1. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Introduction to Engineering Systems Lecture 3 (9/4/2009) Empirical Models: Fitting a Line to Experimental Data Prof. Andrés Tovar

2. EG 10111/10112 Introduction to Engineering Copyright © 2009 University 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

3. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Announcements Homework 1 –Available on Concourse http://concourse.nd.edu/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

4. EG 10111/10112 Introduction to Engineering Copyright © 2009 University 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 pool ball golf ball

5. EG 10111/10112 Introduction to Engineering Copyright © 2009 University 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? pool ball golf ball Empirical Models: Fitting a Line to Experimental Data

6. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Theoretical Model of Colliding Pendulums Given 2 pendulum masses m 1 and m 2 –golf ball initially at h 2i = 0 –pool ball released from h 1i –golf ball bounces up to h 2f –pool ball continues up to h 1f 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 pool ball golf ball

7. EG 10111/10112 Introduction to Engineering Copyright © 2009 University 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

8. EG 10111/10112 Introduction to Engineering Copyright © 2009 University 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

9. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Theoretical Model of Colliding Pendulums Empirical Models: Fitting a Line to Experimental Data 4) Finally 4) In Matlab this is h1i = (h2f*(m1 + m2)^2)/(4*m1^2);

10. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Matlab implementation Empirical Models: Fitting a Line to Experimental Data % 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)

11. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Matlab implementation Empirical Models: Fitting a Line to Experimental Data % collision1.m m1 = 0.165; % mass of pool ball, kg m2 = 0.048; % mass of golf ball, kg 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) plot(h2f,h1i,'o'); xlabel('h2f'); ylabel('h1i') hold on Let us compare the theoretical solution with the experimental result. What happened?!?!

12. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Run the Pendulum Experiment Empirical Models: Fitting a Line to Experimental Data Pool ball release height (m) Golf ball final height (m) 0.00 0.05 0.10 0.15 0.20 0.25

13. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Experimental Results Empirical Models: Fitting a Line to Experimental Data % collision2.m h1ie = 0:0.05:0.25; % heights for pool ball, m h2fe = []; % experimental results for golf ball, m plot(h1ie,h2fe, '*')

14. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame MATLAB GUI for Least Squares Fit Empirical Models: Fitting a Line to Experimental Data

15. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame MATLAB commands for Least Squares Fit Empirical Models: Fitting a Line to Experimental Data % collision2.m h1ie = 0:0.05:0.25; % heights for pool ball, m h2fe = []; % experimental results for golf ball, m plot(h1ie,h2fe, '*') c = polyfit(h1ie, h2fe, 1) m = c(1) % slope b = 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)

16. EG 10111/10112 Introduction to Engineering Copyright © 2009 University 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 Empirical Models: Fitting a Line to Experimental Data

17. EG 10111/10112 Introduction to Engineering Copyright © 2009 University 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

18. EG 10111/10112 Introduction to Engineering Copyright © 2009 University of Notre Dame Matlab implementation Empirical Models: Fitting a Line to Experimental Data % collision3.m m1 = 0.165; % mass of pool ball, kg m2 = 0.048; % mass of golf ball, kg Cr = 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?