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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

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
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Professor
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Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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LITNING Reply
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LITNING Reply
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LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Rafiq
what is differents between GO and RGO?
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Kyle
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Adin
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
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Introduction about quantum dots in nanotechnology
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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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