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User:Eml4500.f08.ateam/hw 1/trusses

EML 4500 Notes:
Trusses are structural systems consisting of long slender members. The members are arranged so that they are loaded axially and do not experience bending or shear forces. For more information on trusses, please refer to the Truss article on Wikipedia.

Global Picture
Before trusses are analyzed with the Matrix Method or any other method, it is important to develop a standardized system for labeling and drawing the systems. First, the truss system is drawn and the components are labeled. Global nodes are labeled with a number circumscribed in a circle and elements are labeled with a number circumscribed in a triangle.

Global Free Body Diagram
The next step in the process is to draw the global free body diagram (FBD). The global FBD is a diagram of the whole structure that shows the applied forces, as well as the reaction forces. The FBD includes known and reactionary forces represented by arrows. The reaction forces are labeled Rna, where a denotes the direction of the force in reference to the global coordinate system and n represents the global node number. For example, R1X represents the reaction force R applied along the X axis at node 1.

Element Free Body Diagrams
After the global FBD is created, an element FBD must be created. The element FBD is a diagram of each element of the whole structure that includes known, reaction, and internal forces applied to that element. The elemental diagram is labeled with the element number as stated above. Each node of the element is labeled with its global node number as stated above, as well as a local node number that is a number circumscribed in a square. Each internal force of the element is labeled fi(e), where i is the force number and e is the element number. Force numbers are assigned first by local node number and then orientation related to the coordinate system. For example, a force f1(1) is the first degree of freedom (DOF) in element one.

Force-Displacement Relation
The next "big step" in applying the matrix method to a truss problem is to develop a relation between the applied force and the resulting displacement. The deformation of an elastic truss member can be modeled as the deformation of a spring. Therefore, mass-spring relations will be examined.

Spring with One End Constrained
First, consider a one dimensional spring with one end fixed. The force-displacement relation for this system is represented by the equation $$f=kd$$, where f is the force, d is the displacement, and k is the spring constant that it a property of the spring. This relation shows that there is linear relationship between the applied load and the resulting displacement.

Spring with Both Ends Free
Another spring system to consider is a one dimensional spring with both ends free. This system has two degrees of freedom so there is a system of two equations that govern the relationship between the applied force and the resulting displacement. Consider the spring system shown in the figure. Force 1, F1, acts at node 1, causing a displacement, d1 and force 2 acts at node 2, resulting in a displacement d2. The equations of motion for this system are developed for two cases. For the first case, consider an observer sitting on node 1. The resulting equation of motion is $$f_2 = k(d_2-d_1)$$. The second case is when an observer sits on node 2, resulting in the equation $$f_1 = k(d_1-d_2)$$. Therefore, the resulting force-displacement relation for this system is represented by the system of equations below.

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

Steps to solve simple truss system described previously
1. Global picture

The global picture consists of a diagram of the entire structure detailing all of the nodes and nodal forces relevant to the problem. A global coordinate system is selected. At the structure level there are two components to be concerned about. These are the global degrees of freedom, or displacement degrees of freedom, and the global forces. Both are divided into known parts and unknown parts. In the previous truss example, the known parts consist of the applied force and the fixed degrees of freedom for nodes 1 and 3. These two nodes are fixed, therefore the degrees of freedom for each is constrained to zero. The unknown part is solved using the Finite Element Method (FEM). FEM is a numerical method used to obtain solutions to ordinary and partial differential equations. In the previous truss system, the reactions forces are unknown.

2. Element picture

An element picture is made of each individual part of the structure. These elements are considered "free" and a free-body diagram with a local coordinate system is made for each. Included in the element picture are both the element degrees of freedom and the element forces. These can be described either in the global coordinate system or the local coordinate system.
 * Element degrees of freedom: $$d_1,\ d_2,\ d_3,\ d_4,$$
 * Element forces: $$f_1,\ f_2,\ f_3,\ f_4$$

3. Global force-displacement relationship

The global force-displacement relationship is shown through matrices. These matrices consist of:
 * Element stiffness matrix in global coordinates ($$k$$) . The dimensions of this matrix are $$n$$ x $$n$$ where $$n$$ is the number of both known and unknown displacement degrees of freedom.
 * Element displacement matrix in global coordinates ( $$d$$) . The dimensions of this matrix are $$n$$ x $$1$$.
 * Element force matrix in global coordinates ($$f$$) . The dimensions of this matrix are $$n$$ x $$1$$.

To assemble the element stiffness matrices, the element displacement matrices, and the element force matrices into global force-displacement relationship matrix multiplication is applied in the form of:

 * $$kd= f$$

This is considered a free-free system where $$k$$ is singular, meaning $$k$$ is not invertible, or $$k^{-1}$$ does not exist.

4. Elimination of known degrees of freedom

In order to make the method more manageable, the global force-displacement relationship is reduced by eliminating the known degrees of freedom. This makes the stiffness matrix non-singular, or invertible. The form is:

 * $$KD= F$$

$$K$$ is the new stiffness matrix with dimensions $$m$$ x $$m$$, where $$m$$ is the number of unknown displacement degrees of freedom. Therefore, $$m$$ < $$n$$. Similarly, $$D$$ is the new displacement matrix with dimensions $$m$$ x $$1$$ and $$F$$ is the new force matrix with dimensions $$m$$ x $$1$$.

To compute the displacement, the equation is rearranged:

 * $$D = K^{-1}F$$

5. Compute element forces

Solving for $$D$$ above, the element stresses can now be determined.

6. Compute reactions

From the element stresses, the reaction forces can be determined.

Example: Truss problem

 Data

Element length:

L(1) = 4

L(2) = 2

Young's Modulus:

E(1) = 3

E(2) = 5

Cross section area:

A(1) = 1

A(2) = 2

Inclination angle:

θ(1) = 30°

θ(2) = -45°

1. Global picture

Global degrees of freedom: $$d_1,\ d_2,\ d_3,\ d_4,\ d_5,\ d_6$$

n = global node number

Note: When numbering the displacement degrees of freedom, it is useful to follow the order of the global node number. For each individual node, number the displacement degrees of freedom in the order of the global coordinate axes.

Global forces: $$f_1,\ f_2,\ f_3,\ f_4,\ f_5,\ f_6$$

Global force matrix

 * $$\begin{Bmatrix} F_1 \\ F_2 \\ F_3 \\ F_4 \\ F_5 \\ F_6 \end{Bmatrix}$$

Global stiffness matrix

 * $$\begin{bmatrix}

\; \; K & -K & \; \; K & -K & \; \; K & -K \\ -K & \; \; K & -K & \; \; K & -K & \; \; K \\ \; \; K & -K & \; \; K & -K & \; \; K & -K \\ -K & \; \; K & -K & \; \; K & -K & \; \; K \\ \; \; K & -K & \; \; K & -K & \; \; K & -K \\ -K & \; \; K & -K & \; \; K & -K & \; \; K \end{bmatrix} $$

Global displacement matrix

 * $$\begin{Bmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5 \\ d_6 \end{Bmatrix}$$

To solve for the unknowns, the matrices are arranged according to the equation above.

 * $$\begin{Bmatrix} F_1 \\ F_2 \\ F_3 \\ F_4 \\ F_5 \\ F_6 \end{Bmatrix} = \begin{bmatrix}

\; \; K & -K & \; \; K & -K & \; \; K & -K \\ -K & \; \; K & -K & \; \; K & -K & \; \; K \\ \; \; K & -K & \; \; K & -K & \; \; K & -K \\ -K & \; \; K & -K & \; \; K & -K & \; \; K \\ \; \; K & -K & \; \; K & -K & \; \; K & -K \\ -K & \; \; K & -K & \; \; K & -K & \; \; K \end{bmatrix} \begin{Bmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5 \\ d_6 \end{Bmatrix}$$

title: User:Eml4500.f08.ateam/hw 1/trusses


text:
User:Eml4500.f08.ateam/hw 1/tutorial

MATLAB Tutorial: Part1

MATLAB Tutorial: Part 2

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

title: User:Eml4500.f08.ateam/hw 1/tutorial


text:
User:Eml4500.f08.wiki.hood

 See my comments below. Eml4500.f08 11:03, 21 September 2008 (UTC)

= EML 4500: An Introduction =

The University of Florida course EML4500 Finite Element Analysis and Design (FEAD) will emphasize an understanding of the Finite Element Analysis method and its applications when working with structural, solid, and bio-mechanics. During this course we will work with 3-D truss systems, beams and solid structures. To help solve these systems, we will use matrix algebra and its application to the finite element model, along with a finite element codes derived in MATLAB to provide for a complete analysis of the truss and beam structures. Fundamental concepts of Liner Algebra are also taught along with coordinate transformations. The Finite Element Method (FEM) will be employed in order to solve the Partial Differential Equations that govern aerospace structure mechanics. FEM is also widely used to solve other challenges in engineering and applied math. A specific example is the use of FEM to aid in the exploration of oil reservoirs.

Course Information
Throughout the semester there will be many resources implemented that will encourage learning through lecture notes, assigned readings, cooperative learning assignments as well as in class exams.

Exam Guidelines
= Chapters 1 & 4 - Methods for Solving Simple Trusses using the Matrix Method =

Developing a "Recipe" to Simplify Problem Solving
The following sections will help in developing a recipe that will help us solve Truss problems with the use of FEA requirements and notation. Learning this scheme now will serve as an introduction to the remainder of the course.

Statically Determinate: Problem 1.2
The following is an example of a two member truss system which is known to be statically determinate. Given the specified information, and since it is statically determinate, this problem can be setup and solved using simple statics to determine any unknown forces, and moments. Solving this problem will also provide us with further exposure to FEAD notation in problem solving.

Figure 7: Two Member Truss System

 * This problem can be solved using the knowledge of two forces members. It simplifies the unknowns so we have the same number of equations and unknowns. This means that we would not have to use the Finite Element Method (FEM). This problem can be solved by simply using a statics method learned previously. The system will generate forces that act along the members themselves which will reduce unknowns

Figure 8: Global Forces of Two Member Truss System

 * The global forces are shown here in the global coordinate system. We see both the applied forces and the reactions forces. We see the additional unknowns (reaction forces) added to the system by removing the constraints of the ground connections.

Figure 9: First Element Forces

 * The element is broken down and the element forces are displayed. With this we can now determine the magnitudes and direction of the combined forces. We see here that at the pin connection we display the reaction forces created by the separation of the two elements. With the two force member method we can solve the system

Figure 10: Second Element Forces

 * The second element is broken down and the element forces are displayed. With this we can now determine the magnitudes and direction of the combined forces. We see here that at the pin connection we display the reaction forces created by the separation of the two elements. With the two force member method we can solve the system

Statically Indeterminate: Problem 1.3
For this problem we will use the "recipe" to explain why this truss system of members is considered statically indeterminate.

Figure 11: Statically Indeterminate Examples

 * The two systems shown above are statically indeterminate using the standard static methods. They must be solved using Finite Element Method (FEM).

 * The system to the left will generate 5 equations with 6 unknowns. That qualifies it as statically indeterminate by the standard statics method. We can solve for the reaction forces in the truss system. This forces use to use the "recipe" to solve the system.
 * The system to the right will generate 3 equations with 6 unknowns. This once again forces us to use the "recipe" to solve for the system.

 Error 2: No, please review the lecture presentation on this topic: Using statics, there are only 3 unknowns (due to 2-force bodies) with only 2 usable equations (the moment equation is useless). Also review the lecture presentation on "Euler cut principle". Eml4500.f08 10:59, 21 September 2008 (UTC)

Figure 12: Global Force Breakdown of an Indeterminate System

Working with Element Equations, Solutions and Plane Trusses,
= Matlab Primer =

Introduction
The matlab primer helps introduce a beginner to Matlab. Matlab is a powerful programming language that is useful in math applications.

1.Accessing MATLAB
MATLAB can be accessed by typing in "matlab", or an icon maybe available on most computers.

2.Entering Matrices
MATLAB uses matrices as the basis of operation. All variables (elements) are represented as matrices. There are several methods used to enter matrices into MATLAB: Example:

>>A=[1 2 3; 4 5 6; 7 8 9] A = 1    2     3   4     5     6   7     8     9  Complex Matrices A=[1 2;3 4]+i*[5 6;7 8] or A=[1+5i 2+6i; 3+7i 4+8i] MATLAB automatically recognizes i or j as imaginary numbers unless they are defined differently A new imaginary variable can be created if assigned as follows:  ii=sqrt(-1)

The best way to enter large matrices is to create an extension file '.ext' in the editor. The data is opened in the command window by typing the name of the file followed by '.ext'. If a data file is named 'data' it can be opened by typing 'data.ext' into the command window. Additional matrix generating functions are 'rand, magic, and hilb'. Individual entries of a matrix can be referenced. For example, the matrix A shown above can have the value '4' from row 2 column 1 referenced by typing in A(2,1).

3. Matrix operations, array operations
Common math operations can be performed on matrices using the following syntax: + addition - subtraction ^ power ' conjugate transpose \ left division / right division Array operations The operations above can be performed on entry level values within a matrix by following the entered array with a period (.).For example: >>[1 2 3 4].*[1 2 3 4] ans 1 4 9 16 >>
 * multiplication

4. Statements, expressions, and variables, saving a session
MATLAB reads expressions in the form of variables, functions, and operations. Typically a variable is assigned an expression in the format: 'variable = expression'. These statements are usually terminated with the carriage return key. Expressions can be carried over to another line using three consecutive periods in a row (...). Several expressions can be placed on one line using semicolons between expressions All variables assigned in the MATLAB command window are lost when the program is closed. To prevent the loss of these variables type the command 'save' and the variables will be saved to a file named 'matlab.mat'. To reload these variables type 'load' the next time these variables are to be used.

5. Matrix building functions
There are many types of matrix building functions. Some examples of the uses of the zeros, diag, and eye functions are shown below.

6. For, while,- and relations
Example of a 'For' statement:

Example of a 'While' statement: Examples of a 'If' statements: 1. simple 'If' statement 2. Use of relation operators: <, >, <=, >=, ==, ~=

7. Scalar Functions
Some functions such as scalar functions apply element-wise when applied to a matrix. Some examples of these functions are: asin, acos, atan, exp, log, rem, abs, sqrt, etc.

8. Vector Functions
Other MATLAB functions can perform row or column calculations acting in a column-by-column fashion to produce a row vector of results. Row-by-row calculations can be obtained by using the transpose function. Functions: (Use the help for more information) Example: Code: Output:
 * max
 * min
 * sort
 * sum
 * prod
 * median
 * mean
 * std
 * any
 * all

9. Matrix Functions
MATLAB aslso has matrix functions with single or multiple outputs. Functions:
 * eig - Eigenvalues and EigenVectors
 * chol - Cholesky Factorization
 * svd - Singular Value Decomposition
 * inv - Inverse
 * lu - LU Factoriztion
 * qr - QA Factorization
 * hess - Hessenberg Form
 * schur - Schur Decomposition
 * rref - Reduced Row Echelon Form
 * expm - Matrix Exponential
 * sqrtm - Matrix Square Root
 * poly - Characteristic Polynomial
 * det - Determinate
 * size - Size
 * norm - N-Norm
 * cond - Condition Number in the 2-norm
 * rank - Rank

Example: Code: Output:

10. Command Line Editing and Recall
The command line in MATLAB can be editing using typical word processor methods. Type help cedit or cedit for more information. You can recall previous command lines by using the up/down arrow keys. For example, you can compair sine waves with different frequencies in the command line.

Example: Code: Output:

11. Submatrices and Colon Notation
Vectors and submatrices are used to calculate complex data. "Colon notation" is key to efficiently manipulate these objects. It is higly recommended that you are familiar with them The expresstion "1:5" is actaully a row vector of "[1 2 3 4 5]". The numbers nor the increment do not have to be integers. For example, "0.2:0.2:1.2" gives "[0.2 0.4 0.6 0.8 1.0 1.2]" and "5:-1:1" gives "[5 4 3 2 1]". Example: Code: Output: Since "sin" operates entry-wise, it produces a vector y from the vector x. The colon notation can be used to access parts of the matrix. "A(1:4,3)" is a column vector consisting of the first four values of the third column. A colon by itself means an entire row. "A(:,3) is the third column of A, and "A(1:4,:) is the first four rows. Arbitrary intergral vectors can be used as subscripts. "A(:,[2 4])" is a column with the columns 2 and 4 of A. Subscripting can be used on both side of the "=". "A(:,[2 3 4]) = B(:,1:3)" repaces columns 2,4,5 of A with the first three columns of B. Matrices can also be multiplied using subscripting. "A(:,[2,4]) = A(:,[2,4])*[1 2;3 4]" multiplies columns 2 and 4 of A by the 2x2 matrix. During these operations, The altered A matrix is permanently written. Example: Code: Output:

12. M-files
MATLAB can execute lines of code stored in files with the file type ".m". They can be editied using the MATLAB

Editor or any other local text editor. There are two types of M-files: script files and function

files Script Files Script files are files with normal MATLAB codes. If the file has the filename, "test.m", and the MATLAB command

"test" in inputed, all the code in "test.m" is executed. Function files Function files extend the use of MATLAB by allowing you to create your own MATLAB functions. The variables in the

function file can either be declared local or global. Type help global for more information. Example: Code: This should be placed in filename randint.m, which corresponds to the function name. MATLAB statements like

"z = randint(4,5)" can be used. Functions can also have multiple outputs. Code: This should be placed in filename stat.m. MATLAB statements like "[xm, xd] = stat(x)" can be used. Single

assignments can also be used. "xm = stat(x)" will assign the mean of x to xm. The "%" symbol is used to add comments to your code. MATLAB will not run or recognise and input with the "%" in

front of it. The following code is an example of an advanced function. Some MATLAB functions are built in, while other are in M-files.

13. Text Strings, Error Messages, Input
Text strings can be entered in MATLAB. They can either be assigned to a varibale, displayed as a message or error, or used to promt the user to input a value. The code shows them respectively. Code:

14. Managing M-files
Any system command beginning with ! is executed without leaving MATLAB. The !-feature can be used to create or edit an M-file with the local editor then return to MATLAB without deactivating it (which would delete all variables). For example, the code !-ed rotate.m with allow you to edit rotate.m using the local editor, then return back to MATLAB.

On systems allowing multiple processes, such as Unix or MS Windows, it may be easiest to keep both MATLAB and the local editor active while working.

Debugging tools are available. The command help dbtype provides more information.

Some system commands do not require the use of an !:
 * pwd - returns the name of the present working directory
 * cd - changes the working directory
 * dir or ls - lists the contents of the working directory
 * what - lists only M-files from the directory
 * delete - deletes the specified diskfile
 * type - prints the specified M-file to the screen

M-files must be stored in a directory that is directly accessible by MATLAB. Most mainframe or workstation network installations place personal M-files in a subdirectory of the home directory named matlab. The command path displays a current list of directories obtained by MATLAB's search path. See help path for more details.

15. Comparing efficiency of algorithms: flops, tic and toc
The command flops keeps a running total of the number of flops performed. The command flops(0) resets this count to zero. Thus, to count the number of flops in an algorithm, enter flops(0) immediately before and flops immediately after the algorithm is performed. NOTE: the command flops is a floating point operation count that is now obsolete. Earlier version of MATLAB included it, but the incorporation of the function LAPACK in MATLAB 6, it no longer had a practical use.

Stopwatch timers tic and toc calculate the elapsed time from an execution. For example, if A is a matrix [n x n] and b is a vector [n x 1], the following code will display the time for solving the linear system x=A/b:

Code: Display:

Another useful feature is comparing the execution time of two different statements solving the same equation. For example, x=A/b vs. x=inv(A)*b.

On timesharing machines, elapsed time may not be extremely accurate since other running processes may slow down execution time.

16. Output Format
All MATLAB computations are performed in double precision. However, the following commands control the output format:

 * format short - fixed with 4 decimal places (default)
 * format long - fixed with 14 decimal places
 * format short e - scientific notation with 4 decimal places
 * format long e - scientific notation with 4 decimal places
 * format rat - approximation by ratio of small integers
 * format hex - hexadecimal format
 * format bank - fixed dollars and cents
 * format + - +,-,blank
 * format compact - supresses most blank lines; allows more information to fit on the page
 * format loose - returns to non-compact format

The last two are independent of the other format commands. The new format will then ensue until changed.

17. Hardcopy
The command diary filename sends what follows on the screen (excluding graphics) to the specified diskfile. If no name is given, it is written to a default file named "diary". The command diary off stops the recording; disk on turns it back on. When finished writing to the file, it can be printed from the system.

The "!-feature" also allows the editing and printing of code without leaving MATLAB (refer to section 14).

18. Graphics
Several types of graphics can be produced by MATLAB, including:
 * planar plots (primary command: plot)
 * 3-D plots (primary command: plot3)
 * 3-D mesh surface plots (primary command: mesh)
 * 3-D faced surface plots (primary command: surf)

It may be useful to keep the graphics window- "Figure"- open and moved to the side while keeping the command window active. It is possible to have multiple graphics windows open at a time; one will be the "current" figure (where subsequent commands are plotted). To change which figure is "current", one may type figure(X) [X being any number]. This will open the Xth, and now current, figure. he command gcf returns the number of the current figure.

Graphics Hardcopy

The command print will print a high-resolution copy of the current figure to the default printer. The M-file printopt specifies the default setting used by the print command, and can be edited to change the printing defaults.

The command print filename saves the current figure to filename in the default file format. If filename has no extension, a suitable extension will be added (ie: .ps, .eps, or .jet).

The following information, or the command demo, may be used to preview the graphics options.

Planar Plots

The command plot forms linear x-y plots. Assuming x and y are vectors of the same length, plot(x,y) graphs a 2-dimensional x-y plot of x vs. y in a new window. (See Code 1 and Code 2).

The command fplot can be used to plot a defined function, including from an M-file. (See Code 3).

Graphics can be formatted with the following commands:
 * title - graph title
 * xlabel - x-axis label
 * ylabel - y-axis label
 * gtext - place text on graph using mouse
 * text - position text on graph at specified coordinates
 * grid - places grid lines on the plot

The axes are auto-scaled, but can be overwritten using the command axis, which has the following uses: (should be given AFTER the plot command)
 * axis([xmin,xmax,ymin,ymmax]) - scales axes to these limits
 * axis(axis) - freezes scaling of axes for following graphs
 * axis auto - returns to default axis scaling
 * v = axis - returns vector v showing current scaling
 * axis square - sets same scale on both axes
 * axis equal - sets same scale and tic marks on both axes
 * axis off - turns off axis scaling an tic marks
 * axis on - turns on axis scaling an tic marks

More than one plot can be represented on a single graph. (See Code 5a and 5b). The command hold accomplishes the same thing by freezing the current plot on the graph and layer subsequent plots on top of it. Entering hold off will end this command.

The default linetypes, marktypes, and colors can be reformatted. (See Code 6 and Code 7). Use the following guidelines:
 * Linetypes: solid (-), dashed (--), dotted, dashdot (-.)
 * Marktypes: point (.), plus (+), star ("*"), circle (o), x-mark (x)
 * Colors: yellow (y), magenta (m), cyan (c), red (r), green (g), blue (b), white (w), black (k)

The command subplot breaks the screen up in order to place several small plots in one figure. The command help subplot can be used for further instruction.

Other 2-D plotting functions (that can be explored using help):
 * polar
 * bar
 * hist
 * quiver
 * compass
 * feather
 * rose
 * stairs
 * fill

Examples:

Code 1: x = -4:.01:4; y = sin(x); plot(x,y) –Plots the sine function over the interval -4 to 4; x is a division of the domain with meshsize 0.01; y is a vector of the values of sine at the nodes of x Display 1:

Code 2: x = -1.5:.01:1.5; y = exp(-x.^2); plot(x,y) –Plots y=e-x 2 from -1.5 to 1.5 Display 2:

Code 3: sphere -Plots a parametrically defined sphere Display 3:

Code 4: t = 0:.001:2*pi; x = cos(3*t); y = sin(2*t); plot(x,y) Display 4:

Code 5a: x = 0:.01:2*pi; y1 = sin(x); y2 = sin(2*x); y3 = sin(4*x); plot(x,y1,x,y2,x,y3) Code 5b (uses matrix Y; produces same display): x = 0:.01:2*pi; Y=[sin(x)', sin(2*x)', sin(4*x)']; plot(x,Y) Display 5:

Code 6: x = 0:.01:2*pi; y1 = sin(x); y2 = sin(2*x); y3 = sin(4*x); plot(x,y1,'--',x,y2,':',x,y3,'+') Displays same plot as Code 5 with different line formats. Display 6:

Code 7: plot(x,y,'r--') Displays a red dashed line. Display 7:

3D Line Plots

Similar to plot, the command plot3 displays curves in 3 dimensions. Assuming x, y, and z are vectors of the same length, plot3(x,y,z) graphs a perspective plot of the linear curve in 3-D space of the respective elements of x, y,, and z, which are often defined parametrically. (See Code 1).

Using the same commands as for planar plots, a title and axis labels can be added (including zlabel). The axis command features also work as explained earlier, except now the axis scaling limits require a 6-vector.

Code 1: t = .01:.01:20*pi; x = cos(t); y = sin(t); z = t.^3; plot3(x,y,z) Displays a helix that is compressed near the x-y plane. Display 1:

3D Mesh and Surface Plots

The command mesh produces 3-dimensional wire mesh plots. (See Code 1 and 2).

Similarly, the command surf produces 3-dimensional faceted surface plots. (See Code 3).

Below are more examples of mesh and surface plots. (See Code 4 and Code 5).

Formatting for 3-D surface and mesh plots is similar to other plots, including commands to reformat the title, axes and their labels as well as hold.

More commands:

1. shading: Shades the surfaces of plots different colors. These commands should be entered AFTER the surf command. The three settings for shading are:
 * shading faceted (default)
 * shading interp (interpolated- removes superimposed meshlines on surf plots)
 * shading flat (flat- removes superimposed meshlines on surf plots)

2. colormap: Controls a surface's color profile. Predefined colormaps include: hsv (default), hot, cool, jet, pink, copper, flag, gray, and bone. For example, typing colormap(cool) will set a certain color profile for the current figure. (See Code 6).

3. view: Specifies the viewpoint from which is object is to be viewed in cartesian or spherical coordinates. (See help view for more details).

4. peaks: Produces plot good for experimenting with shading, colormap, and view. (See Code 7, Code 8, and Code 9).

Parametrically defined surfaces can also be used in producing plots. (See Code 10).

More 3-D plotting functions to explore via help:
 * meshz
 * surfc
 * surf1
 * contour
 * pcolor

Code 1: mesh(z) Displays a 3-D perspective plot of the elements of pre-defined matrix "z" Display 1:

Code 2: mesh(eye(10)) Display 2:

Code 3: surf(eye(10)) Display 3:

Code 4: xx = -2:.2:2; yy = xx; [x,y] = meshgrid(xx,yy); z = exp(-x.^2-y.^2); mesh(z) Displays the MESH plot of z=e-x 2-y2 over the square [-2,2]X[-2,2] Display 4:

Code 5: xx = -2:.2:2; yy = xx; [x,y] = meshgrid(xx,yy); z = exp(-x.^2-y.^2); surf(z) Displays the SURFACE plot of z=e-x 2-y2 over the square [-2,2]X[-2,2] Display 5:

Code 6: colormap(cool) Display 6:

Initial Code 7: peaks

z = 3*(1-x).^2.*exp(-(x.^2) - (y+1).^2) ... - 10*(x/5 - x.^3 - y.^5).*exp(-x.^2-y.^2) ... - 1/3*exp(-(x+1).^2 - y.^2)

Code 7a: shading flat Display 7a:

Code 7b: colormap(flag) Display 7b:

Code 7c: view Display 7c:

Handle Graphics

The system Handle Graphics allows for the production of sophisticated plots; it provides functions to control all aspects of the graphics environment. For example:
 * set(1)
 * gca,set(ans)

Please refer to the MATLAB User's Guide for more information on Handle Graphics.

19. Sparse Matrix Computations
MATLAB has two matrix storage modes: full (default) and sparse. The functions full and sparse convert between the two modes. Since MATLAB assumes matrices are dense with non-zero entries, computation time can be reduced and memory use can be minimized by storing only the non-zero entries of a sparse matrix. For any matrix A, nnz(A) returns the number of nonzero elements in A.

Sparse matrices are stored as linear arrays of their nonzero entries along with their row and column indices. See the following example:

Example Code:Creates a full tridiagonal matrix, F and converts it to sparse mode. Display:

Related Code: Converts S to full storage mode and displays the storage mode of matrix A.

The command spdiags creates a sparse matrix by specifying the diagonals. See the following example and help spdiags for more information.

Example Code:

More Commands:
 * speye - sparse analog for full matrix of eye
 * sparse - sparse analog for full matrix of zeroes
 * spones - sparse analog for full matrix of ones; takes matrix argument and replaces only nonzero entries with ones
 * sprandn - sparse analog for full matrix of randn; takes matrix argument and replaces only nonzero entries with normally distributed random numbers
 * sparse(m,n) - sparse zero matrix

The command sparse can also create a sparse matrix via listing its nonzero entries. See following example:

Example Code 1: Vector s lists nonzero entries of S, integral vectors i and j list their row and column indices respectively, then creates sparse matrix S with dimensions m x n).

Example Code 2:

Operations and functions can be performed on matrices in both full and sparse modes. Resulting storage mode guidelines are as follows:
 * Opeations on FULL matrices give FULL results
 * FULL: S+F, S*F, S\F, F\S
 * SPARSE: S+S, S*S, S.*S, S.*F, S^n, S.^n, S\S; inv(S), chol(S), lu(S), diag(S), max(S), sum(S)

More examples:
 * If S is SPARSE, eig(S) is full if S is symmetric but undefined if S is unsymmetric
 * svd - a FULL argument
 * Matrices built from blocks (ie: [A,B;C,D] is SPARSE if any block is sparse

Example: The following code compares the efficiency of the two storage modes for solving a tridiagonal system of equations for n=20,50,500,1000: Code:

Display:

20. Reference
MATLAB has many features that have not been included in this introduction. It contains many functions and operators that can be referenced via the on-line help facility or the Reference Guide.

There also exists "toolboxes" of functions for specific purposes, including:
 * Signal processing
 * Control systems
 * Robust-control
 * System identification
 * Optimization
 * Splines
 * Chemometrics
 * μ-analysis and synthesis
 * State-space identification
 * Neural networks
 * Image processing
 * Symbolic math (Maple kernel)
 * Statistics

These toolboxes can be further explored using help. }

= Contributing Members = The following Team Aero members contributed to this report.

Gonzalo Barcia --Eas4200c.f08.aero.barcia and Eml4500.f08.wiki.barcia 06:47, 18 September 2008 (UTC)

Rodney Dagulo --Eml4500.f08.wiki.dagulo.r 19:18, 18 September 2008 (UTC)

Renee Hood -- Eml4500.f08.wiki.hood

Cortland Glowacki -- Eml4500.f08.wiki.cort 12:01, 19 September 2008 (UTC)

Ricardo Lopez -- Eml4500.f08.wiki.Lopez 12:12, 19 September 2008 (UTC)

Thomas Kehoe -- Eml4500.f08.wiki.Kehoe 12:14, 19 Septempber 2008 (UTC)

title: User:Eml4500.f08.wiki.hood


text:
User:Dr Denelder

Dr. Jason Denelder is a professor of historical studies at the Royal Academy at St. George, with a primary focus on 20th Century international relations. Aside from his focus, Dr. Denelder also lectures on the history of Christianity, the Soviet Union, general military history, United States history, and general European history after the fall of Rome.

Denelder Homeschooling Series
For the Wikiversity, Dr. Denelder will prepare a series of lessons on historical topics with homeschoolers in mind. They will be specially marked with the following userbox for easy navigation in the order they are to be completed.

Courses Offered

 * HS-DH001: Introduction to Diplomatic History - In development currently; an introductory lesson on diplomatic history that familiarizes students with general concepts of diplomacy and the history of international relations.

title: User:Dr Denelder


text:
User talk:Eml4500.f08.echo.sell.br

 Hello Eml4500.f08.echo.sell.br, 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 04:08, 9 September 2008 (UTC)

title: User talk:Eml4500.f08.echo.sell.br


text:
User:Eas4200c.f08.blue.a/Lecture 2

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

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

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

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

title: User:Eas4200c.f08.blue.a/Lecture 2


text:
User:Eas4200c.f08.spars.stoute

[[Image:Lockheed_SR-71_Blackbird.jpg|thumb|right|]

=Professor's Notes= The forces governing aerospace structures can be described by partial differential equations which can be solved through implementation of finite elemental method (FEM). It is therefore the goals of EAS 4200C to; develop problem solving skills stemming from an understanding of aerospace mechanics, instill the ability to formulate aerospace mechanics problems, and judge the correctness of the arrived at solution all while avoiding the older ad hoc techniques of structural analysis. This class also hopes to implement cooperative learning in a manner that reduces the stress on the individual while also promoting the communication skills essential to work in the aerospace field.

=Class Notes= The vision for the class is broken up over several topics. These include the technique of solving partial differential equations known as the finite element method-which is used in every field of engineering to date-as well as aerospace structures, which is governed by partial differential equations. The emphasis of this course will comprise of developing both the skills needed to become effective engineers and the methodologies that can be used to understand, formulate, and judge the solutions to the problems we will be working on. Students are broken up into 5-6 member teams, that collaboratively work on and submit homework assignments. Students will be using the features of mediawiki, textbooks from wikiversity, information from MIT's Open Courseware and wikipedia, all of which are a part of the big picture that makes up the course to accomplish this. The goal of the configuration of the course is to be a collaboration between the team of students, the whole of the class, as well as the professor. Each team member has a username that can be used to identify them by the instructor, and allows the instructor to keep track of each member's contribution to the overall effort. By increasing the incentive to learn collaboratively, the instructor hopes to have an overall positive impact on improving the learning method in which each student participates.

=Text Book= Mechanics of Aircraft Structures, written by C.T. Sun, is intended for Aeronautical Engineering students with existing knowledge in courses such as Mechanics of Materials.

Preface
The second edition preface simply names the purpose of creating a second edition. The second edition's purpose is to "correct a number of typographical errors", "add more examples and problems for the student, and introduce a few new topics." The author then thanks those who contributed to the changes. The preface then includes the first edition's preface, which gives the overall introduction to the book. The author explains that because of the development of finite element codes, mathematical techniques which were highly emphasized in the engineering curriculum in the past, are now obsolete. Because of this, the book's material is presented with a greater emphasis on the understanding of the theoretical concepts, how to structure problems, and then judge their solutions. This also means that the text places a minor emphasis on the mathematical steps for finding the solutions to the problems.

The text also includes introductory information for two recent, important developments in the industry. The first is fracture mechanics, which the book describes as "the most important tool in the study of aircraft structure damage tolerance and durability in the past thirty years." (p. XV) It is explained that this material is vital for us to know upon graduation since it is an industry standard, having previously been knowledge reserved specifically for graduate school. As such, the text is attempting to correct this. The second topic the book also incorporates, in addition to the information found in rival texts that cover this subject, information on composite materials.

1.1: Introduction
"'The main difference(s) between aircraft structure materials and civil engineering structure materials lies in their weight.' (p. 1)"Weight is the number one consideration in the design of an aircraft. Because one can't afford to fly around with excess weight, not only does the aircraft have to be designed out of materials with high structural property values with low weights, but the individual components have to be pushed to their limits in order to ensure every pound of material is used to its fullest capability. "“Aircraft structures must be designed to ensure that every part of the material is used to its full capability. This requirement leads to the use of shell-like structures (monocoque constructions) and stiffened shell structures (semimonocoque constructions).” (Sun, Page 1)" Along with material characteristics, nonstructural considerations (i.e. aircraft size, shape, lift, drag) have to be considered. Since there are so many factors to consider, special materials are developed for these special applications. The dominant aircraft structural materials have been aluminum and titanium due to their high stiffness/weight and strength/weight ratios. However, in recent years, engineers have realized that they can obtain 30-40% weight savings by switching to composites such as carbon fiber, and have slowly been doing so. According to the text (p. 2) composites account for 50% of the weight of the Boeing 787.

Additionally, on top of pushing the components' materials to the limit, the design is further complicated because often, a component is designed with application in mind, rather than structural consideration. For example, an airfoil is designed specifically to provide lift, not necessarily to be extremely rigid. This means that the solutions to problems encountered during the design process are extremely limited, and because of this we tend to use exotic building materials. (p. 1)

1.2: Basic Structural Elements in Aircraft Structure
"“Major components of aircraft structures are assemblages of a number of basic structural elements, each of which is designed to take a specific type of load, such as axial,bending, or torsional load.”(Sun, Page 2)" This section covers the basic equations and background information for understanding the purpose with which certain parts of the airplane are designed. Every component has a very specific application, and as such, has certain characteristics whose abilities have to be maximized by the part's design.

1.2.1 Axial Member
Axial members are used to carry extensional or compressive loads applied in the direction of the axial direction of the member. The resulting stress is uniaxial:

 * $$\ \sigma = \mathbf{E}\varepsilon \,$$

where E and ε are Young's modulus and normal strain, respectively, in the loading direction. The total axial force F provided by the member is

 * $$\ \mathbf{F} = \mathbf{A}\sigma = \mathbf{E}\mathbf{A}\varepsilon \, $$

where A is the cross-sectional area of the member. (p. 2)

"EA" is known as the axial stiffness of the component, and as one can see, its value is determined by both the modulus of the material the part is made of, and the cross-sectional area at a given point. It should be noted that as long as two different components have the same material, and the same cross-sectional area (regardless of shape) then they have the same axial stiffness.

Axial members tend to be slender, and because of this, they're particularly vulnerable to buckling when under compression. "Buckling strength can be increased by increasing bending stiffness and by shortening the length of the buckle mode." (p. 2) Also, proper selection of the axial member may also help in reducing buckling. For example, a channel section has a higher bending stiffness than a circular one, so it tends to prevent buckling more effectively. However, in practical applications, most axial members, such as stringers are so slender that no design can provide a buckling strength great enough to avoid buckling. In order to remedy this, lateral supports are placed along the member to provide more strength. Some of these are ribs in wings, and frames in the fuselage. (p. 3-4)

1.2.2 Shear Panel
A shear panel is a thin sheet of material used to carry in-plane shear load. Consider a shear panel of uniform thickness t under uniform shear stress τ ... The total shear force in the x-direction provided by the panel is given by

 * $$\ \mathbf{V_x} = \tau\mathbf{t}\mathbf{a} = \mathbf{G}\gamma\mathbf{t}\mathbf{a} \, $$

where G is the shear modulus, and γ is the shear strain. Thus, for a flat panel, the shear force Vx is proportional to its thickness and the lateral dimension a. (p. 4)

The shear force in a curved panel under stress τ can be broken up into a horizontal component, Vx, and a vertical component, Vy. These resulting forces only depend on the dimensions of the curved panel, and not the actual contour of it. If the contour doesn't affect the capacity of a member to handle shear stress, then a more efficient use of material would be to use a straight panel. (p. 5)

1.2.3 Bending Member (Beam)
A structural member that can carry bending moments is called a beam. A beam can also act as an axial member carrying longitudinal tension and compression. According to simple beam theory, bending moment M is related to beam deflection w as

 * $$\ \mathbf{M} = \mathbf{-E}\mathbf{I}\frac{d^2\omega}{dx^2} \, $$

where EI is the bending stiffness of the beam. The area moment of inertia I depends on the geometry of the cross-section. (p. 5)

Since transverse shear forces tend to produce bending moments, beams tend to be designed with the capability to sustain both kinds of forces in mind. (p. 5) Additionally, for beams with large span/depth ratios, bending stress usually tends to be a more critical design criterion than transverse shear stress. (p. 5-6)

For a beam supported by a wall at one end, with a shear force acting on the opposite end, of cross-sectional height h and width b, we can define the maximum normal stress as:

 * $$\ \sigma_\max = \frac{\mathbf{M_\max (h/2)}}{\mathbf{I}} = \frac{\mathbf{V}\mathbf{L}\mathbf{(h/2)}}{\mathbf{bh^3/12}}= \frac{\mathbf{6VL}}{\mathbf{bh^2}} \, $$ (p. 6, eqn. 1.7)

The profile of the magnitude of the transverse shear stress distribution follows a parabola along the depth of the beam with a maximum value along the neutral plane of:

 * $$\ \tau_\max = \frac{3}{2} \frac{\mathbf{V}}{\mathbf{bh}}\, $$ (eqn. 1.8)

From the ratio

 * $$\ \frac{\sigma_\max}{\tau_\max} = \frac{\mathbf{4L}}{\mathbf{h}} \, $$ (eqn. 1.9)

one can see that for large span-to-depth ratios, such as in wings, the bending stress if far more important than transverse shear stress. For these kinds of beams then, the goal of the design is to maximize the bending stiffness by having the most appropriate cross-section possible. (p. 6)

In order to utilize the full capacity of the material in a beam, the material must be set as far apart from the neutral axis as possible. (p. 6)

1.2.4 Torsion Member
Torque, formed by shear stress acting in the plane of a member's cross section, is an important load to consider when designing aircraft components. (p. 7)

Consider a hollow cylinder subjected to a torque T ... The torque-induced shear stress τ is linearly distributed along the radial direction. The torque is related to the twist angle Θ per unit length as

 * $$\ \mathbf{T} = \mathbf{G}\mathbf{J}\theta \, $$ (eqn. 1.10)

where J is the torsional constant. For hollow cylinders, J is equal to the polar moment of inertia of the cross-section, i.e.,

 * $$\ \mathbf{J} = \mathbf{I_p} = \frac{1}{2}\pi\mathbf{(b^4-a^4)}= \frac{1}{2}\pi\mathbf{(b-a)(b+a)(b^2+a^2)} \, $$ (eqn. 1.11)

The term GJ is usually referred to as torsional stiffness.

If the wall thickness t=b-a is small compared with the inner radius, then an approximate expression of J is given by

 * $$\ \mathbf{J} = 2\mathbf{t}\pi\mathbf{r^3} \, $$ (eqn. 1.12)

where r=(a+b)/2 is the average value of the outer and inner radii. Thus, for a thin-walled cylinder, the torsional stiffness is proportional to the 3/2 power of the area (πr2) enclosed by the wall. (p. 7-8)

Since the torsional shear that a beam undergoes is proportional to the radius, the material towards the inside of the beam isn't used to its full capacity. Take for example using a solid circular beam and a hollow tube, both of the same mass, with the beam of a diameter of 2cm, and the tube of an inner radius of 5cm with a wall thickness of 0.1cm, the hollow tube undergoes 50 times more shear than the beam. This proves that thin-walled structures may be used as extremely efficient torsion members. (p. 8)

1.3 Wing and Fuselage
"'The wing and fuselage are the two major airframe components of an airplane. The horizontal and vertical tails bear close resemblance to the wing. Hence, these two components are taken for discussion to exemplify the principles of structural mechanics employed in aircraft structures.' (Sun Page 8)"

1.3.1 Load Transfer
Load transfer is the "passing on" of high loading from flexible members to stiffer members. Stiffeners can be added to the skin of an aircraft to avoid excessive deflection in the skin. When the skin of an aircraft body encounters an external pressure, the load can be transferred to these stiffeners, which in turn may transfer the load to other more rigid members. Understanding of load transfer is important when dealing with non-ideal situations where a bending moment or air pressure is not uniform. If the load from these things are not transferred correctly from a weak member to a stronger member, fracture may occur.

1.3.2 Wing Structure
"'The main function of the wing is to pick up the air and power plant loads and transmit them to the fuselage.' (Sun, Page 10)"

The wing, along with most other parts of the aircraft body, acts as a beam and as a torsion member. The individual members of the wing include: ribs, spars, flanges, wing skin, vertical stiffeners, and so on.

Wing ribs run chordwise inside the wing and carry in-plane loads. They help keep the wing's shape and also serve to transfer loads. Spars run spanwise along the wing and are designed to take bending and shear loads. It is usually shaped with a web and flange configuration, with the flange or heavy cap to take bending.

1.3.3 Fuselage
"'The fuselage structure is a semimonocoque construction] consisting of a thin shell stiffened by longitudinal axial elements (stringers]and longerons) supported by many transverse frames or rings along its length.' (Sun, Page 12)|undefined"

The fuselage is the body of an aircraft. Its purpose is for position control and stabilization of surfaces in specific relationships to lifting surfaces, required for aircraft stability and maneuverability. The primary forces acting on the fuselage are wing reactions, pay loads, reactions of the landing gear, and internal pressures due to passengers on board. To combat these forces the fuselage often has a circular cross section. This ring-like shape, along with added support (stringers), increases the buckling strength of the material used and creates small, self-equilibriated loads.

1.4: Aircraft Materials
"'Selection of aircraft materials depends on many considerations...Seldom is a single material able to deliver all desired properties in all components of the aircraft structure. A combination of various materials is often necessary.' (Sun, Page 14)"

In the last 30 years, there have been many advances to replace the traditional metallic materials used in aircraft structures today. Those advances have brought about materials which allow for less weight and more strength; two things that are key in the structural industry. Some of the major materials are steel, aluminum, and titanium alloys, along with fiber-reinforced composites.

Steel alloys are used when there is a need for a material with high yield stress and high stress because of its high density. Steel alloys are used in landing gear units and highly loaded fittings. (Sun, 14) The only downfall with using steel is that it is extremely heavy and must be plated due to its poor corrosion resistance. Aluminum alloys are used because they have a good fatigue life as well as fracture toughness and a very slow crack growth rate. They are commonly found in the fuselage and lower wing skins due to the constant cyclic tensile and compressive stresses. Titanium alloys are used because they have superior corrosion resistance along with ultimate and yield stresses much higher than the best aluminum alloys. Titanium alloys can also be used in high heat conditions (i.e. up to 1000° F). This alloy has been used sparingly because of its high machining cost but more effective methods, like near net shape forming, have been employed to help make using titanium more economic."'Materials made into fiber forms can achieve significantly better mechanical properties than their bulk counterparts.' (Sun, 15)" Fiber-reinforced materials can be categorized into the three categories of polymers, metals, and ceramics. These composites are much stiffer, stronger, and lighter than the major materials and are most suitable for aircraft structures. Through various application processes, they can provide multi-directional load capabilities, corrosion resistance, damage tolerance, and high fatigue life.

Team Member Contributions
The following students contributed to this report:

Eas4200c.f08.spars.stoute 00:44:31, 12 September 2008 (UTC)

Eduardo Rondon Eas4200c.f08.spars.rondon 00:58, 17 September 2008 (UTC)

Eas4200c.f08.spars.prey 03:12, 18 September 2008 (UTC)

Michael Lee Eas4200c.f08.spars.lee

Thomas McGilvray --Eas4200C.f08.spars.mcgilvray 21:11, 12 October 2008 (UTC)

=References= 1. Sun, C.T. (2006). Mechanics of Aircraft Structures John Wiley & Sons, New York. ISBN 0471699667.

2. Vu-Quoc, Loc (2008) Lecture Notes University of Florida

title: User:Eas4200c.f08.spars.stoute


text: File:Metals graph.JPG

title: File:Metals graph.JPG


text:
Topic talk:GIS

the information is too general.

i feel, more subcatagories should be added.

title: Topic talk:GIS


text:
User talk:Eas4200c.f08.spars.prey

 Hello Eas4200c.f08.spars.prey, 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 06:27, 9 September 2008 (UTC)

title: User talk:Eas4200c.f08.spars.prey


text:
User talk:Onlyramram

Welcome
 Hello Onlyramram, 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 06:28, 9 September 2008 (UTC)

Images
Hi, feel free to let me know if you want a hand. I see you had a play in the sandbox with images. I recently posted to some of my students' user pages on getting started with using images. e.g., if interested, see User talk:Khayne. Hope a little unsoliticed help is OK! :) -- Jtneill - Talk - c 12:44, 9 September 2008 (UTC)

Navigation box
Hi onlyramram. By "navigation box" I'm thinking perhaps you mean the Table of Contents box? If so, this will appear automatically I think if you have three four or more headings (e.g., this talk page has several headings now, so a Table of Contents box should appear). You can force the display or vertical posititioning by using. If I'm off-track here, let me know some more about what you're trying to achieve. There's a bit of a learning curve, but there are lots of helpful people around. -- Jtneill - Talk - c 07:09, 10 September 2008 (UTC)

title: User talk:Onlyramram


text:
User talk:Dr Denelder

 Hello Dr Denelder, 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 06:31, 9 September 2008 (UTC)

though I am not a homeschooler I am interested in your course about diplomatic history. What would have me begin reading and what would be a first assignment? --Jolie 22:25, 9 October 2008 (UTC)

title: User talk:Dr Denelder


text:
Meta-analysis/To do

This page is for listing tasks to be completed for the meta-analysis learning resources. Anyone can help, but in practice this is essentially a to do list currently for and by Jtneill:
 * 1) Create a template for a proposal for a meta-analytic study
 * 2) Create a list of dedicated meta-analysis software, with pros and cons
 * 3) Create/find a list of experimental design/quality criteria (e.g., from Campbell and Stanley)
 * 4) Make a list of typical criteria used to judge study "quality"
 * 5) Link to electronic copies of exemplar meta-analyses
 * 6) Find an article for example coding
 * 7) Add a link to an example database

title: Meta-analysis/To do


text:
User talk:Wing

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

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

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

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! Jtneill - Talk - c 11:24, 9 September 2008 (UTC)

 Hello , we are the Baah and Bahh twins and we'd both like to welcome you to Wikiversity. While you sit back and explore Wikiversity, enjoy some of our fresh goat milk. You can help us learn by creating new learning resources or help us by improving the guided tours and other exploration tools. We look forward to seeing you around. Baaaahbye for now.

title: User talk:Wing


text: Category:Meta-analysis

title: Category:Meta-analysis


text:
Real estate/Development processes

In this course you will learn about the over all development process of real estate, from Project identification to project completion.

Important aspects of real estate development process.

 * Fund management
 * Asset managemant
 * Accounts and Finance
 * Business development
 * Marketing
 * Customer services
 * Planning and Architecture
 * Construction
 * Operation and Maintainance

Onlyramram 12:16, 9 September 2008 (UTC)

title: Real estate/Development processes


text: Category:Confidence interval

title: Category:Confidence interval


text:
Confidence interval

Confidence intervals (CIs) (or confidence limits) provide information about the "reliability" of a statistical parameter - or "range of certainty" or "area of confidence".

Rather than rely on a single estimate for a parameter (such as a measure of central tendency, a CI approach specifies a likely range within which the population central tendency for a variable of interest is likely to be.

For example, if we measure the heights of a sample of human beings, the mean might be 170cm. From this sample, we might conclude that the mean height for the entire population is 95% likely to be between 168cm and 172cm (a lower and upper confidence level respectively).

CIs are an alternative or adjunct approach to statistical inference.

CIs are very useful and are underutilised.

This page provides an undergraduate-level introduction to confidence intervals and their usage.

CI types
Confidence intervals can be reported for any paramater of a sample:
 * Ms
 * M differences (M2 – M1)
 * Effect sizes
 * β (standardised regression coefficient) in multiple linear regression

CIs can be examined statistically and graphically (e.g., error-bar graphs)

CI for a mean

 * e.g., true M is 95% likely to lie between -1.96 SD and +1.96 of the sample M
 * CI is based on the M, SD, N, and critical ά, calculate:
 * Lower-limit (e.g., .025 for 95% CI)
 * Upper-limit (e.g., .975 for 95% CI)

CIs and error bar graphs

 * CIs around means can be presented as error bar graphs
 * More informative alternatives to bar graphs or line graphs
 * For representing the central tendency and distribution of continuous data for different groups

Review question
If I have a sample M = 5, with 95% CI of 2.5 to 7.5, what would I conclude?
 * Q
 * 1) Not reject H0 that the M is equal to 0.
 * 2) Reject H0 that the M is equal to 0.

 * A
 * 2

If I have a sample M = 5, with 95% CI of -.5 to 11.5, what would I conclude?
 * Q
 * 1) Not reject H0 that the M is equal to 0.
 * 2) Reject H0 that the M is equal to 0.

 * A
 * 1

title: Confidence interval


text:
Central tendency

 refers to descriptive statistics for identifying the "middle" points in data distributions. Common measures of central tendency include:
 * 1) (Arithmetic) mean (or average)
 * 2) Mode (the most commonly occuring score)
 * 3) Median (the 50th percentile)

title: Central tendency


text:
User talk:Consandram

 Hello Consandram, 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:Consandram


text:
User:Eml4500.f08.wiki1.handy

$${\hat{\mathbf{W}}}_{2x1}$$ is the virtual axial displacement which corresponds to $${\mathbf{q}}^{(e)}_{2x1}$$. On the other hand $${\mathbf{W}}_{4x1}$$ is the virtual displacement in the global coordinate system which corresponds to $${\mathbf{d}}^{(e)}_{4x1}$$. If we combine equations (2), (3) and (4) we get

$$(\mathbf{T^{(e)}}\mathbf{W})[\hat{\mathbf{k^{(e)}}}(\mathbf{T^{(e)}}\mathbf{d^{(e)}})-\mathbf{P^{(e)}}]=0$$ (5)

Which applies for all $${\mathbf{W}}_{4x1}$$. If we recall that

$$\mathbf{(A_{pxq}}\mathbf{B_{qxr})^T}=\mathbf{B^TA^T}$$ (6)

and

$$\mathbf{(a_{nx1}}\mathbf{b_{nx1})}=\mathbf{a_{1xn}^Tb_{nx1}}$$ (7)

we can manipulate (5) using (7). This gives us

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

Then by substituting in (6) we get

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

which can be simplified to

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

and finally

$$\mathbf{k^{(e)}d^{(e)}=f^{(e)}}$$

Up until now we have dealt with the discrete case with matrices. From here forward we will consider the continuous case which involves partial differential equations. The model problem we will use consists of an elastic bar with varying area and modulus. It will be subjected to varying concentrated and axial loads, as well as inertial forces.

title: User:Eml4500.f08.wiki1.handy