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Translations of this page English [en] Afrikaans [af] العربية [ar] Azərbaycanca [az] български [bg] বাংলা [bn] Bosanski [bs] Català [ca] Česky [cs] Dansk [da] Deutsch [de] Ελληνικά [el] English [en] Esperanto [eo] Español [es] فارسی [fa] Français [fr] Galego [gl] עברית [he] Hrvatski [hr] Magyar [hu] Bahasa Indonesia [id] Italiano [it] 日本語 [ja] 한국어 [ko] Norsk (Bokmål) [nb] Nederlands [nl] Polski [pl] português do Brasil [pt-br] Română [ro] Русский [ru] Slovinsko [sl] Српски [sr] Svenska [sv] தமிழ் [ta] Tagalog [tl] Türkçe [tr] 简体中文 [zh-cn] 繁體中文 [zh-tw] Help:Contents From Meta, a Wikimedia project coordination wiki Jump to: navigation, search MediaWiki Handbook: Contents, Readers, Editors, Moderators, System admins +/-

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This is a list of guides and single-topic help files for the MediaWiki software. It is compiled at meta.wikimedia.org. You may wish to contribute to this project and import this help into your site. This page is sorted by user type. This help does not cover site-specific conventions, it is for the software only. Enjoy!

Contents [hide] 1 Introduction 2 For Installers 3 For readers 4 For editors 5 For moderators 6 For administrators 7 Appendix 8 See also 9 Links to other help pages

Introduction Detailed Introduction | Help overview

MediaWiki is a free software wiki engine licensed under the GNU General Public License. It is used by many websites, including this one.

For Installers See mw:Category:Installation

For readers Navigating Search, [ Go ] What links here Special pages Printable version

Tracking changes Recent changes {enhanced} Related changes Watching pages Page history Diff page Image page Contributions

Accounts Logging in Preferences User style View this alone

For editors Editing Start new pages Page name

Referencing Links  URLs Piped links Interwiki linking Footnotes (References, Citations)

Formatting Wikitext  CSS Lists  Tables Images and File uploads Formula HTML in wikitext

Organizing Sections and the Table of Contents (TOC) Categories Redirects (forward) Namespaces Moving a page Page size

Fixing mistakes Show preview Testing Reverting edits -- Saving effort Editing shortcuts Edit toolbar Magic words Templates Variables Calculation Embed page -- Communicating Edit summary Talk page Edit conflict Minor edit -- Other Characters Sandbox Legal issues for editors (copyright) For moderators Administration Special:Import Common.css Navigation bar System messages Special:CheckUser -- Help pages Guidelines Help policy Help page footers View this alone

-- For administrators View this alone

Admin's Guide MediaWiki architecture Database layout Job queue -- Installation Working with SVN Installing the SVN tree OS specific help on Sourceforge.net Timezone adjustments -- Security techy Security Security Checklist Paranoid Permissions User rights management Reset admin´s password --

Configuration Preventing Access Configuration Configuration settings Apache config Robots.txt Shorter URLs PHP config MySQL config Postgres config Tips and tricks --

Customization Layout customization Set Global Timezone Localisation Interwiki links Custom navigation bar Custom skins Custom namespaces MediaWiki extensions --

Manage the database Move a database Reduce the size Lock the database --

Performance PHP caching File cache Squid caching Cache strategy Wikimedia servers As your site grows -- Maintenance Maintenance scripts Upgrade Mediawiki Math errors LocalSettings.php

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Appendix ISBN links Common stop words Keyboard shortcuts How to move a wiki to another server Miscellany -- Sites using MediaWiki Browser issues with MediaWiki Page metadata Contact -- irc.freenode.net #mediawiki #wikipedia Mailing lists MediaWiki feature request See also w:Wikipedia:Glossary Check the privacy policy, if any, of the site you use; for Wikimedia: Privacy policy wikicities:Category:Help mw:Category:MediaWiki Documentation mw:Category:PD Help Links from the MediaWiki wiki to the Help namespace here on Metahttp://www.mediawiki.org/wiki/Special:Linksearch/meta.wikimedia.org/wiki/Help: - not complete, requires external link style Links from the English Wikipedia to the Help namespace here on Metahttp://en.wikipedia.org/wiki/Special:Linksearch/en.wikipedia.org/wiki/Help: - not complete, requires external link style or a link produced by w:template:H:mlm ( talk edit history links )

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User:Eml4500.f08.Ateam.paquette/Homework6

Lecture Notes 29, November 3rd, 2008
The initial conditions for the system at t = 0 can be defined below.

$$ u\left(x,t=0\right) = \overline{u}(x) $$ for the given function of x, a known displacement function,

and

$$\frac{\partial u}{\partial t}(x,t=0)={\ddot{u}}(x,t=0)=\overline{v}(x) $$ for a known velocity function.

The principle of virtual work (continuous) of dynamics of the elastic bar can be shown as a partial differential equation:

$$\frac{\partial}{\partial x}\left[(EA)\frac {\partial u}{\partial x}\right]+ f = m{\ddot{u}} $$ (1)

The Discrete Equation of Motion can then be represented as

$$-\underline{K}\underline{d} + \underline{F} = \underline{M}\underline{\ddot{d}}$$

or $$\underline{M}\underline{\ddot{d}} + \underline{K}\underline{d} = \underline{F} $$ (2)

Where $$\mathbf\underline{M}$$ is a mass matrix.

$$\frac{\partial }{\partial x}\left[\left(EA \right)\frac{\partial }{\partial x} \right] = -Kd$$

$$f = F$$

$$m\ddot{u} = M\ddot{d} $$

Note that equation (2) is the equation of motion for a system with multiple degrees of freedom. A system with a single degree of freedom will be shown in equation 3.

For a single degree of freedom

$$\underline{m}\underline{\ddot{d}} + \underline{k}\underline{d} = \underline{F}$$

All of these equations have been derived with the help of the following integral:

$$\int_{x=0}^{x=L} W(x)\left({\frac{\partial}{\partial x}\left[(EA)\frac{\partial u}{\partial x}\right] + f - m\ddot{u}}\right)dx = 0 $$ (3)

for all possible $$\mathbf {W(x)}$$, where $$\mathbf {W(x)}$$ is the weighting coefficient. As seen before, going from equation 1 to equation 3 is trivial but it is important to know how to get from equation 3 to equation 1. Equation 3 can be rewritten as:

$$\int_{x=0}^{x=L} w(x)g(x)dx = 0$$ for all $$\mathbf {w(x)}$$

Since equation 3 holds true for all values of $$\mathbf{w(x)}$$ by the principle of virtual work, we can choose $$ \mathbf {w(x) = g(x)}$$. This changes equation 3 to

$$\int_{x=0}^{x=L} g^2(x)dx = 0$$

Integrating by Parts
Integrating by parts is needed when trying to integrate two functions multiplied together. The formula for integrating by parts can be seen below. The symbols 'r' and 's' will represent the two different functions of x in this example.

$$ \left(rs\right)^'=r^'s+rs^'$$

Where: $$ r^' = \frac{dr}{dx} $$ and $$ s^'=\frac{ds}{dx} $$

Integrating both sides forms the equation:

$$\int \left(rs\right)^' = \int r^'s + \int rs^'$$

This leads to the final equation of:

$$ \int r^'s = rs - \int rs^'$$

Now to continue with the Principle of Virtual Work, recall equation 3:

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

For the first term, let $$r(x) = (EA){\frac{\partial u}{\partial x}}$$ and $${s\left(x\right) = W\left(x\right)}$$

After integrating by parts, the equation yields:

$$ \int_{x=0}^{x=L} W(x)\frac{\partial}{\partial x}\left[(EA)\frac{\partial u}{\partial x}\right]dx = \left[W(EA)\frac{\partial u}{\partial x}\right]_{x=0}^{x=L} - \int_{x=0}^{x=L}\frac{\partial W}{\partial x}(EA)\frac{\partial u}{\partial x}dx$$

Where the minus sign on the right side of the equation relates to the - Kd in the step between equations 1 and 2 above.

Replacing the x's with either L or 0, sets the right side equal to:

$$ = W(L)(EA)(L)\frac{\partial u}{\partial x}(L,t) - W(0)(EA)(0)\frac{\partial u}{\partial x}(0,t) - \int_{0}^{L}\frac{\partial w}{\partial x}(EA)\frac{\partial u}{\partial x}dx $$

Now consider the model equation with two boundary conditions:

At x = 0, select W(x) such that $$W\left(0\right) = 0$$ or so that it is kinematically admissible.

Motivation: Discrete PVW applied to equation below:

$$ \mathbf{{W}_{6x1}\left(\left[K \right]_{6x2}\begin{Bmatrix} d_{3} \\ d_{4}\end{Bmatrix}_{2x1} - {F}_{6x1}\right)=0_{1x1}}$$

Where $$\left(\left[K \right]_{6x2}\begin{Bmatrix} d_{3} \\ d_{4}\end{Bmatrix}_{2x1} - {F}_{6x1}\right)$$ becomes a 6x1 matrix and $$ \mathbf{F}^{T}=\left[F_{1}F_{2}F_{3}F_{4}F_{5}F_{6}\right] $$ where F1, F2, F5, and F6 are unknown reactions.

Note that W can be selected arbitrarily, but for simpliciity select W such that W1 = W2 = W5 = W6 = 0. This is so all equations involving unknown reactions can be eliminated, or rows 1,2,5 and 6 are eliminated. Equation 1 shown below can then be formed.

$$ \mathbf{\overline{K}_{2x2}\overline{d}_{2x1}=\overline{F}_{2x1}} $$

Where $$\mathbf\overline{d} = \begin{Bmatrix} d_{3} \\ d_{4}\end{Bmatrix}$$ and $$\mathbf\overline{F} = \begin{bmatrix}F_{3} \\ F_{4}\end{bmatrix}$$.

Note that $$ \mathbf{\overline{W}\left(\overline{K}\overline{d}-\overline{F}\right)}=0 $$ (2) is actually the step before equation 1.

$$ W(L)F(t) - \int_{0}^{L}\frac{\partial w}{\partial x}(EA)\frac{\partial u}{\partial x}dx+\int w(x)[f - m\ddot{u}]dx=0 $$

This becomes the final equation:

$$ \int_{0}^{L}w(m\ddot{u})dx+\int_{0}^{L} \frac{\partial w}{\partial x}(EA)\frac{\partial u}{\partial x}dx

= W(L)F(t) + \int_{0}^{L}wfdx $$

These equations are for all $$ W\left(x\right)$$ such that $$ W\left(0\right) = 0 $$

Two cases of PVW

For the rest of this section, we will be focusing on element i in the above figures.

Assume the displacement for u(x) for $$x_i \le x \le x_{i+1}$$. (i.e., x $$\epsilon [x_i, x_{i+1}]$$)

The motivation for the linear interpolation of u(x) is shown below using a 2-bar truss.

The deformed shape is a straight line. This means there was an implicit assumption of the linear interpolation of the displacement between two nodes. Consider the case where there is only axial displacement, or zero transverse displacements. The question that needs to be solved is:

Express u(x) in terms of di = u(xi) and di+1 = u(xi+1) as a linear function in x.

The answer to this question is defined by linear interpolation, or:

$$u(x) = N_i\left(x\right)d_i + N_{i+1}\left(x\right)d_{i+1}$$

where $$N_i\left(x\right)$$ and $$N_{i+1}\left(x\right)$$ are linear functions of x.

$$N_i\left(x\right) = $$

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

Continuing PVW to Discrete PVW
Lagrangian interpolation The Motivation for the form of $$\mathbf{N_i(x)}$$ and $$\mathbf{N_{i+1}(x)}$$: 1) $$\mathbf{N_i(x)}$$ and $$\mathbf{N_{i+1}(x)}$$ are linear, straight lines, thus any linear combination of $$\mathbf{N_i(x)}$$ and $$\mathbf{N_{i+1}(x)}$$ are also linear, and in particular, the expression for u(x).


text:
Creating imagemaps for use on wikis

Imagemaps are images which contain links to URLs or files that pop up when you move the mouse over various sections of the image. This project aims at learning how to create imagemaps for use on websites using the mediawiki software (such as this site and the other wikimedia wikis).

Software
The Gimp is an open-source program that has an imagemap extension.

Projects

 * 1) Make imagemaps for use on the Wikiversity help pages, to provide a graphical interface on how to edit on wikis in general, and on wikiversity in particular.

title: Creating imagemaps for use on wikis


text: File:Element1 Qwiki 2BarTruss.jpg

title: File:Element1 Qwiki 2BarTruss.jpg


text: File:Element2 Qwiki 2BarTruss.jpg

title: File:Element2 Qwiki 2BarTruss.jpg


text:
User:Eas4200c.f08.gator.reger

Robert Reger  1521-3324 Aerospace Structures

 * Week 1,2 Homework
 * Week 3,4 Homework
 * Week 5,6 Homework
 * Week 7,8 Homework
 * Week 9,10 Homework
 * Week 11,12 Homework
 * FINAL

title: User:Eas4200c.f08.gator.reger


text: File:Books.jpg

title: File:Books.jpg


text:
User:Uf.team.aero/HW6

=Plate Buckling= A plate under uniform stress in one dimension (see Figure 1) undergoes deformation according to the following equation:
 * $$u_z(x,y)= \sum_{m=1}^{ \infty }{}\sum_{n=1}^{\infty}\psi_{mn} (x,y) $$ where $$\psi$$ is expressed as
 * $$\psi_z(x,y)=csin(\frac{m\pi x}{a})sin(\frac{n\pi x}{b})$$

and $$u$$ is the out of plane deformation. (Can be thought of as deformation in z direction.) Please note this is for a simply supported plate.

The resulting shape when looking at the plate normal to the xy plane is called the buckling mode shape.

The critical loading, $$P_x$$ is given by the following equation:
 * $$(P_x)=k_c\frac{\pi ^{2}D}{b}$$

 * $$k_c(m,\frac{a}{b})=(\frac{mb}{a}+\frac{a}{mb})^2$$

and

 * $$D=\frac{Eh^3}{12(1-\upsilon ^2)}$$

Where h is the thickness of the plate in the z direction.

The critical buckling stress is:
 * $$(\sigma_{xx})_{cr}=\frac{(P_x)_{cr}}{bh}=k_c \frac{\pi^2D}{b^2h}$$

Components related to the minimum critical value are denoted with a star, e.g. $$\sigma^\star_{cr}$$ The minimum critical buckling stress is dependent on $$k^\star_c$$, which in turn is dependent on $$m^\star$$ by the following equation:
 * $$k_c^\star(a/b)=\min_{m = 1, 2, \cdots}\ k_c(m, a/b)=\min_{m = 1, 2, \cdots}\left( \frac {m b}{a} +\frac{a}{m b}\right)^2 =

\left( \frac{m^\star b}{a}+\frac{a}{m^\star b}\right)^2$$ where
 * $$ m^\star(a/b)=\underset{m=1,2, \cdots}{\rm argmin}\ k_c(m, a/b) =\underset{m=1,2, \cdots}{\rm argmin}\left(\frac{m b}{a}

+ \frac {a}{m b}\right)^2$$ The transition from m to m+1 waves occurs when:
 * $$\frac{a}{b}=\sqrt{m(m+1)}$$

For a square plate with all 4 edges clamped, the buckling mode shape approximates to:
 * $$ \psi(x,y)=\frac{c}{4}\left(1 - \cos \frac{2 \pi x}{a}\right)\left(1 - \cos \frac{2 \pi y}{b} \right)$$

This gives a “bubble” like shape, as seen in the matlab figure below. It makes logical sense since the slope of the buckled shape must approach zero (dz/dx and dz/dy) at all clamped edges.

Shear Load, Simply Supported
Load is denoted as $$N_{xy}$$, which is force per unit length, or stress times h, or plate thickness.

Stored strain energy is equated to the work done by shear load U=W.

The Nxy term is in the equation for strain energy, and the critical Nxy is expressed as:
 * $$\displaystyle

(N_{xy})_{cr} =-\frac{ab D}{32}\frac{\displaystyle\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}C_{mn}^2\left(\frac{m^2 \pi^2}{a^2}+       \frac{n^2 \pi^2}{b^2}\right)^2}{\displaystyle\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\sum_{p=1}^{\infty}\sum_{q=1}^{\infty} C_{mn}C_{pq}\frac{mnpq}{(m^2 - p^2)(n^2 - q^2)}} $$ After further analysis and derivation, crit. buckling stress is:
 * $$(\sigma_{xy})^{\star,5}_{cr}=\pm k^{\star, 5}_c \frac{\pi^2D}{b^2h}, k^{\star,5}_c = \frac{\pi^2}{32}\frac{1}{\lambda\vartheta }$$

=A note on dimensional analysis=

A bracket notation (ex. [m]) is used to indicate the dimensions associated with a variable. Here are some examples:

 * $$[f]=\frac{F}{L}$$

this means the dimensions of "f" are force per length.

Since

 * $$[A]=L^2$$

then

 * $$\left[\frac{f}{A} \right]= \frac{F}{L^{3}}$$

meaning the dimensions of "f" per unit area are force per volume, or length cubed. Note, L is only a unit of measure, not a specific dimension.

The following describes a body force:

 * $$\frac{d\sigma }{dx}+\frac{f(z)}{A}=0\Rightarrow [\sigma ]=\frac{F}{L^2}\Rightarrow \left[\frac{d \sigma}{dx} \right]= \frac{[d \sigma ]}{[dx]}=\frac{F/L^2}{L} = \frac{F}{L^3}$$

To show that axial strain is nondimensional:

 * $$[\epsilon ] = \frac{du}{dx} = \frac{L}{L} = 1...nondimensional...$$

Shear strain is also nondimensional:

 * $$[\gamma ]=\frac{[\epsilon _{yy}]}{[\epsilon _{xx}]}=1...nondimensional...$$

Also note
 * $$\left[\frac{d \sigma_{ij}}{dx_i} \right]=\frac{F}{L^3}=\frac{Force}{Volume}$$

Intro
We introduce a function, $$\phi$$, which plays the role of a potential function. We call this the Prandtl stress function. $$\sigma_{yx}$$ and $$\sigma_{zx}$$ are components of the gradient of $$\phi$$ with respect to y and z.

In wikipedia, a gradient is defined as is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change. In notation, it is defined as
 * $$\bigtriangledown f(x,y,z)= \frac{\partial f}{\partial x}i + \frac{\partial f}{\partial y}j + \frac{\partial f}{\partial z}k$$

where f is the potential.

For our case,
 * $$\sigma_{yx}=\frac{\partial \phi}{\partial z}$$ and
 * $$\sigma_{zx}=-\frac{\partial \phi}{\partial y}$$

We can plug these values into the following equation:
 * $$\frac{\partial \sigma_{yx}}{\partial y}+ \frac{\partial \sigma_{zx}}{\partial z}=0$$

to obtain
 * $$\frac{\partial }{\partial y}\frac{\partial \phi}{\partial z}+\frac{\partial }{\partial z} \frac{(-\partial \phi)}{\partial y}=\frac{\partial^2 \phi}{\partial y \partial z}- \frac{\partial^2 \phi}{\partial z \partial y}=0$$

Phi is continuous and smooth, making the 2nd mixed derivative interchangeable, i.e.
 * $$\frac{\partial^2 \phi}{\partial y \partial z}= \frac{\partial^2 \phi}{\partial z \partial y}$$

These equations can be compared with Mechanics of Aircraft Structures(Sun) Ch. 3 eqs 3.15-3.19. Using the author's notation:
 * $$\frac{\partial \tau_{yz}}{\partial x}-\frac{\partial \tau_{xz}}{\partial y}=2G\theta$$

and
 * $$\frac{\partial^2 \phi}{\partial x^2}+ \frac{\partial^2 \phi}{\partial y^2}=-2G\theta$$

The remainder of section 3.2 brings us to our roadmap equation for J (Roadmap eq 7) and notes that once the Prandtl stress function is solved, GJ, the torsional rigidity (the bar's ability to resist deformation due to torsion) is also found. Torsional analysis of structures fall into two cases - that of structures with a thin walled cross section (also referred to as closed cross section) and those of structures with a solid cross section.

Case 1: Thin Walled Cross Section
A prime example of this case is that of the NACA airfoils we have been studying throughout the semester. For thin walled cross section the boundary conditions for Prandtl stress function, $$ \phi $$, are that $$\phi$$ is a constant on inner and outer surfaces of the cross section. The figure below depicts the inner and outer surfaces, $$ S_1 $$ and $$ S_0 $$, respectively.

The boundary conditions for this case are $$\phi=C_0$$ on $$S_0$$ and $$\phi=C_1$$ on $$S_1$$.

Case 2: Solid Cross Section
This case consists of any bar or beam that has a solid cross section such as the one depicted below.

The constant value of $$\phi$$ at the boundary solid cross sections is arbitrary and so it can be set to zero on that surface. Thus the boundary in this case is $$\phi = 0$$ on $$S_0$$

Uniform Bar With Solid Circular Cross Section
As an example, we will analyze the following bar with a circular cross section as shown below.

For this case, $$\phi$$ can be defined as

 * $$\phi(y,z) = C\left(\frac{y^2}{a^2}+\frac{z^2}{b^2}-1\right)$$

It might also be noted that for the case of a circular cross section a=b. As mentioned before for the case of a solid cross section $$\phi=0$$ on $$S_0$$. Recalling the relation:

 * $$\nabla^2\phi=-2G\Theta$$

an expression for C can be obtained by solving for the laplacian of $$\phi$$.

 * $$\nabla^2\phi = \frac{4C}{a^2} = -2G\Theta$$
 * $$\Rightarrow C = -\frac{1}{2}a^2G\Theta$$

To analyze the Prandtl stress function, $$ \phi $$, as a function of the radius of a circular cross section of diameter $$ 2\alpha $$, consider the definition of the radius as

 * $$ r = \left(y^2 + z^2 \right)^{1/2} $$

Then, a plot of $$ \phi(r) $$ vs. r would yield the figure below

Now, recall from previous lectures the following equations

 * $$ T = 2\int_{A}^{}{\phi dA} = 2C\left(\frac{J}{a^2}-A\right) $$

 * $$ J = \int_{A}^{}{r^2 dA} = \frac{1}{2} \pi a^4 $$

 * $$ \tau = \frac{Tr}{J} $$

Combine these equations with the following

 * $$ \sigma_{yx} = \frac{\partial \phi}{\partial z} = -G \theta z $$

 * $$ \sigma_{zx} = -\frac{\partial \phi}{\partial y} = G \theta y $$

to obtain

 * $$ u_x\left(y,z\right) = 0 $$

This implies that there is no axial displacement at any point along a bar with uniform, solid, circular cross-section. This was assumed to be true during ad hoc derivations of twist; however, we see now that the Theory of Elasticity proves it to be so. A rigorous proof is presented below.

= Traction Force Equation: Derivation =

Geometric Manipulation Using an Infinitesimal Element
Goal: Using the traction force equation to derive the equation:

$$\displaystyle[t_i]_{3x1}=[\sigma_{ij}]_{3x3}[n_i]_{3x1} $$

Consider the following figure below of an infinitesimal element to be able to be able to derive geometrically, the traction force from the stress tensor and normal vector at the surface of the element.

From the figure the vector $$\boldsymbol\vec{n}$$ has the vector equation: $$\displaystyle \boldsymbol \vec{n}=n_y \boldsymbol \vec{j}+n_z \boldsymbol \vec{k}$$

From this equation, $$\boldsymbol\vec{n}$$ is the unit normal vector and is given as $$\displaystyle ||\boldsymbol \vec{n}||=1$$

Looking at the figure, the elemental geometry helps in the establishing the following relationship between dy and dz:

$$\displaystyle dy=ds*sin(\theta)$$ $$\displaystyle dz=ds*cos(\theta)$$

Additionally, the normal vector equation can be used to rewrite the vector in terms of angle $$\theta$$ and conclude also that:

$$\displaystyle n_y=cos(\theta)$$ $$\displaystyle n_z=sin(\theta)$$

Looking at Figure and continuing through this derivation, the next logical choice would be to take the sum the forces in the y and z directions using the equilibrium equations. It is important to note that the depth into the page is taken as the x-direction and is neglected for this case since the depth is considered to be unity, or 1.

As given in class, the derivation for the equilibrium in the y direction of the sum of forces is given as:

$$\sum {F}_y=0=-\sigma_{yy}*(dz*1)-\sigma_{yz}*(dy*1) +t_y*(d_s*1)$$

Substituting values for dy and dz into the equilibrium equation above, the following equation is produced:

$$\sum {F}_y=0=-\sigma_{yy}ds n_y-\sigma_{yz}ds n_z +t_yd_s$$

Further simplification of the equilibrium equation above and solving for $$\displaystyle t_y : $$

$$\displaystyle t_y= \sigma_{yy}n_y+\sigma_{yz}n_z(1)$$

As part of HW 6, the equilibrium equation for the sum of forces in the z-direction are given below:

Using equations (1) and (2) for tractional force components, we can arrange the equations in the form of a matrix, reflected below to produce equation 3:

$$\begin{Bmatrix} t_y\\ t_z\\ \end{Bmatrix} = \begin{bmatrix} \sigma_{yy} & \sigma_{yz}\\ \sigma_{zy} & \sigma_{zz}\\ \end{bmatrix} \begin{Bmatrix} n_y\\ n_z\\ \end{Bmatrix}(3) $$ Similarly for a general 3D case, a similar matrix may be constructed using indicial notation, shown below:

$$\begin{Bmatrix} t_1\\ t_2\\ t_3\\ \end{Bmatrix} = \begin{bmatrix} \sigma_{11} & \sigma_{12}& \sigma_{13}\\ \sigma_{21} & \sigma_{22}& \sigma_{23}\\ \sigma_{31} & \sigma_{32}& \sigma_{33}\\ \end{bmatrix} \begin{Bmatrix} n_1\\ n_2\\ n_3\\ \end{Bmatrix}(4) $$

An advantage of using this indicial notation allows us to rewrite the entire matrix equation in the form of a single equation that is more compact. This compact notation is shown below:

$$t_i=\sum_{j=1}^3{\sigma_{ij}n_j},i=1,2,3$$

In this equation looking at the stress value, where the subscript i, and j designates the row, and column of the matrix respectively.

Generalizing this formula, produces our end goal and solves for the desired equation: $$\displaystyle [t_i]_{3x1}=[\sigma_{ij}]_{3x3}[n_i]_{3x1} $$

Roadmap Incorporation
Looking at the roadmap on lecture page 16-2, we recall the following three equations:

$$\displaystyle T=2\int\int_{A}^{}{}\phi dA(5)$$ $$\displaystyle T = GJ\theta~(6) $$ $$\displaystyle J=\frac{-4}{\nabla^{2}\phi }\int \int _{A}\phi dA(7)$$

These three equations can be combined, and the equation (8) shown below can be derived by solving the three equations shown above for the Lagrangian of phi.

$$\frac{\partial^2{\phi}}{\partial{y}^2}+\frac{\partial^2{\phi}}{\partial{z}^2}=-2G\theta(8)$$

This becomes important because $$\phi$$ is taken to be constant on the lateral surface of the bar.

= Flexural Shear in Thin-walled Sections =

Motivation
It is known that for a generic loading condition of a beam, there exists not only a bending moment but also a transverse shear load. Consider the beam in the figure below. Some generic loading condition is applied to the straight beam on the left. A cut can be made in the beam at a distance x from the wall. The bending moment and shear loads are drawn for each side of the exposed section, where M(x) represents the moment as a function of distance x and V(x) represents the shear load as a function of distance x.

= Unsymmetrical Thin-walled Cross Section =

General Equation for unsymmetrical cross sections $$\displaystyle \int_{A_s} \frac{\sigma_{xx}}{dx} dA = -q_s $$ Eq. 5.1

Symmetric Cross Section (about y- axis)
$$ \displaystyle \sigma_{xx} = \frac{M_y z}{I_y}$$

$$ \displaystyle q(s) = \frac{-V_z Q_y}{I_y}$$ Eq. 5.2

$$ \displaystyle Q_y = \int_{A_s} z dA = z_c A_s$$

Unsymmetric Cross Section
$$ \displaystyle \sigma_{xx} = (k_y M_z - k_{yz} M_y)y + (k_z M_y - k_{yz} M_z)z$$ → Eq. 5.3

This is true for the following k values from p.26-2 of notes: $$ \displaystyle k_y = \frac{I_y}{D}$$ $$k_{yz} = \frac{I_{yz}}{D}$$ $$k_z = \frac{I_z}{D}$$

The expression for $$\displaystyle \sigma_{xx}$$ can also be put into matrix form:

$$\displaystyle \sigma_{xx} = \begin{bmatrix} z & y\\ \end{bmatrix} \begin{bmatrix} k_z & -k_{yz}\\ -k_{yz} & k_y\\ \end{bmatrix} \begin{Bmatrix} M_y\\ M_z\\ \end{Bmatrix} $$

$$ \displaystyle q(s) = -(k_y V_y - k_{yz} V_z) Q_z - (k_z V_z - k_{yz} V_y) Q_y$$ → Eq. 5.5

$$ \displaystyle => Q_z = \int_{A_s} y dA$$ and $$ \displaystyle Q_y = \int_{A_s} z dA $$

= Matlab Analysis =

The main purpose of this weeks homework problem is to examine the bending stress in the skin of the NACA airfoil. In order to determine this, we will use exact approximations as well as approximations for simply supported and clamped boundary conditions since the airfoil's spars and stringers cause the skin to fall somewhere in between the two.

The first part involves using the previous homework's inertia matrix, centroid, and applied moments to determine the normal bending stress in the skin between the two spars as a function of the y-coordinate using the equations:

 * $$\sigma_{xx}=E\epsilon_{xx}=\frac{I_yM_z-I_{yz}M_y}{I_yI_z-(I_{yz})^2}y+\frac{I_zM_y-I_{yz}M_z}{I_yI_z-(I_{yz})^2}z$$

\begin{bmatrix} I_{22} \\ I_{33} \\ I_{23} \\ \end{bmatrix}=

\begin{bmatrix} 1.298\ x\ 10^{-5}\ m^4 \\ 6.346\ x\ 10^{-7}\ m^4 \\ -9.494\ x\ 10^{-8}\ m^4 \\ \end{bmatrix}$$

 * $$Centroid(y,z)=(0.210\ m,\ 0.210\ m)$$

From this we find that the average stresses are $$\sigma_{xx,FB}=26.5\ MPa$$ and $$\sigma_{xx,EH}=32.1\ MPa$$.

Next we wish to explore what shapes the skin would undergo due to these bending stress. For an aspect ratio of 1.5, we can see in Figures XXX and XXX that depending on m and n, different modes of buckling occur. The variables m and n determine the number of half wave lengths in the x and y direction respectively, which is readily apparent in the figures.

We are interested in finding the period T of this function that determines the shape of the buckled plate to prove the m is indeed the number of half waves. This involves finding the period of $$ \sin ( \frac{m \pi x}{a} )$$  by considering   $$\sin ( \frac{m \pi (x + T)}{a} ) = \sin ( \frac{m \pi x}{a} )$$. We know that:

 * $$\sin( \theta ) = \sin( \theta + 2\pi k)=\sin(\frac{m \pi x}{a}+\frac{m \pi T}{a})$$

 * $$Period=T=\frac{2\pi}{\frac{m\pi}{a}}=\frac{2a}{m}$$

For constant a=1, we see that the period is 2/m. For m=1, we see the period is 2 and therefore there is one half of the wave at a=1. For m=2, the period is 1 and two half waves at a=1, so on and so forth.

In the determining of critical stresses, a variable of large importance is k_c. This number is a function purely of m, the number of half wave lengths, and the aspect ratio of the plate. The figure to the right shows how k_c varies with these variables. This factor is used in critical stress calculations for simply supported plates.

As we discussed earlier, there are stringers and spars connect to the skin so it is not just simply supported. The other approximation for bending of a flat plate is clamped boundary condition. Here the two ends are not allowed to rotate, making the slope of plate zero at the boundary. The figure to the right shows how this sort of boundary condition affects the buckling mode of the plate.

The final section is interested in modeling the skin section in between the two spars as flat plates and determining the critical stresses in this plate. We will compare the value reached earlier for average stress to each of the two boundary conditions. In this figure, we vary the aspect ratio of plate while holding the width constant to that of the actual skin section. The two cases shown are for the clamped and simply supported boundaries mentioned earlier. We can see that at small aspect ratios, our average stress for the top plate, which the figure was based upon, falls in between the two curves. At higher aspect ratio's however (which is what we would expect for an aircraft wing), the stress is higher the than the critical stress, implying that the skin does not work well to resist bending, which is what we expected.

= Contributing team members =

 * Gonzalo Barcia--Eas4200c.f08.aero.barcia 03:05, 21 November 2008 (UTC)
 * William Kurth--Eas4200c.f08.aero.kurth 14:56, 21 November 2008 (UTC)
 * Melisa Gaar--Eas4200c.f08.aero.gaar 19:21, 21 November 2008 (UTC)
 * Thomas McGilvray--Eas4200c.f08.aero.mcgilvray 19:38, 21 November 2008 (UTC)
 * Jared Lee--Eas4200c.f08.aero.lee 19:43, 21 November 2008 (UTC)
 * Oliver Oyama--70.168.185.148 19:54, 21 November 2008 (UTC)
 * Ray Strods--[User:Eas4200c.f08.aero.strods|Eas4200c.f08.aero.strods]] 01:02, 22 November 2008 (UTC)

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CSS

CSS, short for Cascading Style Sheet, is a language used in web design to apply styles to web pages. CSS has a somewhat similar syntax to JavaScript. CSS can be used to easily edit and create styles for web pages and can be either directly incorporated into the web page or can be used by the web page externally. CSS is very efficient for sites, especially sites with many pages as it can be used to edit the style of multiple web pages, from easily editing only one document, in under a minute which would take sometimes more than 5 minutes by editing each web page one by one.

Lessons

 * /Syntax/
 * /Grid/
 * /Exam Questions/

Activities

 * 1) Complete one or more of the following tutorials:
 * 2) * Mozilla: CSS Basics
 * 3) * HTML.net: CSS Tutorial

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User:BennuDaPhoenix

BennuDaPhoenix

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Ordinary differential equation

Ordinary differential equation
In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable.

A simple example is Newton's second law of motion, which leads to the differential equation

$$ m \frac{d^2 x(t)}{dt^2} = F(x(t)),\, $$

for the motion of a particle of mass m. In general, the force F depends upon the position of the particle x(t) at time t, and thus the unknown function x(t) appears on both sides of the differential equation, as is indicated in the notation F(x(t)).

Ordinary differential equations are distinguished from partial differential equations, which involve partial derivatives of several variables.

Ordinary differential equations arise in many different contexts including geometry, mechanics, astronomy and population modelling. Many famous mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler.

Much study has been devoted to the solution of ordinary differential equations. In the case where the equation is linear, it can be solved by analytical methods. Unfortunately, most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions are arrived at using computer approximations (see numerical ordinary differential equations). The trajectory of a projectile launched from a cannon follows a curve determined by an ordinary differential equation that is derived from Newton's second law. Contents

* 1 Definitions o 1.1 Ordinary differential equation o 1.2 Solutions * 2 Examples * 3 Reduction to a first order system * 4 Linear ordinary differential equations o 4.1 Homogeneous equations o 4.2 Nonhomogeneous equations o 4.3 Fundamental systems for homogeneous equations with constant coefficients * 5 Theories of ODEs o 5.1 Singular solutions o 5.2 Reduction to quadratures o 5.3 Fuchsian theory o 5.4 Lie's theory o 5.5 Sturm-Liouville theory * 6 See also * 7 Bibliography * 8 External links

[edit] Definitions

[edit] Ordinary differential equation

Let y be an unknown function

$$y: \mathbb{R} \to \mathbb{R}$$

in x with y(n) the nth derivative of y, then an equation of the form

$$F(x,y,y',\ \dots,\ y^{(n-1)})=y^{(n)}$$

is called an ordinary differential equation (ODE) of order n; for vector valued functions,

$$y: \mathbb{R} \to \mathbb{R}^m$$,

it is called a system of ordinary differential equations of dimension m.

When a differential equation of order n has the form

$$ F\left(x, y, y', y'',\ \dots,\ y^{(n)}\right) = 0$$

it is called an implicit differential equation whereas the form

$$ F\left(x, y, y', y'',\ \dots,\ y^{(n-1)}\right) = y^{(n)}$$

is called an explicit differential equation.

A differential equation not depending on x is called autonomous.

A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y

$$ y^{(n)} = \sum_{i=0}^{n-1} a_i(x) y^{(i)} + r(x)$$

with ai(x) and r(x) continuous functions in x. The function r(x) is called the source term; if r(x)=0 then the linear differential equation is called homogeneous, otherwise it is called non-homogeneous or inhomogeneous.

[edit] Solutions

Given a differential equation

$$F(x,y,y',\ \dots,\ y^{(n)})=0$$

a function

$$ u: I \subset \mathbb{R} \to \mathbb{R} $$ is called the solution or integral curve for F, if u is n-times differentiable on I, F is defined for all

$$ (x,u,u',\ \dots,\ u^{(n)}) \quad x \in I$$

and

$$ F(x,u,u',\ \dots,\ u^{(n)})=0 \quad x \in I.$$

Given two solutions

$$u: J \subset \mathbb{R} \to \mathbb{R}$$

and

$$v: I \subset \mathbb{R} \to \mathbb{R}$$

u is called an extension of v if I ⊂ J and

$$ u(x) = v(x) \quad x \in I.\,$$

A solution which has no extension is called a global solution.

A general solution of an n-th order equation is a solution containing n arbitrary variables, corresponding to n constants of integration. A particular solution is derived from the general solution by setting the constants to particular values, often chosen to fulfill set 'initial conditions or boundary conditions'. A singular solution is a solution that can't be derived from the general solution.

[edit] Examples

Main article: Examples of differential equations

[edit] Reduction to a first order system

Any differential equation of order n can be written as a system of n first-order differential equations. Given an explicit ordinary differential equation of order n and dimension 1,

$$F\left(x, y, y', y'',\ \dots,\ y^{(n-1)}\right) = y^{(n)}$$

we define a new family of unknown functions

$$ y_n := y^{(n-1)}.\!$$

We can then rewrite the original differential equation as a system of differential equations with order 1 and dimension n.

$$y_1^' = y_2$$

$$\vdots$$

$$y_n^' = F(y_n, \dots, y_1, x).$$

which can be written concisely in vector notation as

$$ \mathbf{y}^'=\mathbf{F}(\mathbf{y}, x)$$

with

$$ \mathbf{y}:=(y,\ldots,y_n).$$

[edit] Linear ordinary differential equations

Main article: Linear differential equation

A well understood particular class of differential equations is linear differential equations. We can always reduce an explicit linear differential equation of any order to a system of differential equation of order 1

$$y_i'(x) = \sum_{j=1}^n a_{i,j}(x) y_j + b_i(x) \, \mathrm{,} \quad i = 1,\ldots,n$$

which we can write concisely using matrix and vector notation as

$$\mathbf{y}^'(x) = \mathbf{A}(x) \mathbf{y}(x) + \mathbf{b}(x)$$

with

$$\mathbf{y}(x):=(y_1(x),\ldots,y_n(x)) \mathbf{b}(x):=(b_1(x),\ldots,b_n(x)) \mathbf{A}(x):=(a_{i,j}(x)) \, \mathrm{,} \quad i,j = 1,\ldots,n.$$

[edit] Homogeneous equations

The set of solutions for a system of homogeneous linear differential equations of order 1 and dimension n

\mathbf{y}^'(x) = \mathbf{A}(x) \mathbf{y}(x)

forms an n-dimensional vector space. Given a basis for this vector space \mathbf{z}_1(x), \ldots, \mathbf{z}_n(x), which is called a fundamental system, every solution \mathbf{s}(x) can be written as

\mathbf{s}(x) = \sum_{i=1}^{n} c_i \mathbf{z}_i(x).

The n × n matrix

\mathbf{Z}(x) := (\mathbf{z}_1(x), \ldots, \mathbf{z}_n(x))

is called fundamental matrix. In general there is no method to explicitly construct a fundamental system, but if one solution is known d'Alembert reduction can be used to reduce the dimension of the differential equation by one.

[edit] Nonhomogeneous equations

The set of solutions for a system of inhomogeneous linear differential equations of order 1 and dimension n

\mathbf{y}^'(x) = \mathbf{A}(x) \mathbf{y}(x) + \mathbf{b}(x)

can be constructed by finding the fundamental system \mathbf{z}_1(x), \ldots, \mathbf{z}_n(x) to the corresponding homogeneous equation and one particular solution \mathbf{p}(x) to the inhomogeneous equation. Every solution \mathbf{s}(x) to nonhomogeneous equation can then be written as

\mathbf{s}(x) = \sum_{i=1}^{n} c_i \mathbf{z}_i(x) + \mathbf{p}(x).

A particular solution to the nonhomogeneous equation can be found by the method of undetermined coefficients or the method of variation of parameters.

[edit] Fundamental systems for homogeneous equations with constant coefficients

If a system of homogeneous linear differential equations has constant coefficients

\mathbf{y}^'(x) = \mathbf{A} \mathbf{y}(x)

then we can explicitly construct a fundamental system. The fundamental system can be written as a matrix differential equation

\mathbf{Y}^' = \mathbf{A} \mathbf{Y}

with solution as a matrix exponential

e^{x \mathbf{A}}

which is a fundamental matrix for the original differential equation. To explicitly calculate this expression we first transform A into Jordan normal form

e^{x \mathbf{A}} = e^{x \mathbf{C}^{-1} \mathbf{J} \mathbf{C}^{1}} = \mathbf{C}^{-1} e^{x \mathbf{J}} \mathbf{C}^{1}

and then evaluate the Jordan blocks

J_i = \begin{bmatrix} \lambda_i & 1 & \; & \; \\ \; & \ddots & \ddots & \; \\ \; & \; & \ddots & 1 \\ \; & \; & \; & \lambda_i \end{bmatrix}

of J separately as

e^{x \mathbf{J_i}} = e^{\lambda_i x} \begin{bmatrix} 1 & x & \frac{x^2}{2} & \dots & \frac{x^{n-1}}{(n-1)!} \\ \; & \ddots & \ddots & \ddots & \vdots \\ \; & \; & \ddots & \ddots & \frac{x^2}{2} \\ \; & \; & \; & \ddots & x \\ \; & \; & \; & \; & 1 \end{bmatrix}.

[edit] Theories of ODEs

[edit] Singular solutions

The theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century did it receive special attention. A valuable but little-known work on the subject is that of Houtain (1854). Darboux (starting in 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field which was worked by various writers, notably Casorati and Cayley. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.

[edit] Reduction to quadratures

The primitive attempt in dealing with differential equations had in view a reduction to quadratures. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the nth degree, so it was the hope of analysts to find a general method for integrating any differential equation. Gauss (1799) showed, however, that the differential equation meets its limitations very soon unless complex numbers are introduced. Hence analysts began to substitute the study of functions, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter the real question was to be, not whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and if so, what are the characteristic properties of this function.

[edit] Fuchsian theory

Two memoirs by Fuchs (Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and Frobenius. Collet was a prominent contributor beginning in 1869, although his method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those followed in his theory of Abelian integrals. As the latter can be classified according to the properties of the fundamental curve which remains unchanged under a rational transformation, so Clebsch proposed to classify the transcendent functions defined by the differential equations according to the invariant properties of the corresponding surfaces f = 0 under rational one-to-one transformations.

[edit] Lie's theory

From 1870 Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.

Sturm-Liouville theory
Sturm-Liouville theory is a general method for resolution of second order linear equations with variable coefficients.

1st ordered ordinary differential equation
1st ordered ordinary differential equation of general form
 * $$A \frac{d}{dt} f(x) + B f(t) = 0$$

Rearrange equation above,
 * $$\frac{d}{dt} f(t) = - s f(t)$$ . With, $$s=\frac{B}{A}$$

Root of equation
 * $$f(t) = Ae^{-st} = Ae^{ - \frac{B}{A} t}$$

2nd ordered ordinary differential equation

 * $$A \frac{d^2}{dx^2} f(x) + B \frac{d}{dx} f(x) + C f(x) = 0$$

 * $$\frac{d^2}{dx^2} f(x) + \frac{B}{A} \frac{d}{dx} f(x) + \frac{C}{A} f(x) = 0$$
 * $$\frac{d^2}{dx^2} f(x) = - \frac{B}{A} \frac{d}{dx} f(x) - \frac{C}{A} f(x) $$
 * $$\frac{d^2}{dx^2} f(x) = - 2 \alpha \frac{d}{dx} f(x) - \beta f(x) $$
 * $$\beta=\frac{C}{A}$$
 * $$\alpha=\beta \gamma= \frac{C}{A} \gamma = \frac{B}{2A}$$
 * $$\gamma=\frac{B}{2C}$$

Roots of differential equations
 * 1 real root . $$f(x)=Ae^{-\alpha t}$$ . $$\alpha=\beta$$
 * 2 real roots . $$f(x)=Ae^{(-\alpha \pm \sqrt{\alpha-\beta}) t}$$ . $$\alpha>\beta$$
 * 2 complex roots . $$f(x)=Ae^{(-\alpha \pm \sqrt{\beta-\alpha}) t}$$ . $$\alpha<\beta$$

 * $$f(x)=Ae^{(-\alpha \pm \sqrt{\beta-\alpha}) t} = A(\alpha)Sin \omega t$$

With
 * $$A(\alpha)=Ae^{-\alpha t}$$
 * $$\omega=\sqrt{\beta-\alpha}$$

title: Ordinary differential equation


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User:Eml4500.f08.qwiki.bishop/Lecture 33

=Meeting 33 Wednesday November 12, 2008=

FEM via PVW (continued)
As we saw in Lecture ___, the assumed solution u(x) for a two-node element is a linear interpolation between the nodal unknowns, and is expressed as $$ u(x) = N_i(x) d_i + N_{i+1}(x) d_{i+1}$$. To obtain a solution for the linear displacement at $$x_{i+1} $$, we must evaluate $$ u(x) $$ for $$ u(x_{i+1})$$. Doing so yields:
 * $$ u(x_{i+1}) = N_i(x_{i+1}) d_i + N_{i+1}(x_{i+1}) d_{i+1} = d_{i+1} $$
 * $$ u(x_{i+1}) = \Bigl(\frac{x - x_{i+1}}{x_i - x_{i+1}}*d_i + \frac{x - x_i}{x_{i+1} - x_i} * d_{i+1}\Bigr) \Bigl\vert_{x = x_{i+1}} $$
 * $$ u(x_{i+1}) = \Bigl(\frac{x_{i+1} - x_{i+1}}{x_i - x_{i+1}}*d_i + \frac{x_{i+1} - x_i}{x_{i+1} - x_i} * d_{i+1}\Bigr) = \Bigl(\frac{0}{x_i - x_{i+1}}*d_i + \frac{x_{i+1} - x_i}{x_{i+1} - x_i} * d_{i+1}\Bigr)

= 0 * d_i + 1 * d_{i+1} $$ Therefore, one can see that $$ u(x_{i+1}) = d_{i+1} $$.

Since the weighting function w(x) is a linear function similar to $$ u(x) $$, one can linearly interpolate between $$ x_i \text{ and } x_{i+1} $$ to solve for $$w(x) \text{ : } $$
 * $$ w(x) = \Bigl(\frac{x - x_{i+1}}{x_i - x_{i+1}}*w_i + \frac{x - x_i}{x_{i+1} - x_i} * w_{i+1}\Bigr)

= N_i(x)w_i + N_{i+1}(x)w_{i+1} $$.

Element Stiffness Matrix for Element i
In practice, most finite element solutions are written in matrix notation to ease understanding of the system. For simple elements such as the one pictured above from Lecture ___, matrix notation is not necessary. However, for complicated elements it is nearly impossible to express element solutions without using matrix notation. In this section, we will show how to transform from the continuous stiffness equation $$ \beta = \int_{0}^{L} \frac{\delta w}{\delta x}(EA)\frac{\delta u}{\delta x}dx $$ to a discrete stiffness matrix $$ \mathbf{w}*(\mathbf{K} \mathbf{d}) $$. To begin, we must first note that
 * $$ \frac{\delta w}{\delta x} = [N_i^{'}w_i + N_{i+1}^{'}w_{i+1}] \text { and }

\frac{\delta u}{\delta x} = [N_i^{'}d_i + N_{i+1}^{'}d_{i+1}] \text{.}$$

$$\text{Therefore, } \beta = \int_{0}^{L} \frac{\delta w}{\delta x}(EA)\frac{\delta u}{\delta x}dx  \text{ can be expressed as } $$
 * $$ \beta = \int_{x_i}^{x_{i+1}} [N_i^{'}w_i + N_{i+1}^{'}w_{i+1}](EA)[N_i^{'}d_i + N_{i+1}^{'}d_{i+1}]dx \text{, where } N_i^{'} = \frac {\delta N_i(x)}{\delta x} \text{ and } N_{i+1}^{'} = \frac {\delta N_{i+1}(x)}{\delta x}.$$

Note that u(x) and $$ \frac{\delta u(x)}{\delta x} \quad $$can also be expressed in matrix form, where
 * $$ u(x) =

\lfloor N_i(x) \quad N_{i+1}(x)\rfloor \begin{bmatrix} d_i \\ d_{i+1} \\ \end{bmatrix} = \mathbf{N(x)}_{1x2} \begin{bmatrix} d_i \\ d_{i+1} \\ \end{bmatrix} \text{, and }$$

 * $$\frac{\delta u(x)}{\delta x} =

\lfloor N_i^{'}(x) \quad N_{i+1}^{'}(x)\rfloor \begin{bmatrix} d_i \\ d_{i+1} \\ \end{bmatrix} = \mathbf{B(x)}_{1x2} \begin{bmatrix} d_i \\ d_{i+1} \\ \end{bmatrix} $$

Similarly, w(x) and $$ \frac{\delta w(x)}{\delta x} \quad $$can also be expressed in matrix form, where
 * $$ w(x) = \mathbf{N(x)}_{1x2}

\begin{bmatrix} w_i \\ w_{i+1} \\ \end{bmatrix} \text{, and } $$

 * $$ \frac{\delta w(x)}{\delta x} = \mathbf{B(x)}_{1x2}

\begin{bmatrix} w_i \\ w_{i+1} \\ \end{bmatrix} $$

The element DOFs for element i are shown in the picture below:

The picture above shows that: Therefore, we may express equivalent DOFs in matrix format:
 * Node 1 of element i is equivalent to node i
 * Node 2 of element i is equivalent to node i+1
 * $$ \begin{bmatrix}

d_i \\ d_{i+1} \\ \end{bmatrix} = \begin{bmatrix} d_1^{(i)} \\ d_2^{(i)} \\ \end{bmatrix} = \mathbf{d}^{(i)} $$

 * $$ \begin{bmatrix}

w_i \\ w_{i+1} \\ \end{bmatrix} = \begin{bmatrix} w_1^{(i)} \\ w_2^{(i)} \\ \end{bmatrix} = \mathbf{w}^{(i)} $$

By substituting equivalent values for $$\frac{\delta u(x)}{\delta x} \quad$$ and $$ \frac{\delta w(x)}{\delta x} \quad $$ into $$ \beta = \int_{x_i}^{x_{i+1}} \frac{\delta w}{\delta x}(EA) \frac{\delta u}{\delta x}dx \quad $$, the continuous stiffness equation reduces to
 * $$ \beta = \int_{x_i}^{x_{i+1}} (\mathbf{B} \mathbf{w}^{(i)})(EA)(\mathbf{B} \mathbf{d}^{(i)})dx $$

(Move this and/or expain) Goal: $$ \beta = \mathbf{w}^{(i)} * (\mathbf{k}^{(i)} \mathbf{d}^{(i)}) $$

Since $$(EA) $$, $$(\mathbf{B} \mathbf{w}^{(i)})  $$ , and $$(\mathbf{B} \mathbf{d}^{(i)}) $$ are scalars, we may rearrange the terms in any fashion. Doing so allows us to express the continuous stiffness equation as

 * $$ \beta = \int_{x_i}^{x_{i+1}} (EA)(\mathbf{B} \mathbf{w}^{(i)})(\mathbf{B} \mathbf{d}^{(i)})dx $$

Recall that by the definition of a transpose:
 * $$ (\mathbf{B} \mathbf{w}^{(i)})^T = (\mathbf{w}^{(i)T} \mathbf{B}^T) = (\mathbf{w}^{(i)} \mathbf{B}^T)$$
 * $$ (\mathbf{B} \mathbf{d}^{(i)})^T = (\mathbf{d}^{(i)T} \mathbf{B}^T) = (\mathbf{d}^{(i)} \mathbf{B}^T)$$

Due to the fact that $$\mathbf{w}^{(i)} $$ and $$ \mathbf{d}^{(i)} $$ are independent from x, we may remove them from the integral:
 * $$ \beta = \mathbf{w}^{(i)} * \Bigl (\int_{x_i}^{x_{i+1}} \mathbf{B}^T(EA) \mathbf{B} dx \Bigr) \mathbf{d}^{(i)} $$

From the above equation, one may easily see that the stiffness equation $$ \beta = \mathbf{w}^{(i)} * \Bigl(\int_{x_i}^{x_{i+1}} \mathbf{B}^T(EA) \mathbf{B} dx \Bigr) \mathbf{d}^{(i)} $$ reduces to $$ \beta = \mathbf{w}^{(i)} * (\mathbf{k}^{(i)} \mathbf{d}^{(i)})$$. Therefore, we have solved for the stiffness matrix $$ \mathbf{k}^{(i)} $$:
 * $$ \mathbf{k}^{(i)} = \int_{x_i}^{x_{i+1}} \mathbf{B}^T(EA) \mathbf{B} dx $$

It is also known that
 * $$ B(x) = \lfloor N_i^{'}(x) \quad N_{i+1}^{'}(x)\rfloor = \Bigl \lfloor \frac{1}{x_i - x_{i+1}} \quad \frac{1}{x_{i+1} - x_i} \Bigr \rfloor = \Bigl \lfloor \frac{1}{-L^{(i)}} \quad \frac{1}{L^{(i)}} \Bigr \rfloor \quad$$, where the length of element i equals $$\quad L^{(i)} = x_{i+1} - x_i $$


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User:Eml4500.f08.team.guan/hw6117

The Continuous FEM and PVW continued
The following table compares the different terms of the FEM equations of the continuous and discrete methods: inertia, stiffness, and the applied force terms.
 * {| border=1 class="wikitable" style="margin: 1em auto 1em auto"

 * style="text-align: center;" |Continuous setting (PVW)
 * style="text-align: center;" |Discrete setting (PVW)

 * style="text-align: center;" |inertia: $$\displaystyle \alpha=\int_0^Lwm \ddot {u}dx$$
 * style="text-align: center;" |inertia: $$\displaystyle \bar {w} \cdot (\bar{M}\bar{d})$$
 * style="text-align: center;" |inertia: $$\displaystyle \bar {w} \cdot (\bar{M}\bar{d})$$

 * style="text-align: center;" |stiffness term: $$\displaystyle \beta=\int_0^L\frac{dw}{dx}(EA)\frac{\partial{u}}{\partial{x}}dx$$
 * style="text-align: center;" |stiffness term: $$\displaystyle \bar {w} \cdot (\bar{K}\bar{d})$$
 * style="text-align: center;" |stiffness term: $$\displaystyle \bar {w} \cdot (\bar{K}\bar{d})$$

 * style="text-align: center;" |applied force term: $$\displaystyle \gamma=w(L)F(t)+\int_0^L{wfdx}$$
 * style="text-align: center;" |applied force term: $$\displaystyle \bar {w} \cdot \bar{F}$$
 * style="text-align: center;" |applied force term: $$\displaystyle \bar {w} \cdot \bar{F}$$

 * style="text-align: center;" |for all $$\displaystyle w(x)$$ such that $$\displaystyle w(0)=0$$, the boundary condition.
 * style="text-align: center;" |for all $$\bar{w}$$, the boundary conditions eliminated.
 * style="text-align: center;" |for all $$\bar{w}$$, the boundary conditions eliminated.

 * }

Now, the stiffness term of the continuous finite element method will be examined by discretizing the clamped bar at its axis. It will be discretized into "n,nodes" nodes and "n,nodes-1" elements. The nodes need not be equidistant from each other and consequently the elements will not need to be the same length. The displacements will be only in the direction of the axis and will be denoted by d. No transverse forces will be taken into consideration. The following figure will describe the system being analyzed.  u(x) is the value of the axial displacement and is linearly interpolated along the element i. Assume displacement u(x) for $$x_i \le x \le x_{i+1}$$ (i.e. x $$\in [x_i,x_{i+1}]$$). The symbol $$\in$$ means "belongs to" or "is an element of a set". The local nodes are labeled in order of increasing x. x tilda is the incremented x coordinate where x tilda equals x plus x_i. Motivation for the linear interpolation of u(x) of the 2 bar truss is as follows. Let's consider an element in the figure below. The dotted line is the undeformed truss and the solid line is the deformed truss. The deformed shape is a straight line. In other words, there was an implicit assumption of linear interpolation of displacement between 2 nodes.  This is true considering the case where there are only axial displacements (i.e. zero transverse displacements). (31.2)    The question is how to express u(x) in terms of $$d_i=u(x_i)$$ as a linear function in x (i.e. using linear interpolation). Let u(x) be written in the following linear form. $$\displaystyle u(x)=N_i(x)d_i + N_{i+1}d_{i+1}$$ where $$N_i$$ and $$N_{i+1}$$ are linear functions in the x domain.  Writing the equation for N_i+1, we have the following. $$N_{i+1}(x)=\frac{x-x_i}{x_{i+1}-x_i}$$ Checking this to be the correct equation, we plug in x_i in the equation to get zero and plug in x_i+1 to get one. This parallels the figure so the equation is correct. The equation for N_i is given in the collapsible box below.

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User talk:Metal.lunchbox

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French Mentoring
Salut, Metal.lunchbox. You are listed at the French stream as being a proficient student of French who is interested in learning French. The French Mentoring project has recently been launched, and if you add your name to that list, I will do my best to put you in contact with an appropriate Mentor who would be glad to help you. The Jade Knight (d'viser) 07:00, 13 November 2008 (UTC)

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User talk:Obsideus~enwikiversity

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

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

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

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --The Jade Knight (d'viser) 07:42, 13 November 2008 (UTC)

French Mentoring
Salut, Obsideus. You are listed at the French stream as being an intermediate student of French who is interested in learning French. The French Mentoring project has recently been launched, and if you add your name to that list, I will do my best to put you in contact with an appropriate Mentor who would be glad to help you. The Jade Knight (d'viser) 07:47, 13 November 2008 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Obsideus. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Obsideus~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:40, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:17, 19 April 2015 (UTC)

title: User talk:Obsideus~enwikiversity


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User:Amk~enwikiversity

Collection of material relating to my study of physics. It would consist of mechanics, electronics and thermodynamics.

title: User:Amk~enwikiversity


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User talk:Daniela S.

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

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

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

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! Jtneill - Talk - c 12:39, 13 November 2008 (UTC)

 Hi! Our name is Vicky-Verity. We are twins and are Wikiversity mascots. We think Wikiversity is a great place to learn and make friends (when we aren't sleeping). Leave us a message!

title: User talk:Daniela S.


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Digital media and democracy

Link to any subpages this page might have

There has been much discussion in the news lately about how digital media (such as texting, peer-to-peer, youtube, etc.) was used by Barack Obama's presidential campaign in 2008. The use of the internet in general has been growing in political campaigns, and Obama's use of the media is considered by most to be quite effective.

Ideas for subtopics

 * History of digital media in presidential campaigns
 * Presidential campaigns before digital media
 * Describe various campaigns (in the US and elsewhere) in terms of how they used and were affected by digital media
 * "Non-affiliated" activists (Wikipedia editors, YouTube, etc.)

 * How is digital media used in governance
 * What's next (pull out your crystal ball!)
 * Start an imaginary campaign (mayor of Lunar Boom Town, maybe?)
 * Think about a new political system based upon the speed and "openess", "root" intellect of the Internet?

title: Digital media and democracy


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User:JoliePA

I am JoliePA (formerly the user user:Jolie )...

I am a chemist at an gasoline refinery. My goal is to be a lifelong student and to explore wiki as a forum for personal learning, and social interaction.

Schools

 * School of History  School:History
 * and I am enrolled in a class of US history Introduction_to_US_History
 * We are revising the authoritative textbook US_History/Contents

 * School of Strategy School:Strategic_studies
 * and I am enrolled in a self-taught class Introduction_to_Strategic_Studies

 * School of Chemistry School:Chemistry
 * Topic Chemistry_and_its_socioeconomic_impact

 * School of Astronomy School:Physics_and_Astronomy
 * Topic Topic:Astronomy
 * I am willing to teach the course Initial_experiences
 * based on the content of Stargazing

Wikiversity learning (and teaching)

 * Wikiversity_Courses
 * Editing
 * Wiki_101
 * Learning
 * Welcome,_newcomers
 * Learning_by_doing
 * teaching
 * philosophy
 * How_to_be_a_Wikimedia_sysop

Hobbies

 * Observing Topic:Amateur_astronomy
 * Observational_astronomy/Planning
 * comets an initial observing project

 * Pages that need attention
 * Skygazing
 * The_visible_sky

Interactions with others

 * interesting reading group Worldpolitics
 * interesting idea Think_tank
 * Project on wikiversity ethics and specifically what are uncivil comments Learning_from_conflict_and_incivility
 * also relatively interesting to understand editing in a wiki environment Editing_Applied_ethics
 * Topic:Editing_disagreements

Inquiries

 * School of Relosophy School_talk:Relosophy
 * School of Political Science School_talk:Political_science

 * School of Diplomatic History Topic:Diplomatic_History
 * possible course? DIPL_1001:_Introduction_to_Diplomatic_History

Help,questions, other wikis

 * Admin request Request_custodian_action
 * wikibooks b:User:JoliePA
 * wikipedia w:User:JoliePA

title: User:JoliePA


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User:Eml4500.f08.ramrod.D/HW6

Principle of Virtual Work (the continuous case) of the dynamics of an elastic bar (Monday Nov 3)
The partial differential equation that describes this phenomena is below:

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

The representative equation of motion to the above equation is:

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

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

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

Deriving (equation 2) from (equation 1)

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

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

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

Equation 3 can be rewritten as:

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

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

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

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

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

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

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

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

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

PlaneTrussElement.m

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

NodalSoln.m

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

PlaneTrussResults.m

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

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

title: User:Eml4500.f08.ramrod.D/HW6