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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