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User:Ffdssa/Mechation

content moved from Colloquium. --Abd 21:29, 12 September 2010 (UTC)

--Ffdssa 19:02, 12 September 2010 (UTC)Robert Franklin

--Ffdssa 18:53, 12 September 2010 (UTC)

Consider this: organic evolution has peaked as we enter the sixth mass extinction on Earth. Revolutionary changes are still going on, but they are in another realm. From above, the Earth looks like it is rapidly being covered by roads, machines, factories, and mechanisms. From below, our lives are rapidly being consumed by our use of computers, cellphones, audiovisual images, and mechanisms which make our lives easier or more fun. Darwin's theory of evolution applies to changes in the organic molecules of life. He established that living things have cells, respond to change, reproduce, have DNA, use energy, and grow or develop. I suggest that machines have infrastructures, respond to change, reproduce, have symbolic codes, use energy, and grow or develop. Rather than being built with organic molecules, they are built with inorganic materials, codes, and systematization. Let's say that mechanisms live too. As this epoch or era comes to an end, mechanisms are dominating life on earth.

Mechation
Let mechation be a new term used to denote a form of evolution related to machine growth, development, and stabilization. While "evolution" has historically been linked to organic life forms based on carbon molecules, the dominant growth and development on our planet over the last century has been that of machines.

Machines, mechanisms, and tools all derive from human symbolic activity. They are neither human nor organic, however. They do not exhibit the same functions as living organisms, but they change in their own way. While humans do not give "birth" to machines, they generate codes which enable machines to reproduce and multiply. At this moment, we are in a symbiotic relationship. The world automobile population doubles every other decade. This is made possible by engineering manuals, factories, finance, and distribution techniques. Slowly, with the rapid development of computer capacities, these functions are accomplished with less and less human support. In the near future, automobile design and reproduction will be entirely machine-controlled.

Although humans claim responsibility for successes and problems caused by machines, it is only because they are caught up in the transitional nature of organic evolution transforming into mechanical evolution: signs and symbols. Humans, the most advanced mammals, create thought and code. The code develops a "life" of its own and has become the "DNA" of mechanisms. Any mechanical device can be constructed or reconstructed by using codes and materials.

What force guides the transition of evolution towards mechation? Throughout evolutionary time, there has always been intention of change, adaptation, and reproduction. As organic life approaches the current mass extinction, this intention carries us on. Humans have rapidly been creating more and more machines. Even when the economy stops functioning in a recession, systems rapidly work to restore progress. There is no possibility of return: most land animals never returned to the oceans. While reduced machine dependence and usage often creates comfortable lifestyles for adventurous participants, the collective intention of the human species wills that mechation take place. Human museums may help preserve the ways of our ancestors and the beauties of their culture, but these settings will not be able to compete with the rapid innovations of lifestyle derived from machine use applied on a mass scale. Modern life choices change the nature of being human.

Why do humans seem blind to their relinquishment of planetary dominance? Most humans would not agree to give up their human ways for replacement by a mechanical device, but a subtle set of transitional stages exist which hide mechation from popular view. Over millenia, humans created tools and signs. Slowly, leadership became attached to the mastery of language. From Aristotle (reputed to have read every book of his day) to a modern Doctor of Philosophy or Master of Business Administration, there has been a progression of abstraction by use of signs that has been carried out by leaders in many societies. Highly abstracted leaders are rooted in the ideas, publications, and peer groups which constitute their identities. They may enjoy wilderness living as well, but their self esteem and self promotion are rooted in their abstracted codifications. There is often excitement as a newly published author becomes popular by rebelling from the technological society or some aspect of the status quo. With success, however, comes enmeshment into the technological society. The author is drawn into it and undergoes a shift in personal identity. This shift aligns his or her intentions with the intentions of mechation. As the current natural process of mechation overtakes evolution in importance, human leaders are trapped in their codes, royalties, status, and peer reviews; they are unable to take actions to restore human-centered lifestyles--ways of living characterized by oral cultures over many millenniums.

Below the level of leadership, the worldwide process of gaining literacy helps change intentions in populations. While highly literate humans do not sense separation from their codes, their machines, and their progress, the average world citizens differ in their worldview. They currently see the Earth being rapidly altered by the leading countries. The intentions of these citizens have been altered as well with the influx of first world media. Global economic forces mechate the economies of these countries and the populations receive immediate economic benefits that make the process seem attractive. However, as larger scales of development ensue, the consequences of urbanization, mechanization, pollution, and climate change alter the social patterns of the country. In these ways, both leaders and followers bring about the rapid mechation of Earth.

Signs and Symbols
Mammals have evolved into humans, and with this progression has come the use of signs and symbols. While a chimpanzee may have drawn a circle in the dirt with a stick and then crossed a line through it, it took millions of years for this figure to hold meaning. Now humans living in most urban areas recognize that the picture of any object with a circle around it and a single line crossing through it indicates that the object or its use is prohibited. While alphabets and the abstractions that they made possible (words like "museums" or "polls", for example) slowly codified the real world humans inhabited, the deeper functions of mechation involve identity changes in which a human's personal meaning system is manipulated by signs. The science of semiotics carefully describes these processes. Advertising can take a simply curved line (a "swoosh") and weave a person's self esteem and self image into it. The swoosh then carries a new meaning: invented by the advertiser it was invested with human intention. Now, it acts like a command embedded inside a product which prompts the use of the product. If a human understands this manipulative process, there is no "hook" and the created meanings are false and unattached. Most humans have no clear idea about the process and simply promote these meanings. Use of the product spreads, new factories open up, and suddenly the shoe with the swoosh isn't worn just to protect the feet until the shoe wears out. The shoe establishes identity for the human. Alphabets, graphic images, and logos all share in similar movements of meaning.

Technique
Half a century ago, Jacques Ellul described a concept that he called technique as a process encompassing “rationality, self-augmentation, monism, universalism, and autonomy.” He correctly saw technique as a force which "elicits and conditions social, political, and economic change." In trying to describe it, he sensed its overwhelming importance. However his writings appeared before the processes of computing and mechation radically changed all aspects of global interaction. The term he chose, technique, already had a meaning and did not serve to define and describe the later changes in mechanical culture that occurred after his books were written. Also, Ellul was a law professor unversed in evolutionary science. His primary focus was on changes brought about in social systems. Though a pioneer in thinking about these phenomena, he was unable to see the complete picture that was to unfold after he wrote his many books.

The complete picture
--Martin Lenoar 23:04, 12 September 2010 (UTC)

One way to explain what is going on today is by understanding the past. Specifically how the world we see today has evolved. If this leads to the understanding that to create machines or computers is the continuum of the evolution than it will be easier to understand that intelligent machines are not the downfall of humans. Since humans create the code which make machines intelligent then it is also up to humans to make intelligent machines their friends and accomplice in life. But in order not to be surprised by his own intelligent creation he needs to know the real world. One way to know the real world is attempted by the Action/Reaction theory which tries to establish that nothing with what evolution comes up, no matter if it is created by nature or human, is new. It can only be build on something which aready exists. Therefore by knowing the past we can explain the present and control the future. Specifically if it is understood that data processing, communication or fantasy did not start with humans. If they would not have existed before humans then there would have been no evolution.

Media
While primitives may have scratched signs in the sand, moderns have given permanence to their signs and symbols. Oral cultures passed on their symbols by word of mouth: what Buddha said was repeated over and over again for centuries. When the words were written down, they lasted millenia. Jesus Christ probably wrote very little, but his words were remembered and printed; they are now found throughout the world. The transmission of data, thought, and meaning has slowly formed into media and marketing (intentional data). Ellul was sensitive to this domain as well. Having seen how intentions were shaped by propaganda during World War II, he carefully analyzed media and its effects. Since he was primarily concerned with the intentions that Hitler created before and during the war, the word "propaganda" carries mostly political meanings in his writings.

"It is the emergence of mass media which makes possible the use of propaganda techniques on a societal scale. The orchestration of press, radio and television to create a continuous, lasting and total environment renders the influence of propaganda virtually unnoticed precisely because it creates a constant environment. Mass media provides the essential link between the individual and the demands of the technological society. . . . [I]ntelligent people can be made to swallow professed intentions by well-executed propaganda." (Propaganda: The Formation of Men's Attitudes. Trans. Konrad Kellen & Jean Lerner. New York: Knopf, 1965, p.58).

Since Ellul's writings appeared, cable television and the internet have been invented. The media now affect the consciousness of billions of humans. Instead of a poster with a provocative picture and a few carefully chosen words, we have studiously crafted audiovisual messages that rapidly change and generate intentions amongst huge populations. Carefully constructed personal representations ("activated profiles") are now beginning to shift human identity.

Infrastructure
Evolving organisms are dependent on light, water, oxygen, and organic foods, but mechating machines are not. Instead, machines need infrastructures such as roads, electric wires, libraries, optic cables, airports, or satellites and dishes. While scientists have generated adequate descriptions of processes and structures of organic life, they have not yet coherently analyzed mechation. In the realm of cultural anthropology, scientists have described the relative permanence of artifacts, codes, and physical structures that make cultural evolution possible. When a primate or hominid dies next to a stone tool, the tool lives on and serves another creature that may find it and put it to use. Ancient cities are dug up and life patterns are reconstructed from insights that scientists develop in teams supported by extensive research funding. It is likely that this same investigative energy will soon be applied to infrastructures and machine life. At this time, however, most scientists do not understand the importance and significance of planetary dominance by mechanisms. Their symbiotic relationship with mechanisms seems to hide the big picture. Like fish becoming frogs, they don't see the drastic difference between aquatic and terrestrial life. Paradoxically, scientists and the media have recently concluded that humans are causing climate change and the current mass extinction. This is incorrect: machines are causing the acceleration of global warming. Collective human intention has created highways and vehicles, coal-fired plants and electricity, and deforested farms in the interest of moving around, staying warm/cool, and eating excess protein. For millions of years, human patterns of moving, staying warm, and eating were radically different, but satisfied perceived needs. These new intentions have been generated by media, advertising, politics, economics and the technological society. The technological system has supplanted lifestyles that individual humans personally chose for themselves during the last few million years.

Global Computing
At a conference composed of scientists studying computers and the internet, a conclusion was reached that the internet itself had a “consciousness of its own.” [Markoff, John. "Scientists Worry that Machines May Outsmart Man." New York Times, July 25, 2009.] What does this mean? My own “consciousness” includes all of the codes and abstractions that I carry in my brain, the memories that can be summoned, and my perceptions or judgments of the experience around me. How does Google compare? Mechation has always been dependent on human-generated codes in libraries and places where books were preserved. The construction and service manuals for the Model T Ford were carefully created and distributed to guarantee the economic success of this early automobile. Times have changed. Such information is now stored as electronic data and its pattern of use or availability is completely different. Until recently, human thought was in complete control of data access. Since the last century this has changed. At the end of the 20th century, humans thrilled in the fact that chess masters could always beat a computer. Then “Deep Blue” beat the world chess master and there was a shift in perceptions. [Johnson, Kirk. "Endgame: It's All Work Now for Deep Blue, Chess Champ." New York Times, September 24, 1997.] Currently, no person can grasp what some mechanisms “know”. How does this relate to “human consciousness?” Highly functional computers have been with us for a few decades. How will human consciousness compare to "machine consciousness" in a few more decades? The internet has become the global computing domain, and techniques have gradually been established that have slowly modified the process the internet's function. Of primary significance, of course, is the modification of the human brain to fit the patterns of the internet: synapse and neural functions change as we invest our minds and meanings in modern multitasking and information update [Carr, Nickolas. The Shallows: How the Internet is Changing Our Brains. W.W. Norton: New York. 2010.]. With our constant investment in time on a computer, our brains change and begin to function at a "shallow" level. We cannot deeply reflect and go through the higher thinking functions that literacy has imposed on our species. Traditionally, human mental abstraction has been ordered in its complexity: know, comprehend, apply, analyze, synthesize, and evaluate. On this continuum from perceiving to judging, generations of deep thinkers have been developed. With the "shallowing down" of our mental input we can be expected to expand our perception functions, but lose those of judgment. In just a few years, meaning in human writing has rapidly shifted from complex paragraphs (including summaries) to texting (currently a dozen times per youth waking hour). As Carr notes, hyperlinks, interruptions, and embedded distractions have reduced our ability to concentrate. "Our ability to make the rich mental connections that form when we read deeply and without distraction remains largely disengaged." [Ibid. p. 122.]

With this obvious shift in our perception and judgment (who cannot notice how distracting it is?), comes another unseen change that is having a rapid effect on human abstract thinking: a subtle feedback mechanism is making constant adjustments to global intentions. Human intention, as mentioned above, is the reason we support mechation. Advertising generates most of this force. Until recently, there was a way for us to avoid most advertising: don't watch television, learn to shut off all advertising stimuli when reading a newspaper or magazine, and avoid commercial radio stations. Billboard advertising still penetrates, but can also be shut off or reduced with a little practice. Now, however, a feedback loop of advertising is shifting mass intention in a surprising way. Carr notes that idealistic Google relented (in 2002) and began to insert contextual ads which it auctioned to the highest bidder. In order to remain in place, however, the ad had to prove itself popular and get many clicks of the mouse. If no one read the ad, it would just disappear. Global computing has also established well-known records of most clicks. These records can be used to pinpoint advertising and shift intentions. Google's profits, of course, come from the advertising, and it is easy to see how this feedback loop (only a few years old) will gradually reconstruct human intention in a way that all the previous advertising in the last two centuries has not been able to do. It is hard to picture what this will look like a few decades from now, but it is easy to see the enormity of the change.

Finance and Economics
With globalization and the strong interdependence of world trade markets, business and venture capital create a background for planetary mechation. Data and business structures that controlled the establishment and spread of automobile production were once localized in a few countries; now these structures have moved around the world. When banks collapse in New York and London, worldwide economic practices change. Without stable financial and economic systems, machines are limited in their growth and ability to spread throughout the world. Paradoxically, as the world markets have recently collapsed and created a deep recession there has been a parallel development of “green” political movements whose primary agenda is the control of mechation.

Forecasts of global warming (which most scientists denied for years) were recently determined to be much too slow when compared with actual data collected since 2000. Many humans from all over the planet have united in an effort to control production of mechanisms which are currently bringing about this sixth mass extinction. It would seem obvious that this "Great Recession" is a perfect time to stop changing Earth's conditions. Every Thursday could be a “walk, bike, or stay day.” The first week of the month could be a “natural climate day” with drastically reduced heating or air conditioning. Could humans gradually establish control over the financial and economic processes that have driven the last century? Many humans were able to turn off their night lights together for five minutes a few years ago and an hour last year. Is this significant? Why can’t this reversal of lifestyle be accomplished? As mentioned earlier, intention in evolution and mechation is a force which guides development. There can be no intention of retreating to 14th century lifestyles in the modern era. Only a few species of marine mammals gave up a viable habitat on land to return to the sea. Now that machines have established patterns of servicing all human needs, mass markets cannot grind to a halt. No person or country can enforce a “walk, bike, or stay” Thursday. No person or security force can stand guard over the oil wells and force the discontinuation of their use. No humans can even stop the process of prospecting for new oil sources. As the Arctic ice disappears, many new oil wells will replace it to take advantage of the 25% of Earth’s oil deposits thought to be found there. There are just over a trillion barrels of oil still expected to be found; all of these will be consumed. What force could stop this process?

Personal Mechation
How did symbolic development transition from primates to robots? Humans played the pivotal role by incorporating symbols into their being and establishing the mechating world. Oral cultures did not have imprints of the spoken words. Without a storage facility (other than human memory), the symbolic infrastructures of mechation could not be established. With printed letters and ideographic script, however, symbolic ideas gained a life of their own in established cultural patterns. For thousands of years the progression was slow. Still photography, audio, and audiovisual imprints followed. Those humans with the most social power planted their identity into these cultural imprints to become famous. Finally, however, the human mechated identity of formated representation became possible. While "publishing" was once restricted to a privileged few, this technique now serves to spread ideas from any of a billion "publishers". Currently, half a billion humans have stored information that is print/audio/visual in formated segments that can be revealed to others in varying degrees. Embedded in the intention of creating these formats are core elements of human self esteem and values. As deep meaning is embedded into pictures and captions, personal identity shifts. People have committed suicide because of mechated identity abuse. This was uncommon when someone read another's diary in the past. As these formatted representations gain wider power over the human mind, humans will no longer be centering their self esteem and identity concepts around the non-virtual experiences of their daily lives. Previously, old journals became yellow and disintegrated, but now new emails and pages of formatted representation have different storage and different control. While a diary from decades past may not have survived, we are entering "the end of forgetting" when transparent personal data is becoming fixed in time and space. The new time permanence of mechation and the new identity that we are projecting to virtual "friends" will change the way we look at ourselves and the world.

Living with Mechation
Finally, there must be a spiritual or religious approach established by humans to adapt to a planet controlled by machines. Buddhists are able to withdraw from many aspects of experience: this provides them with a ready adaptation to a changing planet. Moslems struggle to re-establish lifestyle norms laid down many centuries ago: this will be problematic. Christians, the primary supporters of worldwide mechation, are recently noticing that humans must struggle to maintain “stewardship of the Earth.” A book in the Bible describes apocalyptic circumstances in which the air turns to smoke, floods prevail, and mass extinction occurs. As it is written, many are willing to believe. A person may drive a huge tundra-melting truck but may also believe that one day a heaven will open up to allow immortal happiness. This approach helps millions of people to be happy in today's world.

Another paradox allows mechation to slowly spread through human culture: primitives or humans steeped in oral culture do not develop the abstractions required to view the massive evolutionary changes taking place on the planet. They notice the demise of life's quality, but cannot explain it. Advanced literates, however, have wedded themselves to abstract thinking. Ludwig Wittgenstein was able to leave this thinking, donate his fortune, and withdraw to a garden; later he returned to abstract thinking. Once this canopy of thought is developed, a human rarely strays from its cover. What is a satisfactory personal approach that can provide happiness in a human’s work and play? Each person must come up with a personal formula, of course. To avoid regret or guilt humans must find ways of making money (if it is used) that does not contribute to destruction. Leisure, recreation, and lifestyle must be carefully chosen as well. If someone is accustomed to a semi-annual cross-continental flight to visit their mother, for example, the person’s spiritual awareness will be influenced by a negative emotion since the jet fuel is so toxic to Earth. Fighting mass intention and a mass extinction, of course, seems hopeless. But small personal successes can still bring small satisfactions.

Practically, humans can still find comfortable human lifestyles offering rewarding lives. Primitives in the Amazon, of course, are not burdened with abstractions and codes. They still live in close interaction with natural environments. Television often makes entry into their psycho-social patterns, however, and changes their consciousness. Economics intrudes as well when forests are removed to provide land for grazing cattle or growing soybeans for beef (or protein bars).

For humans in urban areas who are literate or post-literate, possibilities did recently exist to regain primitive awareness. Until this century, it was quite easy to step out of the technological system to see it clearly--as the first lizards which stopped entering the water must have done. Over the last few years, however, this has become extremely difficult. Previously, by merely avoiding any reading or tv/monitor use for half a year, a person could leave her or his abstracted world and enter into an alternative way of seeing things. The life in signs fades away and the life in experience brightens. Any non-virtual experience (walking, sport, sewing, weeding, etc.) becomes reality and the symbols seem to intrude on it. Many dedicated spiritual people still embark on three-month retreats. Nonetheless, with the recent advent of formated representations of your identity, your email monitoring, and your increased audiovisual device monitoring over the last decade, this discipline has become much more difficult. How can you go without looking at the screen of a cellphone or some monitor for half a year? In 1970, yes; in 2010, no.

Once the electronic and print information is bypassed, a human can further delete the creeping intentions of the mechating world by learning to avoid looking at any billboard or advertisement. This last step is hard, but takes you completely out of the grip of mass intention. Once a period of abstinence is established over months or years, reentry into the virtual world is different. Careful, meticulous use of electronic devices or print can be accomplished in ways that help preserve aspects of natural human consciousness. Advertising, somehow, no longer deserves any place in your virtual world. A cell phone, email address, or wristwatch had been considered “essential” for functioning in today’s world. After this deep reflection period, a person can decide to use each machine carefully in a way that allows a human-centered natural order to unfold. This conscientization creates an altered worldview. Is it necessary to know what time it is on Saturday? Could you spend the day without using a coin or credit card? Some of the schools in the coming century will offer new ways to examine and approach the challenges of modern lifestyle choices. Strong, intimate, human relationships that are not infiltrated by machine use (or at least excessive machine use) will continue to create satisfying lives for many people.

Personal Example
By inputing these words (thoughts I have had for four decades but have never written down), my identity changes and I become invested in exchanging codes. Responses to these words now have meaning for me. Others might share my thoughts. I am preparing to spend large numbers of hours typing into a computer, adding words to the trillions of electronic words being produced. I am creating abstractions upon which my sense of identity and self esteem will depend. This experience of data exchange could largely replace the extremely experiential life I currently enjoy!

The Stages of Mechation
Time is a variable -/ Time is a constant

STAGES OF EVOLUTION; TIME VARIABILITY OF ORGANISMS

STAGES OF MECHATION; TIME CONSTANCY OF MECHANISMS

Infrastructure

INFRASTRUCTURE OF ORGANISMS:

LAND, WATER, AIR

INFRASTRUCTURE OF MECHANISMS:

ROADS, INFORMATION CONDUITS, POWER CONDUITS, FINANCIAL CONDUITS, MALLS, COLLEGES

Information Bank

SCROLLS, PAPERS, BOOKS, AND DIRECTORIES; LIBRARIES; COMPUTERS; DATA BANKS

Production Center

FARMS; FACTORIES; LABORATORIES

Market

PEDDLERS, TINKERS, BAZAARS, SHOPS, STORES, DEPARTMENT STORES

Intention Center

SCRIPTORIUM, COUNCIL, AGENCY, PERSONALLY FORMATED REPRESENTATION

What do you think??

title: User:Ffdssa/Mechation


text: File:Natbasisvectors.JPG

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

title: File:Natbasisvectors.JPG


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User:Egm6321.f10.team03.sigillo/Hwk1

=Homework One Summary= Problems assigned in the first two weeks of class.

= Problem 1 - Maglev First and Second Derivative Derivations =

Reading convetion
A function $$\,f^1$$ or $$\,f^1(.)$$ has to be read as: "$$f$$ superscritp 1". A function $$\,(f^1)^2$$ or $$\,(f^1(.))^2$$has to be read as: "$$f$$ superscritp 1, power 2".

Given
Based on the definitions and equations in Maglev Train EOM the axial deformation of the guidance for a High-Speed Train is represented by a function $$\,u^1(s,t)$$. The argument $$\,s$$ itself is also a function of the timevariable $$\,t$$, so that for a general function $$\,f$$ the following valid is:
 * {|style="width:100%" border="0"

$$\,{{\left. f(s,t) \right|}_{s={{y}^{1}}(t)}}=f({{y}^{1}}(t),t)$$
 * $$\,(Eq.1.1)$$
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Find
The first and second derivation of $$\,f({{y}^{1}}(t),t)$$ has to be calculated with respect to $$\,t$$.

Solution
First derivation $$\frac{df}{dt}$$: Let's set
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$$\,z:=f({{y}^{1}}(t),t)$$ $$ and
 * $$\,(Eq. 1.2)
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$$\,s:={{y}^{1}}(t)$$ $$ thus we get for $$z$$ :
 * $$\,(Eq. 1.3)
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$$\,z=f(s,t)$$ $$ The absolute derivation of $$z$$ is:
 * $$\,(Eq. 1.4)
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$$\,dz={{f}_{s}}\ ds+{{f}_{t}}\ dt$$ $$ Following the suggestion in Papula, we devide $$(Eq.1.5)$$ by $$\,dt$$ and get:
 * $$\,(Eq. 1.5)
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$$\frac{dz}{dt}={{f}_{s}}\ \frac{ds}{dt}+{{f}_{t}}$$ $$ substituting $$z$$ gives:
 * $$\,(Eq. 1.6)
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$$\frac{df(s,t)}{dt}=\frac{\partial f(s,t)}{\partial s}\frac{\partial s}{\partial t}+\frac{\partial f(s,t)}{\partial t}$$ $$ substituting $$s$$ this time, gives:
 * $$\,(Eq. 1.7)
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$$\frac{df({{y}^{1}}(t),t)}{dt}=\frac{\partial f({{y}^{1}}(t),t)}{\partial s}\frac{\partial {{y}^{1}}(t)}{\partial t}+\frac{\partial f({{y}^{1}}(t),t)}{\partial t}$$ $$ So that we can finally say:
 * $$\,(Eq. 1.8)
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$$\frac{df}{dt}={{f}_{s}}\cdot \dot{y}^{1}+{{f}_{t}}$$ $$
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 * style="width:2%; padding:1px; border:2px solid #FF0000" |
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 * $$\,(Eq. 1.9)
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'''Second derivation $$\frac{{{d}^{2}}f}{d{{t}^{2}}}$$''': The procedure to calculate the second derivative of $$f$$ is the same used to calculate the first derivative. For a better overview let's assigne $$(Eq. 1.9)$$ to a new variable and also parts of the equation to new variables like follows:
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$$\,q:= {{f}_{s}}\cdot \dot{y}^{1}+{{f}_{t}}$$ $$\,r:= {{f}_{s}}$$ $$\,m:= \dot{y}$$ $$\,h:= {{f}_{t}}$$ Equation $$(Eq. 1.9)$$ can then be written as:
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$$\,q=r\cdot m+h$$ $$ Similarly to the calculation of the first derivative, we calculate the total derivative of $$q$$, paying attention to the fact that now we have to derive the prduct $$(r\cdot m)$$ with $$r$$ being a function of $$s$$ and $$t$$ :
 * $$\,(Eq. 1.10)
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$$dq=({{r}_{s}}\cdot ds+{{r}_{s}}\cdot dt)\cdot m+r\cdot {{m}_{t}}\cdot dt+{{h}_{t}}\cdot dt+{{h}_{s}}\cdot ds$$ $$ multiplying $$m$$ into the braket expression gives:
 * $$\,(Eq. 1.11)
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$$dq={{r}_{s}}\cdot ds\cdot m+{{r}_{s}}\cdot dt\cdot m+r\cdot {{m}_{t}}\cdot dt+{{h}_{t}}\cdot dt+{{h}_{s}}\cdot ds$$ $$ Dividing also here $$dq$$ by $$dt$$ and substituiting $$r, m, h$$ with their original terms we get:
 * $$\,(Eq. 1.12)
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$$\frac{dq}{dt}=\frac{\partial {{f}_{s}}}{\partial s}\ \frac{ds}{dt}\ {{{\dot{y}}}^{1}}+\frac{\partial {{f}_{s}}}{\partial t}\ {{{\dot{y}}}^{1}}+{{f}_{s}}\frac{\partial {{{\dot{y}}}^{1}}}{\partial t}+\frac{\partial {{f}_{t}}}{\partial t}+\frac{\partial {{f}_{t}}}{\partial s}\ \frac{ds}{dt}$$ $$ That in fact after deriving becomes:
 * $$\,(Eq. 1.13)
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 * }
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$$\frac{dq}{dt}={{f}_{ss}}\cdot {{({{{\dot{y}}}^{1}})}^{2}}+{{f}_{st}}\cdot {{{\dot{y}}}^{1}}+{{f}_{s}}\cdot {{{\ddot{y}}}^{1}}+{{f}_{tt}}+{{f}_{ts}}\cdot {{{\dot{y}}}^{1}}$$ $$ Since the derivation sequence doesn't matter for partial derivations, we finally after ordring the terms in $$(Eq. 1.14) $$ get for the second derivative:
 * $$\,(Eq. 1.14)
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$$\frac{{{d}^{2}}f}{d{{t}^{2}}}={{f}_{ss}}\cdot {{({{{\dot{y}}}^{1}})}^{2}}+2{{f}_{st}}\cdot {{{\dot{y}}}^{1}}+{{f}_{s}}\cdot {{{\ddot{y}}}^{1}}+{{f}_{tt}}$$ $$
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 * style="width:2%; padding:1px; border:2px solid #FF0000" |
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 * $$\,(Eq. 1.15)
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= Problem 2 - Dimensional Analysis=

Given
From the Maglev Train EOM we are given the horizontal force on the wheel/magnet $$\,{{c}_{0}}({{y}^{1}},t)$$ :
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$$\,{{c}_{0}}({{y}^{1}},t)=-{{F}^{1}}[1-\bar{R}\cdot u_{ss}^{2}({{y}^{1}},t)]-{{F}^{2}}\cdot u_{s}^{2}-\frac{T}{R}+M[[1-\bar{R}\cdot u_{ss}^{2}][u_{tt}^{1}-\bar{R}\cdot u_{stt}^{2}]+u_{s}^{2}\cdot u_{tt}^{2}]$$ $$ with :
 * $$\,(Eq. 2.1)
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$$\,[{{F}^{1}}]=F(force)$$ $$\,[T]=T(torque)$$ $$\,[M]=M(mass)$$ $$\,[\bar{R}]=L(lenght)$$ $$\,[t]=t(time)$$ $$\,[{{u}^{2}}]=L(length)$$ $$\,[u_{ss}^{2}]=\frac{L}={{L}^{-1}}(1/length)$$ $$\,[1]=1$$
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Find
Dimensional Analysis of the terms in $$(Eq. 2.1)$$ and physical meaning.

Solution
Analysis of the first term $$\,(-{{F}^{1}}[1-\bar{R}\cdot u_{ss}^{2}({{y}^{1}},t)])$$ :
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$$\,F[1-L\cdot {{L}^{-1}}]=F[1-1]\to$$ Dimension results to be a Force Analysis of the second term $$\,(-{{F}^{2}}\cdot u_{s}^{2}-\frac{T}{R})$$ :
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$$F\cdot \frac{L}{L}-\frac{T}{L}=F-F\to $$ Dimension results to be a Force Analysis of third term $$\,(M[[1-\bar{R}\cdot u_{ss}^{2}][u_{tt}^{1}-\bar{R}\cdot u_{stt}^{2}]+u_{s}^{2}\cdot u_{tt}^{2}])$$ :
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$$\,M\left[ \left[ 1-L\cdot {{L}^{-1}} \right]\left[ \frac{L}-L\cdot \frac{L}{L\cdot {{t}^{2}}} \right]+\frac{L}{L}\cdot \frac{L} \right]=M\cdot \frac{L}+M\cdot \frac{L}=F+F\to $$ Dimension results to be a Force
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title: User:Egm6321.f10.team03.sigillo/Hwk1


text:
User:EGM6321.F10.TEAM1.HW1

Problem 1: Derive (3) and (4) on p1-3
This solution is based on the one previously found by Vasquez et al. The primary improvement is the inclusion of a background section, where discussion of why these equations are important is included.

Background A very common use for the time derivative is in consideration of applied forces. Newton's 2nd law is based in an inertial frame, whereas, whereas most physical systems studied are referenced to a non-inertial coordinate system (the rotating earth). This gives rise to several inertial forces, which go to zero when an inertial reference frame is considered. Amongst these are the Coriolis force and the centrifugal force.

The Coriolis force is calculated as

$${{F}_{c}}=-2m\Omega \times v$$

where m represents the mass of the body, $${\Omega}$$ is the angular velocity of the rotating coordinate system, and v is the linear velocity of the body referenced to the rotating frame. The Coriolis effect of the Earth's rotation can often be ignored, but there are certain engineering applications. Long range ballistic missiles tend to veer off from their intended path if the Coriolis effect is not considered. The Earth's Coriolis effect also causes certain types of waves, the rotation of cyclones, and creates the jet streams and western boundary currents in the oceans. Various other systems may have important Coriolis contributions including rolling jets and rotating missile/gun turrets.

Another important force caused by a rotating frame of reference is the centrifugal force. This force can be defined mathematically as:

$$F=-m\Omega \times (\Omega \times r)$$

where $$ \Omega $$ is defined as above and r is the radius from the center of rotation of the frame. The centrifugal force acts outward from the center of rotation. Some examples of use for this force are: slings, artificial gravity through rotation, spin casting where liquid metal is placed inside of a tube and then the tube spun until the metal is dry thus creating a hollow metal cylinder, and centrifuges to separate materials based on their mass.

Given
The equation

$$f({{y}^{1}}(t),t)$$.

can be used to model the motion of a high speed train.

Find

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$$\frac{\partial }{\partial t}f({{y}^{1}}(t),t)=f{{,}_{s}}{{y}^{1}}{{,}_{t}}+f{{,}_{t}}$$

(p1-3.3)
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$$\frac{d{{t}^{2}}}({{y}^{1}},t)=f{{,}_{s}}{{y}^{1}}{{,}_{tt}}+f{{,}_{ss}}{{\left( {{y}^{1}}{{,}_{t}} \right)}^{2}}+2f{{,}_{st}}{{y}^{1}}{{,}_{t}}+f{{,}_{tt}}$$

(p1-3.4)
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Solution
Equation 3 First, for compactness define:

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$$s:={{y}^{1}}(t)$$

(p1-3.3) so that the given term can be re-written as:
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$$f(s,t)$$

(1.2)
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Next, take a derivative with respect to time of this term. This gives the equation:

$$\frac{\partial }{\partial t}f(s,t)=\frac{\partial f}{\partial s}\frac{\partial s}{\partial t}+\frac{\partial f}{\partial t}\frac{\partial t}{\partial t}$$

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$$\frac{\partial }{\partial t}f(s,t)=\frac{\partial f}{\partial s}\frac{\partial s}{\partial t}+\frac{\partial f}{\partial t}$$

(1.3)
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Then substituting back in the definition of s (1):

$$ \frac{d}{dt}f(y^1(t),t)= \frac{\partial f(y^1(t),t)}{\partial s} \frac{\partial y^1}{\partial t}+ \frac{\partial f(y^1(t),t)}{\partial t} $$

Finally, using the notation:

$${{y}^{1}}{{,}_{t}}=\frac{d}{dt}{{y}^{1}}(t)$$

The final equation can be expressed as:

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 * $$\frac{\partial }{\partial t}f({{y}^{1}}(t),t)=f{{,}_{s}}{{y}^{1}}{{,}_{t}}+f{{,}_{t}}$$

(p1-3.3)
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Equation 4 Begin with equation 1.3:

$$\frac{\partial }{\partial t}f(s,t)=\frac{\partial f}{\partial s}\frac{\partial s}{\partial t}+\frac{\partial f}{\partial t}$$

Then, taking the time derivative term-wise will produce:

$$\frac{\partial }{\partial t}\left( \frac{\partial f}{\partial t} \right)=\frac{{{\partial }^{2}}f}{\partial {{s}^{2}}}\frac{\partial s}{\partial t}\frac{\partial s}{\partial t}+\frac{{{\partial }^{2}}f}{\partial t\partial s}\frac{\partial s}{\partial t}\frac{\partial t}{\partial t}+\frac{\partial f}{\partial s}\frac{\partial {{s}^{2}}}{\partial t\partial s}+\frac{\partial f}{\partial s}\frac{{{\partial }^{2}}s}{\partial {{t}^{2}}}\frac{\partial t}{\partial t}$$

$$\frac{{{\partial }^{2}}f}{\partial {{t}^{2}}}=\frac{{{\partial }^{2}}f}{\partial {{s}^{2}}}\frac{\partial s}{\partial t}\frac{\partial s}{\partial t}+\frac{{{\partial }^{2}}f}{\partial t\partial s}\frac{\partial s}{\partial t}+\frac{\partial f}{\partial s}+\frac{\partial f}{\partial s}\frac{{{\partial }^{2}}s}{\partial {{t}^{2}}}$$

and again substituting in (1) and switching to comma notation will give the final equation:

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$$\frac{d^2f}{dt^2}=f,_s(y^1,t) \frac{d^2y^1}{dt^2}+f,_{ss}\left(\frac{dy^1}{dt} \right)^2+2f,_{st} \frac{dy^1}{dt}+f,_{tt} $$ (p1-3.4)
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Author

User:Egm6321.f10.team1.allison

Given

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$${{c}_{o}}({{y}^{1}},t)=-{{F}^{1}}[1-\overline{R}({{u}^{2}}{{,}_{ss}})({{y}^{1}},t)]-{{F}^{2}}{{u}^{2}}{{,}_{s}}-\frac{T}{R}+M[[1-\overline{R}({{u}^{2}}{{,}_{ss}})][{{u}^{1}}{{,}_{tt}}-\overline{R}({{u}^{2}}{{,}_{stt}})]+{{u}^{2}}{{,}_{s}}{{u}^{2}}{{,}_{tt}}]$$

(p 2-2.3)
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Find
Perform a dimensional analysis on all terms in the equation and describe the physical meaning of each.

Solution
Dimensional analysis Individual variables The abbreviations used herein are: L represents length, M for mass, F for force, and t for time $$\begin{align} & \left[ {{F}^{1}} \right]=F \\ & \left[ {\bar{R}} \right]=L \\ & [u,{}_{ss}]={}^{L}\!\!\diagup\!\!{}_\; \\ & \left[ {{y}^{1}} \right]=L \\ & [t]=t \\ & [{{F}_{2}}]=F \\ & \left[ {{u}^{2}}{{,}_{s}} \right]=L \\ & [M]=m \\ & \left[ u{{,}_{tt}} \right]={}^{L}\!\!\diagup\!\!{}_\; \\ & \left[ u{{,}_{stt}} \right]={}^{1}\!\!\diagup\!\!{}_\; \\ & \left[ {{c}_{o}} \right]=F \\ & [T]=F\times L \\ & [R]=L \\ \end{align}$$

Terms Used
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$$[{{F}^{1}}[1-\overline{R}({{u}^{2}}{{,}_{ss}})({{y}^{1}},t)]]=F(1-L{{L}^{-1}})=F$$ (2.1)
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$$[{{F}^{2}}{{u}^{2}}{{,}_{s}}]=F, [\frac{T}{R}]=T{{L}^{-1}}=F$$ (2.2),(2.3)
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$$[M1-\overline{R}({{u}^{2}}{{,}_{ss}})][{{u}^{1}}{{,}_{tt}}-\overline{R}({{u}^{2}}{{,}_{stt}})]+{{u}^{2}}{{,}_{s}}{{u}^{2}}{{,}_{tt}} =M((1-L{{L}^{-1}})(L{{S}^{-2}}-LL{{L}^{-1}}{{S}^{-2}})+(L{{L}^{-1}}L{{S}^{-2}}))=ML{{S}^{-2}}=F$$ (2.4)
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$$\therefore[{{c}_{o}}({{y}^{1}},t)]=F$$ (2.5)
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Physical Meaning

Term (1) : $${{u}^{2}}{{,}_{ss}}$$ is the second derivative of axial deformation of guideway with respect to s. $${{u}^{2}}{{,}_{ss}}$$ means curvature of deformed guideway. Physical meaning of Equation (1) is changed transverse force $${{F}^{1}}$$ by guideway deformation.

Term (2) : $${{u}^{2}}{{,}_{s}}$$ is an approximation of slope of deformed guideway. $${{F}^{2}}$$ is a axial force applied to the wheel. Physical meaning of $${{F}^{2}}{{u}^{2}}{{,}_{s}}$$ is transverse force, which is generated by axial force with deformed guideway.

Term (3) : $${T}$$ is a torque applied to the wheel. $${R}$$ is a radius of wheel. Physical meaning of $$\frac{T}{R}$$ is rotating force applied to wheel.

Term (4) : $${{u}^{2}}{{,}_{stt}}$$ is acceleration of the slope. Generated by the guideway is deforming up and down. $${{u}^{1}}{{,}_{tt}}$$ is the second derivative of transverse displacement of the flexible guideway. $${{u}^{2}}{{,}_{tt}}$$ is the second derivative of axial displacement of the flexible guideway. $${M}$$ is a mass of wheel. Physical meaning of Equation (4) is transverse force generated by acceleration of wheel.

Term (5) : Physical meaning of $${{c}_{o}}({{y}^{1}},t)$$ is sum of transverse force acting on wheel.

Author

User:EGM6321.f10.team1.Kim.TH

Given

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$${{c}_{3}}({{y}^{1}},t){{\ddot{y}}^{1}}+{{c}_{2}}({{y}^{1}},t){{({{\dot{y}}^{1}})}^{2}}+{{c}_{1}}({{y}^{1}},t){{\dot{y}}^{1}}+{{c}_{0}}({{y}^{1}},t)=0$$

(p2-2.3)
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Find
Prove that
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$${{c}_{3}}({{y}^{1}},t){{\ddot{y}}^{1}}$$

is nonlinear with respect to $$ y^1 $$.
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Solution
To prove that $${{c}_{3}}({{y}^{1}},t)\frac{d{{t}^{2}}}$$ is a nonlinear equation with respect to $${{y}^{1}}$$, we will consider a specific case for $${{c}_{3}}({{y}^{1}},t)$$. In an effort to make obvious the nonlinear nature of the previous equation, we choose the trigonometric function sine to be included in our definition of $${{c}_{3}}({{y}^{1}},t)$$.

Let

$${{c}_{3}}(y,t)=\sin (y)+{{t}^{2}}$$

Next we need to define a criterion to determine whether or not a given equation is linear or nonlinear. For L to be considered a linear operator, it must have the following properties:

$$L(\phi +\psi )=L(\phi )+L(\psi )$$

$$L(k\phi )=kL(\phi )$$

where k is any scalar.

In order to test our newly formed rules for linearity (and simplify the writing), lets express the equation

$${{c}_{3}}({{y}^{1}},t)\frac{d{{t}^{2}}}=[\sin ({{y}^{1}})+{{t}^{2}}]\frac{d{{t}^{2}}}$$

as an operator L.

Let

$$L(\cdot )=[\sin (\cdot )+{{t}^{2}}]\frac{{{d}^{2}}(\cdot )}{d{{t}^{2}}}$$

We may now apply the prior definitions for linearity as a test for whether L is linear or nonlinear. First we check if the distributive property holds by computing the LHS of the defining equation for linear operators.

$$L(\phi +\psi )=[\sin (\phi +\psi )+{{t}^{2}}]\frac{{{d}^{2}}(\phi +\psi )}{d{{t}^{2}}}$$

and using the trigonometric identity for sine:

$$\sin (\phi +\psi )=\sin (\phi )\cos (\psi )+\cos (\phi )\sin (\psi )$$

we arrive at the following:

$$=[\sin (\phi )\cos (\psi )+\cos (\phi )\sin (\psi )+{{t}^{2}}][\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\frac{{{d}^{2}}\psi }{d{{t}^{2}}}]$$

which then becomes

$$=[\sin (\phi )\cos (\psi )[\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\frac{{{d}^{2}}\psi }{d{{t}^{2}}}]+\cos (\phi )\sin (\psi )[\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\frac{{{d}^{2}}\psi }{d{{t}^{2}}}]+{{t}^{2}}[\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\frac{{{d}^{2}}\psi }{d{{t}^{2}}}]$$

and expanding into the 6 term expression we have

$$=[\sin (\phi )\cos (\psi )\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\sin (\phi )\cos (\psi )\frac{{{d}^{2}}\psi }{d{{t}^{2}}}+\cos (\phi )\sin (\psi )\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\cos (\phi )\sin (\psi )\frac{{{d}^{2}}\psi }{d{{t}^{2}}}+{{t}^{2}}\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+{{t}^{2}}\frac{{{d}^{2}}\psi }{d{{t}^{2}}}$$

We will see in a moment why we picked the trigonometric function sine to demonstrate a test for linearity of an operator. Now we compute the RHS of the defining equation for linear operators.

$$L(\phi )+L(\psi )=[\sin (\phi )+{{t}^{2}}]\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+[\sin (\psi )+{{t}^{2}}]\frac{{{d}^{2}}\psi }{d{{t}^{2}}}$$

distributing the derivatives over the sums we arrive at the following 4 term expression

$$=\sin (\phi )\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+{{t}^{2}}\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\sin (\psi )\frac{{{d}^{2}}\psi }{d{{t}^{2}}}+{{t}^{2}}\frac{{{d}^{2}}\psi }{d{{t}^{2}}}$$

As shown, the nonlinearities in L with respect to $${{y}^{1}}$$ are obvious in the sine term, and L as defined does not distribute over the given sum. If L did distribute, then we would have seen equivalent expressions between the RHS and LHS of the defining equation for linearity (i.e. the RHS gave 6 terms which were not equivalent to the 4 terms the LHS gave). Therefore, $${{c}_{3}}({{y}^{1}},t)\frac{d{{t}^{2}}}$$ is a nonlinear equation with respect to $${{y}^{1}}$$.

Egm6321.fall10.team1.lang 07:23, 14 September 2010 (UTC)

Given
given the boundary values:

$$\begin{align} & y(a)=\alpha \\ & y(b)=\beta \\ \end{align}$$.

Find
Solve for variables c and d in the following equation:

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$$y(x)=cy_{H}^{1}(x)+dy_{H}^{2}(x)+{{y}_{p}}(x)$$

(p5-3)
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Solution
Substituting the boundary point data into the given equation yields:

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$$y(a)= \alpha = cy_{H}^{1}(a)+dy_{H}^{2}(a)+{{y}_{p}}(a)$$ (4.1),(4.2)
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$$y(b)= \beta = cy_{H}^{1}(b)+dy_{H}^{2}(b)+{{y}_{p}}(b)$$ (4.3),(4.4)
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Using algebra to solve equations (4.2) and (4.4) for variable c yields:

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$$c = \frac{1}{y_{H}^{1}(a)}[\alpha - dy_{H}^{2}(a)-{{y}_{p}}(a)]$$ (4.5)
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$$c = \frac{1}{y_{H}^{1}(b)}[\beta - dy_{H}^{2}(b)-{{y}_{p}}(b)]$$ (4.6)
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Setting equations (4.5) and (4.6) equal to one another by recognizing that c=c yields:

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$$\frac{1}{y_{H}^{1}(a)}[\alpha - dy_{H}^{2}(a)-{{y}_{p}}(a)]= \frac{1}{y_{H}^{1}(b)}[\beta - dy_{H}^{2}(b)-{{y}_{p}}(b)]$$ (4.7)
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Solving equation (4.7) for variable d using algebra:

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$$\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}[\alpha - dy_{H}^{2}(a)-{{y}_{p}}(a)]= \beta - dy_{H}^{2}(b)-{{y}_{p}}(b)$$ (4.8)
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$$\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}[\alpha -{{y}_{p}}(a)]+{{y}_{p}}(b)-\beta = \frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}dy_{H}^{2}(a) - dy_{H}^{2}(b) = d(\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}y_{H}^{2}(a) - y_{H}^{2}(b))$$ (4.9),(4.10)
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$$d = \frac{\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}[\alpha -{{y}_{p}}(a)]+{{y}_{p}}(b)-\beta}{\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}y_{H}^{2}(a) - y_{H}^{2}(b)}$$ (4.11)
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Equation (4.11) is the final answer for variable d.

Now, variable c may be found by substituting equation (4.11) into equation (4.5):

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$$c = \frac{1}{y_{H}^{1}(a)}[\alpha - \frac{\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}[\alpha -{{y}_{p}}(a)]+{{y}_{p}}(b)-\beta}{\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}y_{H}^{2}(a) - y_{H}^{2}(b)}y_{H}^{2}(a)-{{y}_{p}}(a)]$$ (4.12)
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Equation (4.12) is the final answer for variable c, and so the solution is now complete.

EGM6321.f10.team1.russo 22:23, 13 September 2010 (UTC)

Given
Legendre's Equation with n=1 and two solutions
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$$\begin{align} & {(1-{{x}^{2}})}{y_H}''-{2x}{y_H}'+{2}{y_H}=0 \\ \end{align}$$

(5.1)
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$$ y_H^1=x $$ and $$ y_H^2=\frac{x}{2} \log \left( \frac{1+x}{1-x} \right ) - 1 $$

Find
Show that:

$$ y_{H}^{1} $$ and $$ y_{H}^{2} $$ are both valid homogeneous solutions to the given equation.

Solution

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$${{y}_{H}^{1}}=x$$

(5.2)
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$${{y}_{H}^{1}}'=1$$

(5.3)
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$${{y}_{H}^{1}}''=0$$

(5.4) plugging equations (5.2)-(5.4) into equation (5.1):
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results in:

$$\begin{align} & (1-{{x}^{2}})[0]-2x[1]+2[x]=0 \\ & -2x+2x=0 \\ & 0=0 \\ \end{align}$$

$$\begin{align} & \therefore {{y}_{H}^{1}}=x \end{align}$$ is a valid solution.

similarly, the second solution

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$${y_{H}^{2}}=\frac{x}{2}\ln \left( \frac{1+x}{1-x} \right)-1$$

(5.5)
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can be shown to be a homogeneous solution to the solution by:

$$\begin{array}{*{35}{l}} {} & y{{_{H}^{2}}^{\prime }}=\frac{1}{2}\ln \left( \frac{1+x}{1-x} \right)+\frac{x}{2}{{(\ln \left( 1+x \right)-\ln \left( 1-x \right))}^{\prime }} \\ \end{array}$$
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$$=\frac{1}{2}\ln \left( \frac{1+x}{1-x} \right)+\frac{x}{2}\left( \frac{1}{1+x}+\frac{1}{1-x} \right)$$

(5.6)
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$$\begin{array}{*{35}{l}} {} & y{{_{H}^{2}}^{\prime \prime }}=\frac{1}{2}\left( \frac{1}{1+x}+\frac{1}{1-x} \right)+\frac{1}{2}\left( \frac{1}{1+x}+\frac{1}{1-x} \right)+\frac{x}{2}\left( \frac{-1}+\frac{1} \right) \\ \end{array}$$
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$$=\left( \frac{1}{1+x}+\frac{1}{1-x} \right)+\frac{x}{2}\left( \frac{-1}+\frac{1} \right)$$

(5.7)
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$$\begin{array}{*{35}{l}} {} & (1-{{x}^{2}})y{{_{H}^{2}}^{\prime \prime }}=(1-{{x}^{2}})\left( \frac{1}{1+x}+\frac{1}{1-x} \right)+\frac{x(1-{{x}^{2}})}{2}\left( \frac{-1}+\frac{1} \right) \\ {} & =(1-x+1+x)+\frac{1}{2}\left( \frac{-x(1-x)}{(1+x)}+\frac{x(1+x)}{(1-x)} \right) \\ \end{array}$$
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$$=2+\frac{x}{2}\left( \frac{4x}{1-{{x}^{2}}} \right)=2+\left( \frac{2{{x}^{2}}}{1-{{x}^{2}}} \right)$$

(5.8)
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$$\begin{array}{*{35}{l}} {} & -2xy{{_{H}^{2}}^{\prime }}=-2x\left[ \frac{1}{2}\ln \left( \frac{1+x}{1-x} \right)+\frac{x}{2}\left( \frac{1}{1+x}+\frac{1}{1-x} \right) \right] \\ {} & =-2x\left[ \frac{1}{2}\ln \left( \frac{1+x}{1-x} \right)+\left( \frac{x}{1-{{x}^{2}}} \right) \right] \\ \end{array}$$
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$$=-x\ln \left( \frac{1+x}{1-x} \right)-\frac{2{{x}^{2}}}{1-{{x}^{2}}}$$

(5.9)
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$$\begin{array}{*{35}{l}} {} & 2y_{H}^{2}=2\left[ \frac{x}{2}\ln \left( \frac{1+x}{1-x} \right)-1 \right] \\ \end{array}$$
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$$=x\ln \left( \frac{1+x}{1-x}\right)-2$$

(5.10)
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Equations (5.5)-(5.10) can then be substituted into equation (5.1) $$\begin{align} {(1-{{x}^{2}})}{y_{H}^{2}}''-{2x}{y_{H}^{2}}'+{2}{y_{H}^{2}} \\ \end{align}$$ $$={2}+{\frac{{2x}^{2}}{1-{{x}^{2}}}}-{x}\ln \left( \frac{1+x}{1-x} \right)-\frac{{2x}^{2}}{1-{{x}^{2}}}+{{x}\ln \left( \frac{1+x}{1-x} \right)-{2}}=0 $$

$$\begin{align} \therefore {y_{H}^{2}}=\frac{x}{2}\ln \left( \frac{1+x}{1-x} \right)-1

\end{align}$$ is a valid solution.

Author

User:EGM6321,F10.TEAM1.WILKS

Signatures
Solved problem 1 --Bryan Allison (BA)

Solved problem 2 --Taeho Kim (TK)

Solved problem 3 --Paul Lang (PL)

Solved problem 4 --Charles Russo (CR)

Solved problem 5 --192.91.147.35 17:06, 15 September 2010 (UTC)User:EGM6321,F10.TEAM1.WILKS|David Wilks (DW)]]

title: User:EGM6321.F10.TEAM1.HW1


text:
User talk:EGM6321.F10.TEAM1.HW1

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User talk:TMcCormick31

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User talk:Stack.kara

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User:Rastignac~enwikiversity

My name is David Bloom, and I've been around for quite a while. Years ago I was a student of linguistics, in Buffalo, New York, and I still am, but I haven't got a place to be a student at; I have a research project in mind that I'd like to work out in the open and hypertextually--i.e., in a Wiki environment. Wikiversity is an especially nice place for me in that the School of Linguistics is such a neglected piece of real estate anyway and I could help landscape it a little. Wikiversity is a problematic place in that the official ethos of the Wikimedia in general, with its emphasis on properly documented authority, so marvelous in the way in works in Wikipedia (to which I contribute an edit every once in a while), is not quite right for a university, which needs questioning and experimentation. I have a couple of ideas about that, too, which I hope to be exploring.--Rastignac 02:30, 13 September 2010 (UTC)

title: User:Rastignac~enwikiversity


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User:Egm6321.f10.team2.patterson

Homework Contributions
/Homework 1/

/Homework 2/

/Homework 4/

/Homework 5/

/Homework 6/

/Homework 7/

title: User:Egm6321.f10.team2.patterson


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User:Jtneill/Grants/Student-authored open textbooks 2010

About the project

 * 1) Proposal
 * 2) Plan
 * 3) For more information, contact James Neill

The project in action

 * 1) Motivation and emotion - Textbook chapter guidelines
 * 2) Motivation and emotion - Textbook
 * 3) Motivation and emotion - Textbook pedagogy

Other student-authored textbook projects @ UC

 * 1) Robin McConnell (doing a student authored open text book on Wikibooks.org Sem 1, 2010)
 * 2) Lubna Alam (Sem 1, 2010)
 * 3) Ben Rattray (Sem 2, 2011)


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User talk:Rastignac~enwikiversity

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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.

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title: User talk:Rastignac~enwikiversity


text:
Motivation and emotion/Tutorials/Needs/Brain

Motivational & emotional states associated with brain structure: Sub-cortical
Note: Based on Reeve (2018), Table 3.1.

Motivational & emotional states associated with brain structure: Cortical
Note: Based on Reeve (2018) Table 3.1.


text:
User:Jtneill/Grants/Student-authored open textbooks 2010/Proposal

Student-authored open textbooks: Proposal for nomination as an Innovative Teaching and Learning Leader
For participation in the University of Canberra Blended Learning Hothouse, September-November, 2010.

Projects must clearly address the criteria detailed below.


text:
User talk:Pims96

test
test

Capital budgeting
Capital budgeting

title: User talk:Pims96


text:
Ideas in Geometry/Euclid's Axioms

Topic:Ideas_in_Geometry

title: Ideas in Geometry/Euclid's Axioms


text:
User:EGM6321.f10.team6.yoshioka

/HW2/

/HW3/

/HW5/

/HW6/

= Homework #1 = [Due Wed, 15 Sep 10]

Problem 1 - Additional
Developing the Reynolds’ Transport Theorem (taken from Nonlinear Solid Mechanics - A Continuum Approach for Engineering, Gerhard A. Holzapfel) is an application of the first derivative of $$\Phi (\text{x},t)$$, and it is presents as follows:

Suppose we have a spatial scalar field $$\Phi (\text{x},t)$$ describing some physical quantity (for example, mass, internal energy, entropy, heat, or entropy sources) of a particle in space per unit volume at time t.

$$I(t)=\int\limits_{\Omega }{\Phi (\text{x},t)\text{d}\upsilon }$$ (9)

Assume $$\Phi (\text{x},t)$$ to be smooth, so that it is continuously differentiable. Hence, the present status of a continuum body in some three-dimensional region Ω with volume $$\upsilon $$ at given time t may be characterized by the scalar-valued function

The aim is now to compute the material time derivative of the volume integral $$I(t)$$ Since the region of integration Ω depends on time $$t$$, integration and time differentiation do not commute. Therefore, as a first step $$I(t)$$ must be transformed to the reference configuration. By changing variables using the motion $$\text{x}=\chi (\text{X},t)$$ and the relation $$\text{d}\upsilon \text{= }J(\text{X},t)\text{d}V$$ we find the time rate of change of $$I(t)$$ to be

$$\dot{I}(t)=\frac{\text{D}}{\text{Dt}}\int\limits_{\Omega }{\Phi (\text{x},t)\text{d}v=}\frac{\text{D}}{\text{Dt}}\int\limits_{\Phi (\chi (\text{X},t),t)J(\text{X},t)\text{d}V}$$ (10)

Since the region of integration is now time-independent, integration and differentiation commute. Hence, as a second step, from (10)we obtain, using the product rule of differentiation,

$$\frac{\text{D}}{\text{Dt}}\int\limits_{\Omega }{\Phi (\text{x},t)\text{d}\upsilon =}\int\limits_{\left[ \dot{\Phi }(\chi (\text{X},t),t)J(\text{X},t)+\Phi (\chi (\text{X},t),t)\dot{J}(\text{X},t) \right]}\text{d}V$$ (11)

where $${\dot{\Phi }}$$ denotes the material time derivative of the spatial scalar field $$\Phi $$. In a last step we undo the change of variables and convert the volume integral back to the current configuration. By means of the material time derivative of the scalar J, $$\text{d}\upsilon \text{= }J(\text{X},t)\text{d}V$$ and motion $$\text{x}=\chi (\text{X},t)$$, we find finally that

$$\begin{align} & \frac{\text{D}}{\text{Dt}}\int\limits_{\Omega }{\Phi (\text{x},t)\text{d}\upsilon =}\int\limits_{\left[ \dot{\Phi }(\chi (\text{X},t),t)+\Phi (\chi (\text{X},t),t)\frac{\dot{J}(\text{X},t)}{J(\text{X},t)} \right]J(\text{X},t)}\text{d}V \\ & =\int\limits_{\Omega }{\left[ \dot{\Phi }(\text{x},t)+\Phi (\text{x},t)\frac{\dot{J}(\text{X},t)}{J(\text{X},t)} \right]\text{d}\upsilon } \\ & =\int\limits_{\Omega }{\left[ \dot{\Phi }(\text{x},t)+\Phi (\text{x},t)\text{div}\nu (\text{x},t) \right]\text{d}\upsilon } \\ \end{align}$$ (12)

In the following the arguments of the tensor quantities are dropped in order to simplify the notation. However, in cases where additional information is needed, they will be employed. Hence, relation (12) reads as

$$\frac{\text{D}}{\text{Dt}}\int\limits_{\Omega }{\Phi \text{d}\upsilon =}\int\limits_{\Omega }{\left( \dot{\Phi }+\Phi \text{div}\nu \right)}\text{d}\upsilon $$ (13) Where we have assumed smoothness of the spatial velocity field v.

Other forms of the time rate of change of the integral (9) result from (13) by means of the material time derivative of the spatial scalar field $$\Phi $$ can be developed, i.e. we can write the first derivative of the homework equation as

$$\dot{\Phi }(\text{x},t)={{\left( \frac{\partial \Phi (\text{x},t)}{\partial t} \right)}_{\text{x}}}+{{\left( \frac{\partial \Phi (\text{x},t)}{\partial \text{x}} \right)}_{t}}\cdot {{\left( \frac{\partial \chi (\text{X},t)}{\partial t} \right)}_{\text{X}={{\chi }^{-1}}(\text{x},t)}}$$ (14)

$$v(\text{x},t)=\frac{\partial \chi (\text{X},t)}{\partial t}$$ (15)

$$\text{grad}\Phi (\text{x},t)=\frac{\partial \Phi (\text{x},t)}{\partial \text{x}}$$ (16)

$$\text{div(}\Phi \text{u) = }\Phi \text{divu + u}\cdot \text{grad}\Phi $$ (17)

We can write the integral as

$$\dot{\Phi }(\text{x},t)=\frac{\text{D}\Phi (\text{x},t)}{\text{Dt}}=\frac{\partial \Phi (\text{x},t)}{\partial t}+\text{grad}\Phi (\text{x},t)\cdot \nu (\text{x},t)$$

$$=\int\limits_{\Omega }{\left( \text{div}\left( \Phi \nu \right)+\frac{\partial \Phi }{\partial t} \right)}\text{d}\upsilon $$ (18)

And finally, using the divergence theorem

$$\int\limits_{S}{\Phi \text{u}\cdot \text{nd}s=\int\limits_{\upsilon }{\text{div(}\Phi \text{u)d}}}\upsilon $$ (19)

The first term on the right-hand side of eq.(19) characterizes the rate of transport (or the outward normal flux) of $$\Phi \nu $$ across the surface $$\partial \Omega $$ out of region $$\Omega $$, which is assumed to be fixed. This contribution arises from the moving region. The second term denotes the local time rate of change of the spatial scalar field $$\Phi $$ within region $$\Omega $$. In (19) n denotes the outward unit normal field acting along $$\partial \Omega $$. Relation (18) is referred to as Reynolds’ transport theorem.

title: User:EGM6321.f10.team6.yoshioka


text:
User:Egm6321.f10.team2.vrasul/HW1

=Problem 1: Perform Dimensional analysis on Governing Equation of Force acting along Maglev Deformed Guideway=

Given
Governing Equation of force acting along a Maglev deformed guideway
 * {| style="width:100%" border="0"

$$C_0\left ( Y^1,t \right)=-F^1\left [ 1-\overline{R}{u^2_{,ss}}\left ( Y^1,t \right) \right] -F^2 {u^2_{,s}}\left ( Y^1,t \right) - \frac{T}{R} +M\left [ \left [ 1-\overline{R}{u^2_{,ss}}\left ( Y^1,t \right) \right ] \left [ {u^1_{,tt}}\left ( Y^1,t \right)-\overline{R}{u^2_{,stt}}\left ( Y^1,t \right) \right ]+{u^2_{,s}}\left ( Y^1,t \right){u^2_{,tt}}\left ( Y^1,t \right) \right ]
 * style="width:95%" |
 * style="width:95%" |

$$     (Eq 1.1)
 *

 * }

Note: Argument $$\left ( Y^1, t \right)$$ will be omitted for clarity in subsequent steps.

The following are the known variables of (Eq 1.1):

$$F^{1}$$:vertical force

$$F^{2}$$:horizontal force

$$F^{1}$$:vertical force

$$\overline{R}$$:distance from deformed guide way centerline to wheel centerline

$$ u^2_{,s}$$:slope of deform guide way

$$T$$:Torque exerted on wheel

$$R$$:radius of wheel

$$M$$:mass of wheel

The following are the derived variables of (Eq 1.1):

$$ u^2_{,ss}$$

$$ u^1_{,tt}$$

$$ u^2_{,stt}$$

$$ u^2_{,tt}$$

Find
Verify validity of (Eq 1.1) via “dimensional analysis”

Solution
Dimensional analysis of each variable must be completed for each variable in the equation prior to determining the overall dimension of the equation.

 * {| style="width:100%" border="0"

$$[F^{1}]\longrightarrow$$ $$F$$ (Eq 1.2)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

 * {| style="width:100%" border="0"

$$[F^{2}]\longrightarrow$$ $$F$$ (Eq 1.3)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

 * {| style="width:100%" border="0"

$$[\overline{R}]\longrightarrow$$ $$L$$ (Eq 1.4)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

 * {| style="width:100%" border="0"

$$ [u^2_{,s}]=\frac{\partial{u}^2}{\partial s}\longrightarrow $$ $$\frac{L}{L}$$ $$ \longrightarrow [1] $$ (Eq 1.5)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

 * {| style="width:100%" border="0"

$$[T] \longrightarrow$$ $$F\cdot L$$ (Eq 1.6)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

 * {| style="width:100%" border="0"

||  (Eq 1.7)
 * $$[R]\longrightarrow$$ $$L$$
 * $$[R]\longrightarrow$$ $$L$$
 * }

 * {| style="width:100%" border="0"

(Eq 1.8)
 * $$[M]\longrightarrow $$ $$L$$
 *
 *
 * }

 * {| style="width:100%" border="0"

(Eq 1.9)
 * $$ [u^2_{,ss}]=\frac{\partial^2{u}^2}{\partial s^2}\longrightarrow $$$$\frac{L}{{L}^2}$$ $$ \longrightarrow \frac{1}{L}$$
 *
 *
 * }

 * {| style="width:100%" border="0"

(Eq 1.10)
 * $$ [u^1_{,tt}]=\frac{\partial^2{u}^1}{\partial s^2}\longrightarrow $$$$\frac{L}{T}^2$$
 *
 *
 * }

 * {| style="width:100%" border="0"

(Eq 1.11)
 * $$ [u^2_{,stt}]=\frac{\partial {u}^2}{\partial s\partial^2 {t}}\longrightarrow $$$$\frac{L}{L\cdot{T}^2}$$ $$ \longrightarrow \frac{1}{T}^2$$
 *
 *
 * }

 * {| style="width:100%" border="0"

(Eq 1.12)
 * $$ [u^2_{,tt}]=\frac{\partial^2 {u}^2}{\partial {t}^2} \longrightarrow $$$$\frac{L}{T^2}$$
 *
 *

 * }

The dimensional analysis of the entire equation can now be applied to (Eq 1.1) by substitution of (Eq 1.2) to (Eq 1.12).

 * {| style="width:100%" border="0"

$$ [C_0\left ( Y^1,t \right)]=F\left [ 1-L \frac{1}{L}\right] -F \cdot\frac{L}{L} -\frac{F \cdot L}{L} +M\left [ \left [ 1-L \cdot \frac{1}{L} \right] \left [ \frac{L}{T^2} - \frac{L}{T^2} \right] +1 \cdot \frac{L}{T^2} \right]
 * style="width:95%" |
 * style="width:95%" |

$$     (Eq 1.13)
 *

 * }

In conclusion, it can be deduced by simplification of fractions that the components of equation (Eq 1.13) will reduce down to force ($$F$$) dimensions and mass x acceleration ($$M \cdot \tfrac{L}{T^2}$$), which further reduces to force ($$F$$) dimensions.

=Problem 2: Proof of Nonlinarity=

Given
Mass component of equation of motion for Maglev train.

 * {| style="width:100%" border="0"

$$C_0\left ( Y^1,t \right) \ddot{Y}(t) = M\left [1 - \overline{R} {u^2_{,ss}}\left ( Y^1,t \right) \right]
 * style="width:95%" |
 * style="width:95%" |

$$     (Eq 2.1)
 *

 * }

The following are the known variables of (Eq 2.1):

$$\overline{R}$$:distance from deformed guide way centerline to wheel centerline

$$ u^2_{,s}$$:slope of deform guide way

$$M$$:mass of wheel

Find
Prove that $$C_0\left ( Y^1,t \right) \ddot{Y}(t) $$ is nonlinear with respect to $$Y^1$$.

Solution
For (Eq 2.1) to be nonlinear, the following condition must exist:

 * {| style="width:100%" border="0"

$$F\left ( \alpha u + \beta v \right) \ne \alpha F\left ( u \right) + \beta F\left ( v\right)
 * style="width:95%" |
 * style="width:95%" |

$$     (Eq 2.2)
 *

 * }

The left hand condition of (Eq 2.2) is then applied to (Eq 2.1) and the following results are yielded:
 * {| style="width:100%" border="0"

$$ C_0\left ( \alpha Y^1,t \right) \ddot{Y}(t)= M\left [1 - \overline{R} {u^2_{,ss}}\left ( \alpha Y^1,t \right) \right]
 * style="width:95%" |
 * style="width:95%" |

$$     (Eq 2.3)
 *

 * }

The right hand condition of (Eq 2.2) is then applied to (Eq 2.1) and the following results are yielded:
 * {| style="width:100%" border="0"

$$ \alpha C_0\left ( Y^1,t \right) \ddot{Y}(t)= \alpha \left [ M\left [1 - \overline{R} {u^2_{,ss}}\left ( Y^1,t \right) \right]\right]
 * style="width:95%" |
 * style="width:95%" |

$$     (Eq 2.4)
 *

 * }

The results of (Eq 2.3) and Eq 2.4) show that the right sides of each of these equations are indeed not equal to eachother and therefore, this proves that (Eq 2.1) is nonlinear.

title: User:Egm6321.f10.team2.vrasul/HW1


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User talk:Christine Yu

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User talk:EGM6321.f10.team6.yoshioka

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User talk:Haros2008

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 * 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! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 11:30, 13 September 2010 (UTC)

title: User talk:Haros2008


text:
WikipediaOS/Content and process transparency

Hypothesis
Both Wikipedia and Open Source development environments are based in transparency, potential contributors and other people interested in their activities can know the details of its output (being an article or a piece of software) as well as the discussions generated during its development..

Votes/comments

 * This is probably a very general topic, transparency can be also found in many other Internet-mediated projects and processes. What's more remarkable is that the changelog files in open software releases and the history view of wiki pages (apart from open mailing lists, chatrooms or other channels where discussion takes place) have very much in common. Probably the early adoption of a wiki system influenced the Wikipedia project in this direction (making software feature influence social behaviour) --Esenabre 12:25, 13 September 2010 (UTC)
 * When reading the statement I thought mainly about the chat logs, because they explain most often why something was started. Here is the question always, are they publically logged or not. Erkan Yilmaz uses the Chat 18:01, 19 January 2011 (UTC)


text:
User:Egm6321.f10.team5

 Team 5 website

=Problem Assignments=

HW1

 * 1) Sang Min Oh
 * 2) Scott Elliot
 * 3) Michael Steinberg
 * 4) Michael Faraone
 * 5) Sang Min Oh

=Homework Reports= HW1

HW2

HW3

HW4

HW5

HW6

HW7 =Team 5 Members= Raul Riveros

Sang Min Oh

Michael Faraone

Scott Elliot

Rob Carroll

Jo

Michael Steinberg

title: User:Egm6321.f10.team5