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This report summarizes work done as part of the Michell Trusses PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed around the study of a common problem. In this PFUG, we study cost minimizing trusses. We ask what is the structure, constructed of bars and cables, of least cost that will withstand a set of point forces. This work was studied in the Rice University VIGRE class MATH499 in the Spring and Fall of 2008 and by the 2013 Michell research group at Rice University

Introduction

The Michell Truss PFUG studies a variational problem posed by mechanical engineer Anthony George Maldon Michell in the early part of the last century;what configuration of bars and cables is needed to withstand a system of balanced point forces is most economical?Suppose that some collection of forces with their points of application (point forces) is given and you are asked to build a structure which can withstand these forces.Your choice of materials are bars and cables. Bars can withstand compression (forces parallel and pointing inward) but will breakunder extension (forces parallel and pointing outward). Cables can withstand extension but will buckle under compression.You may also choose the strength of the bar or cable; the strength is the largestcompression or extension the bar or cable respectively can withstand. A structure built of such cables and bars will be called a truss.

The first question the PFUG has addressed is whether one can build such a truss at all. Notice that a bar or cable by itself canwithstand only one set of point forces; that of two opposite forces applied to the endpoints of the bar or cable. These point forces have the property that if the points of application were attached to a stationary body, the center ofmass would not accelerate and the body would not rotate. This property is called balanced. Since point forces can be linearly superimposed, a necessary condition that a set ofpoint forces be withstood by a truss is that it be balanced. Below we show that being balanced is also a sufficient condition; if the set of point forces is balanced then there is a truss withstanding it.

Next, the PFUG formulated necessary conditions for a truss to be economical. The cost of a bar or cable is its strength times its length. The cost of a truss is definedto be the sum of the cost of bars and cables constituted by the truss. In general, there are many trusses which withstand a given a set of point forces.The question is, which is most economical, i.e. which truss costs the least? In general this is a very difficult question to answer because a minimizing sequence,which we know to exist from the first part, may converge in a larger class of measures than described here and it is difficult to formulate any class of perturbations of a truss.Below we describe one class of perturbations on trusses with corners which yields a surprising necessary condition for economy. A set of perturbations to be studiedin the future are also described.

Questions & Answers

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Daniel
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s. Reply
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Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
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Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
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s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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Sanket Reply
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Damian Reply
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Cied
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abeetha Reply
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Porter
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Porter
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AMJAD
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AMJAD
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
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silver nanoparticles could handle the job?
Damian
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Azam
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Uday
I'm interested in Nanotube
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Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Michell trusses study, rice u. nsf vigre group, summer 2013. OpenStax CNX. Sep 02, 2013 Download for free at http://cnx.org/content/col11567/1.2
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