EDU>> sum(sum(A))

ans =

36

The next type of MATLAB function is the WHILE statement. For a given number a, the following statement will produce the smallest nonnegative number n such that 2^n>=a.

EDU>> n=0;

EDU>> while 2^n> n

n =

4

The final MATLAB statement is the IF statement. This type of statement will only work if the statement is true. The following example illustrates multiple branching.

EDU>> if n<0

parity = 0;

elseif rem(n,2)==0

parity = 2;

else

parity = 1;

end

EDU>> parity

parity =

2

EDU>> n

n =

4

The remainder function produces the amount left over when using division. The example below shows the remainder of four divided by two.

EDU>> rem(4,2)

ans =

0

This example illustrates the result of using the remainder function without fully completing the argument.

Many relational operators are available for use in MATLAB. A chart has been formed listing the most common operations, and a short description of each.

The following statements are examples of the various relational operators in use. An answer of 1 signifies a true statement, while an answer of 0 illustrates a false statement.

EDU>> 3<5

ans =

1

EDU>> 3>5

ans =

0

EDU>> 3==5

ans =

0

EDU>> 3==3

ans =

1

When applying relational operators to matrices, the true statements are illustrated by a 1 or nonzero. The following examples depict three scenarios of using relational operators on matrices.

EDU>> a=rand(5)

a =

0.0357   0.7431    0.0318    0.6948    0.3816

0.8491   0.3922    0.2769    0.3171    0.7655

0.9340   0.6555    0.0462    0.9502    0.7952

0.6787   0.1712    0.0971    0.0344    0.1869

0.7577   0.7060    0.8235    0.4387    0.4898

EDU>> b=triu(a)

b =

0.0357   0.7431    0.0318    0.6948    0.3816

0   0.3922    0.2769    0.3171    0.7655

0        0    0.0462    0.9502    0.7952

0        0         0    0.0344    0.1869

0        0         0         0    0.4898

EDU>> a==b

ans =

1    1     1     1     1

0    1     1     1     1

0    0     1     1     1

0    0     0     1     1

0    0     0     0     1

Section 7: Scalar Functions
Some functions in MATLAB are scalar in nature, but operate element-wise when used in a matrix. The most common of these functions are found below.

=Section 8: Vector Functions=

MATLAB functions operate essentially on a vector (row or column). A row by row application can be obtained by using the transpose, i.e. mean(A’)’. As shown the mean(A’)’ function determines the mean of the matrix in a row by row fashion.

EDU>> mean(A')'

ans =

5    5     5

Moreover, to find the maximum entry in a matrix the max(max(A)) function should be used rather than max(A). Here it is shown that the maximum value for the matrix A is equal to 9.

EDU>> max(max(A))

ans =

9

=Section 9: Matrix Functions=

The matrix functions in MATLAB offer the programs greatest asset. Some of the most useful functions are tabulated below.

Once a matrix is defined, the eig(A) function is used to find the eigenvalues of a given matrix.

EDU>> A

A = 8    1     6

3    5     7

4    9     2

EDU>> eig(A)

ans =

15.0000

4.8990

-4.8990

The [U,D] = eig(A) function, on the other hand, produces a matrix U where each column signifies an eigenvector for matrix A and a matrix D containing the eigenvalues of matrix A along its diagonal. The matrix A that is used for these computations is the same as above. EDU>> [U,D]= eig(A)

U =

-0.5774  -0.8131   -0.3416

-0.5774   0.4714   -0.4714

-0.5774   0.3416    0.8131

D =

15.0000        0         0

0   4.8990         0

0        0   -4.8990


text:
Talk:MATLAB

This is an old wikipedia page
This page appears to be have copied whole from Wikipedia on 17 May, 2007. From the wikipedia history, it appears that this version was copied: http://en.wikipedia.org/w/index.php?title=MATLAB&oldid=131403607. This is over a year ago, and there have been hundreds of edits since on Wikipedia, but only a couple of very minor edits here on Wikiversity. So, it seems to me that it might better "blank" this page, returning it to a stub-state, with a link the current Wikipedia article. What do you think. I partly raise this by the way because there is a currently considerable editing going on some some user pages, with student notes from a course they are doing on using MATLAB. -- Jtneill - Talk - c 03:48, 11 September 2008 (UTC)

Templates
I propose to create a tutorial that consists of MATLAB code that students can copy, paste, and then modify. Each lesson will consist of three or four "code snippets" embedded in a case structure. As the student reads the lesson, he/she can run the entire code and choose which case to view. The intent is for students to read a snippet and predict the outcome. Running the code will allow students to verify that they understand the code.--guyvan52 (discuss • contribs) 13:06, 25 September 2014 (UTC)

title: Talk:MATLAB


text:
User:Eas4200c.f08.vqcrew.d

 * User:Eas4200c.f08.vqcrew.d/hw1
 * User:Eas4200c.f08.vqcrew.d/hw2
 * User:Eas4200c.f08.vqcrew.d/hw3
 * User:Eas4200c.f08.vqcrew.d/hw4
 * User:Eas4200c.f08.vqcrew.d/hw5
 * User:Eas4200c.f08.vqcrew.d/hw6
 * User:Eas4200c.f08.vqcrew.d/hw7

title: User:Eas4200c.f08.vqcrew.d


text:
User:Eas4200c.f08.vqcrew.d/hw1

Problem 1.1
Given Figure 2, find the optimum ratio $$\frac{b}{a}$$  to maximize the load bearing capability of the thin-walled beam.

The assumptions for this problem are as follows:
 * 1) The cross-sectional perimeter is constant, i.e. $$L=2(a+b)=Constant$$
 * 2) The torque $$T$$ is equal to the bending moment $$M$$, i.e. $$T = M$$
 * 3) The maximum allowable bending normal stress is twice the maximum allowable shear stress, i.e. $$\sigma_{allow} = 2\tau_{allow}$$

The assumption that $$t$$ is very small compared to $$a$$ and $$b$$ is also valid.

Since the cross-section walls are very thin, the shear stress can be assumed to be uniform along the walls, as shown in the shear flow diagram in Figure 3. The shear distribution and resultant shear vector $$V$$ for a small piece of the wall is also shown. This distribution is also mechanically equivalent to the linear distribution shown in Figure 4, where $$\tau =\frac{V}{t}$$.

The total torque ($$T$$) can be described as the sum of the torques on each wall of the box beam as follows: $$T = T_{AB}+T_{BC}+T_{CD}+T_{DA}$$

$$T_{AB}=(\frac{b}{2})Va=\frac{1}{2}\tau tab$$

$$T_{BC}=(\frac{a}{2})Vb=\frac{1}{2}\tau tab$$

$$T_{CD}=(\frac{b}{2})Va=\frac{1}{2}\tau tab$$

$$T_{DA}=(\frac{a}{2})Vb=\frac{1}{2}\tau tab$$

Therefore, $$T=2t\tau ab$$, and $$\tau =\frac{T}{2abt}$$.

There are two cases that must be analyzed in order to solve the problem:
 * 1) Assume the bending normal stress (σ) reaches $$\sigma_{allow}$$ first, then verify that $$\tau \le \tau_{allow}$$.
 * 2) Assume the shear stress (τ) reaches $$\tau_{allow}$$ first, then verify that $$\sigma \le \sigma_{allow}$$.

Case 1

Case 1 begins by recalling the formula for bending normal stress: $$\sigma =\frac{Mz}{I}$$, where $$M$$ is the bending moment, $$z$$ is the ordinate of a point on the axis perpendicular to the bending neutral axis, and I is the second area moment of inertia defined by $$I=\iint_A z^2 \ dy\, dz$$. Figure 5 shows each item pictorially.

title: User:Eas4200c.f08.vqcrew.d/hw1


text: Category:Christian History

title: Category:Christian History


text:
User:Zmadscientist

Bob Chesebrough - zmadscientist - is a senior course architect for Intel Software College (ISC) and has worked at Intel since 2000.

He was the primary editor and compiler of the Blackbelt user's guide to Intel Compiler Switches and has been a manager, a compiler technical consultant, an instructor and a Course Architect during his stint so far at Intel.

ISC's mission is to help faculty and the broader community to bring the powerful message of multi-core and many core computing to students around the globe.

Bob is particulalry instrested in how academia can modify exisiting curriculum to address current industry changes in software and computing especially the changes around the advent of multiple core architectures.

Bob can be contacted directly at robert.a.chesebrough@intel.com

title: User:Zmadscientist


text:
User talk:Zmadscientist

 Hello Zmadscientist, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --Emesee 04:38, 11 September 2008 (UTC)

title: User talk:Zmadscientist


text: Category:History of Albania

title: Category:History of Albania


text:
University of Florida/Eml4500/f08.FEABBQ/HW1/Matlab

Homework 1: Matlab Section
BACK TO FEABBQ/HW 1

Entering matrices
Matlab works completely with matrices.

Matrices are created through 3 different methods. 1. By an entered explicit list of elements 2. By generating a built-in statement or function 3. Loaded from an external file

Matlab also allows for complex numbers. You may use i or j.

If i or j are used as variables. A replacement imaginary unit may be created.

Many built in functions are available in MATLAB. The function rand may be used to created a matrix with randomly generated entries. rand (3) ans = 0.9501   0.4860    0.4565        0.2311    0.8913    0.0185        0.6068    0.7621    0.8214

rand (2,3) ans = 0.4447   0.7919    0.7382        0.6154    0.9218    0.1763

Function magic will create a martix with a magic square.

Function hilb will create a Hilbert matrix.

Individual numbers from a matrix may also be accessed by using a similar command example below.

Matrix operations, array operations
There are multiple matrix operations available. They can be used for scalars and matrices. They are

Operations can be made to operate entry wise by preceding them with a period.

[1,2,3,4].^2   ans = 1    4     9    16

[1,2,3,4].*[1,2,3,4]   ans = 1    4     9    16

Statements, expressions, variables; saving a session
MATLAB statements are usually of the form variable = expression To suppress printing of a statement you must follow the statement with a semicolon. MATLAB IS case sensitive. Therefore when creating variables keep in mind the case of the letter. The statement who will list the variables currently created and available.

To clear a variable simply write clear variable.

A runaway display can be stopped by pressing CTRL-C.

Lastly to save a session simply press the same button or go to file->save. If a session is not saved it will be lost, thus losing all variables created.

Matrix building functions
Here are few examples

B = zeros (size(A)) B = 0    0     0         0     0     0         0     0     0

diag(A) ans = 1        5         9

diag(diag(A)) ans = 1    0     0         0     5     0         0     0     9

B = [A,zeros(3,2); zeros(2,3), eye(2)] B = 1    2     3     0     0         4     5     6     0     0         7     8     9     0     0         0     0     0     1     0         0     0     0     0     1

For, while, if and relations
These are all commands that will generally be used in M-files.

FOR will set a limit to the number of iterations that will run in a file. For instance

This code will make q = x^2 as long x is between 1 and n.

WHILE will set an end to a loop by stating the values needed to exit the loop.

This code will raise n by one as long as 2^n is less then a.

IF will not run a code unless the starting conditions mach the relation given with if.

RELATIONS Here are a list of the relational operators in MATLAB

<   less than

>   greater than

<=  less than or equal

>=  greater than or equal

==  equal

~=  not equal

Relations may be connected by

&   and
 * or

~   not

Scalar functions
built in functions that operate on scalars or individual elements of a matrix. The most commonly used of these functions are as follows:

 * sin
 * asin (arcsin)
 * exp (exponential)
 * abs
 * round
 * cos
 * acos (arccos)
 * log (natural log)
 * sqrt
 * floor
 * tan
 * atan (arctan)
 * rem (remainder from division)
 * sign
 * ceil

Vector functions
functions perform actions on vectors in a "column by column" fashion and produce a row vector. The transpose can be used to operate on a row by row basis. Some of those functions are:

 * max
 * min
 * sort
 * sum
 * prod
 * median
 * mean
 * std
 * any
 * all

Matrix functions
The following is a list of matrix functions for Matlab. These functions can be accessed using different commands.
 * eig                (eigenvalues and eigenvectors)
 * chol               (cholesky factorization)
 * svd                (singular value decomposition)
 * inv                (inverse)
 * lu                 (LU factorization)
 * qr                 (QR factorization)
 * hess               (hessenberg form)
 * schur              (schur decomposition)
 * rref               (reduced row echelon form)
 * expm               (matrix exponential)
 * sqrtm              (matrix square root)
 * poly               (characteristic polynomial)
 * det                (determinant)
 * size               (size)
 * norm               (1-norm, 2-norm F-norm)
 * cond               (condition number in the 2-form)
 * rank               (rank)

Command line editing and recall
Matlab Editing and function keys:

 * left/right arrows scrolls the cursor left and right

 * up/down arrows scrolls up and down from line to line

 * backspace deletes the character left of the cursor

 * delete deletes the character left of the cursor

 * home moves the cursor to the beginning of the line

 * end moves the cursor to the end of the line

 * There is a graph for this section I just have to figure out how to add it.***
 * I guess you use the 'upload file' in the up-left corner, -andy

Submatrices and colon notation
Matlab allows for the simple creation of vectors with it's "colon notation", whose main operator is the ":", which means "everything." In the following example, a row vector is generated with a starting value of 0.2 and includes "everything" in between 0.2 and its final value, 1.2. The 0.2 in between the colons represents the step-size, meaning that the next value from the starting value is 0.2 over, and the next value after that is 0.2 over, and so on. Other examples of the use of the colon follow.

The following are more advanced uses of the colon operator.

M-files: script files, function files
Matlab has a built-in programming language, wherein commands that are usually typed at the prompt can be saved to a file (called an M-file because of its .m filename) and run all at once. There are two types of M-files: script files and function files.

A script is akin to a recipe, where commands are read top-down and executed in that order. Common commands (as well as more advanced commands) can be placed in a script so that they can be eaily executed. Inside of a script file, functions can also be called. Note that the definition of a function must exist in its own M-file, hence the second type of M-file, the function file. An example of function files follow.

Learning to write scripts and functions in M-files is essential to gaining proficiency in Matlab use.

Text strings, error messages, input
Like other programming environments/languages, Matlab has many ways of handling text. To assign a string of text to a variable, enclose the text in single quotes. This method is useful if the text is to be operated on or changed later on. To directly display text, use the  function, which is useful for simple messages. The following is a lighthearted example When programming, it is often desirable to handle errors and use text to display a readable message. The previous example shows the error message if the code was expecting a symmetric matrix and received otherwise. Another common use of text is to take in user input interactively. The previous examples takes in user input and assigns the data to the variable iter. Normal execution of the program then follows.

Managing M-files
The command  opens the M-file editor while simultaneously closing down the Matlab prompt. While this frees up RAM, it may be inconvenient to have to reopen Matlab and possibly redefine variables and other workspace entities. In order to keep Matlab open whenever running a command that would normally close Matlab, simply precede the command with an exclamation,. The following code opens the M-file editor; upon exiting the editor, Matlab is opened up in the same state as it was before closing. To sum up the introduction to M-file programming, we present common tools to use: debugging, setting directories and working with files. The following is the output of the  command.

Comparing efficiency of algorithms: flops, tic, toe
The term flops is used in computer science to designate the number of floating point operations performed (hence the name flops). In Matlab, the command  can be used to gauge how computationally intensive a section of code (or an entire M-file or script) is numerically. The usage is detailed as follows: A special note, with version 7 of Matlab, flop counts are no longer available, though another method of gauging performance of a section of code is to use the  and   commands to time the length of execution. The usage is similar to : An example of this would be
 * 1) set the number of flops to 0 with the command .  This should happen at the beginning of your code.
 * 2) At the terminating point (the point after the code whose flops are desired), simply place the command.
 * 1) At the starting point of the code to test, enter the command.
 * 2) At the point where the desired execution time is located, enter the command.

Output format
Any standard computations performed in Matlab are done in double precision. It is possible to control the appearance of the output with the following commands. The following example used is  and the format option is followed by the output,

Hard Copy
Sometimes it is desirable to obtain a hardcopy of your Matlab work, i.e. a printed version. With the exception of graphics, it is possible to save everything on screen to a file with the following command, The  function can be controlled using the commands   and , which determine when to stop and restart the recording of on-screen contents, respectively.

Graphics
Matlab is capable of producing a variety of mathematical graphs and figures in 2 and 3 dimensions. The basic commands to use these functions are:

Planar 2d Plots

The plot(x,y) function will create a linear graph if x and y are both 1d vectors of the same length. The first variable in the plot function is the horizontal axis, and the second is graphed on the vertical. The easiest way of producing a 1d vector is using colon notation:

The following is a graph of a sin wave from -4 to 4. Adding shown in the code above is the syntax for adding titles and axis labels using the title and xlabel/ylabel commands respectively.

Streamlined code exists for graphing functions as well. First, an M-file must be created such as the following:

Using this code would now create a 2d graph from -1.5 to 1.5:

2d parametric graphs can also be created in Matlab. The following code and Figure 3 demonstrate:

Annotating Graphs

Graphs are easily annotated in Matlab as well. Typing in the following commands will label the active figure. The format is like:

Other visual features are available as well:

All of these visual modifications should be performed after the plot command.

Multiple Graphs on the Same Figure

There are different ways to place multiple graphs onto the same Matlab figure. The first is to overload the  function. The following code and Figure 4 show how this can be done:

The  command can also be used. When hold is on, every plot graphed will go onto the active figure. Turning hold off will create a new figure for each new graph, which is the Matlab default.

Using Different Colors and Line Types on Graphs

Sometimes the default blue solid line in Matlab graphs can get a little boring. To add a little variety to your graphs, there are several commands that can be incorporated into the  function:

These commands are used inside single quotation marks inside the  function. would plot a dashed red line and  would plot a green dotted line.

Other Types of Graphs

There are several other types of graphs which Matlab can produce. Although this guide does not have the details for them, they can all be easily found using the  command.

 * Polar
 * Bar
 * Hist
 * Quiver
 * Compass
 * Feather
 * Rose
 * Stairs
 * Fill
 * Subplot

Graphics Hardcopy

Oftentimes it is necessary to print a graph from Matlab. This is most easily done with the convenient 'Print' menu option on the Figure. It is normally easier though to use 'Save as' and save the graph as a .jpg or other type of graphics file. For the more adventurous, Matlab has a  command which will print the active graphics item. The  option can also be used to append a different saved file to the current graphics item for printing. Just use the format.

3-D Line Plots

There are few more impressive ways to show off Matlab skills than producing colorful 3-D plots. The  command can easily plot in 3 dimensions, just like   plots in 2 dimensions. The following example shows how a 3-D plot (Figure 6) can be parametrically created:

All of the settings applicable to 2-D plots (including axis labels and commands, titles, colors, etc) can be used on a 3-D plot as well.

3-D Mesh and Surface Plots

The mesh and surf functions can create 3 dimensional representations of matrices, either with a wire mesh or solid surface. If a single matrix can be inputed into either function, the z value is the matrix value at point (x,y) (row, col). Figures 7 and 8 show the mesh and surf functions being used to graph the 'eye' (identity) matrix:

3-D functions can also be graphed using mesh and surf in conjunction with the  function. Combined with colon notation,  creates an x-y rectangle in which a function is graphed. The following code and Figure 9 show how a 3-D function can be defined and graphed:

There are several ways to change the appearance of a 3-D graph in Matlab. First of all, the  command will affect the superimposed mesh lines on a surface graph. Various color profiles can also be experimented with. The following commands can be used like this:

Any of the functions to change the appearance should be used after the graph has been made.

Graphs can also be produced parametrically with M-files. Check the guide for further information on this, and Matlab's 'Handle Graphics' which can produce advanced graphs and plots.

Sparse Matrix Computations
Matlab typically handles matrices of which the contents are mostly nonzero. These matrices are considered "dense". A matrix with mostly zero entries is considered "sparse" and is worth defining for the possibility of ignoring the zero values and operating on the nonzero values only. The benefit of this is to reduce the memory footprint of the matrix and the computation time. Suppose a matrix is created by the following command, and that the matrix is filled with mostly zeros by the following command, the output of which looks like The nonzero values of the matrix can be easily found with the command, followed by the output, The following commands will return the matrix to it's previous form and check it's storage mode, respectively Sparse matrices can also be generated directly as opposed to calling the sparse function on a full ("dense") matrix. A sparse diagonal matrix can be generated with the  command. The output is similar to calling the. are the sparse analogs of, respectively. A more specific approach would be to define the nonzero values of the sparse matrix individually. For example (followed by output) The nonzero entities were stored in the row vector. Another example of directly defining the nonzero values of a sparse matrix is The output of Matlab arithmetic and functions will be in one of the two storage modes (either Full(F) or Sparse(S)). The following defines the output results of the manipulation done to either full or sparse matrices


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User talk:Eas4200c.f08.radsam/Structures and Materials

Nice work on 1.2 Diego...looks really good.

I changed the d1-3 refs to link refs, hopefully that'll work/look better.


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University of Florida/Eml4500/f08.FEABBQ/HW1/Notes

Lecture Notes - Weeks 1,2
BACK TO FEABBQ/HW 1

The Following is a summary of the topics and information discussed during the first two weeks of EML4500 - Finite Element Analysis (FEA) at the University of Florida in the fall of 2008. A summary of all administrative information can be found  here .

Wikiversity
Wikiversity is just one of many projects formed under the parent organization of Wikimedia. Since Wikiversity is the platform in which the Professor, Dr. Loc Vu Quoc, plans on using for group collaboration and homework submission, most of the first week was dedicated to familiarizing the students with its structure.

Free Body Diagrams
Although a review from statics, the free body diagram (FBD) is an essential tool in FEA. In addition, because the systems for which FEA is used can be complex, it is necessary to form a rigorous convention for the labeling of all FBDs. Figures 1-3 below are the examples used in class to illustrate this convention.

Labeling Convention
In breaking a structure apart, it is important to differentiate between global FBDs (Fig. 2) and local FBDs (Fig. 3). For this purpose a circle is used when describing global nodes, while a square is used for local nodes. In addition, it is necessary to distinguish each piece, or element. A triangle is used for this purpose.

It is also important to assign a convention for the naming of forces and displacements. In global FBDs, as shown in Fig. 2, each force is titled R where the subscript denotes which node the force is being applied to, and the superscript shows the direction of the force. For local FBDs, as shown in Fig. 3, f denotes a force while d indicates a displacement. Here the subscript defines the particular local force or displacement, while the superscript indicates the element it corresponds to.

Force-Displacement Relation
The relationship between forces and displacements can be modeled by a combination of springs. The most simple example is a 1-D spring with one fixed end. In this case, the relationship between the force applied to the free end and the displacement of that end of the spring is given by:

 * $$f = k \cdot d$$

To illustrate how matrices can be used for systems with more degrees of freedom (dof), a simple example is a 1-D spring with both ends being free. Here the force-displacement relation is given by:

 * $$\left[\begin{array}{c} f_1\\ f_2 \end{array}\right] = \left[\begin{array}{cc} k & -k\\ -k & k \end{array}\right] \left[\begin{array}{c} d_1\\ d_2 \end{array}\right]$$

Recipe: Steps to Solving Simple Truss System Problems

 * 1) The first step of analyzing a truss system is to look at the entire system as a whole, also known as the global picture. The preliminary procedures in this analysis are first creating the global coordinate system, numbering the nodes in the system, as well as noting the global forces and displacement degrees of freedom (dofs) in the system.  As a convention, both the global forces and displacement dofs should be named starting with the first global node in the directions of the global coordinate system (x and y directions) and continued in this fashion for each successive node.  The aforementioned FBDs are good tools for the visualization of systems and the forces and dofs acting upon them.  For both the global dofs and the global forces, each contain some unknown and some known quantities.  The known values are usually given in the problem as contraints or parameters, such as a fixed node (zero displacement) or an applied force.  The unknown values are the values that will be solved for using finite element method (FEM).
 * 2) After the system is correctly identified, the next step is to analyze each element in the system separately. A FBD should be created for each element of the system, identifying the dofs and forces acting upon each particular element.  The nodes, dofs, and forces should be identified by the same convention mentioned in step 1, however either global or local element coordinates may be used.
 * 3) Once all the dofs and forces are identified for each element in the system, the next step is to create the element stiffness and force matrices. These matrices are assembled into the global force-displacement (FD) relation:
 * $$\underline{K} \cdot \underline{d} = \underline{F}$$
 * where K is a nxn stiffness matrix, d is a nx1 displacement dof matrix, F is a nx1 force matrix, and n is the number of both known and unknown dofs.

 * 1) Next, the known dofs are eliminated to reduce the global force displacement relations. The K,d, and F matrices above - that were originally nxn, nx1, and nx1 matrices, respectively - become mxm, mx1, and mx1, respectively, where m is the number of unknown only dofs and is less than n.  Assuming the new stiffness, K, matrix is an invertible matrix, the above equation can be rearranged to solve for the displacement matrix.
 * 2) Compute the element forces from the now known dofs and from this also calculate the element stress.
 * 3) Use these to compute the remaining unknown (reaction) forces.


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Student expectations of lecturers

Overview
The purpose of this page is gather student feedback about lecturers, to help improve lecture quality.

The Deal
OK, so here is the deal guys. I went to a seminar the other day and learnt that lecturers actually care about the message they are sending to the students and the way in which they teach. To that end I think it would be awesome if we could all very nicely alert them to things that make us sleep or day dream while they are still speaking so that it may improve the quality of lectures for us and student attendance for them. I invite you all to think of your good and bad lectures and assess what makes them what they are, what we want them to be and share!

The Good

 * 1) The best thing in lectures(I find anyway), is when the lecturer has a personality and really engages the class.
 * 2) Interacts with the audience
 * 3) Uses real examples and makes links with other content/concepts
 * 4) Having someone who knows what he or she is doing when marking an assignment (There is nothing worse than having your grammar or referencing incorrectly "corrected")
 * 5) The ability to provide constructive feedback

The Bad

 * 1) When the lecturer is unprepared and unable to answer questions.
 * 2) A friend tells me she has a lecturer who just reads complicated notes and when asked to clarify contradicts the original statement.
 * 3) Too many slides
 * 4) Different versions of uploaded lecture notes, handouts and slides presented at the lecture
 * 5) I put it to the wonderful world of Facebook and you would not believe how many response I got saying if the lecturer had no eye contact with anyone through the lecture it made students less inclined to pay attention.
 * 6) Students as tutors

The Ugly

 * 1) While never having experienced it myself, I am told the worst thing in the world is when a lecturer is a professor taking a first year course and annoyed by the apparent stupidity of the students and is subsequently condescending and not afraid to show his frustration or boredom.
 * 2) If we don't understand the content of the lecture, and when we question the teacher about it, the teacher simply says :'U students find out the answer and let me know about it!' How pathetic! The teacher doesn't know the answer and she is not accepting it in front of the students.
 * 3) The teachers teach in a speed as if they have a train to catch. They do not explain properly. They do not care whether the students have properly understood what they have taught, rather they are concerned about completing the syllabus! Phew! What a shame! :(

What we REALLY Want

 * 1) What I really look for and expect from a lecturer is an approachable, friendly and knowledgeable person. Someone who is passionate about what they teach and able to make the content relate to real life. Most of all I love lecturers who always make time for their students, listen to their idea's and treat them as an equal. I know I expect a lot but hey I've already found quite a few of them out there.
 * 2) I really would like the lecturers/teachers to praise us on our strengths first and then only point our weakness politely suggesting for improvement.
 * 3) I would like the teachers to have a good sense of humor. The moody and aggressive teacher is the least I want in my teacher.

title: Student expectations of lecturers


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User talk:Hesperian

 Hello Hesperian, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! SB_Johnny talk 16:23, 11 September 2008 (UTC)

title: User talk:Hesperian


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Student expectations of lecturers/Header

Anyone can edit this page (especially students). Be bold!!!. Click on the "edit this page" tab, add your stuff (don't worry about formatting) and click "Save". Thankyou!