Welcome

ME 482/582 OPTIMAL DESIGN

Rudy J. Eggert, Professor EmeritusMechanical & Biomedical Engineering

http://coen.boisestate.edu/reggerthttp://highpeakpress.com/eggert/

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Today’s lecture

• Optimization• Design• Analysis versus design• Phases of design• Parametric design• Mathematics review

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OPTIMAL DESIGN

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Definition:

The development and use of analytical and computer methods to provide an optimal design of a product or process with minimal computational effort.

That’s right…

The thing we design will be optimal AND the methods we use will be optimal.

.

Product Realization Process

Industrial DesignEngineering Design

Production Design

Manufacturing(Production)

DistributionService

Disposal

CustomerNeed

RealizedProduct

Sales / Marketing

Product Development

Design

Design

controlholdmoveprotectstore 

decision making processes

shapeconfigurationsizematerialsmanufacturing processes

Function

Form

 

 

Set of decision making processes and activities to determine: the form of an object, given the customer’s desired function.

Analysis is not Design

Which of the following is design and which is analysis?

A. Given that the customer wishes to fasten together two steel plates, select appropriate sizes for the bolt, nut and washer.

B. Given the cross-section geometry of a new airplane wing we determine the lift it produces by conducting wind tunnel experiments.

Problem Type Solution

Design Form(size, shape, matls,cnfg, mfg )

Analysis Predicted behavior(performance)

System Evolution (Arora)

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Figure 1.1 System evolution model.

Design Phases

Formulation

Detail

Parametric

Configuration

Concept

Embodiment Design

Preliminary Design

From Customer Needs thru Concept Design

?

FormulationFormulation

Customer Needs

Customer requirementsImportance weightsEng. characteristicsHouse of QualityEng. Design Spec’s

Concept DesignConcept Design

Abstract embodiment Physical principles Material Geometry

Configuration Design

ConfigurationDesign

ConfigurationDesign

Special Purpose Parts: Features Arrangements Relative dimensions Attribute list (variables)Standard Parts: Type Attribute list (variables)

Abstract embodiment Physical principles Material Geometry

Architecture

Design Phases Cont’dSpecial Purpose Parts: Features Arrangements Relative dimensions Variable list Standard Parts: Type Variable list

ParametricDesign

ParametricDesign

Design variable valuese.g. Sizes, dimensions Materials Mfg. processesPerformance predictionsOverall satisfactionPrototype test results

DetailDesignDetailDesign

Product specificationsProduction drawingsPerformance Tests Bills of materials Mfg. specifications

Design Optimal Design

12Figure 1.2 Comparison of (a) conventional design method and (b) optimum design method.

Systematic Parametric Design

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Determine best alternative

Predict Performance Check Feasibility: Functional? Manufacturable ?

Generate Alternatives

Formulate Problem

Analyze Alternatives

Evaluate Alternatives

Re-Design

Re-Specify

Select Design Variables Determine constraints

Select values for Design Variables

all alternatives

feasible alternatives

best alternative

Refine Optimize

refined best alternative

Engineering Design, Eggert, 2010

Tools used in Optimal Design

• Algebra• Calculus• Vector and matrix aritmetic• Excel (computation & graphing)• Graphing (hand)• Computer Programming (any language)• Engineering principles

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Systematic Parametric Design

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Mathematical Notation

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z)y,(x, f

Recall from Calculus, a function of many variables:

We shall use vectors for multiple variables:

x bold note)( xf

Tn

n

xxx

x

x

x

212

1

,

x

The transpose is used to show a row

All vectors arecolumns

Handwritten vectors

The book shows vectors as lower case bolded, for example:

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x bold note)( xf

For handwritten homework and tests… we will use lower case hand-printed with an underscore, for example:

e underscornote)( xf

Tnxxx 21, x

Points P, x(1) and x(2)

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Figure 1.3 Vector representation of a point P that is in 3-dimensional space.

3.0

5.5

1.2

2.1

3.2

3.1

)2(

)1(

x

x

Superscripts (1),(2)

Vector or point?

Is a “vector” a “point” in n-dimensional space denoted as R(n) ?

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Figure 1.4 Image of a geometrical representation for the set S = {x|(x1 – 4)2 + (x2 – 4)2 9}.

S = {x|(x1 – 4)2 + (x2 – 4)2 9}.

Set of Points, S

Dot Product

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n

i ii yx1yxyx T

3322111yxyxyxyx

n

i ii yx

From Engineering Statics:

In optimal Design:

)cos( yxyx

3322111

321321 ,,,,

yxyxyxyx

yyyxxxn

i ii

TT

yxT

How do we know if two vectors are orthogonal (normal) ?

Vector or Scalar?

Is a dot product of two vectors avector or scalar quantity?

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Norm of a vector

The length or magnitude of a vector is called the NORM.

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xxx

n

iix

1

2

Product of vector and matrix

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131333

321

321

321

3

2

1

)2(

)35(

)(

112

135

111

xxx

xxx

xxx

xxx

x

x

x

yxA

Is the product a scalar or vector?

Triple Product

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AxxAxxc T

43)9(1)8(2)6(3

9

8

6

123

)13..()13()13)(33(

)1)(1()2)(1()3)(2(

)1)(1()2)(3()3)(5(

)1)(1()2)(1()3)(1(

123

1

2

3

112

135

111

123

columnxrowseixxx

Rusty? …. Review appendix A, pgs 785-822

Function continuity

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Figure 1.5 Continuous and discontinuous functions: (a) and (b) continuous functions; (c) not a function; (d) discontinuous function.

First Partial Derivatives of a function

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T

n

n

x

f

x

f

x

f

x

f

x

f

x

f

f*21

*

2

1

*)(x

x

x

Gradient vector

We’ll se a lot of these in chapter 4.

Second Partial Deriivatives of a function…

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*

2

2

21

2

2

2

2

2

22

2

22

2

21

2

22

2

2

2

21

2

22

2

21

2

21

2

2 *)(

x

xH

nn

n

n

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

fx

f

x

f

x

f

x

f

x

f

f

Hessian Matrix

What does the x* mean?

Summary

• Design• Optimal design• Design Phases• Systematic Parametric Design• Vector, matrix review

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