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Transcript of 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
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
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
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
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
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
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
Empirical Models: Fitting a Line to Experimental Data
poolball
golfball
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
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
Empirical Models: Fitting a Line to Experimental Data
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
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
Empirical Models: Fitting a Line to Experimental Data
poolball
golfball
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
Theoretical Model of Colliding Pendulums
Empirical Models: Fitting a Line to Experimental Data
Collision model:
Relative velocity
Conservation of momentum
Conservation of energy
Conservation of energy
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
Theoretical Model of Colliding Pendulums
Empirical Models: Fitting a Line to Experimental Data
1) Conservation of energy
2) Collision model: relative velocity and conservation of momentum
3) Conservation of energy
Empirical Models: Fitting a Line to Experimental Data
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
Theoretical Model of Colliding Pendulums
4) Finally
4) In Matlab this is
h1i = (h2f*(m1 + m2)^2)/(4*m1^2);
Empirical Models: Fitting a Line to Experimental Data
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
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)
Empirical Models: Fitting a Line to Experimental Data
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
Matlab implementation
% 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?!?!
Empirical Models: Fitting a Line to Experimental Data
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
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
Empirical Models: Fitting a Line to Experimental Data
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
Experimental Results
% collision2.m
h1ie = 0:0.05:0.25; % heights for pool ball, mh2fe = []; % experimental results for golf ball, mplot(h1ie,h2fe, '*')
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
MATLAB GUI for Least Squares Fit
Empirical Models: Fitting a Line to Experimental Data
Empirical Models: Fitting a Line to Experimental Data
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
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)
Empirical Models: Fitting a Line to Experimental Data
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
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
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
Improved collision model
Empirical Models: Fitting a Line to Experimental Data
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
Empirical Models: Fitting a Line to Experimental Data
EG 10111/10112 Introduction to Engineering
Copyright © 2009University of Notre Dame
Matlab implementation
% 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?