eg 10111/10112 introduction to engineering copyright © 2009 university of notre dame introduction...

EG 10111/10112 Introduction to Engineering

Copyright © 2009University of Notre Dame

Introduction to Engineering Systems

Lecture 3 (9/4/2009)

Empirical Models: Fitting a Line to Experimental Data

Prof. Andrés Tovar



Reading material and videos

• LC1 – Measure: Concourse material• LT1 – Introduction: Sec. 1.1, 1.2, and 1.4• LT2 – Models: Ch. 4• LC2 – Matlab: Ch. 9 and 10, videos 1 to 9• LT3 – Data analysis: Sec. 5.1 to 5.3, videos 13 and 14

For next week• LT4 – Statistics: Sec. 5.4.1 and 5.4.2, video 10• LC3 – SAP Model: Concourse material • LT5 – Probability: Sec. 5.4.3 and 5.4.4, videos 11 and 12

LT: lecture session

LC: learning center session

Using "Laws of Nature" to Model a System



Announcements

• Homework 1– Available on Concourse http://concourse.nd.edu/– Due next week at the beginning of the Learning Center session.

• Learning Center– Do not bring earphones/headphones.– Do not bring your laptop.– Print and read the material before the session.

Using "Laws of Nature" to Model a System

http://concourse.nd.edu/



From last class

• The 4 M paradigm: measure, model, modify, and make.• Empirical models vs. Theoretical models• Models for a falling object

– Aristotle (Greece, 384 BC – 322 BC)– Galileo (Italy, 1564 – 1642)– Newton (England, 1643 – 1727)– Leibniz (Germany, 1646 –1716)

• Models for colliding objects– Descartes (France, 1596-1650)– Huygens (Deutschland, 1629 – 1695)– Newton (England, 1643 – 1727)

• Prediction based on models


poolball

golfball



From last class

• Given 2 pendulums with different masses, initially at rest– Say, a golf ball and a pool

ball• Would you be willing to bet

that you could figure out where to release the larger ball in order to knock the smaller ball to a given height?

• How could you improve your chances?

poolball

golfball




Theoretical Model of Colliding Pendulums

• Given 2 pendulum masses m1 and m2

– golf ball initially at h2i = 0

– pool ball released from h1i

– golf ball bounces up to h2f

– pool ball continues up to h1f

• Galileo’s relationship between height and speed later developed by Newton and Leibniz.

• Huygens’ principle of relative velocity• Newton’s “patched up” version of

Descartes’ conservation of motion—conservation of momentum


poolball

golfball





Collision model:

Relative velocity

Conservation of momentum

Conservation of energy

Conservation of energy





1) Conservation of energy

2) Collision model: relative velocity and conservation of momentum

3) Conservation of energy





4) Finally

4) In Matlab this is

h1i = (h2f*(m1 + m2)^2)/(4*m1^2);




Matlab implementation

% collision.m

m1 = input('Mass of the first (moving) ball m1: ');m2 = input('Mass of the second (static) ball m2: ');h2f = input('Desired final height for the second ball h2f: ');disp('The initial height for the first ball h1i is:')h1i = (h2f*(m1 + m2)^2)/(4*m1^2)





% collision1.m

m1 = 0.165; % mass of pool ball, kgm2 = 0.048; % mass of golf ball, kgh2f = input('Desired final height for the second ball h2f: ');disp('The initial height for the first ball h1i is:')h1i = (h2f*(m1 + m2)^2)/(4*m1^2)plot(h2f,h1i,'o'); xlabel('h2f'); ylabel('h1i')hold on

Let us compare the theoretical solution with the experimental result.

What happened?!?!




Run the Pendulum Experiment

Pool ball release height(m)

Golf ball final height(m)

0.00 0.00

0.05

0.10

0.15

0.20

0.25




Experimental Results

% collision2.m

h1ie = 0:0.05:0.25; % heights for pool ball, mh2fe = []; % experimental results for golf ball, mplot(h1ie,h2fe, '*')



MATLAB GUI for Least Squares Fit





MATLAB commands for Least Squares Fit

% collision2.m

h1ie = 0:0.05:0.25; % heights for pool ball, mh2fe = []; % experimental results for golf ball, mplot(h1ie,h2fe, '*')

c = polyfit(h1ie, h2fe, 1)m = c(1) % slopeb = c(2) % intercept

h2f = input('Desired final height for the second ball h2f: ');disp('The initial height for the first ball h1i is:')h1i = 1/m*(h2f-b)

fit a line (not quadratic, etc)




What About Our Theory

• Is it wrong?

• Understanding the difference between theory and empirical data leads to a better theory

• Evolution of theory leads to a better model



Improved collision model


Huygens’ principle of relative velocity Coefficient of restitution

Improved collision model: COR and conservation of momentum

hi1 = (h2f*(m1 + m2)^2)/(m1^2*(Cr + 1)^2)

The improved theoretical solution is





% collision3.m

m1 = 0.165; % mass of pool ball, kgm2 = 0.048; % mass of golf ball, kgCr = input('Coefficient of restitution: ');h2f = input('Desired final height for the second ball h2f: ');disp('The initial height for the first ball h1i is:')hi1 = (h2f*(m1 + m2)^2)/(m1^2*(Cr + 1)^2)

Let us compare the improved theoretical solution with the experimental result.

What happened now?

eg 10111/10112 introduction to engineering copyright © 2009 university of notre dame introduction...

Documents

models empirical models

theoretical models models

engineering copyright

smaller ball

larger ball

f pool ball

momentum empirical models

experimental data slide