Meeting 9:
Project 1 requires the use of data
modeling with Excel (© Microsoft), this is learned by
doing CW3-CW5, a polynomial or exponential curve-fitting or data modeling. What is data modeling? When certain
input data X is applied to a system, certain output data Y is produced by that
system. A mathematical model of the system can be obtained by relating Y to X:
e.g. Y’=f(X). We have used a Y’ to indicate that it may not be possible to
obtain an equation that relates Y to X for every pair of data, but just a best
model.
Examples of the system could be a catapult
(X=initial height of a weight; Y=range for a clay ball), a pendulum (X=period;
Y=length needed to produce that period), or a car on an inclined ramp (X=ramp
angle; Y=distance traveled in 2s). To simplify the introduction, we discuss
just simple polynomial models, e.g.
Y’=aX2+bX+c
Y’= bX+c
Y’=dX3+aX2+bX+c
Y’=exp(-b1*X)/(b2+b3*X)
How to obtain a model? CW3
can be done by following these 7 steps. The process consists of using Solver (get
it through Tools/Add-ins if needed) to minimize a “standard deviation”
parameter s by allowing the polynomial coefficient to vary. After using Solver,
the final values for a, b, c determines our quadratic model that represents our
pendulum.
How to find the best model?
Suppose we would like to make
instead a linear model Y’=bX+c. This can be done on the same Excel sheet by
setting a=0, and without touching it, use Solver to minimize the parameter s by
changing only cells $C$4:$C$5. Try it yourself, note
that the final s parameter for a linear model is 0.13, larger than that for the
quadratic model 0.024. This means that our data fit better into a quadratic
model than a linear model. In deed, for those who took Fundamentals of Physics
I, the period of an ideal pendulum (mass of string is negligible, bob is not so
large, no friction involved) is given by
To test how ideal is our pendulum
(represented by the pairs of data we started with), let do a quadratic
curve-fitting by setting b=c=0 and letting the coefficient a change so as to
minimize the standard deviation, check
how well the final value for a agrees with g=9.81m/s2.
Why we need to find the best model?
A better model allows you to make
better predictions.
Does the model you found for your particular device apply
to another device?
No, the models are
device-specific, since it was obtained using a specific set of data associated
with a specific device. If getting a different device, new data need to be
measured and the curve-fitting needs to be redone.
What is the difference between science and engineering?
Science extracts the simplicity
and universality that are behind a wide range of devices or situations by
ignoring many specific factors such as frictions, fluid motions, etc.
Engineering includes more parameters into a model that produces excellent
predictions, but that is only valid for that particular device.
Logbook
Just to remind that the logbook is a very important component
of the course and will be evaluated, together with the final exam they count
20% of the course grade.
You should record the logbook on a daily basis, with pages
numbered and dated, about two main topics:
-Discussions
and findings about projects, collectively or individually
-Answers
to questions posted here in the class notes and other learning conclusions you
found by doing activities related to this course.
The next logbook due date is
Logbook questions:
1-What are the data X and Y
2-How many parameters did you use to
implement the exponential model? What are their values? What is the s
parameter?
3-Attach a plot of the data and model. Show
the parameters for your model using Solver and the parameters obtained from:
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