text:
Fundamental entity

Think of time as a dimensionless duration that everything exists within, with dilation a one-dimensional outward motion within this duration as the only motion. The inward motion we detect as energy is a subset within space/time, and all energy, moving outward from a point is what we detect as a single motion we call the present. One is a temporal entity and like all numbers dimensionless. Time does not run fast or slow, think of space as the present, our motion through space describes our own duration in the present. The faster I move through space, the present, the longer my duration through space/time hence the slower my clock ticks.

Energy twisting and bending while interacting with itself makes its own duration intrinsic, or forms a field of temporal motion, we measure this motion as the outward force of gravity. Even the names given to energy define them; Potential, energy of the future; Kinetic, dynamic energy of the present; and Annihilation, energy of the past, all of them together make up our present. When this motion is intrinsic we name it one, which is the only way we can separate time, where it separates itself. With this in mind, the statement “Two objects cannot occupy the same space at the same time” can be finished with "because all objects are separate parts of the same motion we measure as time".

This field of time is our first concept of time as a circular motion, an atom, one object dilating in time as a whole, and starts our continuum with its creation. When a second clock is created we begin to see things as relative, our concept of zero motion begins. This creation of juxtaposition produces an inflating time between atoms which lead to our concept of relative length. We have to wait for the solar system to form and for the appearance of an observer to focus these ideas together as the present. The present we count as a second with a standard clock today, the same second that sets our passive observers Minkowski's reference frame. All of this within a duration determined by the local energy and because of dilation. Now think of our atomic clock, the hyperfine transitions present in the ground state of the cesium-133 isotope that we use to standardize a second, is time dilating. Everything dilates we can see this effect, 'when' an observer is changing his relative position in space/time, Objects getting bigger as each present moment is brought closer together, and get smaller the further into the past they appear. Everything is emerging, from Big bang til now. At the smallest scale, relative to now, it must still be emerging, repulsive, as a focal point stuck in the present. Every object is visible as a worm-hole over time, but in the instant as a point of one. Color from a point, how our minds see photons, makes up our visible universe.

Every single ponderable entity, is its own clock, keeping track of its own duration in its own present, because everything that exists in time occupies space and we can only detect the interactions (duration) between these spaces juxtapositions of time. The motion of time is dilation and it is the duration of all our static present(s) that are dilating, we only receive signals from the out side of objects. On Planck’s scale time is the smallest common denominator, and even with time being so compact, and without measuring it directly, we do detect where its motion masses, in one form or another. Time is the smallest common denominator on Planck's relative scale and it is the juxtaposition of an expanding field of time we call a photon, that is its single quanta. Time is also the largest common denominator, with everything that exists being nothing more that a subset of this one-dimensional outward motion that we visualize with our “arrow of time” that started at big bang. Let us step back in time far enough to see the leading edge of the entire light sphere of our one visible universe, reach out, pick this field of time up, and place it under a microscope. We would see this electromagnetic field that is dilating at the speed of light squared, relative to itself, with all its mass centered relative to our new reference frame as a single duration, an atom, our next repeating juxtaposition in an infinite homologous universe.

That first atom must be in direct temporal contact with the chaos of time from which it was formed, a small wormhole to all that potential energy back through the center connection we call mass, within the ever changing form of an atom. Particle physics looks back through time to Big bang through atoms, which uses less space but large amounts of energy. On the other hand Cosmology looks back in time by gazing out at the stars, less energy required but with more space between. Both paths through time to a big bang follow or trace energy, one potential and the other kinetic. We can see how space appears curved drawing a time line between both paths from deep within every atom, all the way out to the stars and back in again. We are a part of this dilating sphere, riding through time on earth in the present between this one event. All observers are riding Earth’s accelerating frame out of the same gravity well and it is where and when these two forms of the same motion interact, that is the present. Matter is the contained and gravity is the container in space, in time the present is the container and energy is the contained. Duration is the variable of the present in the form of one, that makes the geometry of space that we see, fit with the structure of matter that we feel.

How can you tell time without looking at matter? If the present is emission, we can only "see" the past no matter how close we are therefore "anything you do because of something you see is trying to change the past", which explains why mass resists changes to its position, no matter the direction in space you approach an object in time it is from the past. The little twist that we now call big bang started time, or the dilating area that we see today as our one visible universe. Matter or what we think of as atoms are our tie to the past, created almost 13.7 billion years ago and I think there is still some of that original motion centered within each atom. We exist as an accelerated frame called earth, with a dilation rate of 9.8 meters per second in time but our lying eyes tell us we are static because of the way we see a continuous past combined in our minds as our present and our future. Just as with the earth being flat I think that in time we will begin to think of mass, the dilating momentum of matter, as the past started the little twist we see today as big bang.

To sum up, we measure the dilating momentum of matter as mass which is a juxtaposition, centered in the past, that we see as individual entities in space. Juxtaposition is a measure of time with a beginning that we never see and an end that only happens in the present, we never see the beginning of the photon's duration we only see its end. Think of a Photon as the radius to its own duration's dilating sphere, Planck's length in four dimensions, making time quantized to a photon. This fits with the idea that a photon is the smallest, one dimensional motion, part of my present as far as I can see.

My Consciousness is the clock that keeps me centered in the present.

Space is the present moment we all share as observers.

Mass is my anchor to the past, while this accelerated frame we call Earth is just all of us dilating into the future.

Outward from a point is the only direction in time, dilation it's only motion, the illusion is thinking there is more than this one direction in space.

title: Fundamental entity


text:
User talk:Eml4500.f08.ateam.nobrega

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

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a question for you
Hi! I see that you uploaded Image:Matlabicon.JPG to Wikiversity. Could you take the time to provide some information about the source of the image? What is the source of the image? Do you think it is a copyright-protected part of a software product? Is it an image that is free to copy and distribute? There is useful information about image licensing at Uploading files. --JWSchmidt 04:22, 9 September 2008 (UTC)

I refereneced the image in my article. The hyperlink to the image source can be found in the citation section under the general homework reports tab, then under 'homework report one' of my wikiversity page. I do not believe the software company would have any problems with the use of the icon because I am simply using it to explain how to start and use their software. I am an owner and user of the MATLAB software. Thank you and I hope I've answered your question. --eml4500.f08.ateam.nobrega

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title: User talk:Eml4500.f08.ateam.nobrega


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User talk:Eml4500.f08.delta 6.lopez

 Hello Eml4500.f08.delta 6.lopez, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Eml4500.f08.delta 6.lopez


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User:Eml4500.f08.ateam.mcnally

EML 4500-Finite Element Analysis ATeam WIKIVERSITY
Ateam - Group Members Akash Shah Taylor Boggs Tatiana Nobrega Pedro Rivero Michael Carr Josh McNally

Class Notes:

 * 1) Week 1 Lecture Notes - August 25th - August 29th
 * 2) Week 2 Lecture Notes - September 1st - September 5th
 * 3) Week 3 Lecture Notes - September 8th - September 12th
 * 4) Week 4 Lecture Notes - September 15th - September 19th
 * 5) Week 5 Lecture Notes - September 21st - October 3rd
 * 6) Week 6 Lecture Notes - October 13th - October 17th
 * 7) Week 7 Lecture Notes - October 26th - October 31st
 * 8) Week 8 Lecture Notes - November 10th - November 14th

Links:

Main Class Page

Homework Assignments:

 * 1) HW1 - Matlab Tutorial Sections 3, 7, 11, 15, 19
 * 2) HW2 -
 * 3) HW3 -
 * 4) HW4 -

title: User:Eml4500.f08.ateam.mcnally


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User talk:Eml4500.f08.delta 6.ramirez/Lecture 1

note: if you leave a leading space before text it will look like this:

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If you want to indent text try   :a colon at the start of the line

 * a colon at the start of the line

See Help:Wiki markup examples. --JWSchmidt 20:03, 7 September 2008 (UTC)


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User talk:Eml4500.f08.ateam.mcnally

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

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title: User talk:Eml4500.f08.ateam.mcnally


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User:Eml4500.f08.bottle.barnes/HW1

Command Line Editing and Recall
The left and right arrow keys can be used to move the cursor and the Backspace key can be used to delete a character. On a PC, hitting the Home key will position the cursor at the beginning of the command line, while the End key will position it at the end of the line.

The up and down arrow keys can be used to move through previous commands. A previous command can be recalled, edited and executed over again. For example, typing

a=rand(8)

in the command line yields:

a =

0.8909   0.8143    0.3517    0.3804    0.5688    0.1656    0.2290    0.1067    0.9593    0.2435    0.8308    0.5678    0.4694    0.6020    0.9133    0.9619    0.5472    0.9293    0.5853    0.0759    0.0119    0.2630    0.1524    0.0046    0.1386    0.3500    0.5497    0.0540    0.3371    0.6541    0.8258    0.7749    0.1493    0.1966    0.9172    0.5308    0.1622    0.6892    0.5383    0.8173    0.2575    0.2511    0.2858    0.7792    0.7943    0.7482    0.9961    0.8687    0.8407    0.6160    0.7572    0.9340    0.3112    0.4505    0.0782    0.0844    0.2543    0.4733    0.7537    0.1299    0.5285    0.0838    0.4427    0.3998

Rather than having to move between MATLAB and the editor, the line can be directly recalled yielding a new set of random numbers:

a =

0.2599   0.8693    0.4018    0.9027    0.7803    0.5752    0.6491    0.6868    0.8001    0.5797    0.0760    0.9448    0.3897    0.0598    0.7317    0.1835    0.4314    0.5499    0.2399    0.4909    0.2417    0.2348    0.6477    0.3685    0.9106    0.1450    0.1233    0.4893    0.4039    0.3532    0.4509    0.6256    0.1818    0.8530    0.1839    0.3377    0.0965    0.8212    0.5470    0.7802    0.2638    0.6221    0.2400    0.9001    0.1320    0.0154    0.2963    0.0811    0.1455    0.3510    0.4173    0.3692    0.9421    0.0430    0.7447    0.9294    0.1361    0.5132    0.0497    0.1112    0.9561    0.1690    0.1890    0.7757

Submatrices and Colon Notation
Using “Colon notation” in MATLAB allows for both the generation of vectors and the referencing of submatrices. This along with subscripting by integral vectors are tools that allow the user to cut down on the use of loops, which slows the program and makes it harder to read.

For example, typing in something as simple as:

x=[0.0:0.2:4.0]'; y=sin(x); [x y]

produces a table of sines:

ans =

0        0    0.2000    0.1987    0.4000    0.3894    0.6000    0.5646    0.8000    0.7174    1.0000    0.8415    1.2000    0.9320    1.4000    0.9854    1.6000    0.9996    1.8000    0.9738    2.0000    0.9093    2.2000    0.8085    2.4000    0.6755    2.6000    0.5155    2.8000    0.3350    3.0000    0.1411    3.2000   -0.0584    3.4000   -0.2555    3.6000   -0.4425    3.8000   -0.6119    4.0000   -0.7568

To access particular submatrices of a matrix, colon notation can be used as follows:

x(1:3,1)

produces the first three entries in the first column:

ans =

0   0.2000    0.4000

A single colon refers to an entire row or column: x(:,2) refers to the second column of x, for example.

Integral vectors such as: x(:,[1 2]) includes columns 1 and 2 of x as subcripts.

To replace columns of a matrix, typing: x(:, [1 2]) = B(:,1:2) overwrites the first and second column of x with the first and 	second column of B producing:

x =

0.4868   0.2305    0.4359    0.8443    0.4468    0.1948    0.3063    0.2259    0.5085    0.1707    0.5108    0.2277    0.8176    0.4357    0.7948    0.3111    0.6443    0.9234    0.3786    0.4302    0.8116    0.1848    0.5328    0.9049    0.3507    0.9797    0.9390    0.4389    0.8759    0.1111    0.5502    0.2581    0.6225    0.4087    0.5870    0.5949    0.2077    0.2622    0.3012    0.6028    0.4709    0.7112

When x is and n-vector typing:

x=x(n:-1:1)

produces:

x =

0.4468   0.4359    0.4868

Entering:

y=fliplr(x) and y=flipud(x’)

yields:

y =

0.4868   0.4359    0.4468

and y=flipud(x) flips the matrix vertically.

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


text:
User:Eml4500.f08.ateam.mcnally/Week 1

Topics Covered

 * What is FEA
 * Class Dates, Policies, and Materials
 * Philosophy of Wiki and “Class Vision”

title: User:Eml4500.f08.ateam.mcnally/Week 1


text:
User:Eml4500.f08.echo.sell.cm

http://en.wikiversity.org/wiki/User:Eml4500.f08.echo.mott/ECHO

Drafting Pages
http://en.wikiversity.org/wiki/User:Eml4500.f08.echo.sell.cm/HW1

http://en.wikiversity.org/wiki/User:Eml4500.f08.echo.sell.cm/HW2

http://en.wikiversity.org/wiki/User:Eml4500.f08.echo.sell.cm/HW3

title: User:Eml4500.f08.echo.sell.cm


text: User:Eml4500.f08.ateam.rivero/HomeworkAssignments


text:
User:Eml4500.f08.ateam.shah/Homework

 * Section 5

 * Section 9

 * Section 13

 * Section 17

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


text:
User:Eas4200c.f08.gator.edwards

 Week One Class Notes

 Week Two Class Notes

 Week Three Class Notes

 Week Four Class Notes

 Week Five Class Notes

 Week Six Class Notes

 Week Seven Class Notes

Category:Courses

title: User:Eas4200c.f08.gator.edwards


text:
User:Eml4500.f08.ateam.rivero/HW1

=Homework Assignment 1: MatLab Introduction Matlab Primer, University of Florida =

Section 2: Entering Matrices
Matrices in MATLAB can be entered in several different ways. Below are two examples that output the same matrix.

>> A=[1,2,3;4,5,6;7,8,9]

A = 1    2     3     4     5     6     7     8     9

>> A = [ 1 2 3 4 5 6 7 8 9 ]

A = 1    2     3     4     5     6     7     8     9

Entering Matrices with complex numbers. Below are two different ways to enter complex matrices.

>> B = [1 2;3 4] + i*[5 6;7 8]

B =

1.0000 + 5.0000i  2.0000 + 6.0000i 3.0000 + 7.0000i  4.0000 + 8.0000i

>> B = [1+5i 2+6i;3+7i 4+8i]

B =

1.0000 + 5.0000i  2.0000 + 6.0000i 3.0000 + 7.0000i  4.0000 + 8.0000i

Referencing an element from a matrix. For example in the matrix A above, if you want to reference the value of the element located in row 2, column 3 type A(2,3). View example below.

>> A(2,3)

ans =

6

Section 6: For, while, if - and relations
For.

Using the For. statement to produce a vector. Example for n = 3

>> x = []; for i = 1:n, x=[x,i^2], end

or

>> x = []; >> for i = 1:n x = [x,i^2] end

Will result in:

x =

1

x =

1    4

x =

1    4     9

Producing the same vector as shown above, but in reverse order. Example is shown below.

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

x =

9

x =

9    4

x =

9    4     1

While.

A while loop allows for the statement to be executed continuosly aslong as the relation defined remains true.

General Form:

while relation statement end Example:

>> a =30; >> n = 0; >> while 2^n < a    n = n + 1; end

Will result in:

>> n

n = 5

If.

An if statement is only executed if the relation is true.

General Form:

if relation statement end

Example:

>> if n < 0 parity = 0; elseif rem(n,2) == 0 parity = 2; else parity = 1; end

Relations

MATLAB Relational Operators:

MATLAB Logical Operators:

If the statement is true it is a value of 1. If the statement is false it is a value of 0. See the examples below.

>> 3 < 5

ans =

1

>> 3 > 5

ans =

0

>> 3 == 5

ans =

0

>> 3 == 3

ans =

1

>> a = rand(5)

a =

0.9501   0.7621    0.6154    0.4057    0.0579    0.2311    0.4565    0.7919    0.9355    0.3529    0.6068    0.0185    0.9218    0.9169    0.8132    0.4860    0.8214    0.7382    0.4103    0.0099    0.8913    0.4447    0.1763    0.8936    0.1389

>> b = triu(a)

b =

0.9501   0.7621    0.6154    0.4057    0.0579         0    0.4565    0.7919    0.9355    0.3529         0         0    0.9218    0.9169    0.8132         0         0         0    0.4103    0.0099         0         0         0         0    0.1389

>> a == b

ans =

1    1     1     1     1     0     1     1     1     1     0     0     1     1     1     0     0     0     1     1     0     0     0     0     1

Section 10: Command Line Editing and Recall
Keys and their functions to edit in MATLAB

Section 14: Managing M-Files
It is recommended that if your system allows you to run multiple processes that you keep both the MATLAB Command Window and your m-file editor open simultaneously. At any point if you would like to edit your m-file use the function !ed. For example you would type into the command window as follows: >>!ed filename.m

Note: Make sure to replace filename with the file you are trying to edit.

Commands useful for managing m-files.

Planar Plots
The plot command creates linear x-y plots; if x and y are vectors of the same length, the command plot(x,y) opens a graphics window and draws an x-y plot of the elements of the x versus the elements of y.

For example command: >> x = -4:.01:4; y=sin(x); plot(x,y)

Results in:

One can also refrence a function from an m-file in the given format: >>fplot('expnormal',[-1.5,1.5])

Results in:

Plots of parametrically defined curves can also be made. For example:

Commands for plots:

Manualing Scaling Axes:

Multiple Plots on a Single Graph: Two possible ways

1. Example: Listing them within the same command line >> x=0:.01:2*pi; Y=[sin(x)', sin(2*x)', sin(4*x)']; plot(x,Y)

Result in:

2. Example: Using the hold function >> x=0:1:10; y=x; plot(x,y) >> hold on >> x=0:1:10; y=2*x; plot(x,y) >> x=0:1:10; y=3*x; plot(x,y) >> hold off

Result in:

Editing Linetypes, Marktypes, and Colors of plots:

Commands:

 * Linetypes: solid(-), dashed(--), dotted, dashdot(-.)
 * Marktypes: point(.), plus(+), star(*), circle(o), x-marks(x)
 * Colors: yellow(y), magenta(m), cyan(c), red(r), green(g), blue(b), white(w), black(k)

Example Code: >> x=0:.01:2*pi; y1=sin(x); y2=sin(2*x); y3=sin(4*x); plot(x,y1,'--',x,y2,':',x,y3,'+')

Graphics Hardcopy
To obtain a hardcopy of your plots you would use the commmand: >>print

3-D Line Plots
To plot a 3-D line defined by vectors x, y, z of the same size, you would use the command plot3(x,y,z). Below is an example:

>> t=.01:.01:20*pi; x=cos(t); y=sin(t); z=t.^3; plot3(x,y,z)

Result in:

3-D Mesh and Surface Plots
3-D Surface Mesh plots are drawn with the command mesh

Example Code:

>> mesh(eye(10))

Result in:

Faceted surface plots are drawn with the command surf

Example Code:

>> surf(eye(10))

Result in:

Plotting 3-D Functions: Function:

Code:

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

Resulting in:

One could also use the surf command for the same function resulting in: Code:

>> surf(z)

Shading and Color Profiles

Commands:

*shading faceted *shading interp *shading flat

*colormap(hsv) Note: Default *colormap(hot) *colormap(cool) *colormap(jet) *colormap(pink) *colormap(copper) *colormap(flag)

Other Useful Commands: Can further explore under the help command

*meshz *surfc *surfl *contour *pcolor

*colormap(gray) *colormap(bone)

title: User:Eml4500.f08.ateam.rivero/HW1


text:
User:Eml4500.f08.ateam.mcnally/HW1

Matlab Primer, University of Florida

Matrix and Array Operations in MATLAB: Section 3
MATLAB incorporates the following matrix operations: Care must be taken to ensure that sizes of the matrices are compatible or an error message will result, and the computation will not be performed."It is important to observe that these other operations, *, ^, \, and /, can be made to operate entry-wise by preceding them by a period." See the following example for further clarification.
 * Addition +
 * Subtraction –
 * Multiplication *
 * Exponential Power ^
 * Conjugate Transpose ‘
 * Left Division \
 * Right Division /

>>[6,7,8,9].*[6 7 8 9]

ans =

36   49    64    81 Notice that the operation performed was not matrix multiplication but rather an entry by entry multiplication. See the following example for matrix multiplication.

>>A=[1 2;3 4]

A =

1    2     3     4

>> B=[1 2;3 4]

B =

1    2     3     4

>> A*B

ans =

7   10    15    22

Scalar Functions: Section 7
MATLAB has built in functions that operate on scalars, but can also operate on individual elements of a matrix. The most common of these functions are as follows:

See the following examples:

>>sin(3*pi/2)

ans =

-1

>> log(1)

ans =

0

Submatrices and colon notaion: Section 11
"Vectors and submatrices are often used in MATLAB to achieve fairly complex data manipulation effects. "Colon notation" ( which is used both to generate vectors and reference submatrices) and subscripting by integral vectors are keys to effcient manipulation of these objects." A row vector of length n may be defined as 1:n where n represents the length of the vector. The default increment is taken to be 1 unless otherwise specified. See the following examples for further clarification. A row vector of length n may be defined as 1:n where n represents the length of the vector. The default increment is taken to be 1 unless otherwise specified. See the following examples for further clarification.

>>1:5

ans =

1    2     3     4     5

>> 1:2:5

ans =

1    3     5 Both vectors are row vectors from 1 to 5, but the second vector is from 1 to 5 with an increment of 2."

Colon Notation" may also be used to access submatrices of a matrix, and to replace elements within matrices. See the following example for further clarification.

>>A =

1    2     3     4     5     6     7     8     9    10    11    12    13    14    15    16

>> A(1:2,3)

ans =

3    7 This creates a column vector consisting of the first 2 entries of the 3rd column of matrix A.

Comparing efficency of algorithms flops, tic and toc: Section 15
Older versions of MATLAB had the function flops which kept a running total of flops performed to solve and algorithm. In more resent versions of MATLAB the command function flops is unavailable and an error message will appear if the command is entered. The command function “tic, any statement, toc” can be used to obtain the elapsed time (in seconds) it takes MATLAB to solve the algorithm. This enables the user to compare the efficiency of his / her algorithms. See the example below for further explanation.

>> A=[ 1 2 3; 7 8 9; 4 5 6]

A =

1    2     3     7     8     9     4     5     6

>> b= [1 2; 4 5; 8 9]

b =

1    2     4     5     8     9 >> x=A*b

x =

33   39   111   135    72    87 >> tic, x=A*b; toc

Elapsed time is 0.000000 seconds.

Sparse Matrix Computations: Section 19
A great deal on computation time and memory storage space can saved when performing calculations on a matrix that has many zero entries by using the command sparse. “Matlab has two storage modes, full and sparse with full the default. The functions full and sparse convert between the two modes.” "A sparse matrix is stored as a linear array of it’s nonzero elements along with their row and column indices.” See the following example for further explanation.

F =

9    4     0     0     0     0     2     0     7     0     0     0     0     8     1     0     0     0     0     0     4     3     6     0     0     0     0     8     2     4     0     0     0     0     1     4

>> S=sparse(F)

S =

(1,1)       9   (2,1)        2   (1,2)        4   (3,2)        8   (2,3)        7   (3,3)        1   (4,3)        4   (4,4)        3   (5,4)        8   (4,5)        6   (5,5)        2   (6,5)        1   (5,6)        4   (6,6)        4

>> F=full(S)

F =

9    4     0     0     0     0     2     0     7     0     0     0     0     8     1     0     0     0     0     0     4     3     6     0     0     0     0     8     2     4     0     0     0     0     1     4

The command S=sparse(F) creates the sparse matrix S from the matrix F, and the command F=full(S) returns matrix F back to its original form.

title: User:Eml4500.f08.ateam.mcnally/HW1


text:
User:Eml4500.f08.bike.bernal

Fri Dec 5

Now, a dimensional analysis will be done to the axial as well as the transverse displacements. Axial Displacements

$$ \left[u \right] =L $$ $$ \left[N_1 \right] =[N_4]=1 $$ $$ [\tilde{d}^{(e)}_1]= [\tilde{d}^{(e)}_4]= 1$$ Therefore, $$ [N_1][\tilde{d}^{(e)}_1]= [N_4][\tilde{d}^{(e)}_4]= L $$

Transverse Displacements

$$ \left[v \right] =L $$ The shape functions for the transverse displacements are shown below. $$\left[N_2 \right]=\left[N_3 \right]=\left[N_5 \right]=\left[N_6 \right]=1$$ The displacements corresponding rotation are shown below, which are dimensionless.

$$ [\tilde{d}^{(e)}_3]= [\tilde{d}^{(e)}_6]= 1$$

The rest of the displacements are in the $$ \tilde{Y}$$ direction, and those are shown below.

$$ [\tilde{d}^{(e)}_2]= [\tilde{d}^{(e)}_5]= 1$$

Therefore,

$$ [N_2][\tilde{d}^{(e)}_2]= [N_4][\tilde{d}^{(e)}_5]= L $$

$$ [N_1][\tilde{d}^{(e)}_3]= [N_4][\tilde{d}^{(e)}_6]= 1 $$

Derivation of Beam Shape Functions
First, the governing PDE for beams is analyzed. This PDE is considered without a distributed transverse load and also without an inertia force.

$$ m \ddot{v} $$(static case) is shown below.

$$\frac{\partial^{2} }{\partial x^{2}} = \begin{Bmatrix} (EI)\frac{\partial^{2}v }{\partial x^{2}} \end{Bmatrix}=0$$

Also, let's consider a constant EI, and the final equation is shown below.

$$\frac{\partial^{4} }{\partial x^{4}}=0$$

Solving for this fourth degree derivative, the following equation is obtained.

$$ v(x)= C_0 + C_{1}x+ C_{3}x^{2}+C_{3}x^{3}$$

Now, to get $$ N_2(x) $$, and since it is in the axial direction, $$\tilde{x} = x$$.

This results in the following values using the boundary conditions.

$$ v(0)=1$$ $$ v(L)=0$$ $$ v'(0)=0$$ $$ v'(L)=0$$

title: User:Eml4500.f08.bike.bernal


text:
User:Eml4500.f08.ateam.rivero/w1

Week 1: Sept. 5, 2008 Continued
Example Problem:

The following example below, will show the steps described below in the recipe to solving simple truss problems. Trusses are composed of elements that only allow for axial loads. The elements are connected by nodes. This problem has a total of two elements and three nodes, where each node has two degree's of freedom. This problem will be composed of applying a load P to node two. Once the problem is solved we would be able to see the deflections and force reactions at each node.

Data:

Element Length: *L(1) = 4 *L(2) = 2

Young's Modulus: *E(1) = 3 *E(2) = 5

Cross Sectional Area: *A(1) = 1 *A(2) = 2

Inclination Angle: *Θ(1) = 30o *Θ(2) = 45o

Step 1: Global Picture

Labeling : Global degrees of freedom (dofs) and forces

The number inscribed in a green circle refers to a node value n. For example, in the problem being described the node values range from one through three. When numbering the displacement degrees of freedom (d1, . . . d6), you want to start at node 1 and the x displacement. This displacement degree of freedom would be labeled ( d1 ). This would be followed by node 1, y displacement; labeled ( d2 ). Next would be followed by node 2, x displacement; labeled ( d3 ). Next would be node 2, y displacement; labeled ( d4 ). One would continue this order until all the nodes are labeled with their corresponding degrees of freedom. The same systematic order would be followed to label the global forces ( F1, . . . F6 ).See image above for further explanation on labeling. These displacements and forces are then put into a matrix format as seen in the image below.

title: User:Eml4500.f08.ateam.rivero/w1


text:
User:Shayan~enwikiversity

Shayan Nahrvar is a CIV100 Lecturer at the University of Toronto. He is working on Wikipedia page to support the university's first year Engineering mechanics and statics course.

title: User:Shayan~enwikiversity


text:
User:Eml4500.f08.ateam.shah/Sections

Section 5

Section 9

Section 13

Section 17

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


text:
User:Eml4500.f08.ateam.shah/Section5

Section 5: Matrix Building Function
Matrix building functions are

eye	       identity matrix zeros          matrix of zero ones           matrix of ones diag           create or extract diagonals triu           upper triangular part of the matrix tril           lower triangular part of the matrix rand           randomly generated matrix hilb           Hilbert Matrix magic          magic square topelitz       see help topelitz

Functions like zeros, eye, and ones can be used to create matrices. The functions are followed by the size of the matrix.

Ex:

$$>> zeros(2,3)$$ $$ans =$$ 0    0     0     0     0     0 Functions such as diagonals vectors require a vector as an input.

Ex:

$$>>A =$$ 2    2     2     2

$$>> diag(A)$$ $$ans = $$    2     2

You can combine and create new matrices using multiple functions.

Ex: This function will create a 5x5 matrix shown below.

$$>> 	B= [A, zeros(3,2); zeros(2,3), eye(2)]$$

$$B =$$

0    0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     1     0     0     0     0     0     1

title: User:Eml4500.f08.ateam.shah/Section5


text:
User:Eml4500.f08.bike.gravois

Homework 1
Command line editing and recall

The MATLAB command line can be edited in several ways. The cursor is moved with the left and right arrows on the keyboard, and the backspace key deletes the character to the left of the cursor. Characters can also be deleted using the Home, End or Delete keys. The up and down arrows, on the keyboard, are used to scroll through previous commands.

A command line can be changed and rerun by scrolling up to, editing, and then executing the previous command. The plots of y=sin(mx) and y=sin(nx) from [0 2π] can be compared easily by editing the (mx) to (nx) in the previous command line:

""

Submatrices and colon notation

Colon notation is used in MATLAB to generate vectors and reference submatrices. Colon notation also reduces the use of loops and simplifies codes.

The expression 1:4 is a row vector [1 2 3 4]. Neither the vector numbers nor the increment have to be integers. The example,

yields the vector

To cont down, a negative sign is used in the vector notation.

yields

Colon notation is also used to recall submatrices within a matrix. An example

is a column vector made up of the first three values in the second column of the matrix A.

A lone colon calls an entire row or column:

is the entire second column of the A matrix. Vectors can be used within colon notation to recall multiple columns.

recalls the entire columns two and five of the matrix A.

Columns can be replaced in one matrix from another using colon notation for Example,

replaces columns two, three, and five in matrix A with columns two, three, and four from matrix B.

M-files: script files, function files

MATLAB can run a sequence of statements that are stored in disk files, called “M-files.” M-files end with “.m” in the filename and are usually created with a local editor. The two types of M-files are script and function files.

Script files

Script files contain normal sequences of MATLAB statements. The file  will be run in MATLAB by imputing the command. Variables in the M-file are global and will replace the values of same named variables in the MATLAB environment.

Script files also allow for large matrices to be entered and corrected easily. A matrix can be input, for example, by entering diskfile data.m

A = [ 1 2 3 4 5 6 ];

Running the MATLAB command data will cause the matrix A to be created and stored in the MATLAB environment. An M-file can call on other M-files, or it can run itself repeatedly.

Function files

Function files allow for the creation of new functions, that do not exist in MATLAB already, needed for a specific problem that. Function variables are set as local variables by default.

An example of a function file:

function a = (sin(m))^(-1) %INVSIN computes inverse sin. % invsin(m) returns inverse sin
 * A = 10*invsin(m);

This example should be in a diskfile named  due to the function name. The first line declares the function name, input arguments, and output arguments. Without the first line the function file would be a script file. A MATLAB command  will pass the value 45 into the variables m in the function file, and the output will be put into variable A.

The  symbol is used for note making in an M-file. The symbol tells MATLAB to skip the rest of the line. Comments should always be used in M-files to describe the function.

Loops can be made to run faster by preallocating output vectors or matrices. An example of this can be seen in the second line of the for loop where the matrix E is made one column larger for each iteration.

F = fun(6); D = zeros(6,50); For j = 1:50
 * D(:,j) = det(F^i);
 * End

Most of the MATLAB functions are built into the software, but some are M-files. To see a listing of any non built in M-file the MATLAB command type must be run.

Week 2 Class Notes
Element Labeling

The class begins with an introduction to the nomenclature of element labeling. The figure is first drawn with all members, forces, and constraints in a global view. Each constraint is then labeled globally by an encircled number. The structure is typically numbered from left to right. Each element of the member is then labeled with a number enclosed in a triangle. The constraints are then removed and replaced with force arrows labeled as Rnx or Rny in the x and y axes respectively. The subscript n denotes the constraint node number of the global member.

Once all of the members, nodes, and reaction forces of the global member are fully labeled, a free body diagram for each member of the structure must then be drawn. The member labeled with its global member number within a triangle. The nodes are then labeled with a local node number inscribed within a square. The local reaction forces are then drawn in the x and y axes at each node. These internal forces are then labeled as fi(e) where the subscript i represents the degree of freedom (dof) of that node and the superscript e represents the element number. The displacement of each node is drawn just like the degree of freedom and labeled as di(e) with i representing the degree of freedom of that node and e represents the element number. The labeling nomenclature is illustrated below in a global and local view of a truss system.

[image]

Force-Displacement Relation

A simple 1-D spring element fixed at one end is used to illustrate the force-displacement (FD) relation. This relation is f=kd, where f is the force applied to the spring, k is the spring constant, and d is the displacement of the spring from its natural position. The spring element can be seen below.

[image]

The relation also holds true for a spring with two free ends as seen below.

[image]

The FD relation determined by observing the spring in two ways. The first case is where the observer is sitting on node one, which appears to be stationary. The resulting equation is f2=k(d2-d1). The second case with the observer on node 2 yields the equation f1=k(d1-d2). The matrix of f=kd for the above spring is presented below.

[image]

Steps to Solve Simple Truss Systems

1.     Global Picture

 * Structure level

 * Global dofs

 * o      Known part (fixed dof, constraints)

 * o      Unknown part (solved using finite element method)

 * Global forces

 * o      Known part (applied forces)

 * o      Unknown part (reactions)

2.     Element Picture

 * Element dof

 * Element forces

3.     Global FD relations

 * Element stiffness matrices in global coordinates

 * Element force matrices in global coordinates

 * Assembly of element stiffness and force matrices into global FD relation

4.     Elimination of known dofs to reduce the global FD relations

 * Stiffness matrix non-singular→inveritble

 * To be invertible unknown displacement dofs mus be less than known dofs

5.     Compute element forces from known element stresses

6.     Compute reactions (unknown forces)

Homework 2
Element 1 :

The element stiffness matrix multiplied by the element displacement matrix is k (1) d (1)

$$

=\left[ \begin{array}{cccc} 0.5625 & 0.32476 & -0.5625 & -0.32476 \\ 0.32476 & 0.1875 & -0.32476 & -0.1875 \\ -0.5625 & -0.32476 & 0.5625 & 0.32476 \\ -0.32476 & -0.1875 & 0.32476 & 0.1875 \end{array} \right]\begin{Bmatrix} 0 \\ 0 \\ 4.352 \\ 6.127  \end{Bmatrix}

$$

The matrices can be reduced due to the zeros in the displacement matrix and set equal to the element force matrix k (1) d (1) = f (1)

$$

=\left[ \begin{array}{cc} -0.5625 & -0.32476 \\ -0.32476 & -0.1875 \\ 0.5625 & 0.32476 \\ 0.32476 & 0.1875 \end{array} \right]\begin{Bmatrix} 4.352 \\ 6.127  \end{Bmatrix} = \left[ \begin{array}{c} -4.4378 \\ -2.5622 \\ 4.4378 \\ 2.5622 \end{array} \right]  =  \left[ \begin{array}{c} f_{1}^{(1)} \\ f_{2}^{(1)} \\ f_{3}^{(1)} \\ f_{4}^{(1)} \end{array} \right]

$$

Observation:

Element 1 is in equilibrium therefore:

(1) $$\sum F_x = f_1^{(1)} + f_3^{(1)} = 0 $$

(2) $$\sum F_y = f_2^{(1)} + f_4^{(1)} = 0 $$

(3) $$\sum M_{any pt} = 0 $$

The axial force P1 can be found using the root sum square method (RSS) below.

P1(1) = [(f1(1))2 + (f2(1))2]1/2

The RSS method can be used to verify that Element 1 is in equilibrium and the results from f (1)

-P1(1) = [(f1(1))2 + (f2(1))2]1/2 P2(1) = [(f3(1))2 + (f4(1))2]1/2

-P1(1) = [(-4.4378)2 + (-2.5622)2]1/2 = 5.124 P2(1) = [(4.4378)2 + (2.5622)2]1/2 = 5.124

P2(1) = -P1(1)

Element 2 :

The figure below shows element 2 labeled with axial forces P.

Method 2 to solve for a 2-bar truss system using statics method:

P can be brought back into the big picture by performing an equilibrium balance of node 2.

The equilibrium of global node 2 can be verified using trigonometry. P = 7 P1(1) = 5.1243 P2(2) = 6.2760

$$sum\ F_y = 7 - 5.1243sin(30) - 6.2760sin(45) = 0$$ $$sum\ F_x = 5.1243cos(30) - 6.2760cos(45) = 0 $$

Homework 3
Derivation of Element FD wrt Global Coordinate System

The element force displacement (FD) relation can be determined with respect to the global coordinate system. The following equation shows the general force displacement relation for an element member.

$$\mathbf{k}^{(e)}_{4x4}\,\mathbf{d}^{(e)}_{4x1} = \mathbf{f}^{(e)}_{4x1}$$

The local free body diagram for the element member is pictured in the figure below.

The local axial force displacement diagram for the element member is pictured in the figure below.

The following force displacement equation can is constructed from above axial force displacement diagram.

$$\mathbf{k}^{(e)}\begin{bmatrix}1 & -1\\-1 & 1\end{bmatrix}\begin{Bmatrix}q_1^{(e)}\\q_2^{(e)}\end{Bmatrix}=\begin{Bmatrix}p_1^{(e)}\\ p_2^{(e)}\end{Bmatrix}$$
 * $$ \mathbf{\hat{\; k}}^{(e)}_{2x2}\,\;\;\;\;\;\;\;\; \mathbf{q}^{(e)}_{2x1} \;\; = \;\;\;\; \mathbf{p}^{(e)}_{2x1}$$

Where:
 * qi(e) = axial displacement of element e at local node i.
 * pi(e) = axial force of element e at local node i.

Want to find the relation between $$\begin{cases} \mathbf{q}^{(e)}_{2x1} & \text{ and}\;\;\;\;\;   \mathbf{d}^{(e)}_{4x1}  \\ \mathbf{p}^{(e)}_{2x1} & \text{ and}\;\;\;\;\;  \mathbf{f}^{(e)}_{4x1} \end{cases}$$

The above relation can be expressed in the form below.

$$\mathbf{q}^{(e)}_{2x1} = \mathbf{T}^{(e)}_{2x4}\, \mathbf{d}^{(e)}_{4x1}$$

The figure below depicts the displacement vector of local node 1 as well as a portion of the element member e.

In the above figure, the displacements $$d^{(e)}_1$$ and $$d^{(e)}_2$$ in the $$\vec{i}\,$$ and $$\,\vec{j}$$ directions are shown as components of $$\mathbf{\hat{\; d}}\,$$1(e). The equation of the $$\vec{i}\,$$ and $$\,\vec{j}\,$$ components of $$\mathbf{\hat{\; d}}\,$$1(e) can be seen in the following equation.

$$\bar{d}^{(e)}_1=d^{(e)}_1\vec{i}+d^{(e)}_2\vec{j}$$

q1(e) = axial displacement of node 1 is the orthagonal projection of the displacement vector $$\bar{d}^{(e)}_1$$ of node 1 on the $$\tilde{x}$$ axis of element e. This is represented in the below equation.

$$q^{(e)}_1=\vec{d}^{(e)}_1\cdot \vec{\tilde{i}}$$

The displacement vector of node 1 can be broken down into components. $$q^{(e)}_1= (d^{(e)}_1\vec{i}+d^{(e)}_2\vec{j}) \cdot \vec{\tilde{i}}$$

Which is equivalent to. $$q^{(e)}_1= d^{(e)}_1(\vec{i}\cdot \vec{\tilde{i}})+d^{(e)}_2(\vec{j} \cdot \vec{\tilde{i}})$$
 * $$ \cos \theta _e \;\;\;\;\; \sin \theta _e$$
 * $$ l^{(e)} \;\;\;\;\;\;\;\;\;\; m^{(e)}$$

q1(e) is a 1x1 scalar which can be calculated with the displacements of node 1 and the director cosines as shown in the following equations.

$$q_1^{(e)}=l^{(e)}d_1^{(e)}+m^{(e)}d_2^{(e)}$$

$$q_1^{(e)}=\begin{bmatrix}l^{(e)} & m^{(e)} \end{bmatrix}\begin{Bmatrix}d_1^{(e)}\\ d_2^{(e)}\end{Bmatrix}$$ $$1x1 \;\;\;\;\;\;\;\;\; 1x2 \;\;\;\;\;\;\;\;\; 2x1$$

The matrices that result from combining the displacement equations for nodes 1 and 2 are listed below. $$\begin{Bmatrix}q_1^{(e)}\\q_2^{(e)}\end{Bmatrix}=\begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0\\ 0 & 0 & l^{(e)} & m^{(e)}\end{bmatrix}\begin{Bmatrix} d_1^{(e)}\\ d_2^{(e)}\\ d_3^{(e)}\\ d_4^{(e)}\end{Bmatrix}$$ $$\mathbf{q}^{(e)}=\mathbf{T}^{(e)}\mathbf{d}^{(e)}$$

Homework 4
(10/17/2008)

Equation 1 from above can be rewritten as follows: $$f_3^{(1)}+f_1^{(2)}=0\; \; \;(1a)$$

Equation 2 from above can be rewritten as follows: $$f_4^{(1)}+f_2^{(2)}=0\; \; \;(2a)$$

For Equation 1a, the forces can be written in terms of element stiffness(k) and degrees of freedom(d).$$\mathbf{F=kd}$$: $$\left[k_{31}^{(1)}d_1^{(1)} +k_{32}^{(1)}d_2^{(1)}+k_{33}^{(1)}d_3^{(1)}+k_{34}^{(1)}d_4^{(1)}\right]$$ $$+\left[k_{11}^{(2)}d_1^{(2)} +k_{12}^{(2)}d_2^{(2)}+k_{13}^{(2)}d_3^{(2)}+k_{14}^{(2)}d_4^{(2)}\right]\; \; \;(1b)$$ $$=0$$

Local Degrees of Freedom to Global Degrees of Freedom Relation: $$d_1^{(1)}=d_1$$ $$d_2^{(1)}=d_2$$ $$d_3^{(1)}=d_2^{(2)}=d_3$$ $$d_4^{(1)}=d_1^{(2)}=d_4$$ $$d_3^{(2)}=d_5$$ $$d_4^{(2)}=d_6$$

The local degrees of freedom(dof) of Equation 1b can be replaced with the global degrees of freedom using the above relations.

$$\left[k_{31}^{(1)}d_1 +k_{32}^{(1)}d_2+k_{33}^{(1)}d_3+k_{34}^{(1)}d_4\right]$$ $$+\left[k_{11}^{(2)}d_3 +k_{12}^{(2)}d_4+k_{13}^{(2)}d_5+k_{14}^{(2)}d_6\right]$$ $$=0$$

Assembly of Global Stiffness Matrix from Element Stiffness Matrices

The global stiffness matrix $$\mathbf{k}$$ can be assembled from the element stiffness matrices $$\mathbf{k^{(e)}}, e=1,...,n_{el}$$ where nel is the number of elements:

$$\mathbf{k}_{nxn}=A\mathbf{k}_{n_{el}xn_{el}}^{(e)}$$

New Nomenclature: n: total number of global degrees of freedom before elimination of boundary conditions ned: number of element degrees of freedom
 * (ned << n)

A: assembly operation
 * Principal of Virtual Work (PVW)
 * Used to eliminate the rows for corresponding boundary conditions to obtain $$\mathbf{\bar{k}}_{2x2}$$

Deriving Finite Element Method for Partial Differential Equations

The force displacement (FD) relation for a bar or spring is $$kd=F$$. This implies that $$kd-f=0\; \; \;(3)$$ Which is equivalent to $$w(kd-F)=0\; \; \;(4)$$ for all values of w.

Proof: $$(3)\Rightarrow (4)$$ trivial $$(4)\Rightarrow (3)$$ not trivial

Since Equation 4 is valid for all w, select w=1, then Equation 4 becomes: $$1\left(kd-F \right)=0\Rightarrow (3)$$

Homework 5
$$\hat{\textbf{w}}_{2x1}$$ = virtual axial displacement corresponding to $$\textbf{q}^{(e)}$$. $$\textbf{w}_{4x1}$$ = virtual displacement in global coordinate system, corresponding to $$\textbf{d}^{(e)}$$.

Substituting Equations (3) and (4) into Equation (2) yields the following Equation: $$\left(\textbf{T}^{(e)}\textbf{W} \right)\cdot \left[\mathbf{\hat{k}}^{(e)}\left(\mathbf{T}^{(e)}\mathbf{\bar{d}}^{(e)}-\mathbf{p}^{(e)} \right) \right]=0$$ for all $$\textbf{w}_{4x1} \; \; \; \; \; \; \;(5)$$

In order to solve Equation (5), a few matrix operations must be recalled:

$$\left(\textbf{AB} \right)^T=\textbf{B}^T\textbf{A}^T \; \; \; \; \; \; \; (6)$$

$$\textbf{a}_{nx1}\cdot\mathbf{b}_{nx1} = \textbf{a}^t_{1xn}\mathbf{b}_{nx1} \; \; \; \; \; \; \; (6)$$

To solve Equation (5), Equation (6) and (7) must be applied:

The dot product must be replaced with the transpose as shown below from Equation (7). $$\left(\textbf{T}^{(e)}\textbf{W} \right)^T\left[\mathbf{\hat{k}}^{(e)}\left(\mathbf{T}^{(e)}\mathbf{\bar{d}}^{(e)}-\mathbf{p}^{(e)} \right) \right]=0$$ for all $$\textbf{w}_{4x1}$$

The matrix transpose is replaced with the product of individual transposes from Equation (6). $$\textbf{W}^{T}\textbf{T}^{(e)T}\left[\mathbf{\hat{k}}^{(e)}\left(\mathbf{T}^{(e)}\mathbf{\bar{d}}^{(e)}-\mathbf{p}^{(e)} \right) \right]=0$$ for all $$\textbf{w}_{4x1}$$

After distributing the transpose matrix T the following equation is the result. $$\textbf{W}^{T}\cdot\left[\left(\textbf{T}^{(e)T}\mathbf{\hat{k}}^{(e)}\mathbf{T}^{(e)}\right)\mathbf{\bar{d}}^{(e)}-\left(\textbf{T}^{(e)T}\mathbf{p}^{(e)} \right) \right]=0$$ for all $$\textbf{w}_{4x1}$$

$$\mathbf{k}^{(e)}$$ replaced $$\textbf{T}^{(e)T}\mathbf{\hat{k}}^{(e)}\mathbf{T}^{(e)}$$, and $$\mathbf{f}^{(e)}$$ replaced $$\textbf{T}^{(e)T}\mathbf{p}^{(e)}$$.

The resulting equation is: $$\mathbf{w}\cdot\left[\mathbf{k}^{(e)}\mathbf{d}^{(e)}-\mathbf{f}^{(e)}\right]=0$$ for all $$\mathbf{w}$$

After dividing both sides by w, and moving f(e) to the other side yielded: $$\mathbf{k}^{(e)}\mathbf{d}^{(e)}=\mathbf{f}^{(e)}$$

The previous methods have all been for discrete, non-continuous cases (matrices), and the new analytic methods will be for continuous cases (PDEs).

Motivational Model Problem: An elastic bar with varying A(x) and E(x) is subject to a varying axial load (distributed), concentrated load, and inertia force (dynamic). Both the axial and concentrated loads are time dependent. The figure below depicts the problem.

Homework 6
Elastic Bar Interpolation Example

It is assumed that the displacement u(x) is for $$x_i\leq x\leq x_{i+1}$$.

The following is an example of linear interpolation of u(x) for a 2-bar truss.

The deformed shape is a straight line, which means there was an assumption of linear interpolation of displacement between the two nodes. Below is a case where there is only axial displacement, meaning zero transverse displacement.

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

$$u(x)=N_i(x)d_i+N_{i+1}(x)d_{i+1}$$ where $$N_i(x) and N_{i+1}(x)$$ are linear functions in x.

Homework 7
Motivation: The deformed shape of the truss element id the interpolation of the transverse displacement V(s), where $$s=\tilde{x}$$.

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

Integration by parts of the first term: $$\alpha \equiv \int_{0}^{L}{w(\tilde{x})}\frac{\partial^2}{\partial x^2}\left\{(EI)\frac{\partial^2v}{\partial x^2} \right\}dx$$

where, $$r(x)=\left\{(EI)\frac{\partial^2v}{\partial x^2} \right\}$$

and where, $$r'(x)=\frac{\partial}{\partial x}\left( \frac{\partial}{\partial x}\left\{(EI)\frac{\partial^2v}{\partial x^2} \right\}\right)$$

The equation α can be rewritten as follows: $$\alpha =\left[w\frac{\partial}{\partial x}\left\{(EI)\frac{\partial^2v}{\partial x^2} \right\} \right]^L_0-\int_{0}^{L}\frac{dw}{dx}\frac{\partial}{\partial x}\left\{(EI)\frac{\partial^2v}{\partial x^2} \right\}dx$$

Where, $$\beta_1 =\left[w\frac{\partial}{\partial x}\left\{(EI)\frac{\partial^2v}{\partial x^2} \right\} \right]^L_0$$

then α becomes, $$\alpha =\beta_1-\left[ \frac{dw}{dx}(EI)\frac{\partial ^2v}{\partial x^2}\right]^L_0+\int_{0}^{L}{\frac{d^2w}{dx^2}(EI)\frac{\partial ^2v}{\partial x^2}}dx$$

where, $$\beta_2=\left[ \frac{dw}{dx}(EI)\frac{\partial ^2v}{\partial x^2}\right]^L_0$$

and where, $$\gamma =\int_{0}^{L}{\frac{d^2w}{dx^2}(EI)\frac{\partial ^2v}{\partial x^2}}dx$$

The PVW equation becomes: $$-\beta_1+\beta_2-\gamma +\int_{0}^{L}{wf_tdx}-\int_{0}^{L}{wm\ddot{v}dx}=0$$ for all possible w(x).

The stiffness term γ is now the focus to derive the beam stiffness matrix and to identify the beam shape functions.

For the above beam: $$v(\tilde{x})=N_2(\tilde{x})\tilde{d}_2+N_3(\tilde{x})\tilde{d}_3+N_5(\tilde{x})\tilde{d}_5+N_6(\tilde{x})\tilde{d}_6$$

Recall: $$u(\tilde{x})=N_1(\tilde{x})\tilde{d}_1+N_4(\tilde{x})\tilde{d}_4$$

$$N_2(\tilde{x})=1-\frac{3\tilde{x}^2}{L^2}+\frac{2\tilde{x}^3}{L^3}\;\;\;\;\;\tilde{d}_2 $$

$$N_3(\tilde{x})=\tilde{x}-\frac{2\tilde{x}^2}{L}+\frac{\tilde{x}^3}{L^2}\;\;\;\;\;\tilde{d}_3 $$

$$N_5(\tilde{x})=\frac{3\tilde{x}^2}{L^2}-\frac{2\tilde{x}^3}{L^3}\;\;\;\;\;\tilde{d}_5$$

$$N_6(\tilde{x})=-\frac{\tilde{x}^2}{L}+\frac{\tilde{x}^3}{L^2}\;\;\;\;\;\tilde{d}_6$$

Recommended Software to Improve Productivity When Using Mediawiki
WikED To increase efficiency and productivity in the writing and editing of Mediawiki articles, WikED was used throughout the semester. The WikED software saved time because it inserted Mediawiki code at the click of a button. Without WikED, all Mediawiki format code would have been typed by hand, requiring much excess typing. An example of the typing savings is formatting a simple superscript. To insert a superscript, the code $$undefined$$ must be typed around the characters to be made into superscripts. With WikED, the characters only had to be highlighted and the superscript button clicked, which input the $$undefined$$ automatically. Wik Ed allowed for easy insertion of pictures, changing fonts, as well as formatting all text and symbols. WikED increased efficiency and productivity in the writing and editing of Mediawiki articles.

LaTeX Editor A free online LaTeX editor powered by Codecogs was used to generate the equations made in the homework reports throughout the semester. The website allowed for easy and efficient writing of equations through the use of buttons that generated LaTeX code. Without the LaTeX editor, all of the equation codes would have been manually typed, which would increase the amount of time to write the equaitons. The LaTeX editor is also useful, because it continually updates the typed equaiton, so all formatting changes can be seen instantly. The LaTeX editor also reduces the amount of code that must be remembered, because its buttons show equation formats, and it automatically inserts appropriate code at the click of a button. To insert LaTeX code into Mediawiki, the LaTeX button must be clicked to insert the $$$$ code, and then the code generated by Codecogs can be copied and pasted between the math command. The following hyperlink is a link to the LaTeX editor Codecogs LaTeX Editor.

Drawing Figures Figures for the homework reports were generated in Microsoft PowerPoint. The figures were drawn using the drawing tools of the program, and then when the figures were finished the image was viewed full screen in a sideshow. The print screen command was used to copy the image, which was then pasted into Microsoft paint and saved as a .jpg file. To post the images on Mediawiki, the image button was clicked and the file was uploaded with the appropriate licensing and description.

title: User:Eml4500.f08.bike.gravois


text:
User:Eml4500.f08.echo.mott/HW1draft

THE FINAL HW REPORT PAGE IS LOCATED HERE! User:Eml4500.f08.echo.mott/Echo.HW1

 * To check what we'll be graded by, go here

The NOTATION SECTION still needs pictures of how forces and displacements shall be labeled. The MATLAB TUTORIAL still needs Sparse Matrices, Formatting Output, Mfiles, Common Functions We need to site the TUTORIAL and the CLASS
 * Chris 14:58, 12 September 2008 (UTC)

Common Functions

 * Matrix Creation

The eye(m,n) command generates an identity matrix of size m-by-n. The zeros(n) command creates a n-by-n matrix with zeros and the ones command creates a matrix with ones.

 * Matrix Operation

 * Scalar

 * Vector

Here is a list of commonly used useful functions

eig  - used for eigenvalues and eigenvectors chol - for Cholesky factorization svd  - this is used for singular value decomposition inv  - this will give the inverse of a specified matrix lu   - this is for LU factorization qr   - this is for QR factorization hess - this will give the hessenberg form shur - this will provide the shur decomposition rref - this will provide the reduced row echelon form of a matrix expm - this provides the matrix exponential sqrtm - this gives the square root of a matrix poly - this will give the characteristic polynomial det  - this will give the determinant of a matrix size - returns the size of a matrix norm - provides 1- norm, 2-norm, etc. to infinity cond - condition number rank - provides rank of a matrix

title: User:Eml4500.f08.echo.mott/HW1draft


text:
User:Eml4500.f08.ateam.shah/Section9

Section 9: Matrix Functions
eig       eigenvalues and eigenvectors chol      cholesky factorization svd       singular value decomposition inv       inverse lu        LU factorization qr	       QR factorization hess      hessenberg form schur     schur decomposition rref      reduced row echelon form expm      matrix exponential sqrtm     matrix square root poly      characteristic polynomial det       determinant size      size norm      1-norm, 2-norm F-norm cond      condition number in the 2-form rank      rank

Above is a list of matrix functions that can be used. These functions can be called using a variety of commands.

Ex:

$$Y=eig(A)$$

Or

$$eig(A)$$

title: User:Eml4500.f08.ateam.shah/Section9


text:
User talk:Shayan~enwikiversity

 Hello Shayan, 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! --mikeu talk 21:01, 7 September 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 Shayan. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Shayan~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 23:59, 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) 06:22, 19 April 2015 (UTC)

title: User talk:Shayan~enwikiversity


text:
User:Eml4500.f08.bike.viale

Homework Report 1

Homework Report 2

Homework Report 3

Homework Report 4

Homework Report 5

Homework Report 6

Homework Report 7

title: User:Eml4500.f08.bike.viale


text:
User:Eml4500.f08.ateam.shah/Section17

Section 17: Hardcopy
The easiest way to safe your command file in Matlab is the diary command. The diary command, when followed by a file name will save the data under the new name. However if no file name is provided, the file will be saved under a generic name of diary. The file can be opened later for further editing or printing.

$$diary$$ $$filename$$

title: User:Eml4500.f08.ateam.shah/Section17


text:
User:Eml4500.f08.ateam.nobrega/my homework

Homework 1


text:
User:Eas4200c.f08.gator.edwards/Textbook Chapter One

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

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

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

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

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

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

A high stiffness and low toughness material is glass.

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

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

...FILL IN MORE INFO HERE


text:
User:Eml4500.f08.ateam.shah/Section13

Section 13: Text strings, error message, input
Text strings can be entered into Matlab using single quotations. These strings can be displayed by using the disp command.

Ex:  s= ‘peter pan had an accident’

disp(‘peter pan had an accident’)

Error messages that are placed in M-files can abort the program.

Ex:

error(‘Sorry, this matrix is wrong’)

Strings can also be added into an M-file to make the program interactive.

Ex:

iter = input(‘Enter the number of times you want to enter a number:’)

title: User:Eml4500.f08.ateam.shah/Section13


text:
User:Eml4500.f08.ateam.nobrega/homework 1

What is Matlab?
MATLAB is short for "matrix laboratory". It is a program software that allows users to solve numerical problems using a programming language similar to, yet simpler and faster than, Fortran or C++. It is a matrix based system that allows for quick computation and visualization of scientific and engineering problems such as, but not limited to, plotting data and functions, implementing algorithms, solving algebraic expressions and creating programs.

An Introduction to the Matlab Primer
The MATLAB Primer Tutorial was written based on the 4.0 and 4.1 version of MATLAB. However, it can be used in updated versions of MATLAB as well. The Primer was written by Kermit Sigmon, Department of Mathematics at the University of Florida. It is designed to help one use MATLAB. It is encouraged that the Primer be used hands on with the reader sitting at their computer going through the examples. Users should type the commands into the MATLAB command window and view the MATLAB generated output for each topic.

If the user is unsure of how to perform a task or if an error message arises, he or she should call on the help function. This function can be called by typing the word help into the command window. Doing this will display a list of topics for the user to choose from. To call for a specific help topic type in the word help followed by a space and the name of the topic one is searching for. Another helpful command is to type in the word demo into the command space, allowing the user to preview some of the MATLAB functions.

The MATLAB Primer is a set of instructions for how to perform basic MATLAB functions. The MATLAB User's Guide and Reference Guide will need to be accessed for more advanced applications. To find these documents consult your instructor or visit your local library or computer lab.

"MATLAB is available for a number of environments: Sun/Apollo/VAXstation/HPworkstations, VAX, MicroVAX, Gould, PC and AT compatibles, 80386 and 80486 computers, Apple Macintosh, and several parallel machines. There is a relatively inexpensive Student Edition available from Prentice Hall publishers."

Accessing MATLAB
After logging on to a computer, the easiest way to access MATLAB is to double-click the MATLAB icon on the desktop or search for MATLAB in your program list. The MATLAB icon is featured below:

*


text:
User:Eml4500.f08.ateam.nobrega/Homework Reports

Homework Report One

Homework Report Two

Homework Report Three

Homework Report Four

Homework Report Five

Homework Report Six

Homework Report Seven


text:
User talk:Eas4200c.f08.gator.edwards

 Hello Eas4200c.f08.gator.edwards, 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
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 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 21:20, 7 September 2008 (UTC)

Category Display
Hi! I just wanted to let you know that I changed your Courses category membership to a category display instead. Your user account was appearing as an available course. If you'd like to make this content available to others as a course, I recommend creating an article in the main namespace, something like EAS4200c Aerospace Structures or just Aerospace Structures. Let me know if you have any questions. -- Dave Braunschweig (discuss • contribs) 16:25, 20 July 2013 (UTC)

title: User talk:Eas4200c.f08.gator.edwards


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

 Hello Eml4500.f08.bike.viale, 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
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 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 21:20, 7 September 2008 (UTC)

source code for matlab
Check my diff in your user page for the use of |- | | Wikiversity requires that the work's copyright holder be acknowledged. The copyright holder is usually the work's creator, the creator's employer, or the last person who was transferred ownership rights. Please include this information by placing on the file page and filling it out. |- | | Please signify the copyright information for any other files you have uploaded or will upload. Remember that files without this important information or with incorrect information can be deleted by a custodian. |- | | If you have any questions, feel free to contact me on my talk page or the Wikiversity community at the Colloquium. Thank you. |}Crochet.david 15:34, 2 November 2008 (UTC)

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


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

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

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

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

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 21:21, 7 September 2008 (UTC)

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


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

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

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

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

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 21:22, 7 September 2008 (UTC)

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


text:
User talk:Eas4200c.f08.carbon.booth

 Hello Eas4200c.f08.carbon.booth, 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
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 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 21:23, 7 September 2008 (UTC)

title: User talk:Eas4200c.f08.carbon.booth


text:
User:Eas4200c.f08.wiki.c

Homework Report 1

Homework Report 2

Homework Report 3

Homework Report 4

Homework Report 5

Homework Report 6

Homework Report 7

title: User:Eas4200c.f08.wiki.c


text:
User:Eml4500.f08.ateam.shah/Week 4

Continuation of Lecture from Wednesday October 8, 2008

Note: Consider the case where d4(e) ≠ 0, where d1(e)=d2(e)=d3(e)=0

Multiplying out the equation out results in a 0 matrix as listed below:

f4(e) = k4(e) d4(e) = 0 where f is a 4x1 matrix, k is a 4x4 matrix, d is a 4x1 matrix which creates a 4x1 0 matrix.

The 0 matrix represents the 4th column of the k(e) which is the interpretation of the transverse dof's.

From period 14 slide 3: d4(e) = T4(e) d4(e)

Similarly using the same argument f4(e) = T4(e) d4(e)

Also k4(e) * d4(e) = f4(e)
 * which leads to k4(e) T4(e) d4(e) = T4(e) f4(e)

In the equation listed above the T4(e) is a block diagonal matrix

\begin{bmatrix} R_{2x2}^{(e)} & 0_{2x2}\\ 0_{2x2} & R_{2x2}^{(e)}\\ \end{bmatrix} $$

If T(e) is invertible, then [T(e)-1 k(e) T(e)]d(e) = f(e).

Consider a general block diagonal matrix with $$ A= \begin{bmatrix} D_1& & &  & 0\\ &  .& &  & &\\      &   &  .& & &\\      &   &   & .&&\\      0&   &  &  & D_5\\ \end{bmatrix} $$

Question: What is the inverse of A (A-1? Simple example: Diagonal Matrix

$$ B= \begin{bmatrix} d_{11}& & &  & 0\\ &  d_{12}& &  & &\\ &  &  .& & &\\      &   &   & .&&\\      0&   &  &  & d_{mn}\\ \end{bmatrix} = diag[d_{11}, d_{22}, .... , d_{mn}] $$

thus $$ B^{-1} = Diag \begin{bmatrix} \tfrac{1}{d_{11}},& \tfrac{1}{d_{22}},& .&  .& .& .,&\tfrac{1}{d_{mn}}\\ \end{bmatrix} $$

Assuming that all values of the diagonal coefficients are not equal to 0, i.e. dii ≠ 0 for all values of i=1......n.

For a block diagonal matrix A $$ A = Diag \begin{bmatrix} D_1,& .& .& .&, D_s \end{bmatrix} $$ and $$ A^{-1} = Diag \begin{bmatrix} D_1^{-1}, & .& .& .& .&, D_s^{-1}\\ \end{bmatrix} $$ where D1-1, is the inverse of the matrix D1.

The transport matrix is defined to be
 * T(e)-1=Diag[R(e)-1, R(e)-1]

On Lecture 19 Slide 2 R(e) is defined to be $$ R^{(e)} = \begin{bmatrix} l^{(e)} & m^{(e)}\\ -m^{(e)} & l^{(e)} \end{bmatrix} $$ $$ R^{(e)T} = \begin{bmatrix} l^{(e)} & -m^{(e)}\\ m^{(e)} & l^{(e)} \end{bmatrix} $$ Thus multiplying R(e) by its transpose will result in $$ R^{(e)T}R^{(e)} = \begin{bmatrix} l^{(e)} & -m^{(e)}\\ m^{(e)} & l^{(e)} \end{bmatrix} \begin{bmatrix} l^{(e)} & m^{(e)}\\ -m^{(e)} & l^{(e)} \end{bmatrix} = \begin{bmatrix} 1 & 0\\     0 & l     \end{bmatrix} $$ The resulting matrix represents a 2x2 I or identity matrix.

Thus,
 * R(e)-1=R(e)T
 * T(e)-1=Diag[R(e)T, R(e)T]=(Diag[R(e), R(e)])T

Also
 * T(e)-1=T(e)T

title: User:Eml4500.f08.ateam.shah/Week 4


text:
User:Eml4500.f08.group.cady

'''THIS PAGE IS OBSOLETE: PLEASE GO HERE! User:Eml4500.f08.group'''

 * Matlab is here: User:Eml4500.f08.group/Matlab
 * Homework 1 is here Eml4500.f08.group/Homework1

=Finite Element Analysis=

Course Information

Website - www.mae.ufl.edu/~vql → "teaching" → "my course website" → "EML 4500 Finite Element Anslysis and Design" EML 4500 Course website Password: fall08.wiki

MWF Period 8 (3:00-3:50PM), FLG 270 Office: NEB135

Course Content

Finite Element Analysis (FEM) makes solving partial differential equations (PDEs) convenient.

The emphasis of the course will be to understand the mechanics, formulate problems, judge solution correctness, and avoiding ad-hoc structural analysis methods.

The plan of the course is to maintain confidentiality while using large-scale peer-to-peer domains including Wikiversity and MIT's OpenCourseWare in addition to the textbook. The course will examine the method of work instead of other approaches and such as e-learning and the old approach to EML4500 (10% HW and 30% x 3 for each exam). This format allows students to be introduced to Media Wiki and stimulate collaboration between students with other students, teams, the professor, and the class as a whole.

There are several relevant time zones important to us (UTC, EDT, and EST). HW will be turned in according to the UTC time to maintain consistency with the Wikipedia time stamp format.

HW shortcuts should be created by attaching the URL from the archived version by going to "History" → click on the date to open archived version → copy this address to submit

=Chapter 4: Trusses, Beams, and Frames=

Trusses
The image below is a two-truss system with elastic (deformable) bars.

Global node numbers are denoted by circles while element numbers are denoted by triangles. The elements are separable members of a system which can be individually analyzed. Nodes are points on a system on which forces appear. The x-y coordinate system can be located in any rotation about the z-axis, however in this case the y-axis is chosen to be parallel to the applied force P. Furthermore, the triangles beneath nodes 1 and 2 denote that the node displacement is fixed to zero in both the x- and y-direction.

Forces acting on the system are shown below with a systematic example of how to name the forces on the member. The names given to each of forces are arbitrary, but must be systematic and consistent.

Forces acting on the system are shown below with a systematic example of how to name the forces on the member. The names given to each of forces are arbitrary, but must be systematic and consistent.

The R1 and R3 forces can be replaced with two forces instead of four that act along the line of the trusses. Taking the moment about node 2 equal to zero, gives no conclusion because both R1 and R3 forces run through the node. The figure then has two unknown reactions, and only two equations of equilibrium (the sum of forces about the x- and y-axes. This means that the system is statically indeterminate (equations of statics cannot be used to solve R1 and R3.

To examine the forces, the system can be split into two independent bar elements.

Bar element 1:

This picture is another example of how to name the forces, nodes and elements in a systematic manner, rather than a rigid, consistent manner. The element number is denoted by a triangle again. The local nodes (nodes relative only to the specific FBD section) are denoted by squares as opposed to the global nodes (nodes relative to the entire structure) which are circled. The subscript of the forces are the degrees of freedom (dof) or internal force of the element. Each force is a separate degree of freedom and should have their own number. Also they are numbered in a consistent order: x-, y-, x-, y-axis. The superscript represent the element number in parentheses. For this global system, the dof's are 1, 2, 3, and 4 and the element numbers are 1, 2, which come to eight total numbers in both elements.

Bar element 2:

Force Displacement (FD) Relation

The diagram above shows the relationship between force and displacement in a spring with one end fixed. It both ends are free, the case becomes as follows.

Because there are now two independent variables, the spring must be analyzed with a matrix form of F = kd. Matrices are names by "row x column"

\begin{pmatrix} f_1 \\ f_2 \end{pmatrix}_{2x1} = \begin{bmatrix} k & -k \\ -k & k \end{bmatrix}_{2x2} \begin{pmatrix} d_1 \\ d_2 \end{pmatrix}_{2x1} $$

There are 2 cases which can be examined:

Case 1 is the result with respect to node 1:

$$f_2 = k (d_2 - d_1)$$

Case 2 is the result with respect to node 2:

$$f_1 + f_2 = 0$$

→ $$f_1 = -f_2 = -k (d_2 - d_1) = k (d_1 - d_2)$$

Steps to solve simple truss system
Steps to solve simple truss system described on Wed, 3 Sep 08. (Recipe)

Global picture
(description) at structure level
 * Global degrees of freedom (disp coeffs)
 * Global Forces

Actually the displacement degrees of freedom are partitioned into Simularly for the global forces
 * A known part, eq, fixed defs, constraints
 * an unknown part, solved using FEA
 * a know part: applied force
 * an unknown part: reactions

Element picture
either in global coord system on in local coord system
 * element degrees of freedom
 * element forces

Global Force Displacement relationship

 * Elemenet Stiffness matrix in global coord
 * element force matrices in global coord.
 * Assembly of element stiffness matrix and element force matrix into a global Force Displacement relationship

Elimination of known degrees of freedom
Elimination of known degrees of freedom to reduce the global Force Displacement Relationship (stiffness matrix non-singular => invertible)
 * m = number of unknown displacement degrees of freedom
 * n = number of known and unknown and displacement degrees of freedom

Kbar non singular => Kbar^-1   (Kbar invertible)

Compute Element forces
Compute Element forces from now known dbar => element stresses

Compute Reactions
Compute reactions (unknown forces)

Example
Take a specific example to see how the recipe works (no justification yet)

Element Length - L(1) = 4, L(2) = 2

Youngs Modulus - E(1) = 3, E(2) = 5

Cross Section Area - A(1) = 1, A(2) = 2

(Had to leave class here)

=Mon, 8 Sep 08=

2.) Element Picture

3.) Global Force Displacement at element level

k is the element stiffness matrix for element e

d is the the element displacement matrix

f is the element force matrix

Matrix p 225 book, 2nd equation form the bottom

axial stiffness of bar elem "e". e = 1,2

l^(e), m^(e) = director cosines of ~x axis (goes from [1] to [2]) with respect to global (x,y) coords

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!Contribution title, from team group
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=MATLAB Primer=

Accessing MATLAB
MATLAB can be accessed in a variety of ways. For example, most systems can access MATLAB by using the “matlab” system command. Similarly, MATLAB can be shut down using the internal MATLAB command of “quit” or “exit”. Since a MS Windows operation system is being used, MATLAB is accessed through the Windows start menu and closed via the red “X” icon in the upper right corner of the open MATLAB window.

Again, since Windows is being used, it will be convenient to keep both MATLAB and a text editor open in separate windows. Reasons for this will be discussed in section 14 (Managing M-files).

Entering matrices
MATLAB functions via the use of rectangular matrices. Numbers, vectors, and variables can all be represented as a matrix with numbers coming in the form of a 1x1 matrix, vectors as 1xN / Nx1, and variable as an NxN matrix (N = 1,2,3…). Matrices can be generation by built in MATLAB functions, loaded from document files or external data files, and be entered directly as a list of elements. For example, typing

A = [1 2 3; 4 5 6; 7 8 9]

into MATLAB generates a 3x3 matrix of numbers assigned to variable A.

MATLAB also accommodates the use of imaginary numbers in calculations. Complex matrices can be entering in one of two ways. Entering A = [1 2; 3 4] + i*[5 6; 7 8] or                                                                    A = [1+5i 2+6i; 3+7i 4+8i]

will generate the same complex matrix. The variable i and j denote complex values in MATLAB. New complex variable may be generated by entering ii = sqrt(-1)

into MATLAB. This will assign the variable ii equal to an imaginary number thus allowing for its use as a complex variable.

Loading large arrays into MATLAB is best accomplished by utilizing as ASCII file from a text editor. This file should consist of a rectangular array of matrix values. This file can then be loaded into MATLAB with the command “load file.ext” where file is the name of the file and .ext if the file extension.

Three built in MATALB commands can be used to generate matrices. Entering rand(n) or rand (m,n) will create a (n x n) or (m x n) matrix of random numbers. The magic(n) command will create a (n xn ) matrix in which all columns, rows, and diagonals have the same sum. Finally hilb(n) will create a (n x n) Hilbert matrix.

Individual matrix numbers can be accessed by entering A(2,3)). This command will find the numerical value assigned to the second row of column three in matrix A.

Matrix operations, array operation
There are several matrix operations that are available for use in MATALB and they apply to both matrix and scalar operations. Should there be incompatibilities in the size of the matrices to be operated on, an error message will result. The following are available matrix operators. +	addition -	subtraction *	multiplication ^	power ‘	conjugate transpose \	left division	(where x = A \ b is the solution to A * x = b)                             /	right division	(where x = b / A is the solution to x * A = b)

Some of the above matrix operations do not operate entry-wise. The ones that do not (*, ^, /, and \) can be made to operate entry-wise by preceding them with a period.

Example: [1,2,3,4] .* [1,2,3,4] or [1,2,3,4].^2 = [1,4,9,16]

Statements, expressions, variables; saving a session
MATLAB is an expression based language where expressions are composed of operators, and variable names. These expression usually take the form of “variable = expression.” The resulting matrix is then displayed on the screen and assigned a variable. Should no variable be assigned by the used, a standard variable of ans (answer) is assigned to the matrix.

Statements usually terminate at the end of a single line. In the case that a longer statement is necessary, three periods (…) followed by pressing return will continue the statement onto the next line. Conversely, several statements can be placed on a single line of code provided that each expression is separated by commas or semicolons. By placing a semicolon at the end of a line, the expression is evaluated but not printed to the screen.

It is important to note that MATLAB is case sensitive i.e. the variable “A” is not the same as variable “a”. Should it be necessary, typing the command “who” will list are current variable in the workspace and any variable may be removed by typing “clear” followed by the variable name that the user wishes to clear. Infinite loops can be broken by pressing “ctrl-break” on the keyboard

All data and variable names are lost whenever MATLAB is exited unless the save command is used. Using this command will restore the workspace to its previous version the next time MATALB is opened.

For Loop
For loop syntax example, to produce a certain n-vector for a given n:

x = []; for i=1:n, x=[x,i^2], end

or can be written:

x = []; for i= 1:n x = [x, i^2] end

 * Note: x = [] creates an empty matrix

The for statement permits any matrix to be used instead of 1:n. The variable consecutively assumes the value of each column of the matrix. The following example shows this syntax as it sums all the entries of matrix A by adding its column sums. s = 0; for c = A s = s + sum(c); end

While loop
General form:

while relation statements end

The statements will be repeated continuously while the relation is true. Once the relation is no longer true the loop ends.

An example for a given number a, will compute and display the smallest nonnegative integer n, such that 2^n >= a:

n = 0; while 2^n < a 	n = n + 1; end

 * Note: The semicolon after the statement keeps the values of n from printing on screen until after the loop is over and then the program calls to print n on the screen.

If statement
General form:

if relation statements end

The statements are executed only if the relation is true.

Multiple branching is also possible as shown in the following example:

if n < 0 parity = 0; elseif rem(n,2) == 0 parity = 2; else parity = 1; end

Relations
The relational operators in MATLAB are:

 * Note: = is used in an assignment statement while == is used in a relation

Relations may be connected or qualified by the following logical operators:

When applied to scalars a relation is just the scalar 1 (if true) or 0 (if false). Similarly for matrices of the same size a relation is a matrix of 1’s or 0’s. When a relation between matrices is interpreted by a while or if to be true each entry of the relation matrix must be nonzero.

Examples:
To execute statement 	when matrices A and B are equal:

if A == B        statement end

To execute statement when A and B are not equal:

if any(any(A ~=B)) statement end

Or

if A == B else statement end

 * Note: The above example does not work with the syntax:

if A ~= B, statement, end

This will not give the intended result because statement would only execute if each of the corresponding entries of A and B differ.

Scalar functions
A scalar function is one whose range is one dimensional and is made up of one or more variables. In MATLAB scalar functions generally operate on scalars, but will operate element-wise on matrices. some of the more common functions are:

Vector functions
A vector function is one whose range is three or n-dimensional. In MATLAB these functions generally operate on a vector. When applied to matrices in a column-by-column fashion to give you a row vector. If row-by-row action is desired it can be generated using the transpose. Some of the more common of these functions are:

Command line editing and recall
Editing the command line is very simple in MATLAB. Positioning of the cursor can be done with the left and right arrows. Backspace and Delete can be used to remove characters to the left and right of the cursor respectively. Edits can also be made to previous commands by using the up and down arrows to scroll through the commands to find the one you wish to edit and execute again. These edits work best for small projects and revisions.

Submatrices and colon notation
Colon notation is used to produce vectors and to reference submatrices. Colon Notation in addition to subscripting by key vectors are necessary to effectively manipulate matrices. They can also be used in place of loops to make the code more readable and simple. The formula for the colons is initial value: increment: terminating number. An example of this is: The Incrementing value can also be left out, causing the default of one o be used, such as: Colon Notation can be used to obtain submatrices of a matrix. An example of this is:

M-files: script files, function files
“M-files” are an integral part of MATLAB as they are files which can execute a sequence of statements stored in diskfiles – they are designated by the extension “.m” appended to the end of the filename. “M-files” can be broken down into two distinct forms: script files and function files.

Script files:

Script files are “M-files” which can contain a number MATLAB commands which can be executed by calling the script file by name. The advantage of a script file is evident; a single MATLAB command will actually execute a series of previously developed commands. Script files can be used to easily input data into a function, such as a matrix, which can help reduce entry errors. It is important to note that script file variables are global; and, the effect of the global variables contained in script files is that variables of the same name in the current MATLAB window will be altered by running the script command.

Function files:

Function files are another type of “M-file” similar to a script file, but allowing for (in certain ways) more control over how MATLAB handles an occurrence of this file type. Function files allow users to create their own functions in MATLAB, which can be used as any ordinary MATLAB function would (such as how “sqrt” is a MATLAB function that takes the square root of a number or of the value of a variable). A key difference between function files and script files is that variables in function files can be made either global or local – meaning that local variables can be created that pertain only to the function file (and thus a variable with the same name in a current MATLAB session would be unaffected by a call to the function file).

17 Hard copy
Command:  filename

This command pulls up the hardcopy and copies the content excluding graphics to that file, and will go to a default file if no filename is given. It continues to copy to the file until you command it to  or.

18 Graphics
MATLAB plots curves, 3D plots of curves , #D mesh surface plots , and 3D faceted surface plots. allows you to preview these features.

Planar plots
The command  creates an x-y plot as shown below the format for the range is 'lower rang:meshsize: upper range'.

Figures can be suppressed or exposed as the "current" figure by the command according to its figure number, and command   will give you the number of the current figure.

To make graphs, first put a function in an M-file then command it open with a domain:

Parametric graphs can also be defined as shown:

To enter text in the graph, follow these commands:

title of graph x-axis label y-axis label text on graph (placement defined by mouse) text position at specific coordinates places grid on the graph axis limits makes standard axis for all graphs auto-scaling shows current scaling for vector v puts same scale on both axes puts same scale and tic marks on both axes turns off both axis scaling and tic marks turns on both axis scaling and tic marks

Put the  commands after the   command.

To make multiple plots on a graph the format may be one of the two as follows

You can also command  to freeze the current plot and from there superimpose another plot on the same graph. This can be turned off with. To make line types use the codes: solid(-), dashed (--), dotted, dashdot (-.). To make marktypes, use the codes: point(.), plus (+), star (*), circle (o), x-mark (x). To make colors, use the codes: yellow(y), magent(m), cyan(c), red(r), green(g), blue(b), white(w), black(k). An example is a green dotted line:.

Several other features are:

Graphics hardcopy
To send the graphics to the printer, command, while   filename saves the figure to the specified format or to a default format if not specified.

3-D line plots
An example format of a 3-D line plot is as follows:

A 3-D graph has all the same commands plus.

3-D mesh and surface plots
3-D mesh plots shows a plots of matrix z with the command. A surface plot is controlled by. To obtain a mesh graph of a function over a rectangle, an example is as follows:

The shading can be  and are entered after the   command. Color profiles may be:.

Handle Graphics
To see a more complete list of the options to control a graph enter  and.

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User:Eml4500.f08.ateam

About Us
The A-Team-sean is a group of students working together to create the Homework Reports posted here. This page will be updated throughout the Fall 2008 semester.

Course: EML4500 Finite Element Analysis and Design

Professor: Loc Vu-Quoc

Institution: University of Florida

Team Members

 * Daniel DiDomenico
 * Sean Miller
 * Mark Barry
 * Ian Slevinski
 * Jil Paquette
 * Matt Philbin

title: User:Eml4500.f08.ateam