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Normality of All Nodes

Spring constants

Next we look at the calculated spring constants. First, we look at them individually, drawing from Data Set A. We take experiments pairwise, one from each load, and calculate the spring constants. This gives us 16 samples in R of 52 points each. [link] plots the normality of each spring constant: all seem to be fairly normal. Because of this, we believe that the spring constants are normally distributed.

Normality of Individual Spring Constants

Next we look at the spring constants collectively, with one sample in R 16 . Because we believe that each individual spring constant is normally distributed, we expect the combined spring constants to be normally distributed. [link] is the plot from this sample, suggesting that the spring constants are indeed normally distributed.

Normality of Collective Spring Constants

When we look at our equation, A T diag A x k = f , this makes sense. If A and f are fixed, then the relationship between x and k is simply linear, with no worries about the distributions of A and f . To be sure, A and f are not entirely accurate, but they are close enough to constant that we can expect k to come from the same kind of distribution as x .

Note that this criteria is fairly subjective. Plotting the random distribution helps us to get a feel for how much deviation from the reference line we can expect, but these in no way prove that the samples are from normal distributions. Rather, it suggests that, if they are not normally distributed, they can probably be reasonably approximated by a normal distribution.

Rewriting the problem and the maximum likelihood estimate

Rewriting the problem

In our original approach to use statistical inference to solve the inverse problem (see "Our Question" ) our problem reduced to the equation A T E k = f , where E R 16 × 16 has the elongation of each spring along the diagonal. If we consider the experimental error as well as the error in the model the equation becomes

A T + α E + ϵ k + κ f + γ ,

where α , ϵ , κ , and γ are error due to either the measurements or the model. The error ϵ is the easiest error that we can describe with all our experimental data, specifically Data Set A (see "Notes: Our Data Sets, Measuring Spring Constants, and Error" ). The error of α , γ , and κ we either have no description or very little description from our experimental data. The problem that we can then study using our observed error is

A T E + ϵ k f .

Because ϵ is in the middle of the equation, it is very hard to deal with. We wish our problem to take the form

y = F z + ϵ ,

where z is the unknown variable, y is the observed variable with error ϵ , and F R m × n .

To reach an equation of this form we begin by looking at our original problem A T K A x = f (see "An Inverse Problem" ). If A T is invertible, then our equation becomes A x = K - 1 A - T f , where

K - 1 = c 1 0 0 0 0 c 2 0 0 0 0 c 3 0 0 0 0 c n , c i = 1 / k i .

Because A - T f is a vector and K - 1 is a diagonal matrix, we see that our equation can be rewritten to

e = F c ,

where e = A x , F = diag ( A - T f ) , and c is the compliance, or the vector of inverses of spring constants. Now if we include error of the observed data, e , we obtain a model for the system of the form

Questions & Answers

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
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
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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Daniel
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Maciej
characteristics of micro business
Abigail
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
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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