<< Chapter < Page Chapter >> Page >
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

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
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
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
what is hormones?
Wellington
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?

Ask