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) responses

n 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)

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Questions

n 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.

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Engineering Process or Product Design

ExperimentsAnalysisModel

Analysis

OptimizationToolbox

DesignModel

Design

Engineering Processor Product

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Verification and Validation

n Models need to be validated and verified before they can be used with any confidence

n Verification: Are you building it right?– Is the theory/principle embodied in the model implemented correctly?

F ma

g = 9.81 m/s2

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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

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Questions

n 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.

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Case Study

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Case Study

n 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

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Case Study: Basic Steps

n 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.

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Case Study: Principle/Theory

n Euler-Bernoulli Beam Theory (w/o derivation)2

2

( ) ( )

( ) ( )

d v x M x

dx E x I xDifferential

Equation

Boundary Conditions

( 0) 0

( ) 0

v x

v 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

u

L

M Mdv

dx

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Case Study: Cantilever Beam

( 0) 0

( 0) 0

v x

dvx

dx

Boundary Conditions

2

( ) 36

Pxv x x L

EI

AB

x

y, v

L

P

Integrating twice and using the BCs

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Case Study: The Model

2

( ) 36

Pxv x x L

EI

Para. Remarks

P 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

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Case Study: Modulus of Elasticity

n 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)

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Case Study: Moment of Inertia

n 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

xdx

dy

y

y

2

2

x

A

y

A

I y dA

I x dA

32

32

12

12

x

A

y

A

whI y dA

w hI x dA

h

w

x

y

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

steel plate

tw

z

y

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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

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Raw Measurement Data

Histogram Plot

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Statistical Analysis of Data

Caliper 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

1

1 1

n

ii

n

ii

xn

nx

n n

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Questions

n 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.

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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.

6s

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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

x

F F

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Questions

n 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

( ) 36

Pxv x x L

EI

26

Experiment 2Measure the tip displacement of an

aluminum cantilever beam

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Raw Experimental Data

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Case Study: Model Verification

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Case Study: Model Validation

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

Published Computed

Computed

E EDiff

E

30

Forensic Engineering

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One-Parameter Regression Analysis

n 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

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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