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This report summarizes work done as part of the Physics of Strings 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 round the study of a common problem. This module describes experiments done on a spring system.

A network of springs

Our research is centered on a network of springs, built by Jeff Hokansen and Dr. Mark Embree for use in the CAAM 335 Lab. Over the table, we set up a webcam on a beam and connected it to a computer running MATLAB. Springs are connected to pennies (nodes), two of which are fixed to the table. Along the outside pennies, strings run over pullies set along the edge of the table and are attached to hooks, upon which we hang masses. These masses cause the nodes to move. We use the webcam to capture an image of the network, then use a MATLAB script to find the center of each node; the pennies have been painted red to make it easier for MATLAB to detect them. This gives us the displacement of each node, from which we can compute the elongation of each spring. We also know the force applied to each node ( 9 . 8 * m a s s in units of Newtons) and can calculate the spring constant k for each spring using Hooke's Law, f restoring = - ( elongation ) * k

A forward problem

In the forward problem, we seek to compare results from our physical model to the results predicted by solving a linear system of equations. Specifically, we wish to predict our displacements, given we know the load forces and spring constants in our system of springs.

Let us begin with an easier system of just two springs, three nodes, and two forces. Since only two of the nodes are moving, we will have two horizontal displacements denoted in the vector x . There are two elongations, one for each spring, denoted in the vector e .

2 Spring Network
x = x 1 x 2 , e = e 1 e 2 ,

Each spring elongation is a linear combination of node displacements. The equations can be written in the following manner.

e = e 1 e 2 = x 1 x 2 - x 1 = 1 0 - 1 1 x = A x

Now we have our adjacency matrix, A . This translates us from node displacement to spring elongation. It will have one more property which will we shall see shortly. Now let us consider finding the restoring force, y , which will have one component for each spring.

y = y 1 y 2 ,

We assume that each spring follows Hooke's Law, y = k e , where restoring force is directly proportional to elongation. Each spring has a corresponding stiffness, k i which comprise the the diagonal elements of matrix, K .

y = y 1 y 2 = k 1 e 1 k 2 e 2 = k 1 0 0 k 2 e = K e = K A x

The final step is to translate these restoring forces into the load forces acting on each node, denoted by vector f .

f = f 1 f 2 = y 1 - y 2 y 2 = 1 - 1 0 1 = A T y

Now we can see the second feature of the adjacency martrix. The transpose of A performs the reverse translation from edges to nodes. The final product of this example is the equation just shown: f = A T K A x . Now we can expand the problem to any system of springs for which we can create an adjacency matrix A. For this project we focused on the spring network shown below.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
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characteristics of micro business
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for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
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
what is the actual application of fullerenes nowadays?
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
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
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what is biological synthesis of nanoparticles
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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