# 10.3 The physics of springs  (Page 10/11)

 Page 10 / 11

## 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.

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.

When we look at our equation, ${A}^{T}diag\left(A,x\right)k=f$ , 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

In our original approach to use statistical inference to solve the inverse problem (see "Our Question" ) our problem reduced to the equation ${A}^{T}Ek=f$ , where $E\in {\mathbb{R}}^{16×16}$ 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

$\left({A}^{T},+,\alpha \right)\left(E,+,ϵ\right)\left(k,+,\kappa \right)\approx f+\gamma ,$

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 $\alpha ,$ $\gamma ,$ 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

${A}^{T}\left(E,+,ϵ\right)k\approx f.$

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

$y=Fz+ϵ,$

where z is the unknown variable, y is the observed variable with error ϵ , and $F\in {\mathbb{R}}^{m×n}$ .

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

${K}^{-1}=\left[\begin{array}{ccccc}{c}_{1}& 0& 0& \cdots & 0\\ 0& {c}_{2}& 0& \cdots & 0\\ 0& 0& {c}_{3}& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {c}_{n}\end{array}\right],\phantom{\rule{1.em}{0ex}}{c}_{i}=1/{k}_{i}.$

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

$e=Fc,$

where $e=Ax$ , $F=\mathrm{diag}\left({A}^{-T}f\right)$ , 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

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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