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User:Eas4200c.f08.aeris.mcrae

Problem 1.1
Problem Statement: We have a hollow rectangular beam, of side lengths a and b, and wall thickness t. We wish to find optimal b that will maximize the load-bearing capacity of the beam.

Assumptions:
 * $$\ t << a $$ (given)
 * $$\ t << b $$ (given)
 * $$\ L = 2 (a+b) $$ (given)
 * $$\ M = T $$ (given)
 * $$\ \sigma_{allowable} = 2 \tau_{allowable} $$ (given)

Governing Equations:
 * $$\ \sigma = \frac{T}{2 a b t}\,$$

This problem must be solved for 2 separate cases to arrive at a conclusion. The first case is where $$ \sigma_{max} = \sigma_{allowable}$$, and the second case is where $$\tau_{max} = \tau_{allowable}$$.

Case 1: Assume $$\sigma_{max} = \sigma_{allowable}$$.

Recall that $$\sigma = \frac{M z}{I}\,$$ with $$z = \frac{b}{a}\,$$.

We can now solve for M, which as can be seen below is a function of the ratio of I to b.
 * $$M = \frac{2 I \sigma_{max}}{b}\,$$
 * $$M = \frac{2 I \sigma_{allowable}}{b}\,$$
 * $$M = 2 \sigma_{allowable}* \frac{I}{b}\,$$

In order to maximize M we need to either maximize I, minimize b, or do a combination of both. Before we look at this further, we need to get an expression for I in terms of the given variables.

 * $$ I = \sum_{i=1}^4 \frac{b_i h_i^3}{12}\ + A_i d_i^2$$
 * $$ I = 2[\frac{a t^3}{12} + a t (\frac{b}{2})^2] + 2(\frac{t b^2}{12})$$
 * $$ I = \frac{a t}{12} (t^2 + 3 b^2) + 2(\frac{t b^2}{12})$$

Going back to the given assumptions, we recall that
 * $$\ t << b $$ --> $$\ t^2 << b^2 $$ --> $$\ t^2 << 3 b^2 $$

We can now assume that the wall thickness is negligible, and neglecting the t term we can now substitute our I back into the term we are trying to maximize, $$ \frac{I}{b}$$.

 * $$ \frac{I}{b} = \frac{t b}{6} (3 a + b)$$

Recalling that $$ a = \frac{L}{2} - b$$, we can simplify to just one variable, b.

 * $$ \frac{I}{b} = \frac{t b (3 L - 4 b)}{12}$$

We note that this is a downward opening parabola, and using some simple calculus we solve for the vertex at $$ b = \frac{3 L}{8}$$.

Substituting back into the original assumption $$\ L = 2 (a+b) $$, we find that $$ a = \frac{L}{8}$$.

Thus, in Case 1 ($$ \sigma_{max} = \sigma_{allowable} $$) we find that $$ a = \frac{L}{8}$$ and $$ b = \frac{3 L}{8}$$.

Case 2: Assume $$\tau_{max} = \tau_{allowable}$$ ($$\tau_{max}$$ reaches $$\tau_{allowable}$$ first).

 * $$ \tau = \frac{T}{2abt} = \tau_{max} = \tau_{allow} $$

Solving for T, we find that $$\ T = T_{max} = (2t \tau_{allow})(ab)_{max} $$.

Note that here $$ ab_{max} $$ is a variable, and we may treat $$ 2t /tau_{allow} $$ as a constant.

STILL NEED ---> show a = b = L/4 <---

STILL NEED ---> check to see if Case 2 (and maybe Case 1) is acceptable (see notes) <---

Homework Statement from 9/10
Find the moment of inertia, with respect to y, of a circular cross section and a u-shaped cross section, as shown below.

CASE 1: Beam with circular cross section.

We begin with the definition of moment of inertia, I. Specifically here we are calculating moment of inertia with respece to the y-axis, Iy.

 * $$\ I_{y}=\int r^{2} dm $$

Solving for this integral gives us the moment of inertia of the circle in Case 1.

 * $$\ I_{y}=\int r^{2} dm$$

 * $$\ I_{y}=\int r^{2} \pi r dr$$

 * $$\ I_{y}=\int \pi r^{3} dr$$

 * $$\ I_{y}=\frac{\pi}{4} r^{2}$$

CASE 2: Beam with u-shaped cross section.

As can be seen in the diagram, we can divide the "u-shaped" beam into 3 separate parts, with the symmetrical side piece being equal. We can calculate the moments of inertia of these three parts separately, and then combine them using the Parallel Axis Theorem.

We know that the moment of inertia of a rectangle is equal to $$\frac{b^{3} h}{12}$$.

From here we can calculate the moment of inertia for each of the 3 pieces, centered about each centroid.

For the 2 symmetrical pieces.

 * $$\ I_{y}=\frac{b^{3} h}{12}$$

 * $$\ I_{y}=\frac{c^{3} b}{12}$$

Using the Parallel Axis Theorem, we can move the centroid of these moments to the centroid about the x-axis.

 * $$\ I_{y}= \frac{c^{3} b}{12} + \frac{a + c^{2} b c}{4}$$

Noting we have two of these pieces we can multiply this moment by 2.

 * $$\ I_{y}=2 \frac{c^{3} b}{12} + 2 \frac{a + c^{2} b c}{4}$$

For the bottom piece.

 * $$\ I_{y}=\frac{b^{3} h}{12}$$

 * $$\ I_{y}=\frac{a^{3} c}{12}$$

Because the centroid of this piece is already on the x-axis, we do not need to use the Parallel Axis Theorem here.

Combining the pieces

We can combine the moments of the calculated pieces to obtain the total moment of inertia with respect to the y-axis, Iy.

 * $$\ I_{y,total}=\frac{c^{3} b}{6} + \frac{a^{3} c}{12} + \frac{(a + c)^{2} b c}{2}$$

HW3 - 9/22
Open thin-walled sections are very common members in aircraft structures. Sheer flow travels through these members, as can be seen by the directional arrows in the figure below.

A sheer force is induced through this member due to the sheer flow through the member.

 * $$\ dF=q*dl=q(dl_{x}\hat{i}+dl_{z}\hat{k}) $$

 * $$\ dF=q*dl=q(dl*cos(\theta)*\hat{i}+dl*sin(\theta)*\hat{k}) $$

We now integrate, from A to B as given in the figure above, to find the resultant force.

 * $$\ F=\int_{A}^{B}{dF}=q\left[\left( \int_{A}^{B}{dx}*\hat{i}\right) + \left( \int_{A}^{B}{dz}*\hat{k}\right) \right]$$

 * $$\ F=q\left(a*\hat{i}+b*\hat{k} \right)$$

NOTE: q is a constant with respect to x & z.

Now that we have the resulting sheer force vector we can calculate the resultant magnitude of this vector. Noting that this sheer force is broken down into an x and z component, we use Pythagorean Theorem to find the resulting magnitude.

 * $$\ \left| \left| F\right|\right|=\sqrt{F_{x}^2+F_{z}^2} $$

 * $$\ \left| \left| F\right|\right|=\sqrt{(q*a*\hat{i})^2+(q*b*\hat{k})^2}$$

 * $$\ \left| \left| F\right|\right|=q*\sqrt{a^{2}+b^{2}} $$

Strain Matrix (October 15th)
In a 3-D body under strain, this strain can be broken down into a component matrix, as seen below.

$$\varepsilon =\begin{bmatrix} \varepsilon _{xx} & \varepsilon _{xy} & \varepsilon _{xz}\\ \varepsilon _{yx} & \varepsilon _{yy} & \varepsilon _{yz}\\ \varepsilon _{zx} & \varepsilon _{zy} & \varepsilon _{zz} \end{bmatrix} =\begin{bmatrix} \varepsilon _{11} & \varepsilon _{12} & \varepsilon _{13}\\ \varepsilon _{21} & \varepsilon _{22} & \varepsilon _{23}\\ \varepsilon _{31} & \varepsilon _{32} & \varepsilon _{33} \end{bmatrix} =\left[\varepsilon _{ij} \right]$$

Due to symmetry, there are a several equalities we can make.

$$\varepsilon _{xy}=\varepsilon _{yx}$$

$$\varepsilon _{yz}=\varepsilon _{zy}$$

$$\varepsilon _{xz}=\varepsilon _{zx}$$

We can now see that this 3x3 strain matrix has only 6 components, $$\varepsilon _{xx}, \varepsilon _{yy},, \varepsilon _{zz}, \varepsilon _{xy}, \varepsilon _{yz}, \varepsilon _{xz}$$.

The equation for each component can be expressed as $$\varepsilon _{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)$$

Stress Tensor Matrix

Similarly, we can make some equalities to the stress tensor matrix, which is seen below.

$$\sigma =\begin{bmatrix} \sigma _{xx} & \sigma _{xy} & \sigma _{xz}\\ \sigma _{yx} & \sigma _{yy} & \sigma _{yz}\\ \sigma _{zx} & \sigma _{zy} & \sigma _{zz} \end{bmatrix} =\begin{bmatrix} \sigma _{11} & \sigma _{12} & \sigma _{13}\\ \sigma _{21} & \sigma _{22} & \sigma _{23}\\ \sigma _{31} & \sigma _{32} & \sigma _{33} \end{bmatrix} =\left[\sigma _{ij} \right]$$

$$\sigma _{xy}=\sigma _{yx}$$

$$\sigma _{yz}=\sigma _{zy}$$

$$\sigma _{xz}=\sigma _{zx}$$

We see that the stress tensor matrix, like the strain matrix, also has 6 components.

HW5 MatLAB Problem Statement
In this assignment we will modify the MatLAB code from the previous homework to include stringers attaching the spars to the flange, as can be seen in the figure below.

Given an input NACA airfoil (NACA 2415 shown in the above figure), we are to compute the centroid of the stringers (as well as the airfoil), the components of the moment of inertia tensor, the highest normal bending stress and location of this stress, and the ultimate bending moments of the airfoil.

We first compute this data neglecting the bending effects of the skin and the partition walls (spar webs). We are then to recompute the data, only this time NOT neglecting the bending effects of the skin or partition walls (or the stringers themselves). A comparison of the difference will then be made so we can access the contribution of these bending effects.

In order to make these calculations, we need to know some initial data on the stringers to be used. We first make an assumption on the bending moments, shown below.

$$\ M_{y}=-125,000 N*cm $$

$$\ M_{x}=50,000 N*cm $$

We also need to know the cross sectional area of these stringers, which is given below.

$$\ A_{B}=2 cm^{2} $$

$$\ A_{E}=2 cm^{2} $$

$$\ A_{H}=1 cm^{2} $$

$$\ A_{F}=1 cm^{2} $$

For the second part of our calculations, we are to take into account the bending affects of the skin, partition walls, and stringers themselves. We can note that the stringers are essentially stamped out sheet metal which is then bent into the L-shape as seen in the first figure. Because of this, thickness of the stringers are uniform, and we let this thickness equal 5 mm, or 0.005 m. Knowing this, we can compute the width of the metal sheet used to manufacture the stringers, as seen below.

$$\ \frac{\frac{0.0002}{2}}{0.5}=0.02m=2cm $$

Because the thin skin would buckle under high normal bending stress acting on the plane of the skin, we will next find the highest normal bending stress in any of the stringers, and take a closer look at that stringer. Assuming the ratio of My and Mz are constant, we will then set this maximum normal bending stress to the ultimate stress for our material, steel 300M. Upon doing this, we are now able to compute the ultimate bending moments My and Mz for our full blown airfoil (which we will be imputing as the NACA 2415 airfoil).

11/14 - Meeting 34

General equation for asymmetric cross sections:
 * $$\ \int_{A_{s}}^{}{ \frac{d \sigma_{xx}}{dx}dA} = -q_{s}$$

Symmetric Case:

 * $$\ \sigma_{xx} = \frac{M_{y}z}{I_{y}} $$

 * $$\ q_{s} = - \frac{V_{z}}{I_{y}} \int_{A_{s}}^{}{zdA} = \frac{-V_{z}Q_{y}}{I_{y}} $$        (eq. 5.2)

 * $$\ Q_{y} = \int_{A_{s}}^{}{zdA} = z_{c}A_{s} $$

Asymmetric Case:

 * $$\ \sigma _{xx} = \left(k_{y}M_{z} - k_{yz}M_{y} \right)y + \left(k_{z}M_{y} - k_{yz}M_{z} \right)z $$         (eq. 5.3)

with

 * $$\ k_{y} = \frac{I_{y}}{D} $$

and

 * $$\ k_{yz} = \frac{I_{yz}}{D} $$

and

 * $$\ k_{z} = \frac{I_{z}}{D} $$

This can be also written in matrix form:

 * $$\ \sigma _{xx} = \begin{bmatrix} y & z \end{bmatrix}\begin{bmatrix} k_{y} & -k_{yz}\\ -k_{yz} & k_{z} \end{bmatrix} \begin{Bmatrix} M_{z} \\ M_{y} \end{Bmatrix} $$

 * $$\ \sigma _{xx} = \begin{bmatrix} y & z \end{bmatrix}\begin{bmatrix} k_{y}M_{z} & -k_{yz}M_{y}\\ -k_{yz}M_{z} & k_{z}M_{y} \end{bmatrix} $$

We can also flop the order of variables, and come up with the preferred form of this equation:

 * $$\ \sigma _{xx} = \begin{bmatrix} z & y \end{bmatrix}\begin{bmatrix} k_{z} & -k_{yz}\\ -k_{yz} & k_{y} \end{bmatrix} \begin{Bmatrix} M_{y} \\ M_{z} \end{Bmatrix} $$

We particularize to symmetric cross section, and find that $$\ I_{yz} = 0 $$. Proof of this is shown in the tab below.

Consider $$\ M_{z} = 0 $$

We know the following relations.

 * $$\ I_{yz} = 0 $$
 * $$\ D = I_{y}I_{z} $$
 * $$\ k_{yz} = 0 $$
 * $$\ k_{z} = \frac{1}{I_{y}} $$

Combining these equations we find that $$\ \sigma_{xx} = \frac{M_{y}z}{I_{y}} $$.

Asymmetric Case

 * $$\ q_{s} = -\left( k_{y}V_{y} - k_{yz}V_{z} \right) Q_{z} - \left( k_{z}V_{z} - k_{yz}V_{y} \right) Q_{y} $$          (eq. 5.5)

 * $$\ Q_{z} = \int_{A_{s}}^{}{y*dA} $$

 * $$\ Q_{y} = \int_{A_{s}}^{}{z*dA} $$

For stringer web sections, if the thickness (t) of the skin and spar webs is very small we can neglect the computations of Iy, Iz, Iyz, Qy, and Qz. We use only the areas of the stringers.

Unsymmetric Thin-Walled Cross Sections

We know that for a symmetric thin-walled cross section that $$\ q_{s} = -\int\int_{A_{s}}^{}{\frac{d\sigma_{xx}}{dx}dA} $$

For unsymmetric thin-walled cross sections under bidirectional bending, the bending stress $$\left( \sigma_{xx}\right)$$ must be taken into account. Doing so, we arrive at the following equation for sheer flow.

 * $$\ q_{s} = -\left(k_{y}V_{y} - k_{yz}V_{z} \right)Q_{z} + \left(k_{z}V_{z} - k_{yz}V_{y} \right)Q_{y} $$

We can begin our analysis of the figures to the right using the Mean Value Theorem to find the first moments of area around the z and y axes.

 * $$\ Q_{z} = \int \int_{A_{s}}^{}{ydA} = \bar{y}A $$

 * $$\ Q_{y} = \int \int_{A_{s}}^{}{zdA} = \bar{z}A $$

For the case of stringers of different areas, as shown in Figure 1, the area to be used in this equation is $$\ A = \sum_{i=1}^{4}{A_{i}} $$. For the case shown in Figure 2, the area used in this equation is $$\ A = A_{\hat{s}} + A_{\hat{s},L} $$.

Using this, as well as equations from the previous homework, we can break down analysis of unsymmetric thin-walled cross sections, such as the two shown on the right in Figure 1 and Figure 2, into the following four steps.

 * 1. Find $$ \left(\bar{y}, \bar{z} \right) $$.
 * 2. Find $$ I_{y} $$, $$ I_{z} $$, and $$ I_{yz} $$.
 * 3. Find $$ k_{y} $$, $$ k_{z} $$, and $$ k_{yz} $$.
 * 4. Follow path $$ s $$ to find $$ q_{12} $$, $$ q_{23} $$, and $$ q_{34} $$.

Shear Flow
In order to plot the buckling shape, we express the coefficients $$\left\{\begin{matrix} C_{22} & C_{13} & C_{31} & C_{33} \end{matrix} \right\}$$ in terms of $$ C_{11} $$. Using an aspect ratio $$\left( \vartheta \right)$$ equal to 1.5 we can break our analysis down to the following four steps.

 * 1. Find $$ \lambda $$.

 * $$ \displaystyle \lambda = \left[ \frac{\vartheta^4}{81 (1 + \vartheta^2)^4} \left\{1 + \frac{81}{625} + \frac{81}{25} \left( \frac{1 + \vartheta^2}{1 + 9 \vartheta^2} \right)^2 + \frac{81}{25} \left( \frac{1 + \vartheta^2}{9 + \vartheta^2} \right)^2 \right\} \right]^{1/2} = 0.0288 $$

 * 2. Evaluate $$ K_{5x5} $$ numerically.

\left[ \begin{array}{lllll} \frac{\lambda (1 + \vartheta^2)^2}{\vartheta^2} &	 \frac{4}{9} &	 0	 &	 0	 &	 0	 \\	 \frac{4}{9} &	 \frac{16 \lambda (1 + \vartheta^2)^2}{\vartheta^2} &	 - \frac{4}{5} &	 - \frac{4}{5} &	 \frac{36}{25} \\	 0	 &	 - \frac{4}{5} &	 \frac{\lambda (1 + 9 \vartheta^2)^2}{\vartheta^2} &	 0	 &	 0	 \\	 0	 &	 - \frac{4}{5} &	 0	 &	 \frac{\lambda (9 + \vartheta^2)^2}{\vartheta^2} &	 0	 \\	 0	 &	 \frac{36}{25} &	 0	 &	 0	 &	 \frac{\lambda (9 + 9 \vartheta^2)^2}{\vartheta^2} \end{array} \right] \left\{ \begin{array}{l} C_{11} \\	 C_{22} \\	 C_{13} \\	 C_{31} \\	 C_{33} \end{array} \right\} =  \left\{ \begin{array}{l} 0	 \\	 0	 \\	 0	 \\	 0	 \\	 0     \end{array} \right\} $$

 * 3. Truncate $$ K_{5x5} $$ into $$ K_{4x4} $$.

\mathbf \bar K _{4 \times 4} = \left[ \begin{array}{llll} \frac{16 \lambda (1 + \vartheta^2)^2}{\vartheta^2} &	 - \frac{4}{5} &	 - \frac{4}{5} &	 \frac{36}{25} \\	 - \frac{4}{5} &	 \frac{\lambda (1 + 9 \vartheta^2)^2}{\vartheta^2} &	 0	 &	 0	 \\	 - \frac{4}{5} &	 0	 &	 \frac{\lambda (9 + \vartheta^2)^2}{\vartheta^2} &	 0	 \\	 \frac{36}{25} &	 0	 &	 0	 &	 \frac{\lambda (9 + 9 \vartheta^2)^2}{\vartheta^2} \end{array} \right] \left\{ \begin{array}{l} C_{22} \\	 C_{13} \\	 C_{31} \\	 C_{33} \end{array} \right\} =  \left\{ \begin{array}{l} \frac{-4 C_{11}}{9} \\	 0	 \\	 0	 \\	 0     \end{array} \right\} $$

 * 4. Solve for $$\left\{\begin{matrix} C_{22} & C_{13} & C_{31} & C_{33} \end{matrix} \right\}$$ in terms of $$ C_{11} $$, noting that $$ K^{-1} $$ is the inverse of $$ K $$.

 * $$\begin{Bmatrix}

C_{22} \\ C_{13} \\ C_{31} \\ C_{33} \end{Bmatrix} = K^{-1} \begin{Bmatrix} -\frac{4}{9} C_{11} \\ 0 \\ 0 \\ 0 \end{Bmatrix}$$

 * $$u_{z} = C_{11} sin(\frac{\pi x}{a}) * sin (\frac{\pi y}{b}) + C_{22} sin(\frac{2 \pi x}{a}) * sin (\frac{2 \pi y}{b}) + C_{31} sin(\frac{\pi x}{a}) * sin (\frac{3 \pi y}{b}) + C_{31} sin(\frac{3\pi x}{a}) * sin (\frac{\pi y}{b})+ C_{33} sin(\frac{3\pi x}{a}) * sin (\frac{3\pi y}{b})$$

Stability Equations

 * $$\ \dot{x} = Ax + B \eta $$

 * $$\ \begin{bmatrix}

\Delta \dot{u}\\ \Delta \dot{\beta}\\ \Delta \dot{\alpha}\\ \Delta \dot{q}\\ \Delta \dot{r}\\ \Delta \dot{\theta}\\ \Delta \dot{\psi}\\ \end{bmatrix} = A \begin{bmatrix} u\\ \beta\\ \alpha\\ q\\ r\\ \theta\\ \psi\\ \end{bmatrix} + B \begin{bmatrix} pitch flap\\ yaw flap\\ \end{bmatrix} $$

 * $$\ A = \begin{bmatrix}

-0.40617261082 & 0 & 0 & 0 & 0 & -0.00807575185 & 0\\ 0 & -7.34602885379 & 18.84954000000 & 0 & -1 & 0 & -0.00807575185\\ 0 & 18.84954000000 & -9.38398809293 & 1 & 0 & -0.00807574113 & 0\\ 0 & 0 & -11351.24919089994 & -16.41759631431 & 17.95643978307 & 0 & 0\\ 0 & 31751.40272182580 & 0 & -17.95643978307 & -15.42011712577 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{bmatrix} $$

 * $$\ B = \begin{bmatrix}

0 & 0\\ 0 & -0.000167397975141\\ -0.000164984586141 & 0\\ -2.899278907682155 & 0\\ 0 & 2.198365483499154\\ 0 & 0\\ 0 & 0 \end{bmatrix} $$

 * $$\ C = \begin{bmatrix}

1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} $$

 * $$\ D = \begin{bmatrix}

0 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 0 \end{bmatrix} $$

title: User:Eas4200c.f08.aeris.mcrae


license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: File:Structeq-8-1.JPG


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Talk:Outdoor education

''Feel free to post about outdoor education on this page. After you write something, add an automatic signature by signing with four tildes ( ~ ).''

any thoughts on why this space is unused? I think its a pretty interesting subject. Are people in the industry all technophobes? Any ideas on why most of the papers i read on outdoor ed fail to mention fun? I agree (That Guy;1997) when he said something about the gap between 'academic types' and 'field types'. <<(wow look i just created two new words for things that have existed a long time i must deserve a wiki:office;) One theory i have about why this page is 'dead' is because of a lack of 'community'. Users will be reluctant to contribute to a page that doesnt feel 'alive'. Outdoor education does some interesting and i feel important reaserch and many people are passionate about it I am amazed at the lack of online discussion about the topic. Am i looking in the wrong place for free useful info?Takeelos 10:05, 5 October 2010 (UTC)
 * My sense is that people in the outdoor education field are quite disparate; there isn't much central organisation. Even the outdoor education discussion lists are pretty quiet these days, but you can find activity on some. It is also difficult to work in outdoor field settings and also have some tech skills and confidence to contribute to social media. And there is some selfishness (not wanting to share openly). But, hey, there's you and me! Let's start. I put most of my content initially up at http://wilderdom.com but am now gradually converting to wiki format. Sincerely, James. -- Jtneill - Talk - c 11:30, 5 October 2010 (UTC)
 * Thanks for the advice, sorry about the sarcastic tone in my messages so far its hard for my generation to communicate online without it (bad habit i know.. i am working on it).. Im just getting started in the industry and was just a little frustrated about the lack quantity and quality of info online. Ive been using/viewing social meadia since early to mid 90's (BBS 9600 modems etc) my tech skills are pretty rusty at the moment but they will come back with practice. I totally agree with your point that people are unwilling to share information these days (i don't classify this action as the same as selfishness ed.) (applies especially on a online forums!). Im sure there is plenty of evidence out there to support this theory, some examples of lack of sharing are; this site, me using a gaming alt (common practice in most online realms), Colleges and Universitys not collaborating. On the other hand I can quickly think of a few billion examples of people sharing information (sometimes too much according to the law/social norms!) eg, P2P file sharing, flitter, flacebook, a large percentage of blogs and content that can be found (easily) in the major search engines (drowned with advertising ;. So i think the spirit of sharing is alive and well on the net it just has no focus (<this is inevitable is probably a good thing). To summarize if I were asked the question (love how easy it is to answer my own questions!); What are the 3 primary 'negative motivations' behind the lack of contribution to this page among outdoor educators (supporters and critics, field, office and everyone in between)? Id answer 1. Lack of knowledge/awareness about tech and social media and potential benefits 2. Lack of trust in technology, people and organizations. I think these two points are probably the cause of the lack of confidence you have observed in the social media. Id like to think selfishness is somewhere further down on the list of peoples motavations (I hope). Thanks again for the information Id love to do some editing someday I am just too busy with my current task of procrastination. Takeelos 00:24, 6 October 2010 (UTC)

title: Talk:Outdoor education


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User:Eml4500.f08.team.Bengtson

Homework 1

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title: User:Eml4500.f08.team.Bengtson


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Introduction to Abstract Algebra/PS1

Associated problem set for exercises:

asante appiah

title: Introduction to Abstract Algebra/PS1


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User:EAS4200C.F08.WIKI.A/HW

=MediaWiki=

MediaWiki is a web-based wiki software application originally written for Wikipedia by Lee Daniel Crocker. Wiki software refers to collaborative software the manages, creates, and edits web pages through common web browser applications as the user interface. The information in wiki software is stored in a file system, while the edits are stored using a relational database management system. The main features of MediaWiki is the rich core feature set and the ability to add extensions, allowing additional functionality. Also, this software is distributed under the terms of the GNU General Public License with some documentation in the public domain; thus, it is a free and open source software, which allows anyone to use and edit the stored content.

The overall idea behind MediaWiki is that by having anyone add information on topics they thoroughly understand, everyone can benefit from the collective knowledge. The let down is that since there is no credibility check before editing a page, people with less than adequate knowledge of a subject or people who's intentions aren't to improve the content can "vandalize" the stored information, whether it be adding incorrect segments or even deleting previous contributions. In order to control this problem, the history of different versions of each web-page is saved, and editors, or custodians, assess each contribution in order to determine the validity. In the case of vandalism, the web-page is simply returned to the last acceptable version.

= Reasons for Using Wikiversity for Homework Submission =

The Aerospace Structures class will be using Wikiversity in order to submit homework due to the versatility of the mediawiki software. Instead of having to meet in one place to collaborate, the group will be able to just post their part in their user web space and everyone else in the group will be able to view and edit in any additional information they have on the subject. Also, any other group will be able to see the other groups pages and share information.

The structure will be even better if it is carried out in future years as any subsequent classes will have the previous years work to look at.

Also, since the student is already on Wikiversity, it will be easier for him or her to investigate the subject further by looking at any related Wikipedia pages.

= Dangers of Using Wikiversity =

A downside to using Wikiversity is that any one will be able to see your work and change it. This includes people who are not in the class, and the student is asked to keep an eye on their page at all times to make sure no vandalism has been done to their work. Also, any other group in the class could steal another groups work and post it as their own since it's freely available on line. One way around this is for TA's to cross check the work of one group against every other group, but this won't guard against a group rewording the work of another group and doing the same thing. The suggested way around this is to wait until right before the homework is due to post it on line.

=Basics of Aerospace Structures=

The Finite Element method (FEM) can be used in a wide variety of fields, from various types of engineering, to applied math, and even geology. Aerospace structures is one of the fields where FEM can be used to solve the partial differential equations that govern this particular field.

The structure of an aircraft must be designed with two main goals in mind: it must be light, and it must be strong. More specifically, the structure of an aircraft should have a high stiffness, high strength, and be composed of light weight materials.

Stiffness refers to a Young’s modulus $$\displaystyle E$$, or the slope, in a linear relation between stress $$\displaystyle \sigma$$ and strain $$\displaystyle \epsilon$$, or $$\displaystyle \sigma = E \epsilon$$. Strength refers to a particular stress, such as the yield stress $$\displaystyle \sigma_Y$$, where a material begins to weaken, and the ultimate stress $$\displaystyle \sigma_U$$, where a material fails. Toughness is simply the capacity to endure fracture or damage.

Examples of materials with high stiffness and high strength include steel and titanium alloys, since they have a large Young’s modulus and large ultimate and yield strengths. Glass is an example of a material with high stiffness, but low toughness. It does not withstand much plastic deformation and does not resist fracture well. Plastics and nylon are materials with a low stiffness, and high toughness, as they are they behave oppositely to glass.

Aircraft structures typically have monocoque or semi-monocoque designs to them. The term monocoque refers to a “single shell” design, while semi-monocoque is a shell that is stiffened by a supporting structure. These designs allow aircraft to have low weight, while also remaining strong and resisting bending. The geometry of an aircraft is, of course, also restricted by aerodynamics such as lift and drag on an airfoil.

=Aircraft Example=

The following figure provides examples of materials used on the F/A-18E Hornet. All of the materials used in this aircraft are light, high in stiffness, and high in strength therefore allowing for optimal performance and long durability. It is desirable however to eliminate all aluminum and titanium and replace them with Fiber Reinforced Composites which would in turn decrease the weight of the aircraft by 30%-40%. In doing so the aircraft would be at its maximum performance. An example of the composites being used in aviation can be seen in the Boeing 787. Fifty percent of this vehicle is made up of these materials making it lighter and more efficient.

The table below gives some examples of parts made of the materials portrayed or not portrayed in the photo.

{| class="toccolours collapsible collapsed" style="width:100%" ! Metal Matrix Composite vs Fiber Metal Composite
 * align=center |
 * align=center |

A metal matrix composite is material made up of at least two parts - one part is a metal and the other can be a different metal or some other type of material. A hybrid composite is composed of at least three materials. A metal matrix composite is formed by inserting a reinforcing material into a metal matrix. Sometimes the reinforcing material will have to be coated to prevent a chemical reaction that could cause the metal to break down. Usually the matrix is made up of a lighter metal, such as aluminum or titanium. The reinforcement material is not always added just to add structural strength, but also to change the properties of the composite material.1 Unlike metal matrix composites, fiber metal composites make use of organic materials and plastic. Natural fibers are refined, mixed, and merged to form high strength composite material in a polymer matrix. An advantage of this material is that it can be recycled up to 20 times.2

title: User:EAS4200C.F08.WIKI.A/HW


text:
User:Eml4500.f08.delta 6.castrillon/HW 2

= Testing the Steps =

It is important to see how the steps to solve truss systems work.

Given

Take this truss example with the following data:

Global Picture
Releasing the truss system from its supports and showing the reactions, we have:

There are 6 unknowns for which we need to solve for, therefore we need to analyze each element. The matrix equation is as follows.

 * $$\begin{pmatrix}

f_1\\ ...\\ f_6\end{pmatrix}$$ = $$\begin{bmatrix} & &  \\  & k &  \\ & &  \end{bmatrix}$$ $$\begin{pmatrix} d_1\\ ...\\ d_6\end{pmatrix}$$

which corresponds to

 * $$\mathbf{F}$$=$$\mathbf{K}$$ $$\mathbf{d}$$.

To solve for the reactions, we have to now examine the whole system by breaking it down into elements.

Element Picture
Analyzing each element with its own force and dofs yields:

It is important to notice the chosen coordinate axis since they will allow for ease of analyzing each element. The tilda on the coordinate direction names designate a non-horizontal or non-vertical direction.

We now want to creat a matrix equation, just like for the global picture, for the element picture. The matrix equation consists on the basic force-distance relationship, however, this principle is applied at the element level. The notation is altered in that each term is designated to belong to the particular element with an upper script number in parenthesis so that we get, $$ F(e) = k(e) d(e) $$, named as the element force matrix, the element stiffness matrix, and the element distance matrix.

Force-Displacement Relationship
To build the element stiffness matrix, we have to follow a "recipe" for calculating each term. This stiffness matrix will end up measuring 4x4.

The recipe is as follows, and in the wise words of Prof. Vu-Quoc, "The notation is beautiful here. It tells you no more, no less."

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

where the axial stiffness fo the bar "e" is

 * $$k^{(e)}=\frac{E^{(e)} A^{(e)}}{L^{(e)}}$$,

where $$E^{(e)}$$ is the element's material Young's Modulus, $$A^{(e)}$$ is the element cross-sectional area, and $$L^{(e)}$$ the element length. Also, $$l^{(e)}$$ and $$m^{(e)}$$ are te director cosines of the rotated, and convenient, coordinate axes with respect to the positive, and horizontal, x-direction (the global axis).

Director Cosines: The director cosines will yield the projections of vectors oriented with an inclination w.r.t. the positive x-axis on the horizontal or vertical directions. These horizontal and vertical directions are more precisely for this course the global x-y coordinates. From this diagram, we can derive our much needed $$l^{(e)}$$ and $$m^{(e)}$$ values. $$l^{(e)}$$ = $$\vec{\tilde{i}}$$ • $$\vec{\hat{i}}$$ = cosθ$$^{(e)}$$ $$m^{(e)}$$ = $$\vec{\tilde{i}}$$ • $$\vec{\hat{j}}$$ = sinθ$$^{(e)}$$ = cos(π/2-θ$$^{(e)})$$ Also, $$\vec{\tilde{i}}$$=cosθ$$^{(e)}$$•$$\vec{\hat{i}}$$+sinθ$$^{(e)}$$•$$\vec{\hat{j}}$$


text:
MATLAB/Vectors

Quick way to show vector algebra in Matlab with matrices:

EDU>> a = [ 1 2; 3 4 ]

a =

1    2     3     4

EDU>> b = [1;0]

b =

1    0

EDU>> a\b

ans =

-2.0000   1.5000

EDU>> quit

title: MATLAB/Vectors


text:
User:Eml4500.f08.RAMROD.A/HW1

=Finite Element Analysis=

Finite Element Analysis, otherwise known as the Finite Element Method, is a technique used to attain solutions to Partial differential equations(ODE's) and integral equations. This method is most often used to solve ODE's over complex domains. This class will focus on learning how to use and apply this method to real world situations in which it is necessary to determine the tendencies of a system.

The goal of this course is to avoid the old ad-hoc structure of analysis and to encourage true learning and judgment of correctness.

This course will not be conducted in the usual manner. Instead of the old method of simple lecture notes and homework assignments, Wikiversity will be used to post the concepts we learn in class as well as the problems that are assigned to us for homework. At first glance this structure seems like a lot of work and as well as a hindrance to the learning experience, but if an in depth look at the new system is taken, one would emerge with a greater appreciation of what is being accomplished.

=Wikiversity and The Class=

What is Wikiversity

Wikiversity is part of the Wikimedia Foundation, which is a foundation in which provides free content to its users. It is a very large server-based software that allows its users to create articles on any topic of their choosing. The incredible aspect of this software is that anyone who has an account with any Mediawiki project can go into that article and make corrections. The ability to edit any article allows for an environment that promotes collaborative learning.

Questions may arise as to the quality of contributions made to an article found on any of the Mediawiki projects. For example, an article may be writen that doesn't contain a single factual statement. In this case, the administrators in charge of overseeing all activities within the domain will quickly eliminate the article. Another example is that of a contribution made to an article already approved by the administrators. The contribution will be reviewed and either accepted or rejected. If rejected, the last acceptable saved copy of the article gets uploaded back onto the Wiki. This brings up another advantageous aspect of the software. When an edit is made and saved to any article, a copy of that article is stored in a database and will remain there until the MediWiki software is shut down. This means that any copy of any article can be retrieved by simply going into the history of that article and searching for the needed copy. This function serves as a solution to vandalism to articles as well. It is almost always safe to assume that information posted on Mediawiki software are correct.

Advantages of Wikiversity

There are numerous advantages to using a Mediawiki software such as Wikiversity, the first of which would be learning through collaboration and teamwork. Teamwork has been proven to be one of the most effective ways of learning and deploying this method in the classroom is absolutely essential. Not only is this a collaboration of the team, but it's also a collaboration of the class because each team can view and learn from other team contribution. Taking it even a step further, this software allows there to be a collaboration between the class and the professor.

Another important advantage of using such a software is that a user will forever have the information they input into the database for future reference. This is a great aspect of the software that allows the user to go back and review the material they learned whenever they need a little reminder of concepts.

=Trusses, Beams, and Frames=

Diagram of Truss
Below is a diagram of a truss with 2 elastic, or deformable, bars. There is a known load P applied where the two bars join. The two bases of the truss are both fixed, or constrained, in both the x and y axes. There will be no displacement at these points. The goal is to find all of the unknown forces and displacements.

Global Free Body Diagram
Below is the Global (the whole structure) Free Body Diagram of the same 2-bar truss. Notice that the constraints have been replaced with reaction forces.

Currently, there are 4 unknown reaction forces (R1x, R1y, R3x, and R3y). However, there are only three equations that can be used (ΣFx=0, ΣFy=0, and ΣM=0). In other words, we have 4 unknowns but only 3 equations to solve for them, therefore this system is statically indeterminant.

Free Body Diagrams of the 2 bar elements
To continue, we will separate the two bars and look at them independently. A new local free body diagram is drawn for each bar element. Forces and the resulting displacements are again drawn and the nodes are renumbered accordingly.

Bar Element 1 Bar Element 2

Notes
 * fdegree of freedomundefined

 * fi(e) = ith eternal force of element e with:

 * i=1,2,3,4

 * e=1,2

 * for this example

There are two possible cases in this situation:

 * Case 1: The observer sits on node 1

 * In this case:

 * f2 = k(d2 -d1)

 * Case 2: The observer sits on node 2 (or equivalently f1+f2=0)

 * In this case

 * f1= -f2= -k(d2 -d1)

 * f1=k(d1 -d2)

Alternate Node Numbering Method
Below is another way to number the local nodes.

Force Displacement Relation
An elastic bar can be considered a spring with a spring constant of EI/L (E is Young's Modulus, I is the moment of inertia, and L is the length of the bar or spring). Recall the two following spring systems:

 * Below is a 1-dimensional spring with two free ends. A force f is applied and a displacement d results. The force-displacement relation is f=kd.

 * Another system is a 1-D (one dimensional) spring with 2 free ends. A force f is applied and a displacement d results. The matrix method (on the right) is used to find the force-displacement relationship. (The 2x2 matrix with the k's in it is known as the stiffness matrix).

= Notes Summary for Friday, September 5 =

Intro
We discussed the ordered steps to solve a the truss problem from September 3. We discussed a generalized format to begin with the diagram of a model system and to work our way down to computing reaction forces.

Step 1: Global Picture
The global picture is the overall structural picture of the system. The picture includes the following:
 * 1) Global Forces
 * 2) Global Degrees of Freedom (aka DOF)

The displacement definition are partitioned into:
 * 1) Known part, eq., fixed DOF, constraints
 * 2) Unknown part solved by using Finite Elemental Analysis (see the truss example above)

Similarly, the global forces are partitioned into:
 * 1) Known part: Applied forces
 * 2) Unknown part: Reactions

Step 2: Element Picture

 * 1) Element definitions
 * 2) Element forces

These can be found in either global coordinate systems or in local coordinate systems

Step 3: Global Force Diagram relationship

 * 1) Element stiffness matrices in global coordinates
 * 2) Element force matrices in global coordinates
 * 3) assembly of element stiffness matrix and element force matrix into global force displacement relation.

K · d = F

(NxN) (Nx1) = (Nx1)

Step 4: Elimination of DOF
Eliminate degrees of known degrees of freedom to reduce the global force displacement relationship. (The Stiffness Matrix is non-singular-invertible)
 * 1) M = the number of unknown DOF
 * 2) N = the number of both known and unknown DOF

K · d = F

(MxN) (Mx1) = (Mx1)                       M>N

Stiffness Matrix K nonsingular => k-1 exists (K is invertible)

K-1 · d = F

Step 5: Compute Element Forces
Compute element forces from known displacements => element stresses

Step 6: Compute Reactions
Compute reaction forces or the unknown forces.

= Matlab Summary =

Intro
Matlab is a software program, developed by Matrix Laboratories, that uses matrix-based mathematics to solve ordinarily difficult scientific and engineering applications. It has several advantages over traditional methods (i.e. Fortran or C++), specifically its ease of use, versatility of capabilities, and its ability to convey results graphically as charts and its ability to model very complex systems visually for ease in conveyance of results. Since its development, Matlab has become very widely used in nearly all scientific fields and, especially in engineering practice, has become an absolutely vital tool for all types of engineering analysis.

Matrices in Matlab
As was stated above, Matlab is a matrix-based program. Thus, all variables, whether scalars, vectors, or much more complex and larger-scale matrices are all stored in the same manner. Because of the significance of matrices in this program, the first, and most important operations to know are those involving matrices.

Creating Matrices
There are several ways in Matlab to make new matrices. One method is to enter the values directly. For example:

 * A = [2,3,4;5,3,6;9,5,2]

 * A = [2 3 4; 5 3 6; 9 5 2]

Both of these entry syntaxes would create a 3x3 rectangular matrix with the indicated values. Other permissible formats are for scientific notation, such as [2.34e-4, 3.53e8], or for complex numbers, entered as [2+4i,3+8i] or [2,3] + i*[4,8]. In both instances (exponential and complex numbers) spaces should be avoided between numbers and commas should be used. When manually entering values into a matrix, as shown in the above examples, commas (or spaces where allowed) separate entries in a row while a colon separates one row from the next.

Another option for creating matrices is to read from an input file, usually an ASCII file from another editor on your computer. This is completed by using the load command. To load input file "input.dat", the command would be load input.dat.

After creating a matrix, it can then be referenced by its name (i.e. A). Also, you can reference individual entries such as A(4,2) for the second entry in row four of matrix A.

Matrix Building Functions
In addition to creating your own matrices, Matlab has several default "special" matrices and matrix references that can be called on using their reserved names. The following are a few of the standard Matlab matrices and uses:

 * eye(m,n) will create the identity matrix of dimensions mxn.
 * zeros(m,n) will create a mxn matrix where all entries are zeros.
 * ones(m,n) will create an mxn matrix where all entries are ones.
 * diag(x) will create a matrix where the diagonal is the predefined vector x.
 * triu(x) or tril(x) references the upper or lower triangle of matrix x, respectively.
 * rand(m,n) will create an mxn matrix with Matlab randomly generating values for each entry that will be uniformly distributed between 1 and 0.
 * magic(m,n) will provide an mxn matrix that is a "magic" matrix in which each column, row, and diagonal totals to the same sum.
 * hilb(m,n) creates an nxn Hilbert Matrix.

It is useful to note also that creating a square matrix of any of the above forms is done by replacing the (m,n)dimensions by simply an (n) dimension.

Also, any of these matrices can be created to match the size of an existing matrix by using the command size(X) where X is a matrix. For example, eye(size(X)) creates an identity matrix with the same dimensions as matrix X.

All of these commands are extremely useful as starting points for building matrices. For example, A = eye(2) could then be edited (A(2,1)=5) to modify it from an identity matrix to the desired matrix without manually entering the value of each entry.

Matrix Operations
Once a matrix is created using the above stated methods, Matlab has several built in operations to manipulate it mathmatically. The following are the basic matrix operations that Matlab can perform:

 * +: addition - adds two matrices term by term to create a new matrix.
 * -: subtraction - subtracts two matrices term by term to create a new matrix.
 * *: multiplication - multiplies two matrices together
 * ^: power - raises a matrix to power indicated after ^.
 * ': conjugate transpose - returns a matrix where each term's indices are swapped (Aij goes to Aji) and all terms are replaced by their complex conjugate (a+bi goes to a-bi).
 * \: left division - A\b = b/A.
 * /: right division - normal division of two matrices.

While addition and subtraction above execute a term by term operation, the others do not. However, Matlab will allow this by placing a period before the operator. For example:

"[2,5,9,7].*[2,5,3,1] = [4,25,18,7]"

This form of multiplication, division, or exponents can be found very helpful when working with Matlab Graphics which will be described later.

Statements, expressions, and variables in Matlab
In addition to being a matrix-based software, the language of Matlab is expressions. Expressions are used for every function that Matlab is capable of. Generally, an expression can take the form of a equation or a simple statement.

Equations
Equations, such as Y = 5X+5, are generally created by stringing together variables, functions, and operators and results in a matrix that is then output to the screen and can be referenced in later functions.

Statements
If a certain variable and equals sign is not specified, 5X+5, this expression results in a newly created variable "ans" whose value is stored until a new expression is evaluated. For this case, the line ans = (value) will be output to the screen. However, if the expression is followed by a semicolon, 5X+5;, this line will not be shown. This is very convenient in long iterations where extra lines of output answers only complicate the code and calculations and are undesired.

Variables
An important thing to note here is that Matlab is case-sensitive, meaning that the variable y is not the same as the variable Y. If at any given time you wish to see a list of the current variables, the command "who" or also whos will print the list and their values to the screen. Additionally, variables can be cleared by the command clear followed by the variable name you wish to clear. The command clear by itself will clear all non-permanent variables in the cache.

Saving a Session
In order to not lose variables when you exit Matlab, it is very useful to know the save command. This will write all variables and their values to a non-accessable file named "matlab.mat". Upon entereing Matlab the next time, simply typing the load command will access the file and will bring back all variables as they were when the file was saved.

Scalar, Vector, and Matrix Functions
Matlab has many predefined functions that are very useful in doing matrix calcluations. Some are solely used for scalars matrices (1x1), some for vectors, and the most powerful are the matrix functions.

Scalar Functions
The below functions are designed to work with scalars only, but can be applied to a matrix. When the following are used in reference to a matrix, the operation is computed term by term in the matrix.

"*sin - returns sin of each entry" "*cos - returns sin of each entry" "*tan - returns sin of each entry" "*asin - returns arcsin of each entry" "*acos - returns arccos of each entry" "*atan - returns arctan of each entry" "*''exp' - returns e to the value of each entry" "*log - returns natural log of each entry" "*rem - returns remainder of each entry" "*abs - returns absolute value of each entry" "*sqrt - returns square root of each entry" "*sign - returns sign of each entry" "*round - rounds each entry to the nearest whole number" "*floor - returns lowest entry value" "*ceiling - returns highest entry value"

Vector Functions
As with scalar functions, the below commands are designed for vector matrices. The command is evaluated column by column and the output is a row matrix corresponding to the result from each evaluation. The vector functions are:

"*max(x) - returns maximum value in vector x" "*min(x) - returns minimum value in vector x" "*sort(x) - returns vector x in rearranged in numeric order" "*sum(x) - returns sum of all values in vector x" "*prod(x) - returns product of all values in vector x" "*median(x) - returns median of values in vector x" "*mean(x) - returns mean of values in vector x" "*std(x) - returns standard deviation of values in vector x" "*any(x) - applies a comparison to any value in vector x" "*all(x) - applies a comparison to all values x"

To apply these for an entire matrix rather than a single column, use them twice. For example, median(median(A)) would return the median of the row matrix which was returned by median(A).

Matrix Functions
Matrix functions are by far the most useful and powerful of Matlab's abilities. The following are the matrix functions and their description.

"*eig - eigenvalues and eigenvectors" "*chol - cholesky factorization" "*svd - singular value decomposistion" "*inv - inverse" "*lu - LU factorization" "*qu - 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, infinite-norm" "*cond - condition number in teh 2-norm" "*rank - rank"

Logic Functions in Matlab
As in many program languages, logic functions such as for, if-and, and while loops are essential assets to understand. In most cases these are implemented very similarly to other languages. The details of each implementation is below.

For Loops
For loops are created using a for condition, an expression, and the end commmand. For example, if you wanted to make a vector y of size n (a predifined scalar) in which each element is the previous element plus one, the code would look like that below.

"y = [];" "for a = 1:n" "y[y,y(n-1)+1]" "end"

The loop will continue as long as the condition (a=1:n) is true. This condition is saying that a is incremented by one each iteration and will continue until a=n. Each time the loop is evaluated, the expression is evaluated and the value of the variable (in this case matrix y) is updated accordingly. If you wish to print the end value of the variable in question, just type the variable name after the end command.

While Loops
While loops also operate much like in other languages. They are generally stated using a while relation, a group of statements and operations, and the end command to finish the loop. These loops differ from For loops in the fact that they will execute until the relation is false, and not simply a preset number of iterations. A simple while loop, creating a matrix of the numbers one through ten, is shown below.

"x=[];" "a=0;" "while a<=10" "x=[x,n]" "end" "x"

This code would then print out the vector x = [0,1,2,3,4,5,6,7,8,9,10].

If Loops
If loops are implemented very similarly to while loops, involving a true/false relationship, a series of statements, and the end command. However, If loops allow the user to create several different branches of elseif and else loops to allow for more than two sets of operations. For example, if there is a vector with 5 entries, you could add 3 to entries 1 and 5 while subtracting 1 from entries 2 and 3 and force entry 4 to be zero using the code below.

"X=[1,1,1,1,1]" "n=1" "5''" "X(n)=X(n)+3"

"3''" "X(n)=X(n)-1"

"else" "X(n)=zero"

"end" "X"

This would then print out the new vector X = [2,0,0,0,2].

Relationship Operators
The following are the preset relationship operators that Matlab uses in the above described loops.

 * < - Less than
 * > - Greater than
 * <= - Less than or equal to
 * >= - Greater than or equal to
 * == - Equal to
 * ~= - Not equal to

Also, the three logical operators below are used to string relationships together.

 * & - and
 * | - or
 * ~ - not

It is important to so note that when an operator is used to compare two scalars, such as 4>5, the resultant answer will be a 1 or a 0, corresponding to either true or false, respectively.

However, comparing two matrices is a bit different. The comparison if A==B is true if every term in A is the same as every term in B. The any command can get around this aspect to compare matrices more generally. For instance, the command if any(any(A==B)) will be true if any element of A is the same as the corresponding term in B and is not a global matrix comparison.

Colon Notation
One very key shortcut in Matlab is what is known as colon notation. To this point, if you desired to create a vector of all even numbers between 2 and 20, you would have had to manually enter each entry. However, in actuality, the command 2:2:20performs the exact same operation. Note that the first number is the starting point, the second the increment, and the third is the ending point. If the middle number is left out, Matlab assumes is to be one.

The colon notation can also be reference certain submatrices within already created matrices. For example, X(1:2,5) references the first two rows of column five of matrix X. If you would like to reference an entire row or column in a matrix, there is a shortcut here as well. Suppose a 200x200 matrix, A, and assume you want to perform an operation to the entire 45th column of this matrix. Rather than referencing (1:200,45), you could simply use the syntax (:,45). Matlab interprets a colon by itself to mean an entire row or column. You can perform operations to specific submatrices to create the submatrix you would like to reference and you can reference more than one row or column in a specific operations. For examples, A([5,7],:) references all columns in both row 5 and row 7 of A. Likewise, the statemnt A([5,7],:) = A([5,7],:)*[1 2; 3 4] would multiply just rows 5 and 7 by the 2x2 matrix shown.

Again, this notation and utility is something unique to Matlab and is another aspect that gives it an enormous advantage over previously used softwares and languages.

Working with Files in Matlab
In Matlab, you are able to create files, much like source files in other languages, that perform a series of operations each time they are accessed. Such files are called M-files due to their file extension, .m. There are two types of M-files in Matlab and each is discussed below.

Script Files
Script files can be thought of in Matlab as sub-programs that can later be accessed for use. A script file is just series of statements that are executed until the file is complete. In order to reference a script file, say solveatriangle.m, simply type the command solveatriangle in your Matlab code and the file will be executed. One very important thing to note and to be cautious of is that variables in script files are global and any changes made to variables during execution will be applied to all variables with the same name in your matlab code. Additionally, it is possible for M-files in Matlab to reference many other M-files, and even itself, if necessary to complete the desired operation.

Function Files
Function Files are very similar to script files, with two major differences. The first, and most important, is that all variables in a function file are local unless specified to be global.

Secondly, script files are just a sequence of operations to be performed at a specified point in the code. However, function files are input/output operations, to which one or more input variables are passed and one or more outputs are retrieved. Because of this, the first line of every function file is the function name, the input arguments, and the output arguments. The following code would be an example of a simple function that takes 5x5 matrix A as an input, adds 3 to every entry in row 5 and outputs the resulting matrix and assigns it to a new 5x5 matrix B.

"Function File"

"function B = add3(A)" "B=A" "B=B(5,;)+3"

"Matlab Code"

"B = add3(A)"

At this point, Matlab would print the new matrix B.

Obviously, function files can be far more complex and, thus, much more utile than this example. However, the basic outline and format would remain the same whether it was four lines or 400.

Text Strings in Matlab
Text strings are very important tools, usually used in M-files. They serve to prompt for input and also to alert if an error has occurred. Strings can be referenced instance by instance or can be assigned to a variable and are specified by text between single quotes ('string'). If you wish to display a message, the disp(string) command will print the string identified in the. Also, the error will behave similarly. However, error will abort execution of the M-file it is a part of. The other function of strings is to prompt the user for input. To do this in an M-file, you need to use an equation, assigning the user's input to a variable as well as the input command. For example:

a = input('Please enter the length of side a: ')

This would prompt the user to "Please enter the length of side a: ". Upon entering a value and pressing enter, the input would then be assigned to variable a and the next statement in the M-file would be executed.

Managing M Files
Matlab has a neat command to allow users to manage/create new files without the need to exit Matlab or losing active data.

This tool is the ! feature.

! feature
When any command that would require the system to exit is preceded with an "!", Matlab will be returned to automatically with all current data - after whatever file management is completed.

Example: !ed Shamrock.m  will allow the user to edit the m-file named Shamrock and then return to Matlab exactly as it was.

More File Management Features
command pwd returns the present working directory name

command cd is used to change the working directory

commands dir or ls will show all contents of the current working directory

command what shows the M-files in the current directory

command delete can delete a diskfile

command type will print an M-file to the screen (PrtSC)

command path shows the current directories list

Note: M-files need to be in a directory accessible to Matlab.

Efficiency of Matlab Algorithms: Commands: Flops, Tic and Toc
Flops or Floating Point Operations are one way to measure the efficiency of your Matlab algorithm.

The total duration of time is also a good measure of the efficiency.

Flops Command
Enter flops(0) prior to algorithm and flops after and Matlab will count the number of flops registered during the algorithm sequence.

Example: flops(0), x7+ey 2 =x3; flops

Tic and Toc
Similarly, if you precede the Algorithm with Tic and follow it with Toc and Matlab will post the time elapsed during calculations.

Example: tic, x7+ey 2 =x3; toc

Output Format Precision
Matlab computations will always be done in double precision but there are commands to format what is displayed.

Display Commands
format short - fixed point with 4 decimal places

format long - fixed point with 14 decimal places

format short e - scientific notation with 4 decimal places

format long e - scientific notation with 15 decimal places

format rat - approximation by ratio of small integers

format hex - hexadecimal format

format bank - fixed dollars and cents

format + - +, -, blank

Format Commands
format compact - eliminates blank lines compressing the page

format loose - returns to the original expanded format, similar to double space

Note: format commands remain in effect until changed

Hardcopy
A hardcopy of the information on the screen can be copied to a disk file with the command diary

diary filename - writes information to screen until diary off is entered

Graphics and 3D Graphing in Matlab
The most important commands are plot, plot3, mesh, and surf. The Demo command will give a brief into to each.

Plot
creates x-y plots of the linear form. The following examples will show how to plot vectors over an interval for x.

Example: x = -10:2:10; y=sin(x); plot(x,y)

Example: x = -1.5:0.1:1.5; y=exp(x.^3); plot(x,y)

Note: A period precedes the superscript for proper entry-wise operation

Fplot
Fplot works just the same as plot command and is used for plotting functions

Graph Commands
tittle - add a graph tittle

xlabel - label the x axis

ylabel - label the y axis

gtext - place text on the graph at desired location

text - position text by coordinates

Axis Commands
axis - set axis scaling to prescribed limits

axis(axis) - freezes scaling for subsequent graphs

axis auto - returns to auto-scaling

v = axis - returns vector v showing current scaling

axis square - same scale on both axes

axis equal - same scale and tic marks on both axes

axis off - turns off axis scaling and tic marks

axis on - turns on axis scaling and tic marks

Line-types, Marks and Colors
Line-types: solid (-), dashed (--), dotted, dashdot(-.)

Mark-types: point(.), plus(+), star(*), circle(o), x-mark(x)

Colors: yellow (y), magenta (m), cyan(c), red(r), green(g), blue(b), white(w), black(k)

Example: plot(x,y,'b:') will plot a blue dotted line

Plot3
Plot3 will produce a 3-Dimensional plot of vectors: x, y and z'

The vectors will usually be defined parametrically

Example: t = 0.05:0.01:10*pi; x = cos(2t); y = sin(0.5*t); z = t.^4; plot(x,y,z)

Title and axis commands are the same as for command Plot with the z axis included.

Mesh and Surf Commands
These commands will plot 3-Dimensional "mesh" graphs of matrices. Example: z = [3,9,1] mesh(z)

Surf creates surface plots

Advanced Example
Plot the graph of z = e-x 3+y2 over the square [-4,4] x [-4.4]

xx = -4:0.4:4; yy = xx; [x,y] = meshgrid(xx,yy); z = exp(-x.^3+y.^2); mesh(z)

Sparse Matrix Computations
The computing efficiency of large matrix based problems can be increased using the Sparse command.

The Command will force Matlab to only run computation on nonzero matrix elements - significantly increasing efficiency if said matrix is dense with zeroes.

Say we have Matrix A

To find the number of nonzero elements in A use command: nnz(A) to return the number.

Now we can convert it to a sparse matrix with S = sparse(A)

F = Full(S) will bring the matrix back to full

Matlab References
Many MATLAB features were not included in the tutorial but they are listed below and organized by their subject area. There are more functions beyond the ones that are included below but these can be located by using the help command. Here are some easy reference charts that will help during the use of MATLAB.

Reference
Sigmon, K. (1993). MATLAB Primer (Third Edition). Available Here Previously Accessed September, 12, 2008.

/HW1 contributing team members

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User:EAS4200C.F08.WIKI.A/My Contribution: HW

=Aircraft Example=

The following figure provides examples of materials used on the F/A-18E Hornet. All of the materials used in this aircraft are light, high in stiffness, and high in strength therefore allowing for optimal performance and long durability. It is desirable however to eliminate all aluminum and titanium and replace them with Fiber Reinforced Composites which would in turn decrease the weight of the aircraft by 30%-40%. In doing so the aircraft would be at its maximum performance. An example of the composites being used in aviation can be seen in the Boeing 787. Fifty percent of this vehicle is made up of these materials making it lighter and more efficient.

The table below gives some examples of parts made of the materials portrayed or not portrayed in the photo.

=Problem 1.1 (Sun 2006)=

Now that some fundamentals have been laid out, it is time to review some concepts that would have been learned in a course known as Mechanics of Materials. It is a course that covers the tendencies of materials under certain forces, bending moments, and torques. To do so a problem from the text "Mechanics of Aircraft Structures" by C.T. Sun will be covered. The problem statement is as follows.

Problem Statement

"The beam of a rectangular thin-walled section (i.e., t is very small) is designed to carry both bending moment M and torque T. If the total wall contour length L = 2(a+b) [(see figure below)] is fixed, find the optimal ratio b/a to achieve the most efficient section if $$M=T$$ and $$\displaystyle\sigma_a=2\displaystyle\tau_a$$ . Note that for closed thin-walled sections such as the one shown below, the shear stress due to torsion is:

$$\displaystyle \tau = T/2abt$$

Problem Solution

The solution to this problem requires the analysis of two separate cases because the question asks for the optimal ratio. The first case assumes that the $$\displaystyle \sigma_m$$ reaches $$\displaystyle \sigma_a$$. In order to judge the correctness of the answers gotten from the analysis, it is imperative that the $$\displaystyle \tau_m $$ < $$\displaystyle \tau_a$$. If for this case $$\displaystyle \tau_m $$ > $$\displaystyle \tau_a $$, than the ratio of b/a is not acceptable.

 * Case 1
 * The first step in solving this problem is to compute the allowable bending moment, M, that corresponds to $$\displaystyle \sigma_m$$ at the location of z = b/2


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User:Eas4200c.f08.gator.edwards/Wikipedia Introduction

Wikipedia is a safe and effective way to view and edit information on nearly every topic imaginable. Though some information should be taken with a grain of salt, in most cases it is fairly accurate. Individual users are able to alter information for specific pages, increasing the breadth of knowledge.

CAUTIONARY NOTE: Some information can be made false or misleading and, in some cases, purposeful misinformation can be included.

Knowing this a user can return to any previously saved information by looking in the wiki log. This log describes every user who added or subtracted from the information and what their contribution was. To deter aggresive wiki vandals from reaching out and targetting EAS 4200 user pages students are encouraged to keep a low profile. Editing other pages under a class username is not recommended as this may attract unwanted attention.

For use in EAS 4200 when turning in a homework file. The archived link should be included, not just the overall link to the main page since the article can possibly change from the time the user submits the work and the Teachers Assistant or Professor views it.

Articles in Wikipedia can be edited easily. The homeworks are created as an article under the team leader's account. From there all members of the team have access to it. The Wikipedia article editor is straightforward but there are some useful shortcuts to note:

alt + shift + e --> edit article alt + shift + s --> save article alt + shift + p --> preview article

Depending on your web browser, these shortchuts can slightly differ. See Wikipedia Keyboard Shortcuts for an extended list.

Homwork reports prepared on wikipedia are to be written in prose and should follow the fomat of this wikipedia page on Stress. The goal of this idea is to follow the format of MIT's Open CourseWare. This is to allow students to be able to understand material on their own, then communicate with others and share it in their group, and then for each group of students to share all of their work with each other. This allows for students to learn from other students and then allows the professor to communicate with the students as well. Our class is a small scale example of what MIT has to offer which is class notes for all of their classes provided to the public. Wikiversity is similar to MIT's site by providing information on many subject for further learning through collaboration.

Class Information

course title: EAS4200c Aerospace Structures

professor e-mail: vu-quoc@ufl.edu

course website: mae.ufl.edu/~vql $$ \Rightarrow \ $$ teaching $$ \Rightarrow \ $$ my course website

team details: team name between four and six characters, member names and UFID numbers, wikipedia username for the class for each member


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User:Eml4500.f08.ramrod.D

TEAM RAMROD
[Amory T.] [Chris P.] [Jeff  H.] [Dan   O.] [Kevin S.] [Chris F.]

/HW1 /HW2

/HW3

/HW4

/HW5

/HW6

/HW7

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User:EML4500.f08.RAMROD.E/HW1

= Matlab Summary =

Intro
Matlab is a software program, developed by Matrix Laboratories, that uses matrix-based mathematics to solve ordinarily difficult scientific and engineering applications. It has several advantages over traditional methods (i.e. Fortran or C++), specifically its ease of use, versatility of capabilities, and its ability to convey results graphically as charts and its ability to model very complex systems visually for ease in conveyance of results. Since its development, Matlab has become very widely used in nearly all scientific fields and, especially in engineering practice, has become an absolutely vital tool for all types of engineering analysis.

Matrices in Matlab
As was stated above, Matlab is a matrix-based program. Thus, all variables, whether scalars, vectors, or much more complex and larger-scale matrices are all stored in the same manner. Because of the significance of matrices in this program, the first, and most important operations to know are those involving matrices.

Creating Matrices
There are several ways in Matlab to make new matrices. One method is to enter the values directly. For example:

 * A = [2,3,4;5,3,6;9,5,2]

 * A = [2 3 4; 5 3 6; 9 5 2]

Both of these entry syntaxes would create a 3x3 rectangular matrix with the indicated values. Other permissible formats are for scientific notation, such as [2.34e-4, 3.53e8], or for complex numbers, entered as [2+4i,3+8i] or [2,3] + i*[4,8]. In both instances (exponential and complex numbers) spaces should be avoided between numbers and commas should be used. When manually entering values into a matrix, as shown in the above examples, commas (or spaces where allowed) separate entries in a row while a colon separates one row from the next.

Another option for creating matrices is to read from an input file, usually an ASCII file from another editor on your computer. This is completed by using the load command. To load input file "input.dat", the command would be load input.dat.

After creating a matrix, it can then be referenced by its name (i.e. A). Also, you can reference individual entries such as A(4,2) for the second entry in row four of matrix A.

Matrix Building Functions
In addition to creating your own matrices, Matlab has several default "special" matrices and matrix references that can be called on using their reserved names. The following are a few of the standard Matlab matrices and uses:

 * eye(m,n) will create the identity matrix of dimensions mxn.
 * zeros(m,n) will create a mxn matrix where all entries are zeros.
 * ones(m,n) will create an mxn matrix where all entries are ones.
 * diag(x) will create a matrix where the diagonal is the predefined vector x.
 * triu(x) or tril(x) references the upper or lower triangle of matrix x, respectively.
 * rand(m,n) will create an mxn matrix with Matlab randomly generating values for each entry that will be uniformly distributed between 1 and 0.
 * magic(m,n) will provide an mxn matrix that is a "magic" matrix in which each column, row, and diagonal totals to the same sum.
 * hilb(m,n) creates an nxn Hilbert Matrix.

It is useful to note also that creating a square matrix of any of the above forms is done by replacing the (m,n)dimensions by simply an (n) dimension.

Also, any of these matrices can be created to match the size of an existing matrix by using the command size(X) where X is a matrix. For example, eye(size(X)) creates an identity matrix with the same dimensions as matrix X.

All of these commands are extremely useful as starting points for building matrices. For example, A = eye(2) could then be edited (A(2,1)=5) to modify it from an identity matrix to the desired matrix without manually entering the value of each entry.

Matrix Operations
Once a matrix is created using the above stated methods, Matlab has several built in operations to manipulate it mathmatically. The following are the basic matrix operations that Matlab can perform:

 * +: addition - adds two matrices term by term to create a new matrix.
 * -: subtraction - subtracts two matrices term by term to create a new matrix.
 * *: multiplication - multiplies two matrices together
 * ^: power - raises a matrix to power indicated after ^.
 * ': conjugate transpose - returns a matrix where each term's indices are swapped (Aij goes to Aji) and all terms are replaced by their complex conjugate (a+bi goes to a-bi).
 * \: left division - A\b = b/A.
 * /: right division - normal division of two matrices.

While addition and subtraction above execute a term by term operation, the others do not. However, Matlab will allow this by placing a period before the operator. For example:

"[2,5,9,7].*[2,5,3,1] = [4,25,18,7]"

This form of multiplication, division, or exponents can be found very helpful when working with Matlab Graphics which will be described later.

Statements, expressions, and variables in Matlab
In addition to being a matrix-based software, the language of Matlab is expressions. Expressions are used for every function that Matlab is capable of. Generally, an expression can take the form of a equation or a simple statement.

Equations
Equations, such as Y = 5X+5, are generally created by stringing together variables, functions, and operators and results in a matrix that is then output to the screen and can be referenced in later functions.

Statements
If a certain variable and equals sign is not specified, 5X+5, this expression results in a newly created variable "ans" whose value is stored until a new expression is evaluated. For this case, the line ans = (value) will be output to the screen. However, if the expression is followed by a semicolon, 5X+5;, this line will not be shown. This is very convenient in long iterations where extra lines of output answers only complicate the code and calculations and are undesired.

Variables
An important thing to note here is that Matlab is case-sensitive, meaning that the variable y is not the same as the variable Y. If at any given time you wish to see a list of the current variables, the command "who" or also whos will print the list and their values to the screen. Additionally, variables can be cleared by the command clear followed by the variable name you wish to clear. The command clear by itself will clear all non-permanent variables in the cache.

Saving a Session
In order to not lose variables when you exit Matlab, it is very useful to know the save command. This will write all variables and their values to a non-accessable file named "matlab.mat". Upon entereing Matlab the next time, simply typing the load command will access the file and will bring back all variables as they were when the file was saved.

Scalar, Vector, and Matrix Functions
Matlab has many predefined functions that are very useful in doing matrix calcluations. Some are solely used for scalars matrices (1x1), some for vectors, and the most powerful are the matrix functions.

Scalar Functions
The below functions are designed to work with scalars only, but can be applied to a matrix. When the following are used in reference to a matrix, the operation is computed term by term in the matrix.

"*sin - returns sin of each entry" "*cos - returns sin of each entry" "*tan - returns sin of each entry" "*asin - returns arcsin of each entry" "*acos - returns arccos of each entry" "*atan - returns arctan of each entry" "*''exp' - returns e to the value of each entry" "*log - returns natural log of each entry" "*rem - returns remainder of each entry" "*abs - returns absolute value of each entry" "*sqrt - returns square root of each entry" "*sign - returns sign of each entry" "*round - rounds each entry to the nearest whole number" "*floor - returns lowest entry value" "*ceiling - returns highest entry value"

Vector Functions
As with scalar functions, the below commands are designed for vector matrices. The command is evaluated column by column and the output is a row matrix corresponding to the result from each evaluation. The vector functions are:

"*max(x) - returns maximum value in vector x" "*min(x) - returns minimum value in vector x" "*sort(x) - returns vector x in rearranged in numeric order" "*sum(x) - returns sum of all values in vector x" "*prod(x) - returns product of all values in vector x" "*median(x) - returns median of values in vector x" "*mean(x) - returns mean of values in vector x" "*std(x) - returns standard deviation of values in vector x" "*any(x) - applies a comparison to any value in vector x" "*all(x) - applies a comparison to all values x"

To apply these for an entire matrix rather than a single column, use them twice. For example, median(median(A)) would return the median of the row matrix which was returned by median(A).

Matrix Functions
Matrix functions are by far the most useful and powerful of Matlab's abilities. The following are the matrix functions and their description.

"*eig - eigenvalues and eigenvectors" "*chol - cholesky factorization" "*svd - singular value decomposistion" "*inv - inverse" "*lu - LU factorization" "*qu - 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, infinite-norm" "*cond - condition number in teh 2-norm" "*rank - rank"

Logic Functions in Matlab
As in many program languages, logic functions such as for, if-and, and while loops are essential assets to understand. In most cases these are implemented very similarly to other languages. The details of each implementation is below.

For Loops
For loops are created using a for condition, an expression, and the end commmand. For example, if you wanted to make a vector y of size n (a predifined scalar) in which each element is the previous element plus one, the code would look like that below.

"y = [];" "for a = 1:n" "y[y,y(n-1)+1]" "end"

The loop will continue as long as the condition (a=1:n) is true. This condition is saying that a is incremented by one each iteration and will continue until a=n. Each time the loop is evaluated, the expression is evaluated and the value of the variable (in this case matrix y) is updated accordingly. If you wish to print the end value of the variable in question, just type the variable name after the end command.

While Loops
While loops also operate much like in other languages. They are generally stated using a while relation, a group of statements and operations, and the end command to finish the loop. These loops differ from For loops in the fact that they will execute until the relation is false, and not simply a preset number of iterations. A simple while loop, creating a matrix of the numbers one through ten, is shown below.

"x=[];" "a=0;" "while a<=10" "x=[x,n]" "end" "x"

This code would then print out the vector x = [0,1,2,3,4,5,6,7,8,9,10].

If Loops
If loops are implemented very similarly to while loops, involving a true/false relationship, a series of statements, and the end command. However, If loops allow the user to create several different branches of elseif and else loops to allow for more than two sets of operations. For example, if there is a vector with 5 entries, you could add 3 to entries 1 and 5 while subtracting 1 from entries 2 and 3 and force entry 4 to be zero using the code below.

"X=[1,1,1,1,1]" "n=1" "5''" "X(n)=X(n)+3"

"3''" "X(n)=X(n)-1"

"else" "X(n)=zero"

"end" "X"

This would then print out the new vector X = [2,0,0,0,2].

Relationship Operators
The following are the preset relationship operators that Matlab uses in the above described loops.

 * < - Less than
 * > - Greater than
 * <= - Less than or equal to
 * >= - Greater than or equal to
 * == - Equal to
 * ~= - Not equal to

Also, the three logical operators below are used to string relationships together.

 * & - and
 * | - or
 * ~ - not

It is important to so note that when an operator is used to compare two scalars, such as 4>5, the resultant answer will be a 1 or a 0, corresponding to either true or false, respectively.

However, comparing two matrices is a bit different. The comparison if A==B is true if every term in A is the same as every term in B. The any command can get around this aspect to compare matrices more generally. For instance, the command if any(any(A==B)) will be true if any element of A is the same as the corresponding term in B and is not a global matrix comparison.

Colon Notation
One very key shortcut in Matlab is what is known as colon notation. To this point, if you desired to create a vector of all even numbers between 2 and 20, you would have had to manually enter each entry. However, in actuality, the command 2:2:20performs the exact same operation. Note that the first number is the starting point, the second the increment, and the third is the ending point. If the middle number is left out, Matlab assumes is to be one.

The colon notation can also be reference certain submatrices within already created matrices. For example, X(1:2,5) references the first two rows of column five of matrix X. If you would like to reference an entire row or column in a matrix, there is a shortcut here as well. Suppose a 200x200 matrix, A, and assume you want to perform an operation to the entire 45th column of this matrix. Rather than referencing (1:200,45), you could simply use the syntax (:,45). Matlab interprets a colon by itself to mean an entire row or column. You can perform operations to specific submatrices to create the submatrix you would like to reference and you can reference more than one row or column in a specific operations. For examples, A([5,7],:) references all columns in both row 5 and row 7 of A. Likewise, the statemnt A([5,7],:) = A([5,7],:)*[1 2; 3 4] would multiply just rows 5 and 7 by the 2x2 matrix shown.

Again, this notation and utility is something unique to Matlab and is another aspect that gives it an enormous advantage over previously used softwares and languages.

Working with Files in Matlab
In Matlab, you are able to create files, much like source files in other languages, that perform a series of operations each time they are accessed. Such files are called M-files due to their file extension, .m. There are two types of M-files in Matlab and each is discussed below.

Script Files
Script files can be thought of in Matlab as sub-programs that can later be accessed for use. A script file is just series of statements that are executed until the file is complete. In order to reference a script file, say solveatriangle.m, simply type the command solveatriangle in your Matlab code and the file will be executed. One very important thing to note and to be cautious of is that variables in script files are global and any changes made to variables during execution will be applied to all variables with the same name in your matlab code. Additionally, it is possible for M-files in Matlab to reference many other M-files, and even itself, if necessary to complete the desired operation.

Function Files
Function Files are very similar to script files, with two major differences. The first, and most important, is that all variables in a function file are local unless specified to be global.

Secondly, script files are just a sequence of operations to be performed at a specified point in the code. However, function files are input/output operations, to which one or more input variables are passed and one or more outputs are retrieved. Because of this, the first line of every function file is the function name, the input arguments, and the output arguments. The following code would be an example of a simple function that takes 5x5 matrix A as an input, adds 3 to every entry in row 5 and outputs the resulting matrix and assigns it to a new 5x5 matrix B.

"Function File"

"function B = add3(A)" "B=A" "B=B(5,;)+3"

"Matlab Code"

"B = add3(A)"

At this point, Matlab would print the new matrix B.

Obviously, function files can be far more complex and, thus, much more utile than this example. However, the basic outline and format would remain the same whether it was four lines or 400.

Text Strings in Matlab
Text strings are very important tools, usually used in M-files. They serve to prompt for input and also to alert if an error has occurred. Strings can be referenced instance by instance or can be assigned to a variable and are specified by text between single quotes ('string'). If you wish to display a message, the disp(string) command will print the string identified in the. Also, the error will behave similarly. However, error will abort execution of the M-file it is a part of. The other function of strings is to prompt the user for input. To do this in an M-file, you need to use an equation, assigning the user's input to a variable as well as the input command. For example:

a = input('Please enter the length of side a: ')

This would prompt the user to "Please enter the length of side a: ". Upon entering a value and pressing enter, the input would then be assigned to variable a and the next statement in the M-file would be executed.

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User:Eas4200c.f08.carbon.orear/Week1Notes

When designing Aerospace Structures it is important that the structures are light weight, but also strong and stiff. Some commonly used aircraft materials are aluminum, steel, titanium, and carbon fiber composites.

Material Characteristics
Stiffness is defined by the slope of the plotted curve representing the stress and strain relationship. Stress is equal to the product of this slope (designated the elastic modulus, E) and strain. $$\displaystyle \sigma = E \epsilon$$

Strength can be characterized in many ways. Two common measurements important in engineering applications are the yield stress and the ultimate stress. Yield stress is the stress level that a material begins to deform and stretch. Ultimate stress is the stress level that the material ruptures.

Toughness is the ability to resist fracture. A material is said to possess "high fracture toughness" if it resists fracture well and "low fracture toughness" if it is less capable of resisting fracture.

 * {| class="toccolours collapsible collapsed" width="60%" style="text-align:left"

!Contribution Metals Stress-Strain Curve, from team Carbon
 * The graph (drawn by Jeff Wright) depicts the general graph of the relationship between stresses and strains in metals.
 * The graph (drawn by Jeff Wright) depicts the general graph of the relationship between stresses and strains in metals.

 * }

Material Examples
We are interested in aerospace materials beyond just the cost and manufacurability of aircraft components. The nature of the materials' engineering properties (like strength, stiffness, and toughness) help us choose materials appropriate to a specific application.

 * {| class="toccolours collapsible collapsed" width="60%" style="text-align:left"

!Contribution Stiffness Table, from team Carbon
 * The table (drawn by Jeff Wright) shows the mechanical properties of metals at room temperature in aircraft structures, and depicts the correlations between density, tensile ultimate stress, tensile yield stress, and stiffness.
 * The table (drawn by Jeff Wright) shows the mechanical properties of metals at room temperature in aircraft structures, and depicts the correlations between density, tensile ultimate stress, tensile yield stress, and stiffness.
 * }

Some material examples are given below:

HIGH stiffness and HIGH strength:
 * Steel Alloy
 * Titanium Alloy

 * {| class="toccolours collapsible collapsed" width="60%" style="text-align:left"

!Contribution Material Properties with high stiffness and strenth, from team Carbon
 * Steel alloys have very high stiffness and strength. Titanium alloys have less high stiffness and strength, and Aluminum alloys still less.  Modern aluminum lithium alloys are 10% stiffer and 10% lighter than conventional aluminum alloys.
 * Steel alloys have very high stiffness and strength. Titanium alloys have less high stiffness and strength, and Aluminum alloys still less.  Modern aluminum lithium alloys are 10% stiffer and 10% lighter than conventional aluminum alloys.

-Aluminum alloys are resistant to fatigue and cyclic tensile stress, and thus are used on the fuselage and the lower wing skins. Aluminum has historically played a major roll as material used in aircraft structures because it is cheap to buy, manufacture and is abundant, as opposed to Titanium which is expensive.

-Titanium is heavier than aluminum and lighter than steel, and double the yield and ultimate stress of aluminum. It is popular for its corrosion resistance and ability to withstand extreme temperatures up to 1000°F.

-Steel Alloys have extremely high stiffness and strength, but is expensive and has poor resistance to corrosion. Thus, despite its unique material property, is general reserved for landing gear and fittings with high loads.

Reference Sun [2006], pg. 14-15
 * }

HIGH stiffness and LOW toughness:
 * Glass

 * {| class="toccolours collapsible collapsed" width="60%" style="text-align:left"

!Contribution Glass Discussion, from team Carbon
 * Because of the very low toughness of glass, it has a very limited use on aircraft. Bulk glass is avoided on aircraft, and other forms of glass are preferred such as glass fiber for its improved mechanical properties.  Because of the extreme forces acting on the fuselage in-flight, windows are designed to be very small and sturdy.  Recent discoveries of new glass-like materials allow for larger windows.  Also, the study of metallic glass has become increasingly significant and could prove to have major importance on aircraft in the future.  The article below describes some of the issues and discoveries of metallic glass:
 * Because of the very low toughness of glass, it has a very limited use on aircraft. Bulk glass is avoided on aircraft, and other forms of glass are preferred such as glass fiber for its improved mechanical properties.  Because of the extreme forces acting on the fuselage in-flight, windows are designed to be very small and sturdy.  Recent discoveries of new glass-like materials allow for larger windows.  Also, the study of metallic glass has become increasingly significant and could prove to have major importance on aircraft in the future.  The article below describes some of the issues and discoveries of metallic glass:

Metallic Glass
 * }

LOW stiffness and HIGH toughness:
 * Nylon
 * Plastic
 * Aluminum

Aluminum has good fracture toughness and fatigue life, and is therefore used on the fuselage skin.

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!Contribution Material Properties of Aluminum and Steel Alloys, from team Carbon
 * This table (drawn by Jeff Wright) shows the material properties of aluminum and steel alloys
 * This table (drawn by Jeff Wright) shows the material properties of aluminum and steel alloys
 * }

HIGH stiffness and HIGH toughness:
 * Composites

 * {| class="toccolours collapsible collapsed" width="60%" style="text-align:left"

!Contribution Composites Discussion, from team Carbon
 * Fiber-reinforced composites allow for complex loads by introducing fibers embedded in a matrix (polymers, metals, ceramics, etc.). They are stiff, strong, and light, and are ideal to withstand high loads experienced during flight while still minimizing weight to increase the aerodynamic properties.  The composite laminates "have excellent fatigue life, damage tolerance, and corrosion resistance".  Fiber composites are becoming very popular for their unique material properties and compose 50% of the new Boeing 787.  More facts on the 787 can be found at:
 * Fiber-reinforced composites allow for complex loads by introducing fibers embedded in a matrix (polymers, metals, ceramics, etc.). They are stiff, strong, and light, and are ideal to withstand high loads experienced during flight while still minimizing weight to increase the aerodynamic properties.  The composite laminates "have excellent fatigue life, damage tolerance, and corrosion resistance".  Fiber composites are becoming very popular for their unique material properties and compose 50% of the new Boeing 787.  More facts on the 787 can be found at:

Boeing 787 Fact Sheet

Reference Sun [2006], pg. 15-16

 * }

Aspects of Structure
It is key to look at both geometry and material in aerospace structure design. Some geometry examples are mentioned below.

Monocoque structure is that of a single shell design or thin-shell structures. The word has the French base 'coque' meaning, literally, shell. Semi-monocoque, or stiffened shell structures can also be worth exploring. It is important to note that stiffness and brittleness in aerospace structures can be either highly valuable or highly problematic. Normally, shell structures are chosen for their appealing strength for a very limited weight.

Goals with geometry and material
In considering geometry and material in an aerospace structure, one should try to optimize and integrate the two to minimize cost and weight. Sometimes geometry is limited by design goals and aerodynamic properties such as lift and drag. Careful consideration must be taken, therefore, not only to provide sufficient use of the materials for one purpose but to look at the integration of the entire system and how to maximize the utility of the available materials and possible geometric configurations.

The following website contains content not created by Team Carbon. The link was obtained in a google image search. It shows the material breakdown of an F/A-18 Hornet. F/A-18 Materials Full credit is due to http://www.military.cz/usa/air/in_service/aircraft/f18/f18pics/f18materials.jpg

HW 1 Contributing Team Members
Jeff O'Rear Eas4200c.f08.carbon.orear 14:45, 18 September 2008 (UTC)

Jeff Wright Eas4200c.f08.carbon.w 16:19, 18 September 2008 (UTC)

Beau Guidry Eas4200c.f08.carbon.guidry 19:59, 18 September 2008 (UTC)

Aaron Clausen Eas4200c.f08.carbon.clausen 14:12, 19 September 2008 (UTC)

Abby Booth Eas4200c.f08.carbon.booth 16:29, 19 September 2008 (UTC)


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User:Eas4200c.f08.carbon.orear/HW1.1

HW PROBLEM 1.1
BELOW IS THE SOLUTION FROM CLASS.

Consider a square tube (seen to the right)

 * wall thickness << height (a)
 * wall thichness << width (b)

ASSUMPTION 1: $$\displaystyle L = 2 * (a + b)$$

Q: Find optimal b/a ratio to maximize load bearing capability of beam assuming:

ASSUMPTION 2: $$\displaystyle M = T$$ ASSUMPTION 3: $$\displaystyle \sigma$$ allowable $$\displaystyle = 2*\tau$$ allowable

In other words, find optimal cross section carrying max bending moment and max torque:

$$\displaystyle \tau = T /(2 * a * b * t)$$

Since cross section walls are very thin, we can assume shear stress distribution to be uniform along the wall. See the wikipedia article on shear flow to learn more. Shear Flow

Proof:

$$\displaystyle T = T$$ab $$\displaystyle + T$$bc $$\displaystyle + T$$cd $$\displaystyle + T$$da $$\displaystyle$$

$$\displaystyle T$$ab $$\displaystyle= b/2 * (\tau t * a ) = \tau *a*b*t/2$$

$$\displaystyle T$$bc $$\displaystyle= a/2 * (\tau * t * b) = \tau *a*b*t/2$$

$$\displaystyle T$$cd $$\displaystyle= b/2 * (\tau * t * a) = \tau *a*b*t/2$$

$$\displaystyle T$$da $$\displaystyle= a/2 * (\tau * t * b) = \tau *a*b*t/2$$

Recalling that $$\displaystyle T = T$$ab $$\displaystyle + T$$bc $$\displaystyle + T$$cd $$\displaystyle + T$$da $$\displaystyle$$, this proves that $$\displaystyle T = 2 \tau a b t$$ and  $$\displaystyle  \tau = T/(2 a b t)$$.

$$\displaystyle $$

Case:1
Assume stress (bending normal stress) reaches stress allowable, first. What to verify that $$\displaystyle \tau << \tau$$allowable

Recall: $$\displaystyle Stress = \sigma = (M *z)/I$$

M = bending moment, Z = ordinate of a point on axis perpendicular to neutral bending axis

I = 2nd area moment of inertia

Assume $$\sigma_{max} = \sigma_{allowable}$$, that the maximum bending normal stress increases to the allowable stress first. The goal is to find the maximum allowable bending moment which corresponds to the max bending stress at z=b/2.

Recall: $$\sigma = \frac {Mz} I, ~ Z = \frac b 2$$

=> $$M = \frac {2 I \sigma_{max}} b \frac {2 I \sigma_{allowable}} b$$ because of the case 1 assumption

Put constants together and make it a function of (a,b)

To maximize the bending moment, simply maximize the ratio I/b. First, find the moment of inertia with respect to 'b',

Recall from assumption 1:

=> $$a = \frac L 2 - b$$

=> (I/b) fuction of b

Maximize M : Max = max, from assumption 2 (T=M)

According to assumption 3, the allowable shear stress is equal to half of the allowable normal stress. Therefore, if the max shear stress is less than the allowable shear stress, then the optimal ratio is acceptable. If not, then it is not acceptable.

Assume that the maximum bending normal stress reaches the allowable stress first. Then the moment of inertia of the cross section is the sum of the moments of inertia of the cross section is the sum of the moments of inertia of the four segments of the cross section. If you use the parallel axis theorem, it is as follows:

α$$=(a/12)[t^3+t(12/4)b^2] = (at/12)[t^2+3b^2]=(3ab^2t)/12$$

if t^3 << t, hence the approximation

$$ = \frac {tb(3L - 4b)} {12} = \beta_0 + \beta_1 b + \beta_2 b^2 $$

$$ \beta_0 = 0, \quad \beta_1 = \frac {3Lt} {12}, \quad \beta_2 = -\frac 4 {12} t = -\frac t 3 $$

$$ \frac {d^2 f} {db^2} (b) = 2\beta_2 = -\frac {2t} 3 < 0 $$

$$ f(b) = 0 => b ~ \begin{cases} 0 \\ \underline{3L} \\ 4 \end{cases} $$

Plot f(b) we get inverted parabola, 0 at 0 and 3L/4. max at 3L/8

We can see that this case can NOT exist! We have to violate a known equation to use this cross-section!

Below is a Mohr's Circle representation of Case 1:

Case 2:
Assume torsional shear stress reaches the allowable shear stress first. Torsional shear stress is constant along the thin wall because the cross section is negligibly thin.

$$\tau^{(1)}_{max} < \tau_{allowable}$$

$$\tau_{max} = \tau_{allowable} = \tau = \frac T {2~a~b~t}$$

$$T=(2t$$τallowable)$$(ab)$$ → $$T_{max}=(2t$$τallowable$$)(ab_{max})$$

Maximize T by maximizing the product of a and b. For a constant perimeter L, the cross section of the box beam is square.

The product of a and b is the area of the cross section and a=b, therefore L/4 = a = b because of four sides of the frame.

We need to check to make sure that the maximum bending stress for the cross section is below the allowable bending stress. If this is true, then the optimal ratio b/a = 1 is acceptable, otherwise it is not acceptable.

$$\frac {a^{(2)}} {b^{(2)}} = 1$$

$$T^{(2)}_{max} = (2 + \tau_{all}) * (\frac L 4) = \frac 1 8 + L_{tall} = M^{(2)}_{max}$$

$$M_{max} = 2 \sigma_{all} (\frac I b) = \frac {3 + L^2} {32} \sigma_{all}$$

$$\tau_{max} = \frac {M_{max}} {2abt} = \frac {M_{max}} {2 (\frac L 8) (3 \frac L 8) t} = \frac {32M_{max}} {3tL^2} = \sigma_{all}$$

σmax(2) = Mmax(2)b(2)/[2I(2)] = Mmax(2)12/[tL2], where Mmax(2)=tL2/16

→σmax(2) = (12/16)σallow < σallow → ACCEPTABLE

title: User:Eas4200c.f08.carbon.orear/HW1.1