# 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

#### Questions & Answers

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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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