20-Jan-2010 electrical, computer and energy engineering

Prof. Subramaniam (“Subby”) D. Rajan, Prof. Narayanan Neithalath and Amie Baisley

Graduate Students: Kirk Vance, Matt Aguayo, Tejas Ashani, Joseph Harrington and Canio Hoffarth

Engineering 101Linking Experiments to Models through the

Bridge Design Exercise

2

What are Experiments?n Tests to determine the relationship between (input) variables

and (output) responsesn Example 1: What is the effect of dowel diameter on the weight of

the bridge?– Model: The entire bridge system– Input Variable: Dowel diameter– Output Response: Weight of the bridge

n Example 2: What is the effect of dowel diameter on the maximum deflection of the bridge deck?– Model: The entire bridge system– Input Variable: Dowel diameter– Output Response: Deflection of the bridge deck at various locations

n

3

What are Models?n Relationship between (input) variables and (output) responses

– Simple equation– Model described by one or more complex equation(s) – differential

equation(s), integral equation(s), …n Example 1: What is the effect of dowel diameter on the weight of

the bridge?

n Example 2: What is the effect of dowel diameter on the maximum deflection of the bridge deck?– Needs a model whose solution can be described by several linear,

algebraic equations

2

1 4

ni

BRIDGE OTHER DOWELS OTHER i ii

dW W W W g L

4

What is a System?n Dictionary definitions

– a set of connected things or parts forming a complex whole, in particular– a set of principles or procedures according to which something is done; an

organized scheme or methodn Traits of a system

– has structure, its parts or components are directly or indirectly interact with each other

– has behavior (where input and output are linked)

5

Questionsn Q1: Draw a diagram that shows the components of the bridge

system, establishes the boundary and identifies the surroundings.

n Q2: Describe the bridge system with particular attention to (a) its functionalities, (b) how the different components interact with each other and (c) how the bridge system behaves.

6

Engineering Process or Product Design

ExperimentsAnalysisModel

Analysis

OptimizationToolbox

DesignModel

DesignEngineering Process

or Product

7

Verification and Validation n Models need to be validated and verified before they can be

used with any confidencen Verification: Are you building it right?

– Is the theory/principle embodied in the model implemented correctly?

F ma

g = 9.81 m/s2

8

Verification and Validation n Validation: Are you building the right thing?

– Do the results from the model correlate well with experimental results?

Trial M(kg)

m(kg)

Exp. a

(m/s2)

Model a

(m/s2)

% error

1

2

3

9

Questionsn Q3: Describe what a bridge model could be, by identifying the

input variables and output responses.n Q4: Identify the characteristics of each input variable. Describe

how you would obtain the values of these variables.n Q5: Identify the characteristics of each output response. What is

the purpose of each output response?n Q6: Give examples of engineering processes and products?n Q7: Describe the linkages between experiments and modeling.

10

Case Study

11

Case Studyn Develop a model to predict the tip deflection (displacement) of a

cantilever beam due to a tip load. Use experiments to validate the model.

AB x

y, v

L

P

B

12

Case Study: Basic Stepsn Use a sound scientific or engineering principle to develop the

model. What parameters will be a part of this model – input and output variables?

n Design experiment(s) to verify the model.n Design experiment(s) to validate the model.

13

Case Study: Principle/Theoryn Euler-Bernoulli Beam Theory (w/o derivation)

2

2

( ) ( )( ) ( )

d v x M xdx E x I x

Differential Equation

Boundary Conditions

( 0) 0( ) 0v xv x L

v(x): vertical displacementM(x): Bending momentE(x): Young’s modulusI(x): Moment of inertiaL: length of the beam

A B x

y, v

v

uL

M Mdvdx

14

Case Study: Cantilever Beam

( 0) 0

( 0) 0

v xdv xdx

Boundary Conditions

2

( ) 36Pxv x x LEI

AB x

y, v

L

P

Integrating twice and using the BCs

15

Case Study: The Model

2

( ) 36Pxv x x LEI

Para. RemarksP The applied load at the tip of the beam

E Material property that needs to be found

I Cross-sectional property that needs to be computed

L Length of the beam that needs to be measured

x Location where the displacement is computed

16

Case Study: Modulus of Elasticityn What is modulus of elasticity or Young’s modulus (E)?

– In a one-dimensional state of stress it is constant of proportionality between the normal stress and the normal strain and has the units of stress.

1

E

23

4

5

Stress-strain curve (ductile material)

17

Case Study: Moment of Inertian What is moment of inertia, I?

– The second moment of area (or, moment of inertia) is a measure of a beam’s cross-sectional shape’s resistance to bending.

X

Xc

YcY

C

O

x

x dxdy

yy

2

2

xA

yA

I y dA

I x dA

32

32

12

12

xA

yA

whI y dA

w hI x dA

h

w

x

y

18

ExperimentMeasure the width, w, and thickness, t, of a

steel plate

tw

z

y

19

Raw Measurement Data

Measurements taken at 11 different locations

Width (W) Thickness (T) Width (W) Thickness (T)(in) (in) (in) (in)

1.114 0.03 1.115 0.0311.1135 0.03 1.115 0.0341.1145 0.0305 1.115 0.0291.1145 0.03 1.115 0.031.114 0.0305 1.115 0.0341.113 0.0305 1.115 0.0331.115 0.0305 1.114 0.0321.114 0.03 1.114 0.0311.113 0.03 1.113 0.0311.113 0.0305 1.113 0.0311.113 0.0305 1.113 0.031

Caliper 1 Caliper 2

20

Raw Measurement DataHistogram Plot

21

Statistical Analysis of DataCaliper 1 Caliper 2

Width (in) Thickness (in) Width (in) Thickness (in)

# of readings (n)

11 11 11 11

Mean 1.1138 0.0303 1.1143 0.0315

Median 1.114 0.0305 1.115 0.031

Standard Deviation

0.0007198 0.0002611 0.000905 0.00157

: mean: standard deviation

1

2 2

1

1

11 1

n

ii

n

ii

xn

nxn n

22

Questionsn Q8: What is sample size? n Q9: What is mean? What is another name for mean?n Q10: What is median?n Q11: What is standard deviation?n Q12: Write a few sentences on the quality of the thickness and

width data for the steel plate.

23

Normal Distribution

2

221, ,2

x

Xf x e

Probability Density Function*

*Excel terminology: Probability Mass Function

68-95-99.7 rule: 1, 2, 3 standard deviations from mean

Function whose graph is a continuous curve over a range of values that x can take. It has the units of probability rate (not probability). x is called random variable.

Area under curve between x1 and x2 gives the probability that x lies in the interval x1 and x2.

6

24

Cumulative Distribution Function

( ) (z)x

X XF x f dz

What is the probability that a random width value is between 1.113 in and 1.114 in?

Pr[1.113 1.114](1.114) (1.113)

0.6 0.15 0.45X X

xF F

25

Questionsn Q13: Normal distribution is often called bell curve. Are there

other types of distribution? n Q14: Identify and rank the effect of the random variables in the

equation for tip deflection.

2

( ) 36Pxv x x LEI

26

Experiment 2Measure the tip displacement of an

aluminum cantilever beam

27

Raw Experimental Data

28

Case Study: Model Verification

29

Case Study: Model Validation

Published Elastic Modulus of Aluminum (6016-T6) = 1.01(107) psiPublished Computed

Computed

E EDiffE

30

Forensic Engineering

31

One-Parameter Regression Analysisn Objective: Use the model and experimental data to determine

the Young’s modulus of aluminum.n

2exp FEA

1

Find

to min ( )n

i ii

L U

E

f E

E E E

32

Referencesn Do an internet search using these keywords – system, model, experiment,

verification, validation, statistical quantities.n Engineering Statistics: http://www.itl.nist.gov/div898/handbook/n http://www.mathsisfun.com/links/curriculum-high-school-statistics.htmln http://www.stevespanglerscience.com/lab/experimentsn http://en.wikipedia.org/wiki/Verification_and_validation