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

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User:Eas4200c.f08.nine.s/Lecture 6

Homework 1

Homework 2

Homework 3

Homework 4

Homework 5

Homework 6

Homework 7

Group nine - Homework 5

title: User:Eas4200c.f08.nine.s/Lecture 6


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User:Eml4500.f08.delta 6.krueger/HW Ni(x)

The formula for the above diagram is: $$N_{i}(x)={x-x_{i+1}\over x_{i}-x_{i+1}}$$ Wasn't sure if we were supposed to find an actual value for N(x). So, below are the values for N(x). The total length of the bar is $$L^{(i)}$$ and at the origin x is equal to zero. To find $$N_{i}(x)$$ the following formula is used: $$N_{i}(x)=1-{x\over L^{(i)}}$$ Now let x=0 and x=$$L^{(i)}$$ and it can be seen that when: x=0 $$\Rightarrow$$ $$N_{i}=1$$ and when x=$$L^{(i)}$$ $$\Rightarrow$$ $$N_{i}=0$$


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User:Eml4500.f08.ateam.carr/HW6

$$N_{1}^{(i)}(\tilde{x})=1-\frac{\tilde{x}}{L^{(i)}}= \begin{cases} 1 & \text{ at } \tilde{x}=0 \\ 0 & \text{ at } \tilde{x}=L^{(i)} \end{cases}$$

$$N_{2}^{(i)}(\tilde{x})=\frac{\tilde{x}}{L^{(i)}}= \begin{cases} 0 & \text{ at } \tilde{x}=0 \\ 1 & \text{ at } \tilde{x}=L^{(i)} \end{cases} $$

Homework:

Book p. 159 ; Set $$ E_{1}=E_{2}=E_{3}$$, let $$A(\tilde{x})$$ be linear as previously defined, and obtain $$\mathbf{k}^{(i)}$$ and compare to expression given in the book

Where:

$$A(\tilde{x})=N_{1}^{(i)}(\tilde{x})A_{1}+N_{2}^{(i)}(\tilde{x})A_{2}$$

and

$$E(\tilde{x})=N_{1}^{(i)}(\tilde{x})E_{1}+N_{2}^{(i)}(\tilde{x})E_{2}$$

Compare general $$\mathbf{k}^{(i)}$$ from before to $$\mathbf{k}_{avg}^{(i)}$$ where,

$$ \frac {(E_{1}+E_{2})}{2}\frac{(A_{1}+A_{2})}{2}\frac{1}{L^{(i)}} \begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}=\mathbf{k}_{avg}^{(i)}$$

Homework: Find, $$\mathbf{k}^{(i)}-\mathbf{k}_{avg}^{(i)}$$

Solution:

Remember: recall the Mean value Theorem (MVT) and its relations to the centroid:

Mean Value Theorem: for $$\bar{x}\epsilon [a,b]$$,     $$a\leq \bar{x}\leq b$$,

$$ \int_{x=a}^{x=b}{f(x)dx}=f(\tilde{x})[b-a] $$

Where the centroid is defined as:

$$\int_{A}^{}{xdA} = \bar{x} \int_{A}^{}{dA} = \bar{x}A$$

Therefore:

$$\int_{x=a}^{x=b}{f(x)g(x)dx} = f(\bar{x})g(\bar{x})[b-a]$$ where  $$ a\leq \bar{x}\leq b$$

In general:

$$f(\bar{x})\neq \frac{1}{b-a} \int_{a}^{b}{f(x)dx}$$

$$g(\bar{x})\neq \frac{1}{b-a} \int_{a}^{b}{g(x)dx} $$

title: User:Eml4500.f08.ateam.carr/HW6


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Vancouver Flower Clock Project

The Vancouver Flower Clock Project (VFCP) was initiated in October 2008 by ProximityArts, a non-profit interdisciplinary arts collective based out of Vancouver, British Columbia,  Canada. The project is based on a garden plan hypothesized by Carolus Linnaeus in his Philosophia Botanica(1751) that would take advatage of certain species of flowers that open their blooms at set times of day to accurately predict the time, in what he described as an horologium florae (floral clock).

As a collaborative naturalist's journal, The VFCP aims to create a collection of information that would aid particpants in the design and maintenance of a flower clock that is unique to the growing conditions and latitude of the Vancouver region and would also reflect the unique cultures of the region and their history with these plants. The project stresses the importance of regional botany, information sharing and the interweaving of aesthetic and scientific observation.

How to Participate
If you live outside of the Greater Vancouver Regional District you can use this page as a template and start a similar project in your region. If you live in the GVRD, there are a number of ways you can help develop this project. If you are new to wiki take the |The original Tour for Newcomers

Add to the list of species
In Philosophia Botanica, Linnaeus described three groups of flowers: Only Aequinoctales are suitable for use in a flower clock. If you are aware of any species with this trait, please add it to the Species List below using this template:
 * Meteorici - flowers which change their opening and closing times according to the weather conditions.
 * Tropici - flowers which change their times for opening and closing according to the length of the day.
 * Aequinoctales - flowers which have fixed times for opening and closing.

 * [[Image:image name here|left|80px|]]

 * Botanical Name

 * Common Name

 * /Plant name/

Add to the Species Journal
Take a walk. Watch the flowers grow. You can use these journals to log a bloom sighting, and post information about each species. You can include measurements, stories, pictures, recipes as well as indicate whether a bloom was open or closed at a specific time.

For a new species journal, copy and paste /Journal Template/ onto a new page with the format "plant name"

News and Events
PROJECT LAUNCH TEA PARTY Sunday JUNE 21, 2009 1:00 to 4:00 pm

Blim 197 17th Ave. E Vancouver

By Donation

Drop by for a wiki tutorial, a flower tour of the Mount Pleasant neighbourhood and to learn more about the project.

Bring your laptops, digital cameras or a pencil and paper.

Free cucumber sandwiches and homemade lemonade!

Project Background Info

 * Carl Linnaeus
 * Linnaeus' flower clock
 * Online text of Philosophia Botanica

 * Garden Design

 * Ethnobotany

 * Flowering Plants

Plant Identification

 * E-Flora BC-Electronic Atlas of the Plants of British Columia

Climate and Local Growing Conditions
Vancouver's hardiness zone ranges from 6b-7b depending on altitude and proximity to the coast.

 * Vancouver's Climate

Participants

 * Proximityarts

Species List
Use /Journal Template/ when adding a new journal.

Aequinoctales
The current list is based on Linneaus observations in Philosophica Botanica.

Local Additions
Include Species not indicated above. Use /Journal Template/ when adding new species.

title: Vancouver Flower Clock Project


text: Internet psychology

title: Internet psychology


text:
Vancouver Flower Clock Project/Journal Template

Back to Vancouver Flower Clock Project

This is a Vancouver Flower Clock Project Journal Page for Species ___, Common Names_____.

General Info

 * Bloom Season

 * Habitat

Notebook
Share a poem, story, memory or anecdote about this plant.

Log a Bloom Observation
To log a bloom observation, sign your name by typing "~". Please indicate if the time and date of observation are different from the time of signing. Be as detailed as possible in your description of the observation. Were the blooms open or closed? How long did you watch? Where were they situated? What were the weather conditions? Were there any surrounding plants, insects? What was your response?. Take a photograph, video or scan an image and upload it using Special:Upload.

9:00pm-10:00pm
Oenothera acaulis aurea ("Dandelion primrose", "moonflower"). One yellow flower bloomed at sunset (9PM) and only lasted the night. They were in a pot on my north-facing patio (open to the night sky). Toronto, Canada. Hotgardener (discuss • contribs) 14:54, 30 June 2015 (UTC)


text:
Vancouver Flower Clock Project/Meadow Salsify

Back to Vancouver Flower Clock Project

This is a Vancouver Flower Clock Project Journal Page for Species ___, Common Names_____.

General Info

 * Bloom Season

 * Habitat

Notebook
Share a poem, story, memory or anecdote about this plant.Include sketches, photographs and videos as well.

Log a Bloom Observation
To log a bloom observation, sign your name by typing "~". Please indicate if the time and date of observation are different from the time of signing. Be as detailed as possible in your description of the observation. Were the blooms open or closed? How long did you watch? Where were they situated? What were the weather conditions? Were there any surrounding plants, insects? What was your response?. Take a photograph, video or scan an image and upload it using Special:Upload.


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Vancouver Flower Clock Project/Iceland Poppy

Back to Vancouver Flower Clock Project

This is a Vancouver Flower Clock Project Journal Page for Species ___, Common Names_____.

General Info

 * Bloom Season:
 * Predicted Bloom Opening Time:
 * Predicted Bloom Closing Time:

Parlour
Share a poem, story, memory or anecdote about this plant.

Log a Bloom Observation
To log a bloom observation, sign your name by typing "~". Please indicate if the time and date of observation are different from the time of signing. Be as detailed as possible in your description of the observation. Were the blooms open or closed? How long did you watch? Where were they situated? What were the weather conditions? Were there any surrounding plants, insects? What was your response?. Take a photograph, video or scan an image and upload it using Special:Upload


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text: Psychology of religion

title: Psychology of religion


text:
Survey design/Quantitative data analysis

t-tests
There are several types of t-tests
 * one-sample t-test
 * independent/unpaired t-test
 * dependent/paired t-test


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User:Ctan

 Hello Ctan, 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 Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

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

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! Jtneill - Talk - c 12:24, 18 November 2008 (UTC)

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

Statistics
Hi Ctan, Great to see you using this for your class materials at Quantitative data analysis - go for it! You may also want to check out Category:Statistics and see what you might link to or add to. Let me know if you want a hand. -- Jtneill - Talk - c 12:27, 18 November 2008 (UTC)

title: User:Ctan


text: Category:Cross-cultural psychology

title: Category:Cross-cultural psychology


text:
Template:Wikipedia link

(Wikipedia)

Usage

 * Creates a link to the PAGENAME on Wikipedia
 * For substitution or transclusion into e.g., See also sections - use  or

ja:Template:Wplink

title: Template:Wikipedia link


text:
Cultural adaptation

Stages of cultural adaptation
Winkelman's (1994) four stages stages of cultural adaptation:
 * 1) Honeymoon phase. Excitement and fascination with the new culture. This is where they will overlook minor problems and look forward to learning new things.
 * 2) Crisis period (culture shock). This is where excitement turns to disappointment and there are more and more differences that occur. Problems start to be overwhelming and irritating and may use the "fight-back" technique by saying rude remarks or making jokes.
 * 3) Adjustment phase. This is where they learn to accept the culture and to change their negative attitude to a positive one.
 * 4) Acceptance and Adaptation phase. This is where they will feel at home and become involved in activities and may enjoy some of that countries customs.
 * 5) Reentry shock. This is experienced upon returning to the home country and the return may follow with initial euphoria, crisis or disenchantment. It may be hard to readjust and may feel like they are not accepted.
 * 6) * This stage is not identified by Winkelman. From: Culture Shock

This is just one model - what other stages might be considered? For example, this article on culture shock uses different labels for the phases and suggests that Stage 3 Adjustment be broken down further into:
 * 1) Reintegration
 * 2) Automony and includes a second cycle of adaptation related to returning to the original culture.

title: Cultural adaptation


text:
Talk:Wikipedia/Quizzes

How you can help
See the to do list at Talk:Wikipedia.

Fact-checking
Are the formulations correct? Mange01 14:40, 18 November 2008 (UTC)

Suggestions for more quiz topics and questions
What wiki terminology is most important to ask questions about? I add some of my own suggestions. Mange01 14:40, 18 November 2008 (UTC)

Suggested quizz

 * 1) Basic wiki terminology
 * 2) How to edit a wiki page
 * 3) Wikipedia policies and guidelines (Partly done)
 * 4) Wikipedia research and criticism (Partly done)
 * 5) Wiki technology

Suggestions for the "Wikipedia policies and guidelines" quiz
In order of importance:
 * w:Wikipedia:The_five_pillars_of_Wikipedia: NOT, NPOV, COPYRIGHT, ETIQ, IGNORE
 * GFDL and legal aspects
 * How to cite wikipedia
 * Common templates
 * w:Wikipedia:Avoiding common mistakes
 * Authorative sources
 * The most important Style guidelines: w:wikipedia:Citing sources, Lead section, Article style naming, English dialects, Reference style, Lists
 * The most important Behavioural guidelines: *w:Wikipedia:Assume good faith, Self-referencing, conflict of interest, own research, w:Wikipedia:search engine optimization, etc
 * w:Wikipedia:Dispute resolution
 * Administrator rolls and tools
 * Dealing with vandalism and link spam.
 * Blocking of users
 * Common sense. (Part of Ignore all rules?)
 * Definition of Essays, Policies, Guidelines, Help pages, Projects and Template Documentation.
 * Other related terms in the Wikipedia glossary

Suggestions for the "Wikipedia research and criticism" quiz

 * Wikipedia statistics
 * Examples of censorship
 * Social science research on Wikis and Wikipedia
 * Other related terms in the Wikipedia glossary

Suggestions for a "Wiki technology" quiz

 * Web 2.0 and other current web technology trends
 * Wiki server software comparisons
 * How to install, configure and administrate an own Mediawiki server
 * Parameter controlled templates
 * Embedded HTML
 * Wiki robots and tools
 * Mediawiki extensions
 * Bugzilla
 * State-of-the-art wiki technology research and development
 * Other related terms in the Wikipedia glossary

Common misconceptions about Wikipedia
I would like to add some common misconceptions about Wikis and Wikipedia as distraction alternatives in true/false questions. Do you know any such misconceptions, besides the ones mentioned in w:Wikipedia:Press_Kit? Mange01 14:40, 18 November 2008 (UTC)

Comments on the quiz
I think the quiz is a good idea, and has been implemented very well, especially with the automagic marking of responses. I've a couple minor queries regarding the answers to the questions. PhilKnight 09:07, 31 May 2009 (UTC)
 * The culinary recipe answer is possibly somewhat misleading. In practice the article would be kept or deleted on the basis of whether the culinary dish was notable. In other words, whether there was significant coverage in reliable sources independent of the subject. If the subject was notable, a rewrite would be expected, but I doubt that deletion of the article would happen in practice.
 * The current price comparisons of products existing on the market today information wouldn't appear in a single article, but the information could be moved to the individual product articles. See for example F-22 Raptor and Eurofighter Typhoon.

Bugs
For the top half of the quiz, there were five choices at the top, but under them were five circles on the left, and five purple squares on the right. Neither the circles nor the squares lined up with the choices at the top. So it took me a long time to guess whether I should check the circles, the squares, both, or neither.

Also, when I got halfway down the page there was a "submit" button. So I clicked it, expecting the bottom half of the page to be separate. Instead, it marked me wrong for not answering any of the questions on the bottom half. Art LaPella 01:32, 19 June 2009 (UTC)

title: Talk:Wikipedia/Quizzes


text:
Fundamental attribution error

In attribution theory, the fundamental attribution error (also known as correspondence bias or overattribution effect) is the tendency for people to over-emphasize dispositional, or personality-based, explanations for behaviors observed in others while under-emphasizing situational explanations. In other words, people have an unjustified tendency to assume that a person's actions depend on what "kind" of person that person is rather than on the social and environmental forces influencing the person. Overattribution is less likely, perhaps even inverted, when people explain their own behavior; this discrepancy is called the actor-observer bias.

Activity
Hand out two copies to each student of the attribution scale. Ask them to complete the scale twice – once for themselves and once for George Bush. After they have completed both forms, have them count the number of times they circled “depends on the situation” on each rating sheet. A show of hands should show that a greater tendency to attribute another person’s behaviour to personal disposition, while attributing their own behaviour to the environment. Ask students why this is the case and discuss.

We tend to attribute causation to the focus of our attention, which is different when we are observing than when we are acting. When another person acts, our focus is on that person, thus they are the cause of what is happening. When we act, the environment commands our attention and thus seems to explain our behaviour.

title: Fundamental attribution error


text: Category:Social cognition

title: Category:Social cognition


text:
Schema (psychology)

A schema is a cognitively-stored, efficient, pre-set way of interpreting information and experiences. However, because schemas tend to be used automatically (unconsciously), they can also lead to perceptual and cognitive errors. Schemas are also learnt, via upringing and enculturation.

title: Schema (psychology)


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text: User:Jestermeister/Categorical Syllogism/More


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User:Menelaos Pappas

Short C.V.
Menelaos Pappas (Greek:  Μενέλαος Παππάς ) holds a Diploma (5-years degree) in Mechanical Engineering (2002, Dept. of Mechanical Engineering and Aeronautics, University of Patras, Greece) and a Ph.D. (Dr.-Eng.) in the field of Digital Human Modeling for Computer-Aided Ergonomics in Product & Process Design (2009, Dept. of Mechanical Engineering and Aeronautics, University of Patras, Greece). He has been Adjunct Lecturer at the University of Western Macedonia (Dept. of Mechanical Engineering, Kozani, Greece) since 2010 where he is teaching courses in Computer Aided Manufacturing for Industrial Production. He has also held the position of Adjunct Assistant Professor at the Technological Educational Institute (T.E.I.) of Western Macedonia, Dept. of Mechanical Engineering, Kozani, Greece (Courses: Computer Aided Design Optimization (CAD/CAM/CAE Systems) & Mechanical Drawing II) and at the Technological Educational Institute (T.E.I.) of Larissa, Dept. of Mechanical Engineering, Larissa, Greece (Courses: Engineering Measurements, CNC Machine Tools, Engineering Design & Design of Steel Structures) since 2009. He has more than 7 years of technical and management experience in National and European research projects, through his position as Research Engineer at the Laboratory for Manufacturing Systems and Automation - LMS (Dept. of Mechanical and Aeronautical Engineering, University of Patras, Greece). His research skills include among others: Virtual Reality Applications in Product Design & Manufacturing, Collaborative Virtual Environments, Digital Human Modeling, Computer Aided Ergonomic Design & Production, Evaluation of Vehicle Safety Systems Effectiveness, Product Design & Manufacturing Process Optimization, Computer Aided Manufacturing as well as Data Analysis and Modeling Techniques, including Statistical Design of Experiments (SDoE), Analysis of Means / Analysis of Variance (ANOM / ANOVA) and Artificial Neural Networks (ANN). He has 32 scientific publications in international scientific journals, chapter's books and refereed conferences, which they have collected more than 100 citations. He has also contributed in the revision of the book Manufacturing Systems: Theory and Practice (2nd Edition). He has performed labs/seminars on Computer-Aided Design (CAD), Virtual Reality (VR), Design For Assembly (DFA) and Statistical Process Control (SPC), under the undergraduate level's course of Manufacturing Technology I & II (lectures given by Prof. G. Chryssolouris). He has also supervised more than 10 undergraduate student's theses. He acts as a reviewer in the IEEE Transactions on Systems, Man and Cybernetics - Part A, since July 2005, in the International Journal of Computer Integrated Manufacturing, since June 2007 and in the Scientific Journals International (SJI), since May 2009. Finally, he is a member of the European Association for Virtual Reality and Augmented Reality (EuroVR).

Virtual Reality (VR) Applications in Product Design and Manufacturing

 * Virtual Prototype Design and Functional Simulation
 * On-line Virtual Product Catalogues (VRML / X3D based)
 * Virtual Assembly Process Design and Simulation
 * Interaction Metaphors for Realistic Virtual Objects Manipulation using Data Glove
 * Immersive Assembly Environments

Collaborative Virtual Environments (CVEs)

 * Shared VEs for Multi-user Navigation and Interaction
 * Virtual Showrooms for On-line Product Demonstration and Customization
 * Web-based Collaborative Product Design

Digital Human Modeling (DHM)

 * Digital Human Motion Reconstruction from Motion Captured Data
 * Analysis of Human Motion based on Motion Captured Data
 * Human Motion Modeling and Simulation using Computer Manikins
 * Human Motion Adaptation for New Characters and Variations in the Working Environment

Computer Aided Ergonomic Design & Production

 * Ergonomic Assessment (Comfort / Discomfort Prediction) Using Neural Networks
 * Human-Centered Product Design Using Computer Manikins
 * Digital Human Motion Modeling for Vehicle Interior Ergonomic Design
 * Ergonomic Evaluation and Biomechanical Analysis of Manual Assembly Tasks / Workstation

Vehicle Safety Systems Effectiveness

 * A-priori evaluation of safety functions effectiveness using Neural Networks

Data Analysis and Modeling

 * Statistical Design of Experiments (SDoE)
 * Analysis of Means / Analysis of Variance (ANOM / ANOVA)
 * Artificial Neural Networks (ANN)
 * Multiple Functional Regression
 * Response Surface

CAD / CAM / CAE / CNC

 * Computer Aided Design (CAD)
 * Computer Aided Manufacturing (CAM)
 * Computer Aided Engineering (CAE)
 * Computerized Numerical Control (CNC) Machine Tools

Manufacturing

 * Manufacturing Systems: Theory and Practice (2nd Edition) (ISBN: 978-0387-25683-2), Chryssolouris G. (Ed.), Springer New York (May 2006)

Collaborative Virtual Environments (CVEs)

 * Collaborative Design and Planning for Digital Manufacturing (ISBN: 978-1-84882-286-3), Wang L. and Nee A.Y.C. (Eds.), Springer London (April 2009)

 * Methods and Tools for Effective Knowledge Life-Cycle-Management (ISBN: 978-3-540-78430-2), Bernard A. and Tichkiewitch S. (Eds.), Springer Berlin Heidelberg (April 2008)

 * Collaborative Product Design and Manufacturing Methodologies and Applications (ISBN: 978-1-84628-801-2), Li W.D., Ong S.K., Nee A.Y.C. and McMahon C. (Eds.), Springer London (June 2007)

 * The Future of Product Development, Proceedings of the 17th CIRP Design Conference, Berlin, Germany (ISBN: 978-3-540-69819-7), Krause F.L. (Ed.), Springer Berlin Heidelberg (April 2007)

 * Burdea G, Coiffet P (2003) Virtual Reality Technology, Second Edition, (ISBN: 0471360899).

 * Churchill EF, Snowdon DN, Munro AJ (2001) CollaborativeVirtual Environments (Digital Places and Spaces forInteraction). Series: Computer Supported Cooperative Work, (ISBN: 1-85233-244-1).

 * Luo Y (2004) Cooperative Design, Visualization, and Engineering. First International Conference CDVE 2004, Palma de Mallorca, Spain, September 19-22, 2004, Proceedings Series: Lecture Notes in Computer Science, Vol. 3190, (ISBN: 3-540-23149-8).

 * Sherman WR, Craig AB (2002) Understanding Virtual Reality: Interface, Application, and Design, (ISBN: 1558603530).

 * Sriram RD, Logcher RD (2002) Distributed and Integrated Collaborative Engineering Design, (ISBN: 0972506403).

 * Stanney KM (2002) Handbook of Virtual Environments: Design, Implementation, and Applications (Human Factors and Ergonomics), (ISBN: 080583270X).

Digital Human Modeling (DHM)

 * 3D Human Modeling and Animation, 2nd Edition, (ISBN: 978-0-471-21548-6), by Ratner, P., Wiley (Ed.), New York, NY, 2003.

 * 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body, (ISBN: 1-59140-299-9), by Sarris, N., and Strintzis, M., IRM Press (Ed.), Hershey, PA, 2005.

 * Biomechanics in ergonomics, 2nd Edition (ISBN: 978-0-8493-7908-6), by Shrawan Kumar, Taylor & Francis (Ed.), 2007.

 * Bodyspace: Anthropometry, Ergonomics and the Design of Work (ISBN: 978-0-415-28520-9), by Pheasant, S., Taylor & Francis (Ed.), 1998.

 * Digital Human Modeling: 1st International Conference on Digital Human Modeling, (ISBN: 978-3-540-73318-8), by Duffy, V., Springer-Verlag (Ed.), 2007.

 * Digital Human Modeling for Vehicle and Workplace Design, (ISBN: 978-0-7680-0687-2), by Chaffin, D., Society of Automotive Engineers (Ed.), Warrendale, PA, 2001.

 * Ergonomic software tools in product and workplace design: a review of recent developments in human modelling and other design aids, (ISBN: 3-932160-11-8), Landau, K. (Ed.), IfAO Institut für Arbeitsorganisation, Stuttgart, Germany, 2000.

 * Handbook of human factors and ergonomics, 3rd Edition, (ISBN: 978-0-4714-4917-1), Salvendy G. (Ed.), 2006.

 * Introduction to Ergonomics, 2nd Edition (ISBN: 978-0-4152-7377-0), by Bridger, R., McGraw-Hill Inc., 2003.

 * SAE Digital Human Modeling for Design and Engineering Conference Proceedings (ISBN: 0768008123), Society of Automotive Engineers (Ed.), Warrendale, PA, 2001.

 * What Engineers and Managers Need to Know About Human Factors, (ISBN: 978-0-7680-0975-0), by Gabriel R., Society of Automotive Engineers (Ed.), 2003.

Virtual Reality (VR) Platforms
Commercial Softwares
 * 3DVIA Virtools (Dassault Systemes)
 * DIVISION MockUp (Parametric Technology Corporation - PTC)
 * EON Studio (EON Reality)
 * Vizard (WorldViz)
 * CAVELib (Mechdyne Corporation)

Open Source Platforms
 * OpenSceneGraph (OSG Community)
 * OpenMASK (IRISA/INRIA)
 * OpenSG (Dirk Reiners & Gerrit Voss)
 * OpenGL Performer (SGI)
 * Open Inventor (SGI)
 * FreeVR (William R. Sherman)
 * Virtual Universe (3Dchat)

Research Prototypes
 * VR Juggler (Iowa State University, Virtual Reality Applications Center)
 * MAVERIK (University of Manchester, Department of Computer Science)

Designs
 * Retail interior design-Related Designs

Digital Human Modeling (DHM) Platforms

 * RAMSIS (Human Solutions)
 * SAFEWORK (Dassault Systemes)
 * Tecnomatix Jack (Siemens PLM Software)
 * AnyBody Modeling System (AnyBody Technology)
 * SAMMIE (SAMMIE CAD)
 * ManneQuinPRO (NexGen Ergonomics)
 * BHMS (Boeing)

Portal:Engineering and Technology

School:Engineering

Topic:Mechanical engineering

title: User:Menelaos Pappas


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User:TimMartin

Tim Martin: about

title: User:TimMartin


text:
User:Eml4500.f08.gravy.jad/Notes6

$$\ \,$$ $$\  \,$$ $$\  \,$$

Lecture 29

HW6

Continuing Case: PVW of Dynamics of Elastic Bars

PDE: $$\ \frac{d}{dx} \left \lbrack (EA) \frac{du}{dx} \right \rbrack + f = m \ddot{u} \,$$ Equation(1)

$$\ \Rightarrow \mathbf{m} \mathbf{\ddot{d}} + \mathbf{k} \mathbf{d} = \mathbf{F} \,$$ Equation(2)

Derive (2) from (1): $$\ \int_{x=0}^{x=L} W(x) \left \lbrace \frac{d}{dx} \left \lbrack EA \frac{du}{dx} \right \rbrack + f - m \ddot{u} \right \rbrace \, dx = 0 \,$$ for all possible W(x). Where w(x) is the weighting function.

(1) $$\ \Rightarrow \,$$(3) trivial (3) $$\ \Rightarrow \,$$(1) not trivial (3) rewritten as: $$\ \int W(x)g(x)dx =0 \,$$for all W(x) Since (3) holds for all W(x), select W(x)=g(x), then (3) becomes $$\ \int g^{2}(x)dx = 0 \,$$

Integration by Parts: r(x), s(x)

$$\ (rs)' = r's + rs' \,$$ where $$\ r'=frac{dr}{dx} \,$$ and $$\ s'=frac{ds}{dx} \,$$

$$\ \int (rs)' = rs - \int rs' \,$$ $$\ \Rightarrow \int r's = rs - \int rs' \,$$

Recall cont. PVW (equation 3)

1st term: $$\ r(x)=(E,A) \frac{du}{dx}, s(x)=w(x) \,$$

By integrating by parts: $$\ \int_{x=0}^{x=L} \underbrace{w(x)}_{s} \frac{d}{dx} \underbrace{\left \lbrack (EA) \frac{du}{dx} \right \rbrack}_{r} dx = \left \lbrack W(EA) \frac{du}{dx} \right \rbrack |_{x=0}^{x=L} \,$$

$$\ = W(L) \left \lbrack (EA)(L) \right \rbrack \frac{du}{dx}(L,t) - W(0) \left \lbrack (EA)(O) \right \rbrack \frac{du}{dx} (0,t) \,$$

Now consider model prob:

At x=0, select W(x) such that W(0)=0 (ie. kinetically dismiss-able)

Motivation: Discrete PVW applied to Eq $$\ \mathbf{W}_{6x1} \cdot \underbrace{ \mathbf{k}_{6x1} \mathbf{d}_{2x1}-\mathbf{F}_{6x1} }_{6x1} = 0_{1x1} \,$$

$$\ F^{T} = \left \lfloor F_{1} F_{2} F_{3} F_{4} F_{5} F_{6} \right \rfloor \,$$

Where $$\ F_{1}, F_{2}, F_{5}, F_{6} \,$$ are unknown reactions. Since $$\ \mathbf{W} \,$$ can be selected arbitraily, select $$\ \mathbf{W} \,$$ such that $$\ W_{1}=W_{2}=W_{5}=W_{6}=0 \,$$, so to eliminate equations involving unknown reactions, thus eliminating 1,2,5,6; $$\ \mathbf{\overline{k}} \mathbf{\overline{d}}=\mathbf{\overline{F}} \,$$

Please note: $$\ \overline{\mathbf{W}} \cdot (\mathbf{\overline{kd}} - \mathbf{\overline{F}})=0 \,$$ for all $$\ \mathbf{W} \,$$

This concludes the motivation.

Unknown reaction term $$\ N(0,t) = \left \lbrack (EA)(0) \right \rbrack \frac{du}{dx} (0,t) \,$$

Continuing PVW: $$\ W(L)F(L) - \int_{0}^{L} \frac{dW}{dx}(EA) \frac{du}{dx} dx + \int_{0}^{L} W(x) \left \lbrack f-m \ddot{u} \right \rbrack dx=0 \,$$ for all W(x) such that W(0)=0.

$$\ \Rightarrow \int_{0}^{L} W(m \ddot{u}) dx + \int_{0}^{L} \frac{dW}{dx} (EA) \frac{du}{dx} dx = W(L)F(t) + \int_{0}^{L} Wf dx \,$$ for all W(x) such that W(0)=0.

Constant Setting (PVW)

Inertia : $$\ \int_{0}^{L} Wm \ddot{u} dx = \alpha \,$$

Stiffness : $$\ \int_{0}^{L} \frac{dW}{dx} (EA) \frac{du}{dx} dx = \beta \,$$

Applied force : $$\ W(L)F(t) + \int_{0}^{L} Wf dx = \gamma \,$$

For all W(x) such that: $$\ \underbrace{W(0)=0}_{b.c.} \,$$

Discrete setting (PVW)

Inertia : $$\ \overline{\mathbf{W}} \cdot ( \overline{\mathbf{md}}) (1) \,$$

Stiffness : $$\ \overline{\mathbf{W}} \cdot ( \overline{\mathbf{kd}}) (2) \,$$

Applied force : $$\ \overline{\mathbf{W}} \cdot \overline{\mathbf{F}} (3) \,$$

For all W(x) such that: b.c.'s eliminated

Stiffness term:

Assume displacement u(x) for $$\ x_{i} \le x \le x_{i+1} \,$$

Motivation for linear interpolation of u(x)

Deformed shape is a straight line. ie. there was an implicit assumption of linear interpolation of displacement between 2 nodes

consider the case where there are only axial-displacement (ie. zero transverse displacement)

Question: Express V(x) in terms of $$\ d_{i} = u(x_{i}) \,$$ and $$\ d_{i+1} = u_(x_{i+1}) \,$$ as a linear function in x (ie. linear interpolation

$$\ u(x) = N_{i}(x) d_{i} + N_{i+1}(x)d_{i+1} \,$$ $$\ where N_{i}(x) and N_{i+1}(x) are linear functions in x \,$$

insert HW

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

title: User:Eml4500.f08.gravy.jad/Notes6


text:
User talk:Menelaos Pappas

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

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 16:32, 18 November 2008 (UTC)

IP edits to your user page.
There are many IP edits to your user page, I just noticed. We might not allow those, because we don't know that this was you. Do you want these changes to stand? Please log in and approve them here. Do you want the IPs to remain visible? Some people would not, for privacy reasons. Let me know if you want them hidden. If you have trouble logging in, ask for help. Thanks. --Abd 14:13, 8 August 2011 (UTC)

There may be an additional problem here. Wikiversity isn't a web host, as such. We do allow users to maintain a personal page like yours, but the understanding generally is that users are here to participate in Wikiversity. With the flow of edits to your user page, including all the IP edits, you do create some small burden on those who watch Recent Changes. With your qualifications, your help could be valuable, with resources in your fields of interest. Will you consider contributing to resources? Absent that, I think of asking the community if it wants to continue hosting your user page. Thanks. --Abd 14:19, 8 August 2011 (UTC)

I do confirm that the latest IP edits have been made by me. I apologize for this inconvenience. I understand the problems that cause the edit of my user page anonymously and I will try to eliminate it. I will log in every time I change my user page. I will try also to use Preview and be reasonably sure before save in order to reduce the number of edits. I also understand that Wikiversity isn't a web host of personal pages, thus I will consider in the future contributing in Wikiversity resources that are related to my fields of interest. Thanks. --Menelaos Pappas 19:27, 8 August 2011 (UTC)
 * Thank you for responding. I assume this is fine for now. I look forward to your participation. It needn't be much. This is not a salt mine. Even if you just occasionally glance over resources and point out problems it would be great! --Abd 20:39, 8 August 2011 (UTC)

title: User talk:Menelaos Pappas


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User:Eml4500.f08.FEABBQ.koby/HW6

November 10, 2008
Continuous Principle of Virtual Work to Discrete Principle of Virtual Work (Continued)

Lagrangian Interpolation

What is the motivation for the form of $$N_i(x)\,\!$$ and $$N_{i+1}(x)\,\!$$? 1) $$N_i(x)\,\!$$ and $$N_{i+1}(x)\,\!$$ are both linear, thus any linear combination of $$N_i\,\!$$ and $$N_{i+1}\,\!$$ is also linear.  In particular, the expression for $$u(x)=N_i(x)d_i+N_{i+1}(x)d_{i+1}\,\!$$ must be linear.  Below is a demonstration:

Let: $$N_i(x)=\alpha_i + \beta_ix\,\!$$ -$$\alpha_i, \beta_i\,\!$$ are real numbers

$$N_{i+1}(x)=\alpha_{i+1} + \beta_{i+1}x\,\!$$ -$$\alpha_{i+1}, \beta_{i+1}\,\!$$ are real numbers

Now we can combine $$N_i(x)\,\!$$ and $$N_{i+1}(x)\,\!$$ :

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

$$=(\alpha_i + \beta_ix)d_i+(\alpha_{i+1}, \beta_{i+1})d_{i+1}\,\!$$

$$=(\alpha_i+\alpha_{i+1}+(\beta_i+\beta_{i+1})x\,\!$$ < is a linear function

2) Recall the equation for $$u(x)\,\!$$, an interpolation of x:

$$u(x_i)=\underbrace{N_i(x_i)d_i}_{1}+N_{i+1}(x_i)d_{i+1} = d_i\,\!$$

title: User:Eml4500.f08.FEABBQ.koby/HW6


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User:Eml4500.f08.team.figuerrez/hw6

Meeting 30 - Principle of Virtual Work using Integration by Parts
Integration by Parts:

The integration by parts formula is used to rewrite an integral of a product of a derivative function as follows:

Give two functions $${\boldsymbol{r(x)}}$$, $${\boldsymbol{s(x)}}$$

and taking the derivatives of each of the two functions yields: $${\boldsymbol{r'}}=\frac{dr}{dx},{\boldsymbol{s'}}=\frac{ds}{dx}$$

From the equation $${\boldsymbol{(rs)'}}={\boldsymbol{r's}}+{\boldsymbol{rs'}}$$, we integrate both sides to get:

$$\int{\boldsymbol{(rs)'}}=\int{\boldsymbol{r's}}+\int{\boldsymbol{rs'}}$$

Since $$\int{\boldsymbol{(rs)'}}={\boldsymbol{(rs)}}$$ and solving for $$\int{\boldsymbol{r's}}$$ the resulting integrand is:

$$\int{\boldsymbol{r's}}={\boldsymbol{rs}}-\int{\boldsymbol{rs'}}$$

Recall the Cont. Principle of Virtual Work (PVW):

1st term: $$r(x)=(EA)\frac{du}{dx},s(x)=w(x)$$

Using integration by parts we get:

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

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

where $$(EA)(L)\frac{\partial u}{\partial x}(L,t)=N(L,t)=F$$

and   $$(EA)(0)\frac{\partial u}{\partial x}(0,t)=N(0,t)$$

Now consider the 2 boundary condition problem shown below:

At $${\boldsymbol{x=0}}$$ select $${\boldsymbol{w(x)}}$$ such that $${\boldsymbol{w(0)=0}}$$. (i.e. kinematically admissible)

Motivation: Discrete PVW applied to the equation shown below:

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

where the transpose of $${\boldsymbol{\underline{F}}}$$ is equal to $$\underline{F}^{T}=\begin{Bmatrix}F_{1} & F_{2} & F_{3} & F_{4} & F_{5} & F_{6}\end{Bmatrix}$$

$${\boldsymbol{F_{3}, F_{4}}}$$ are known reactions and $${\boldsymbol{F_{1}, F_{2}, F_{5}, F_{6}}}$$ are unknown reactions.

Since $${\boldsymbol{\underline{w}}}$$ can be selected arbitrarily, select $${\boldsymbol{\underline{w}}}$$ such that $${\boldsymbol{w_{1} = w_{2} = w_{5} = w_{6} = 0}}$$ in order to eliminate equations involving unknown reactions; rows $${\boldsymbol{1, 2, 5,}}$$ and $${\boldsymbol{6}}$$ are eliminated.

The resulting equation is:

$$\underline{K}_{2x2}*\underline{d}_{2x1}=\underline{F}_{2x1}$$   $${\boldsymbol{(1)}}$$

$$\underline{K}_{2x2}* \begin{Bmatrix} d_{3} \\ d_{4}\end{Bmatrix} = \begin{Bmatrix} F_{3} \\ F_{4}\end{Bmatrix}$$

Back to the Cont. Principle of Virtual Work (PVW), the unknown reaction is:

$$N(0,t)=(EA)(0)\frac{\partial u}{\partial x}(0,t)$$

The equation then becomes:

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

The final equation then becomes:

$$\int_{0}^{L}w(m\ddot{u})dx+\int_{0}^{L}\frac{\partial w}{\partial x}(EA)\frac{\partial u}{\partial x}dx=W(L)F(t)+\int_{0}^{L}wfdx$$ for all $${\boldsymbol{w(x)}}$$ such that $${\boldsymbol{w(0)=0}}$$

title: User:Eml4500.f08.team.figuerrez/hw6


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User:Eml4500.f08.delta 6.krueger/HW K^i

Finding the element stiffness matrices for the 2 bar system The following is an equation that was derived earlier. It will ultimately be used to find the two element stiffness matrices for the system. Eqn (0) $$\mathbf{k^{(i)}}=\int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}}{\mathbf{B^T(\tilde{x})}}EA(\tilde{x})\mathbf{B(\tilde{x}})d(\tilde{x}) $$ Eqn (1)

$$ N_{i}(x)=N_1^{(i)}(\tilde{x}) $$

Eqn (2)

$$ N_{i+1}(x)=N_2^{(i)}(\tilde{x}) $$ Eqn (3)

$$ A(\tilde{x})=N_1^{(i)}(\tilde{x})A_1+N_2^{(i)}(\tilde{x})A_2 $$

Eqn (4)

$$ E(\tilde{x})=N_1^{(i)}(\tilde{x})E_1+N_2^{(i)}(\tilde{x})E_2 $$

Eqn (5) $$B(x)=\begin{bmatrix} N_i(x) & N_{i+1}(x) \end{bmatrix}$$ Eqn (6) $$ B(x)^{T}=\begin{Bmatrix} N_{i}(x)\\ N_{i+1}(x) \end{Bmatrix}$$ To transform the matrices of B(x) and $$B(x)^{T}$$ into a function of $$\tilde{x}$$ Eqn (1) and (2) must be substituted into the B(x) and the $$B(x)^T$$ equation. Then the $$B(\tilde{x})$$, $$B(\tilde{x})^T$$, Eqn (3) and Eqn (4) have to be substituted into Eqn (0).This gives the following equations: Eqn (7) $$k^{(i)}=(N_{1}^{(i)}(\tilde{x})E_{1}+N_{2}^{(i)}(\tilde{x})E_{2})\begin{bmatrix}N_{1}^{(i)}(\tilde{x}) & N_{2}^{(i)}(\tilde{x})\end{bmatrix}\begin{Bmatrix}N_{1}^{(i)}(\tilde{x}) \\ N_{2}^{(i)}(\tilde{x}) \end{Bmatrix}(N_{1}^{(i)}(\tilde{x})A_{1}+N_{2}^{(i)}(\tilde{x})A_{2})$$ $$-(N_{1}^{(i)}(\tilde{x})E_{1}+N_{2}^{(i)}(\tilde{x})E_{2})\begin{bmatrix}N_{1}^{(i)}(\tilde{x}) & N_{2}^{(i)}(\tilde{x})\end{bmatrix}\begin{Bmatrix}N_{1}^{(i)}(\tilde{x}) \\ N_{2}^{(i)}(\tilde{x}) \end{Bmatrix}(N_{1}^{(i)}(\tilde{x})A_{1}+N_{2}^{(i)}(\tilde{x})A_{2})$$ The following values are known for the given variables. $$ (1)N_1^{(i)}(\tilde{x})= 1 $$ at $$\tilde{x}=0$$ $$ (1)N_2^{(i)}(\tilde{x})= 0 $$ at $$\tilde{x}=0$$ $$ (2)N_2^{(i)}(\tilde{x})= 1 $$ at $$\tilde{x}=L^{(i)}$$ $$ (2)N_1^{(i)}(\tilde{x})= 0 $$ at $$\tilde{x}=L^{(i)}$$ The first term in the element stiffness formula, Eqn (7), is evaluated at$$\tilde{x}=L^{(i)}$$ and the second term is evaluated at $$\tilde{x}=0$$. The above values are substituted into Eqn (7) and the $$B(\tilde{x})$$ and $$B(\tilde{x})^{T}$$ matrices are multiplied out to simplify the equation. This gives: Eqn (8) $$k^{(i)}=(E_{2}^{(i)}A_{2}^{(i)})\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}-(E_{1}^{(i)}A_{1}^{(i)})\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$$ For element 1 the following values are substituted into the element stiffness matrix, Eqn (8). $$k^{(i)}$$. $$E_{1}^{(1)}=2$$ $$E_{2}^{(1)}=4$$ $$A_{1}^{(1)}=.5$$ $$A_{2}^{(1)}=1.5$$ This gives the element stiffness matrix for element 1. $$k^{(1)}=\begin{bmatrix}-1 & 0 \\ 0 & 6\end{bmatrix}$$ For element 2 the same process is used, but the following is substituted into Eqn (8). $$E_{1}^{(2)}=3$$ $$E_{2}^{(2)}=7$$ $$A_{1}^{(2)}=1$$ $$A_{2}^{(2)}=3$$

This gives the element stiffness matrix for element 2. $$k^{(1)}=\begin{bmatrix}-3 & 0 \\ 0 & 21\end{bmatrix}$$

title: User:Eml4500.f08.delta 6.krueger/HW K^i


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User talk:Jeffri139

 Hello Jeffri139, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --The Jade Knight (d'viser) 21:04, 18 November 2008 (UTC)

title: User talk:Jeffri139


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Talk:Ancient Mesopotamia: The Cradle of Civilization

Really starting to look good here—I like how you're integrating Wikipedia sources, Wikisource, and other online sources, asking questions, giving recommend reading, the whole lot—I think one could really learn from this. BTW, have you read Hamblin's book Warfare in the Ancient Near East to 1600 BC? It's the definitive text for that time period. The Jade Knight (d'viser) 21:24, 18 November 2008 (UTC)

 * Thank you for the comments. I have not read the book. I have read good reviews though and I have seen it in a library. If you feel that it would be a possible good reading, added to one of the Further Reading sections of the History of Europe.

 * I believe that in the spirit of Wikiversity projects every and anybody should add s’thing useful to a project began by s’body else as long as there is consensus. I am not preparing this so that it will be mine and only mine. However, there must be communication and consensus before major changes. Dilos1 08:55, 19 November 2008 (UTC)
 * I really appreciat that kind of cooperative attitude. When I finish reading Hamblin's book myself, I'll add it if it seems appropriate (I've glanced through it, but not read most of it yet.  I own it, though).  The Jade Knight (d'viser) 09:16, 20 November 2008 (UTC)

I decided to simplify this part as it became too detailed and boring. Dilos1 07:11, 24 November 2008 (UTC)
 * That's fine, though I'm logging a record of your previos work here so that anyone who wants to explore this area further can start with what you already had. The Jade Knight (d'viser) 07:35, 24 November 2008 (UTC)

This material has rather become too much. As you suggested we better minimize it. However, I want to work a bit with it. Add and remove material and see what would be the optimum reading material etc for this section. The student should definitely be given some idea of the existence of civilization before the Greeks but on the other hand if too material is added here it may become boring and intimidating for the one who will start this course. Let me experiment with a little more, a little less of reading etc and see what it will look like. Sometimes it is better to come back at a later time to something you have worked on and see it anew. This may eventually help you find the optimum point. Dilos1 06:45, 25 November 2008 (UTC)
 * That is very true. The Jade Knight (d'viser) 10:31, 25 November 2008 (UTC)


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A course in troll sockpuppets/Hacker troll

The hacker troll takes pride in terrible spelling and a minor criminal record. He would thrive and survive in an actual prison environment, but with two lawyers for parents that isn't likely to happen. He's preparing for a career in systems engineering. Actually he's fundamentally in demand, as idiots like those who write wiki articles need someone to run their systems and anonymous will not.

His goal in life is to create a virus. he is an 3l33t interwebz haxor on steriods, just like fuax news warned us about.


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User:Bluejaspario14

Salut, je suis néo zélandais et j'avais étudié français pendant environs 7 ans. J'aimerais améliorer ma langue française, mais maintenant j'étudie l'ingénierie à l'université donc je n'ai pas le temps de prendre un course française.

title: User:Bluejaspario14


text:
User:Tomaschwutz

Just discovering wikiversity, I find it a good idea to capture a learning process and to share it with others.

I am new to wikis. Be patient with me. Constructuve criticism and guidance are welcome.

Intro

 * I will not view this page or its discussion very often, if want to come in contact with me, please, use email: thomas.wutzler web.de.
 * Browse Wikiversity
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Instaview
function ScrollToElement(theElement){ var selectedPosX = 0; var selectedPosY = 0; while(theElement != null){ selectedPosX += theElement.offsetLeft; selectedPosY += theElement.offsetTop; theElement = theElement.offsetParent; } //alert(selectedPosY); //window.scrollTo(selectedPosX,selectedPosY); //window.scrollTo(0,selectedPosY); window.content.document.documentElement.scrollTop = selectedPosY; }
 * deprecated: instead using wikEd now
 * InstaView
 * Topic in the Colloquium
 * /instaview_config.js
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 * scrolling to the InstaView priview label by javascript

var b = window.content.document.getElementById("InstaViewDump") ScrollToElement(b)

title: User:Tomaschwutz


text:
User talk:Tomaschwutz

 Hello Tomaschwutz, 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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 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! Jtneill - Talk - c 23:43, 18 November 2008 (UTC)

 Hi! My name is WikiversityJack. I am a pumpkin and a Wikiversity mascot. My aunt told me that fun + learning = Wikiversity. Let me know if I can help you out.

Invitation to Wikipedia course
Hello,

I have noticed that you are an active contributor and participant here at wikiversity. I would like ot invite you as beta tester or developer of the Wikipedia learning resources, for example the multiple-choice self-tests. Mange01 07:43, 24 November 2008 (UTC)

title: User talk:Tomaschwutz


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User:Eml4500.f08.team.figuerrez2

hw6

title: User:Eml4500.f08.team.figuerrez2


text:
User:Eml4500.f08.team.figuerrez2/hw6

Meeting 30 - Principle of Virtual Work using Integration by Parts
The integration by parts formula is used to rewrite an integral of a product of a derivative function as follows:

Give two functions $${\boldsymbol{r(x)}}$$, $${\boldsymbol{s(x)}}$$

and taking the derivatives of each of the two functions yields: $${\boldsymbol{r'}}=\frac{dr}{dx},{\boldsymbol{s'}}=\frac{ds}{dx}$$

From the equation $${\boldsymbol{(rs)'}}={\boldsymbol{r's}}+{\boldsymbol{rs'}}$$, we integrate both sides to get:

$$\int{\boldsymbol{(rs)'}}=\int{\boldsymbol{r's}}+\int{\boldsymbol{rs'}}$$

Since $$\int{\boldsymbol{(rs)'}}={\boldsymbol{(rs)}}$$ and solving for $$\int{\boldsymbol{r's}}$$ the resulting integrand is:

$$\int{\boldsymbol{r's}}={\boldsymbol{rs}}-\int{\boldsymbol{rs'}}$$

Recall the Cont. Principle of Virtual Work (PVW):

1st term: $$r(x)=(EA)\frac{du}{dx},s(x)=w(x)$$

Using integration by parts we get:

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

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

where $$(EA)(L)\frac{\partial u}{\partial x}(L,t)=N(L,t)=F$$

and   $$(EA)(0)\frac{\partial u}{\partial x}(0,t)=N(0,t)$$

Now consider the 2 boundary condition problem shown below:

At $${\boldsymbol{x=0}}$$ select $${\boldsymbol{w(x)}}$$ such that $${\boldsymbol{w(0)=0}}$$. (i.e. kinematically admissible)

Motivation: Discrete PVW applied to the equation shown below:

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

where the transpose of $${\boldsymbol{\underline{F}}}$$ is equal to $$\underline{F}^{T}=\begin{Bmatrix}F_{1} & F_{2} & F_{3} & F_{4} & F_{5} & F_{6}\end{Bmatrix}$$

$${\boldsymbol{F_{3}, F_{4}}}$$ are known reactions and $${\boldsymbol{F_{1}, F_{2}, F_{5}, F_{6}}}$$ are unknown reactions.

Since $${\boldsymbol{\underline{w}}}$$ can be selected arbitrarily, select $${\boldsymbol{\underline{w}}}$$ such that $${\boldsymbol{w_{1} = w_{2} = w_{5} = w_{6} = 0}}$$ in order to eliminate equations involving unknown reactions; rows $${\boldsymbol{1, 2, 5,}}$$ and $${\boldsymbol{6}}$$ are eliminated.

The resulting equation is:

$$\underline{K}_{2x2}*\underline{d}_{2x1}=\underline{F}_{2x1}$$   $${\boldsymbol{(1)}}$$

$$\underline{K}_{2x2}* \begin{Bmatrix} d_{3} \\ d_{4}\end{Bmatrix} = \begin{Bmatrix} F_{3} \\ F_{4}\end{Bmatrix}$$

Back to the Cont. Principle of Virtual Work (PVW), the unknown reaction is:

$$N(0,t)=(EA)(0)\frac{\partial u}{\partial x}(0,t)$$

The equation then becomes:

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

The final equation then becomes:

$$\int_{0}^{L}w(m\ddot{u})dx+\int_{0}^{L}\frac{\partial w}{\partial x}(EA)\frac{\partial u}{\partial x}dx=W(L)F(t)+\int_{0}^{L}wfdx$$ for all $${\boldsymbol{w(x)}}$$ such that $${\boldsymbol{w(0)=0}}$$

title: User:Eml4500.f08.team.figuerrez2/hw6


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Standard deviation

A measure of variance - actually the SD is the square root of the variance.

title: Standard deviation


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User:Eml4500.f08.Ateam.philbin/HW6

Continuous PVW to Discrete PVW (continued)
Langrangian Interpolation

Motivation for form of $$N_i\left(x\right)$$ and $$N_{i+1}\left(x\right)$$

1) $$N_i\left(x\right)$$ and $$N_{i+1}\left(x\right)$$ are linear (straight lines), thus any linear combination of $$N_i\left(x\right)$$ and $$N_{i+1}\left(x\right)$$ is also linear, and in particular the expression for $$u\left(x\right)$$.

$$N_i\left(x\right) = \alpha_i + \beta_i x \qquad \quad \quad \ \ (with \ \alpha_i\, +\, \beta_i\, being\ real\ numbers) $$

$$N_{i+1}\left(x\right) = \alpha_{i+1} + \beta_{i+1} x \qquad (with \ \alpha_{i+1}\ +\, \beta_{i+1}\,  being\ real\ numbers) $$

Linear combination of $$N_i\left(x\right)$$ and $$N_{i+1}\left(x\right)$$:

$$N_i\left(x\right)d_i + N_{i+1}\left(x\right)d_{i+1} = (\alpha_i + \beta_i x)d_i + (\alpha_{i+1} + \beta_{i+1} x)d_{i+1}$$

$$=(\alpha_{i}\ d_i + \alpha_{i+1}\ d_{i+1})+ (\beta_{i}\ d_{i} + \beta_{i+1}\ d_{i+1}) x$$

Since the linear combination of this function is also a function of x (as shown in the last line), the initial assertion has been confirmed.

2) Recall the equation for $$ u(x)$$ (the interpolation of $$ u(x)$$) from the previous meeting:

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

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

Using this relationship and plugging in $$x\, =\, x_i$$]

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

$$N_{i+1}\left(x\right) = \frac {x_{i} - x_{i}}{x_{i+1} - x_{i}}\ = \ 0$$

Plugging these values into the equation for u(x):

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

$$u(x_i) = (1)\, d_i + (0)\, d_{i+1}$$

title: User:Eml4500.f08.Ateam.philbin/HW6


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User talk:Ohara~enwikiversity

 Hello Ohara, 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 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! --– Mike.lifeguard | @en.wb 04:13, 19 November 2008 (UTC) – Mike.lifeguard | @en.wb 04:13, 19 November 2008 (UTC)

Your account will be renamed
Hello,

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

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

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

Sorry for the inconvenience.

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

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

title: User talk:Ohara~enwikiversity


text: File:Pitdegree.jpg

title: File:Pitdegree.jpg


title: User:Eas4200c.f08.gator.reger/Week9,10


text:
User:EAS4200c.f08.Blue.E/nov12th

$$r=(y^2+z^2)^{1/2}$$

$$T=2\int \phi dA=2c(\frac{J}{a^2}-\pi a^2)$$

$$J=\int_A r^2 dA=\frac{1}{2}\pi a^4$$

$$T= GJ\theta$$

$$\sigma_{yz}=\frac{\delta \phi}{\delta z}=-G \theta_z$$

$$\sigma_{zy}=\frac{\delta \phi}{\delta y}= -G\theta_y$$

$$\tau=\frac{T r}{J}$$

$$u_x(x,y)=0$$

i,e no warping

since

$$\gamma_{yx}=\frac{\delta_{yx}}{G}=\frac{\delta_u}{\delta_y}-\theta_z$$

$$\gamma_{zx}=\frac{\delta_{zx}}{G}=\frac{\delta_u}{\delta_z}-\theta_y$$

$$\sigma_{yx}=-G\theta_z$$

$$\sigma_{zx}=G\theta_y$$

therefore

$$\delta u=(G+1)\theta_z\ \delta y=(G+1)\theta_y \delta z$$

since there is no warping

$$\delta y= /delta z = 0$$

therefore

$$u=o$$

title: User:EAS4200c.f08.Blue.E/nov12th


text:
User:Eas4200c.f08.blue.a/Lecture 31

$$r=(y^2+z^2)^{1/2}$$

$$T=2\int \phi dA=2c(\frac{J}{a^2}-\pi a^2)$$

$$J=\int_A r^2 dA=\frac{1}{2}\pi a^4$$

$$T= GJ\theta$$

$$\sigma_{yz}=\frac{\delta \phi}{\delta z}=-G \theta_z$$

$$\sigma_{zy}=\frac{\delta \phi}{\delta y}= -G\theta_y$$

$$\tau=\frac{T r}{J}$$

$$u_x(x,y)=0$$

i,e no warping

since

$$\gamma_{yx}=\frac{\delta_{yx}}{G}=\frac{\delta_u}{\delta_y}-\theta_z$$

$$\gamma_{zx}=\frac{\delta_{zx}}{G}=\frac{\delta_u}{\delta_z}-\theta_y$$

$$\sigma_{yx}=-G\theta_z$$

$$\sigma_{zx}=G\theta_y$$

therefore

$$\delta u=(G+1)\theta_z\ \delta y=(G+1)\theta_y \delta z$$

since there is no warping

$$\delta y= /delta z = 0$$

therefore

$$u=o$$

title: User:Eas4200c.f08.blue.a/Lecture 31


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Vancouver Flower Clock Project/Dandelion

Back to Vancouver Flower Clock Project

This is a Vancouver Flower Clock Project Journal Page for Species Taraxacum officinale, Common Names: Dandelion.

General Info

 * Bloom Season

Anywhere in the city that shows signs of human neglect. Cracks in the pavement, next to dumpsters,in front lawns that have not been mowed, empty lots.
 * Habitat

Stems
Releases milky substance when cut.

Flower
Yellow, narrow, long-ish petals.

Cultivation
Grows and spreads with no effort or assistance.

Medicinal/Food Use
Greens make an excellent salad. A little bitter but still good

Notebook
Share a poem, story, memory or anecdote about this plant. Include sketches, photographs and videos as well.

Chemcials used to kill dandelions: 2,4-D, Mecoprop, glyphosphate

Grab a dandelion and pinch just beneath the flower between your thumb nail and index finger. Say: "Moma Had a Baby and Her Head Popped Off" and flick the flower head as hard as you can. Their carcasses lie drying on the sidewalk, scattered over chalk drawings of the sun, dogs and secret messages.

The number of blows it takes to rid the dandelion of the fluffy bits corresponds to the time of day (elementary school-aged children, 1990's).

Log a Bloom Observation
To log a bloom observation, sign your name by typing "~". Please indicate if the time and date of observation are different from the time of signing. Be as detailed as possible in your description of the observation. Were the blooms open or closed? How long did you watch? Where were they situated? What were the weather conditions? Were there any surrounding plants, insects? What was your response?. Take a photograph, video or scan an image and upload it using Special:Upload.

7:00am-8:00am
75.157.141.10 14:49, 26 April 2009 (UTC) Closed for the most part- just starting to open. Bright, cloudless cool spring day. Flowers in full sun.

11:00am-12:00pm
64.180.70.105 18:38, 26 April 2009 (UTC)

Open-looked out front window, about 13 c slightly overcast with sunny breaks

9:00pm-10:00pm
75.157.141.10 06:38, 2 May 2009 (UTC) Looked out bus window, Petals closed. Warm evening after a clear warm summer day. There is a group of teenage girls trying to set the seeding blumes on fire with their lighter.

10:00pm-11:00pm
75.157.141.10 05:24, 21 April 2009 (UTC)

Petals Closed


text:
Talk:A course in troll sockpuppets/Interview with a female troll

Female troll vs. trollop
Durova or Krystal, I'm curious to know about the difference between a female troll and a trollop. -- Jtneill - Talk - c 06:31, 19 November 2008 (UTC)


text:
Poetry

Poetry is a form of literary art in which language is used for its aesthetic and evocative qualities in addition to, or in lieu of, its apparent meaning. Poetry may be written independently, as discrete poems, or may occur in conjunction with other arts, as in poetic drama, hymns, lyrics, or prose poetry. It is published in dedicated magazines (the longest established being Poetry and Oxford Poetry), individual collections and wider anthologies.

Poetry and discussions of it have a long history. Early attempts to define poetry, such as Aristotle's Poetics, focused on the uses of speech in rhetoric, drama, song, and comedy. Later attempts concentrated on features such as repetition, verse form and rhyme, and emphasized the aesthetics which distinguish poetry from more objectively informative, prosaic forms of writing, such as manifestos, biographies, essays, and novels. From the mid-20th century, poetry has sometimes been more loosely defined as a fundamental creative act using language.

Poetry often uses particular forms and conventions to suggest alternative meanings in the words, or to evoke emotional or sensual responses. Devices such as assonance, alliteration, onomatopoeia, and rhythm are sometimes used to achieve musical or incantatory effects. The use of ambiguity, symbolism, irony, and other stylistic elements of poetic diction often leaves a poem open to multiple interpretations. Similarly, metaphor, simile, and metonymy create a resonance between otherwise disparate images—a layering of meanings, forming connections previously not perceived. Kindred forms of resonance may exist, between individual verses, in their patterns of rhyme or rhythm.

Some forms of poetry are specific to particular cultures and genres, responding to the characteristics of the language in which the poet writes. While readers accustomed to identifying poetry with Dante, Goethe, Mickiewicz and Rumi may think of it as being written in lines based upon rhyme and regular meter, there are traditions, such as Biblical poetry, that use other approaches to achieve rhythm and euphony. Much of modern British and American poetry is to some extent a critique of poetic tradition, playing with and testing (among other things) the principle of euphony itself, to the extent that sometimes it deliberately does not rhyme or keep to set rhythms at all. In today's globalized world poets often borrow styles, techniques and forms from diverse cultures and languages.

Forms of Poetry

 * Sonnet
 * Jintishi
 * Sestina
 * Villanelle
 * Pantoum
 * Rondeau
 * Roundel
 * Tanka
 * Haiku
 * Ruba'i
 * Sijo
 * Ode
 * Ghazal
 * Acrostic
 * Canzone
 * Cinquain
 * Other Forms

title: Poetry


text: User:Allenchainda

title: User:Allenchainda


text:
User:Mange01

A/s/l: 44, male, Sundsvall

Family: Yes.

Project: Currently giving a university level course on Wikis and Wikipedia. See Wikipedia.

title: User:Mange01


text:
Kinetic respiration analysis

under construction

following the Wikiversity:Adding content tutorial

Welcome to the learning project on Kinetic respiration analysis!

The following is a suggested list of headings for a learning project. For more information, please read the directions. Please organise the page as you see fit.

Todo

 * find similar links in soil science topics/schools

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 * Project code:
 * Suggested Prerequisites:
 * Time investment:
 * Assessment suggestions:
 * Portal: Portal:...(List main portal/s here)
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 * Department: Topic:...
 * Stream:
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In this learning project we aim to...

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Add learning materials here. See also the box below.

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Each project/lesson/activity may have a suggested reading selection. For example:
 *  ...
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 * Lesson 1:
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 * Quiz 1:
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Learning materials
Learning materials and learning projects are located in the main Wikiversity namespace (i.e. without any prefix such as 'Topic:' or 'School:'). Simply make a link to the name of the learning resource, lesson, or learning project (these are independent pages in the main namespace, and sometimes subpages when appropriate) and start writing!

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 * }

title: Kinetic respiration analysis


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User:Eas4200c.f08.wiki.b/HW6

=Mtg 33 Wednesday, November 12=

Torque can be expressed as

 $$\displaystyle T = 2 \int_A \phi dA $$

From the theory of elasticity, the torque becomes

 $$\displaystyle T = 2C ( {J \over a^2} - A) $$

Where the area, $$\displaystyle A $$, since the radius is $$\displaystyle a $$, is simply

 $$\displaystyle A = \pi a^2 $$

And where $$\displaystyle J $$ is

 $$\displaystyle J = \int_A r^2 dA = {1 \over 2} \pi a^4 $$

Using the ad-hoc method

 $$\displaystyle T = GJ \theta $$

Recall that the Prandtl Stress function of $$\displaystyle \phi $$ states that the stress in the x-direction normal to the y-axis is

 $$\displaystyle \sigma_{yx} = {\partial \phi \over \partial z} $$

Taking this partial derivative, you obtain

 $$\displaystyle \sigma_{yx} = 2C {z \over a^2} $$

Where substitution for $$\displaystyle C $$ yields

 $$\displaystyle \sigma_{yx} = - G \theta z $$  (1.1)

Similarly, the stress in the x-direction normal to the z-axis is

 $$\displaystyle \sigma_{zx} = - {\partial \phi \over \partial y} $$

Again, taking this partial derivative, you obtain

 $$\displaystyle \sigma_{zx} = -2C {y \over a^2} $$

Where, again, substitution for $$\displaystyle C $$ yields

 $$\displaystyle \sigma_{zx} = G \theta y $$  (1.2)

Also note that shear stress, $$\displaystyle \tau $$, is

 $$\displaystyle \tau = {Tr \over J} $$

Proof of No Warping
Now, using the above equations in addition to the stress strain relation, we can prove that no warping takes place.

First, recall that

 $$\displaystyle \gamma = {\sigma \over G} $$

Or more specifically in the context of this problem

 $$\displaystyle \gamma_{yx} = {\sigma_{yx} \over G} $$

 $$\displaystyle \gamma_{zx} = {\sigma_{zx} \over G} $$

Also recall that

 $$\displaystyle \gamma_{yx} = {\partial u_x \over \partial y} - \theta z $$

 $$\displaystyle \gamma_{zx} = {\partial u_x \over \partial z} + \theta y $$

Thus,

 $$\displaystyle {\sigma_{yx} \over G} = {\partial u_x \over \partial y} - \theta z $$  (2.1)

 $$\displaystyle {\sigma_{zx} \over G} = {\partial u_x \over \partial z} + \theta y $$  (2.2)

Now, substituting equation (1.1) into equation (2.1), and similarly substituting equation (1.2) into equation (2.2) the result is

 $$\displaystyle {-G \theta z \over G} = {\partial u_x \over \partial y} - \theta z $$

 $$\displaystyle {G \theta y \over G} = {\partial u_x \over \partial z} + \theta y $$

In both equations, each $$\displaystyle G $$ cancels out. Also, $$\displaystyle \theta x $$ and $$\displaystyle \theta y $$ cancel out when subtracted from each side of the respective equation, thus leaving

 $$\displaystyle {\partial u_x \over \partial y} = 0 $$

 $$\displaystyle {\partial u_x \over \partial z} = 0 $$

Thus, we can conclude that there is no warping since

 $$\displaystyle u_x (y,z) = 0 $$

title: User:Eas4200c.f08.wiki.b/HW6


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Algebra/Trinomials

Trinomials are a special kind of algebraic equation that looks like this: x²+4x+4=0

There are two main things you can do with trinomials: create them and factor them.

Creating trinomials

If you have the following expressions, you can multiply them into a trinomial following the FOIL method. The letters all stand for something different:

First Outer Inner Last

This is the method that I learned last year and it always works well for me. For more help, you may email me at isabellamasters@hotmail.com. Here is how to do the foil method:

(x+6)(x+5) Multiply the first term in each expression. x*x=x² Multiply the outer terms. x*5=5x. Multiply the inner terms. 6*x=6x. Multiply the last terms. 6*5=30

Now take all that (hope you had it written down) and add it all together like this:

x²+5x+6x+30

Since 5x and 6x both are like terms, because they both are of the same degree of x, we can add them together to get 11x.Do NOT add 5x and x², you can never, ever do that!

Factoring Trinomials You can factor trinomials by basically taking everything we just did and doing it backwards. I'll show you.

x²+13x+42

You have to find two integers that multiply up to 42 and add up to 13. What about 2 and 21? Multiplies up to 42, but together, you come up with 23. What about 6 and 7? That works! So then you just write it like before:

(x+6)(x+7)

Now let's try another, but a bit different:

(x+2)(x-1)

This one has a negative, but it isn't much harder. You should get:

x2+2x-x-2

Which adds up to...

x2+x-2

Notice how the 2 is now negative, instead of positive. That negative sign indicates that the factors have to subtract to 1 instead of add up to one. Reversing this would be the exact same way as shown above, only instead of adding the factors, you subtract them to get the middle term.

See? Math is easy!

title: Algebra/Trinomials


text:
User:Eml4500.f08.jamama.adam/homework6

Meeting 30
Integration by parts:$$ r(x), s(x) $$

$$ (rs)' = r's + rs'$$

$$ r' = \frac{dr}{dx}  ;   s' = \frac{ds}{dx}$$

$$ \underbrace{\int {(rs)'}}_{(rs)}=\int{r's} + \int{rs'}$$

$$ => \underline{\int{r's} = rs - \int{rs'}}$$

We can recall cont. PVW (Principal of Virtual Work) Eq. (3) p. 29-1

First term: $$ r(x)=(EA)\frac{\partial{u}}{\partial{x}}  ;   s(x) = W(x)$$

By intergrating by parts we cen get:

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

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

Now lets consider the following model problem: p.28-3  2 b.c.'s

At x=o, select W(x) so that W(0)=0 (i.e. Kinematically admissible)

Motivation : Discrete PVW applied to Equation on  p. 10-1

$$\mathbf{W}_{6x1}. \underbrace{\left(\left[      \right]_{6x2}\begin{Bmatrix}d3\\d4\end{Bmatrix}_{2x1} - \mathbf{F}_{6x1}\right)}_{6x1} = 0 $$     for all W

$$ \mathbf{F}^T = \begin{bmatrix} F_1&F_2&F_3&F_4&F_5&F_6\end{bmatrix}$$  with $$F_1, F_2, F_5, F_6$$: unknown reactions

Since W can be selectedarbitrarily, select W so that:

$$ W_1 = W_2 = W_5 = W_6 = 0$$; so to eliminate equations involving unknown reactions => eliminate rows 1,2,5,6;

Let equation (1) be: $$ \mathbf{K}_{2x2} \mathbf{d}_{2x1} = \mathbf{F}_{2x1}$$

with: $$ \mathbf{d}_{2x1} = \begin{Bmatrix} d_3 \\ d_4 \end{Bmatrix}$$ and: $$ \mathbf{F}_{2x1} = \begin{Bmatrix} F_3 \\ F_4 \end{Bmatrix}$$

Let us note that for the step before eq.(1)

 * $$ \mathbf{W}.\left(\mathbf{K}\mathbf{d}-\mathbf{F}\right) = 0$$ with $$ \mathbf{W} = \begin{Bmatrix} W_3 \\ W_4 \end{Bmatrix}$$

Now back to the PVW:

unknown reaction $$ N(o,t) = (EA)(0)\frac{\partial{u}}{\partial{x}}(0,t)$$

Cont. PVW :

$$W(L)f(t)- \int_0^L{\frac{dw}{dx}(EA)\frac{\partial{u}}{\partial{x}}dx} + \int_0^L{W(x)\left[f - m\ddot{u}\right]dx} = 0$$

So for all W(x) so that W(0) = 0,

$$=> \int_0^L{W(m\ddot{u})}dx + \int_0^L{\frac{dW}{dx}(EA)\frac{\partial{u}}{\partial{x}}}dx = W(L)f(t) + \int_0^L{Wf}dx$$
 * for all W so that W(0) = 0.

title: User:Eml4500.f08.jamama.adam/homework6


text:
User:Eas4200c.f08.carbon.orear/HW6/MATLAB

Fri, 07 Nov 2008, 07:33:59 EST Dear class (and TAs):

As lectured in class on Mon, 3 Nov 08, part of the HW6 problem is on plate buckling as explained in my wiki page:

http://en.wikiversity.org/wiki/User:Eas4200c.f08/Plate_buckling

Read and report in HW6 the content of this wiki page (with proper reference to this page) and do the following plate buckling analysis on the NACA airfoil.

Refer to the conceptual figure of airfoil with the points B,E,H,F defined on the airfoil as done in my lecture p.22-1 and see also the figure in HW4 by Team Aero_(Eelman):

http://en.wikiversity.org/w/index.php?title=User:Eas4200c.f08.aero.e/Week_8&oldid=356604

Code Expansion
Add the following features to your matlab code.

 * compute the normal bending stress sigma_xx in the skin panels FB (upper) and EH (lower); identify which one is compressive and which one is tensile.

averageTop = -1.0401e+006 N/m^2

Top is compressive

averageBottom =  5.3318e+005 N/m^2

 * plot the stress sigma_xx as a function of the coordinate y of the airfoil for both skin panels FB and EH.

 * average the compressive stress sigma_xx to be compare with the compressive buckling stress to be computed below. Plot the average line of sigma_xx on the same plot of sigma_xx vs y for skin panels FB and EH.

averageTop = -1.0401e+006 N/m^2

averageBottom =  5.3318e+005 N/m^2

=Buckling Code=

Plate
Next, for your report on plate buckling as mentioned above and do the following:

 * replot the figure at the top of p.55 of the MIT lecture notes with clearly defined coordinate axes, plate dimensions, and simply-supported boundary conditions.

 * use matlab to plot the perspective view of the buckling shape of a simply-supported rectangular plate, taking the coefficient c_{mn}=1 in the formula for psi_{mn}. In addition, select a = 1.5 m, b = 1.0 m, and thus a/b = 1.5.

 * Plot two figures:

 * m = 1 and n = 1

 * m = 2 and n = 1

 * consider the factor with x as variable in the expression for u_z, i.e.,

 * $$ \sin ( \frac{m \pi x}{a} )$$

 * Find the period T of this function by considering the following equation:

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

 * Show that m is indeed the number of half wave-lengths.

 * use matlab to reproduce the figure of k_c versus a/b at the top of p.57 in the MIT lecture notes; use different line styles for different curves.

Airfoil

 * consider the NACA skin panel with compressive stress sigma_xx (either skin panel FB or skin panel EH) as a flat rectangular plate, with the width b along the y axis of the rectangular plate (as shown in the figure at the top of p.55 of the MIT lecture notes) being the length FB or EH of the skin panels (you can just add the length of the infinitesimal pieces of the skin in your matlab code to find the total skin panel width).

 * The thickness t_s of the skin is now the "height" h of the plate.

 * Consider a range of aspect ratio of the plate as represented by the ratio a/b, which is to vary between 0.5 and 2.

 * Find the values of critical buckling stress sigma_xx_critical for this range of aspect ratio, and plot sigma_xx_critical versus the aspect ratio a/b.

 * compare the compressive stress sigma_xx of the skin panel FB or EH with the critical buckling stress sigma_xx_critical found above for the same range of aspect ratio of the skin panel

The airfoil will not buckle for the values shown in this graph, and is safest beyond an a/b ~ 0.3.

=Clamped= Next, since the skin panel is supported by stringers and ribs (as explained in class), the boundary conditions can be neither purely simply-supported nor purely clamped; they are somewhere in between these two types of boundary conditions.

So for the rectangular plate with clamped boundary conditions, do the following:

 * use matlab to plot the perspective view of the buckling shape u_z of the clamped rectangular plate with aspect ratio a/b = 1.5 using the expression for u_z as shown in my wiki page (with reference to Timoshenko and Gere's book); use c=1.

 * use the excel file downloaded at the web address==

 * http://www.excelcalcs.com/repository/Strength/Buckling/Flat-plate-buckling.xls/
 * (Be sure to save this excel file right away, since you may not be able to download it again within one day.)

 * to compute the critical buckling stress for the same range of aspect ratio, i.e., from 0.5 to 2, and plot sigma_xx_critical versus a/b for clamped boundary conditions on the same plot as for sigma_xx_critical versus a/b for simply-supported boundary conditions.

 * now for each aspect ratio a/b, the true critical buckling stress for the aircraft skin panel would be bounded (below and above) by the critical stress for simply-supported plate and the critical stress for clamped plate.

 * Compare this range of critical buckling stress with the average compressive stress for the airfoil skin as obtained above, and draw some conclusion about the participation of the airfoil skin to resist bending.

Sincerely, Prof. Vu-Quoc

=Code=


text:
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title: User talk:Mackenzie112195


text:
User:Eml4500.f08.group/HW6/Matlab

Fri, 07 Nov 2008, 14:00:23 EST Dear class:

As part of HW6, consider the electric pylon with the FE mesh described in the matlab code in:

Electric pylon: http://apollo.mae.ufl.edu/~vql/courses/fead/2006.fall/code.web.pages/electric_pylon.m

Scale the electric pylon so that its height is 60 m. Consider a downward vertical applied load of (- 1000 N) to be applied on the far right tip of the arm of the pylon.

Consider the pylon material made of 300 M steel with Young's modulus equal to 200 GPa (See Sun [2006], Mechanics of Aircraft Structures, Wiley, 2006, p.14; or search for info on the internet, and refer to the web page in your HW report). The cross-sectional area of each bar element is taken to be 4 cm^2.

Do the following:

Massless Pylon

 * develop a matlab code to solve the above electric-pylon problem

 * plot the underformed shape (dotted line) and the deformed shape (solid line); you may need to multiply the displacements by a magnification factor to clearly distinguish the deformed shape from the underformed shape.

 * compute the axial stress in each bar; give the highest tensile stress and the highest compressive stress.

 * Identify the element number for the bar element with the highest tensile stress and the element number for the bar element with the highest compressive stress. Indicate with an arrow on the figure of underformed and deformed shapes above the element with highest tensile stress and the element with highest compressive stress.

MaxTensileStress = 3630.5 @ Element 81 (Marked with Green Below)

MaxCompressiveStress = -3478.3 @ Element 55 (Marked with Red Below)

results =

1.0781e-005 2.1562e+006       862.48 -6.5356e-006 -1.3071e+006     -522.85 6.7339e-006 1.3468e+006       538.71 6.4556e-006 1.2911e+006       516.45 -3.5152e-006 -7.0304e+005     -281.22 -2.3798e-005 -4.7596e+006     -1903.8 -1.2325e-005 -2.465e+006      -986.01 7.1182e-006 1.4236e+006       569.46 1.9269e-005 3.8538e+006       1541.5 -2.8828e-005 -5.7655e+006     -2306.2 -1.8941e-007      -37882      -15.153 8.778e-009      1755.6      0.70224 1.9328e-005 3.8655e+006       1546.2 1.4698e-005 2.9395e+006       1175.8 -1.8494e-005 -3.6988e+006     -1479.5 -2.883e-005 -5.7661e+006     -2306.4 1.7948e-006 3.5896e+005       143.58 2.621e-005  5.242e+006       2096.8 9.8986e-006 1.9797e+006       791.89 -1.8874e-005 -3.7749e+006       -1510 -3.258e-005 -6.5159e+006     -2606.4 2.7892e-006 5.5784e+005       223.14 3.2194e-006 6.4388e+005       257.55 2.9828e-005 5.9656e+006       2386.3 -4.0852e-006 -8.1704e+005     -326.82 -1.9808e-005 -3.9616e+006     -1584.6 1.16e-005   2.32e+006       928.02 -4.0653e-006 -8.1307e+005     -325.23 -2.8778e-005 -5.7556e+006     -2302.2 -1.4237e-007      -28474       -11.39 -4.9677e-006 -9.9353e+005     -397.41 -7.6072e-008      -15214      -6.0857 2.9608e-005 5.9216e+006       2368.6 2.7907e-007       55813       22.325 -2.4358e-005 -4.8717e+006     -1948.7 6.9573e-006 1.3915e+006       556.59 -1.5116e-008     -3023.3      -1.2093 -2.8794e-005 -5.7588e+006     -2303.5 -2.427e-005 -4.854e+006      -1941.6 6.9522e-006 1.3904e+006       556.17 -9.4154e-007 -1.8831e+005     -75.323 1.129e-005  2.258e+006       903.22 -3.16e-006  -6.32e+005       -252.8 -2.0051e-005 -4.0102e+006     -1604.1 0           0            0  2.4411e-019  4.8822e-008  1.9529e-011 3.2073e-007       64147       25.659 1.3501e-006 2.7001e+005       108.01 1.3575e-006 2.7149e+005        108.6 -1.0389e-005 -2.0778e+006     -831.13 -2.2136e-005 -4.4271e+006     -1770.9 -2.9388e-005 -5.8777e+006     -2351.1 -3.6774e-005 -7.3549e+006     -2941.9 -4.331e-005 -8.662e+006      -3464.8 -4.3478e-005 -8.6957e+006     -3478.3 -1.2903e-019 -2.5807e-008 -1.0323e-011 -3.0942e-019 -6.1883e-008 -2.4753e-011 2.6985e-019  5.397e-008  2.1588e-011 1.2205e-019 2.4409e-008  9.7637e-012 -8.8522e-007 -1.7704e+005     -70.818 1.1034e-006 2.2069e+005       88.276 -9.5965e-007 -1.9193e+005     -76.772 9.9436e-007 1.9887e+005       79.549 1.0732e-005 2.1464e+006       858.56 -1.0415e-005 -2.083e+006      -833.19 1.1448e-005 2.2895e+006       915.81 -1.1321e-005 -2.2643e+006     -905.71 1.0517e-005 2.1035e+006       841.38 -1.0732e-005 -2.1464e+006     -858.56 7.0689e-006 1.4138e+006       565.51 -6.8859e-006 -1.3772e+006     -550.87 7.9177e-006 1.5835e+006       633.42 -1.8464e-005 -3.6928e+006     -1477.1 -4.8696e-007      -97393      -38.957 -1.1845e-007      -23690      -9.4762 1.5604e-007       31208       12.483 4.5239e-005 9.0479e+006       3619.2 4.6963e-019 9.3927e-008  3.7571e-011 4.5158e-005 9.0317e+006       3612.7 0           0            0  4.5256e-005  9.0511e+006       3620.5 -6.5877e-007 -1.3175e+005     -52.702 -1.4822e-006 -2.9645e+005     -118.58 8.9829e-006 1.7966e+006       718.63 2.2438e-005 4.4876e+006         1795 3.3117e-005 6.6233e+006       2649.3 3.8878e-005 7.7755e+006       3110.2 0           0            0  4.8034e-020  9.6069e-009  3.8427e-012 -2.4017e-020 -4.8034e-009 -1.9214e-012 -5.3704e-020 -1.0741e-008 -4.2963e-012

MaxTensileStress =

3620.5

MaxCompressiveStress =

-3478.3

ElementMaxTensile =

81

ElementMaxCompressive =

55

i =

81

i =

55

Statics
The truss system is statically indeterminate. It requires the use of the Principle of Virtual Work or Mechanics of Materials to solve.

Consider isolating an arbitrary node located within the truss system. Also, determine each reaction at a node as a single transverse reaction along the line of action of a beam rather than two reactions along the vertical and horizontal 2-D coordinate system.

Now begin with the node on the far right tip of the arm of the pylon as shown below:

As you can see there are two unknowns and three possible statics equations. Taking a moment however at the node in consideration reveals no conclusion and cannot be considered a possible statics equation in this example.

Using $$\sum{F_{y}} = 0$$, it is easily found that the y-component of R2 = 1000N.

Using $$\sum{F_{x}} = 0$$, it is also easily found that $$R_{1} = R_{2}cos\theta $$

From here, the final reactions cannot be computed and equations of Mechanics must be introduced (if not using FEM). The node shown above is the simplest FBD in the system.

Consider the node at (25, 30):

From this node, it is clear to see why six reactions cannot be computed with only two equations of statics.

With each node there are two, three, four, five or six unknown reactions. There are six nodes with two unknown reactions at the extreme locations of the truss system which cannot be solved as shown above. The nodes with four, five or six unknown reactions can't be computed using only two equations of statics.

The reason we cannot use the moment equation as a third equation of statics is because if the moment is taken at any point the moment equals zero. At each node, each isolated truss intersects. If the moment is taken at the node, it is clear why the moment equals zero, however at any other point it does as well (as shown in previous homework assignments). Each member is a two-force body and there is no resulting moment arm and no moment. This shows that the nodes with three unknown reactions cannot be solve as well.

Mass Considerations

 * the mass density of the 300 M steel is rho = 7.8 g/cm^3. Consider a lumped mass model in which the mass of each bar element is divided equally by two and is lumped at each of the two nodes of the bar.

 * Construct the lumped mass matrix of the electric pylon.

lumpedMassMatrix =

-391.99     -388.32      -613.61      -757.65      -610.71      -362.59      -856.16      -360.97      -545.38      -544.57      -287.87       -503.9      -285.61      -299.35      -359.77      -357.94      -297.47      -205.49      -204.06      -461.98      -460.64      -233.87      -277.46      -310.28      -212.83      -346.52      -264.92      -266.02      -345.41      -211.59      -312.25      -276.54      -232.63       -202.3      -200.19      -193.54      -194.28       -338.2      -376.58      -249.65      -267.69      -249.01      -374.79      -339.39      -261.75      -261.75

 * Use matlab to solve the generalized eigenvalue problem

 * $$K_bar v = lambda M_bar v$$

 * where K_bar is the reduced stiffness matrix, M_bar the associated mass matrix, v the eigenvector and lambda the eigenvalue.

 * Find the lowest 3 eigenpairs (lambda_i, v_i), for i=1,2,3.

 * In 3 separate plots, plot the eigenvectors v_1, v_2, v_3 as deformed shapes (solid line) superposed onto the underdeformed shape (dotted line). Again, use a magnification factor to clearly distinguish each eigenvector from the undeformed shape.

results =

1.0781e-005 2.1562e+006       862.48 -6.5356e-006 -1.3071e+006     -522.85 6.7339e-006 1.3468e+006       538.71 6.4556e-006 1.2911e+006       516.45 -3.5152e-006 -7.0304e+005     -281.22 -2.3798e-005 -4.7596e+006     -1903.8 -1.2325e-005 -2.465e+006      -986.01 7.1182e-006 1.4236e+006       569.46 1.9269e-005 3.8538e+006       1541.5 -2.8828e-005 -5.7655e+006     -2306.2 -1.8941e-007      -37882      -15.153 8.778e-009      1755.6      0.70224 1.9328e-005 3.8655e+006       1546.2 1.4698e-005 2.9395e+006       1175.8 -1.8494e-005 -3.6988e+006     -1479.5 -2.883e-005 -5.7661e+006     -2306.4 1.7948e-006 3.5896e+005       143.58 2.621e-005  5.242e+006       2096.8 9.8986e-006 1.9797e+006       791.89 -1.8874e-005 -3.7749e+006       -1510 -3.258e-005 -6.5159e+006     -2606.4 2.7892e-006 5.5784e+005       223.14 3.2194e-006 6.4388e+005       257.55 2.9828e-005 5.9656e+006       2386.3 -4.0852e-006 -8.1704e+005     -326.82 -1.9808e-005 -3.9616e+006     -1584.6 1.16e-005   2.32e+006       928.02 -4.0653e-006 -8.1307e+005     -325.23 -2.8778e-005 -5.7556e+006     -2302.2 -1.4237e-007      -28474       -11.39 -4.9677e-006 -9.9353e+005     -397.41 -7.6072e-008      -15214      -6.0857 2.9608e-005 5.9216e+006       2368.6 2.7907e-007       55813       22.325 -2.4358e-005 -4.8717e+006     -1948.7 6.9573e-006 1.3915e+006       556.59 -1.5116e-008     -3023.3      -1.2093 -2.8794e-005 -5.7588e+006     -2303.5 -2.427e-005 -4.854e+006      -1941.6 6.9522e-006 1.3904e+006       556.17 -9.4154e-007 -1.8831e+005     -75.323 1.129e-005  2.258e+006       903.22 -3.16e-006  -6.32e+005       -252.8 -2.0051e-005 -4.0102e+006     -1604.1 0           0            0  2.4411e-019  4.8822e-008  1.9529e-011 3.2073e-007       64147       25.659 1.3501e-006 2.7001e+005       108.01 1.3575e-006 2.7149e+005        108.6 -1.0389e-005 -2.0778e+006     -831.13 -2.2136e-005 -4.4271e+006     -1770.9 -2.9388e-005 -5.8777e+006     -2351.1 -3.6774e-005 -7.3549e+006     -2941.9 -4.331e-005 -8.662e+006      -3464.8 -4.3478e-005 -8.6957e+006     -3478.3 -1.2903e-019 -2.5807e-008 -1.0323e-011 -3.0942e-019 -6.1883e-008 -2.4753e-011 2.6985e-019  5.397e-008  2.1588e-011 1.2205e-019 2.4409e-008  9.7637e-012 -8.8522e-007 -1.7704e+005     -70.818 1.1034e-006 2.2069e+005       88.276 -9.5965e-007 -1.9193e+005     -76.772 9.9436e-007 1.9887e+005       79.549 1.0732e-005 2.1464e+006       858.56 -1.0415e-005 -2.083e+006      -833.19 1.1448e-005 2.2895e+006       915.81 -1.1321e-005 -2.2643e+006     -905.71 1.0517e-005 2.1035e+006       841.38 -1.0732e-005 -2.1464e+006     -858.56 7.0689e-006 1.4138e+006       565.51 -6.8859e-006 -1.3772e+006     -550.87 7.9177e-006 1.5835e+006       633.42 -1.8464e-005 -3.6928e+006     -1477.1 -4.8696e-007      -97393      -38.957 -1.1845e-007      -23690      -9.4762 1.5604e-007       31208       12.483 4.5239e-005 9.0479e+006       3619.2 4.6963e-019 9.3927e-008  3.7571e-011 4.5158e-005 9.0317e+006       3612.7 0           0            0  4.5256e-005  9.0511e+006       3620.5 -6.5877e-007 -1.3175e+005     -52.702 -1.4822e-006 -2.9645e+005     -118.58 8.9829e-006 1.7966e+006       718.63 2.2438e-005 4.4876e+006         1795 3.3117e-005 6.6233e+006       2649.3 3.8878e-005 7.7755e+006       3110.2 0           0            0  4.8034e-020  9.6069e-009  3.8427e-012 -2.4017e-020 -4.8034e-009 -1.9214e-012 -5.3704e-020 -1.0741e-008 -4.2963e-012

MaxTensileStress =

3620.5

MaxCompressiveStress =

-3478.3

ElementMaxTensile =

81

ElementMaxCompressive =

55

i =

81

i =

55

\

Vibrational Considerations

 * given that

 * $$lambda = omega^2$$

 * where omega is the circular frequency of vibration. The vibrational frequency is given by

 * $$f = omega / (2 pi)$$

 * and the vibrational period is given by

 * $$T = 1 / f = 2 pi / omega$$

 * Find the 3 lowest vibrational periods of the electric pylon.

Sincerely, Prof. Vu-Quoc

=Code=

title: User:Eml4500.f08.group/HW6/Matlab


text: Object Oriented Enterprise Modelling/Object Oriented Business Process Modelling


text:
User:Eas4200c.f08.vqcrew.g

And thus I begin anew my Aerospace Structures Wikiversity Extravaganzas!

title: User:Eas4200c.f08.vqcrew.g


text:
User:Eml4500.f08.bike.viale/homework report 6

For the sixth homework, the class was asked to evaluate the function:

$$N_i(\tilde{x})=$$

Which equals:

$$N_i(\tilde{x})=\frac{\tilde{x}-\tilde{x}_{i+1}}{\tilde{x}_{i}-\tilde{x}_{i+1}}$$

Where:

$$\tilde{x} = x - x_{i}$$

Using the linear interpolation method described in an earlier lecture and defining $$\tilde{x}$$ as the length from the beginning of the interpolation to the value that is to be found.

A description of the shape function $$N_{2}^{(i)}$$ using the $$\tilde{x}$$ notation is as follows:

$$N_{2}^{(i)}(\tilde{x}) = \frac{\tilde{x}}{L^{(i)}} = \begin{cases} & \text{0 if } \tilde{x}= 0\\ & \text{1 if } \tilde{x}= L \end{cases}$$

Where the boundary conditions are provided, at $$\tilde{x} = 0$$ and $$\tilde{x} = L$$ respectively.

The class was also asked to complete the example provided in the book on page 159 while setting $$E_{1} = E_{2} = E$$ and to let $$A(\tilde{x})$$ be linear as it was described in the previous lecture. $$\mathbf{k}^{(i)}$$ is found from the problem given in the previous lecture and compared to the expression given in the book, which is:

$$\frac{E(A_{1}+A_{2})}{L^{(i)}2}\begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix} = \mathbf{k}^{(i)}$$

As seen from the previous lecture, this result for the stiffness matrix is identically the same.

Next, the general $$\mathbf{k}^{(i)}$$ matrix from the previous lecture is to be compared with the matrix obtained by using $$\frac{(A_{1}+A_{2})}{2}$$ and $$\frac{(E_{1}+E_{2})}{2}$$ while noting that in this case $$E_{1}\neq E_{2}$$. This produces the stiffness matrix:

$$\frac{(E_{1}+E_{2})(A_{1}+A_{2})}{4L^{(i)}}\begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix} = \mathbf{k}_{ave}^{(i)}$$

The average stiffness matrix is then subtracted from the general stiffness matrix to find that:

$$\frac{(2E - E_{1}+E_{2})(A_{1}+A_{2})}{4L^{(i)}}\begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix} = \mathbf{k}^{(i)} - \mathbf{k}_{ave}^{(i)}$$

In order to interpret this, it is useful to review the Mean Value Theorem from calculus.


text: File:Bar with changing E and A.JPG

title: File:Bar with changing E and A.JPG


text: File:HW 6 mtg 33 1.JPG

title: File:HW 6 mtg 33 1.JPG