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Empirical Models: Fitting a Line to Experimental Data

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Empirical Models: Fitting a Line to Experimental Data

Introduction to Engineering Systems

Lecture 3 (9/4/2009)

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

From last class

pool

ball

golf

ball

- 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

From last class

pool

ball

golf

ball

- 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?

Empirical Models: Fitting a Line to Experimental Data

Theoretical Model of Colliding Pendulums

pool

ball

golf

ball

- 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

Theoretical Model of Colliding Pendulums

Collision model:

Relative

velocity

Conservation of momentum

Conservation of energy

Conservation of energy

Empirical Models: Fitting a Line to Experimental Data

Theoretical Model of Colliding Pendulums

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

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

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

Matlab implementation

% 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?!?!

Empirical Models: Fitting a Line to Experimental Data

Run the Pendulum Experiment

Empirical Models: Fitting a Line to Experimental Data

Experimental Results

% collision2.m

h1ie = 0:0.05:0.25; % heights for pool ball, m

h2fe = []; % experimental results for golf ball, m

plot(h1ie,h2fe,'*')

Empirical Models: Fitting a Line to Experimental Data

MATLAB GUI for Least Squares Fit

Empirical Models: Fitting a Line to Experimental Data

MATLAB commands for Least Squares Fit

% 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)

Empirical Models: Fitting a Line to Experimental Data

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

Improved collision model

Huygens’ principle of relative velocity

Coefficient of restitution

Improved collision model: COR and conservation of momentum

The improved theoretical solution is

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

Empirical Models: Fitting a Line to Experimental Data

Matlab implementation

% 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?

Empirical Models: Fitting a Line to Experimental Data

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