Conflicting objectives in design Common design objectives: - - PowerPoint PPT Presentation

ME 474-674 Spring 2008 Slides 11 -1

Conflicting objectives in design

Each defines a performance metric. Example

mass, m we wish to minimize both cost, C (all constraints being met)

Conflict : the choice that optimizes one does not optimize the other. Best choice is a compromise. Common design objectives:

Minimizing mass (sprint bike; satellite components) Minimizing volume (mobile phone; minidisk player) Minimizing environmental impact (packaging, cars) Maximizing performance (speed, acceleration of a car) Minimizing cost (everything) Objectives

More info: “Materials Selection in Mechanical Design”, Chapters 9 and 10

ME 474-674 Spring 2008 Slides 11 -2

Multiple Constraints and Conflicting Objectives

Solution: a viable choice, meeting constraints, but not necessarily optimum by either criterion.

• Plot solutions as function of

performance metrics.

• Convention: express objectives to be

minimized

• Dominated solution: one that is

unambiguously non-optimal (as A)

• Non-dominated solution: one that is
• ptimal by one metric (as B: optimal by
• ne criterion but not necessarily by both

Trade-off surface: the surface on which the non-dominated solutions lie (also called the Pareto Front)

Light

Metric 1: Mass m

Heavy Cheap

Metric 2: Cost C

Expensive

Trade-off surface

A Dominated solution B Non-dominated solution

There are several possible strategies for trading off or compromising among conflicting objectives

ME 474-674 Spring 2008 Slides 11 -3

Finding a compromise: Strategy 1

Make a plot of the conflicting

• bjectives
• Usually drawn in a manner

that both metrics have to be minimized

Sketch the trade off surface Use intuition to select among the

solutions closest to the trade off surface

But then what is the relative

value of the two variables

• How important is cost

compared to mass?

• Do you pick solutions that

are light but more expensive or those that are heavy, but less expensive?

Light

Metric 1: Mass m

Heavy Cheap

Metric 2: Cost C Expensive

Trade-off surface

ME 474-674 Spring 2008 Slides 11 -4

Cars: Cost-Performance Trade-off

Pence per mile

10 20 50 100 200

1/Top speed

4e-3 6e-3 8e-3 0.01

Skoda Octavia (98-) Toyota Corolla Peugeot 307 Toyota Land-cruiser Land Rover Defender Land Rover Range Rover Isuzu Trooper Renault Clio Citroën C2 Jaguar 4.2 V8 SE Toyota Yaris Fiat Punto 1.2 Smart Fortwo Mercedes-Benz CL 600 Mercedes-Benz C320 SE Porsche Boxster

Cost of ownership (cents/mile) Reciprocal of performance (1/Top speed) Cost-performance trade-off: cars

Trade-off surface

Your car

Cheaper and faster Slower and more expensive Cheaper but slower Faster but more expensive

4-quadrant Plot

ME 474-674 Spring 2008 Slides 11 -5

Finding a compromise – Strategy 2

Reformulate all but one of the

• bjectives as constraints, setting

an upper limit for it

Good if budget limit Trade-off surface gives the best

choice within budget

Not true optimization -- cost

treated as constraint, not

• bjective.

Optimum solution minimising m Light

Metric 1: Mass m

Heavy Cheap

Metric 2: Cost C

Expensive

Trade-off surface

Upper limit on C

Best choice

ME 474-674 Spring 2008 Slides 11 -6

Finding a compromise – Strategy 3 Systematic Methods for multiple constraints

Each constraint gives rise to an equation that must be maximized or

minimized

Consider a tie rod

• Light stiff tie rod

Light strong tie rod

The material parameters to be minimized are different

E M E S L E FL L LA m AE FL ρ ρ δ ρ ρ δ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = = = =

1 2 1

f f f f f f f

M LF LA m F A A F σ ρ σ ρ ρ σ σ = = = = =

2 2

ME 474-674 Spring 2008 Slides 11 -7

Metals

Density / Yield strength (elastic limit) 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000 Density / Young's modulus 200 400 600 800 1000 1200 1400 1600 1800 2000

Thallium, Commercial Purity Indium, Commercial Purity, min 99.97% Corroding Lead Chemical Lead Lead, Arsenical, F-3 alloy, extruded and air cooled Bismuth Metal, Commercial Purity Antimony metal, Commercial Purity, "Regulus" Calcium Lead with 0.3% tin, cast

ME 474-674 Spring 2008 Slides 11 -8

Density / Yield strength (elastic limit) 5 10 15 20 25 30 35 40 45 50 55 Density / Young's modulus 5 10 15 20 25 30 35 40 45 50

AlBeMet 162 Beryllium, grade 0-50, hot isostatically pressed Beryllium Aluminum Casting Alloy, Beralcast 191 Molybdenum, 360 grade, wrought, 150 micron wire

ME 474-674 Spring 2008 Slides 11 -9

Density / Yield strength (elastic limit) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Density / Young's modulus 20 22 24 26 28 30 32 34 36 38 40

Wrought aluminum alloy, 6061, T651 Carbon steel, AISI 1020 (annealed) Titanium alpha-beta alloy, Ti-6Al-4V, Annealed (generic)

ME 474-674 Spring 2008 Slides 11 -10

Finding a compromise – Strategy 3

Consider a tie rod with

• L= 1m
• S = 3 x 107 N/m and
• Ff = 105 N

Then the two criteria may give rise to different materials with different values of m1

and m2

For each material, the larger of m1 and m2 is the better choice m*, and if all materials

are considered, then the one with the smallest of m* is the optimum choice 1.15 0.46 1.15 950 115 4400 Ti-6-4 2.25 2.25 1.16 120 70 2700 AA 6061 2.45 2.45 1.15 320 205 7850 1020 steel m* (kg) m2(kg) m1 (kg) σy (MPa) E (GPa) ρ (kg/m3) Material

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = E S L m ρ

2 1

f f

LF m σ ρ =

2

ME 474-674 Spring 2008 Slides 11 -11

Density / Yield strength (elastic limit) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Density / Young's modulus 5 10 15 20 25 30 35 40 45 50

Titanium alpha-beta alloy, Ti-6Al-4V, Aged Wrought aluminum alloy, 6061, T652 Carbon steel, AISI 1020 (as-rolled)

ME 474-674 Spring 2008 Slides 11 -12

Finding a compromise – Strategy 3

In general, if we can have several equations for minimizing mass For the problem of the tie rod, where we have two equations for mass that

must be minimized. If both equations are satisfied at the same time, then

Cc is called the coupling constant between the two material parameters

etc M G F m M G F m

2 2 2 2 1 1 1 1

= =

1 2 1 2 2 1 1 2 2 2 2 1 1 1 2 1

M C M M G F G F M M G F M G F m m

c

= = = =

ME 474-674 Spring 2008 Slides 11 -13

Finding a compromise – Strategy 3

Mass m1

Heavier Heavier Lighter Lighter

Mass m2

Best Choice

Index M1

Smaller Smaller Larger Larger

Index M2 Large Cc Small Cc

Constraint 1 dominant Constraint 2 dominant

The coupling constant depends upon the particular problem

M2 = CcM1

ME 474-674 Spring 2008 Slides 11 -14

Finding a compromise – Strategy 4

If the two conflicting requirements are

minimum mass m and minimum cost C

Define a locally linear penalty function Z

Z = αm + C

Seek material with smallest Z:

• Either evaluate Z for each solution,

and rank,

Or make a trade-off plot with contours of

constant Z C = – αm + Z

Lines of constant Z all have slopes of – α Read off the solution with the smallest Z. This gives the best solution for a given

value of α

But what is α? Light

Metric 1: Mass m

Heavy Cheap

Metric 2: Cost C

Expensive Optimum solution, minimising Z

Z1 Z2 Z3 Z4

Contours of constant Z Decreasing values of Z

α −

ME 474-674 Spring 2008 Slides 11 -15

The exchange constant α

The constant α is called the exchange constant since it is a conversion

factor between two objective variables, the mass m and the cost C.

It is a measure of the value of saving unit mass The value of a can be obtained: From historical data Performing an analysis of the full life cost, or By interviewing experts

C

m Z C m Z ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = ⇒ + = α α

Z

m C m Z C ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − = ⇒ − = α α

ME 474-674 Spring 2008 Slides 11 -16

The exchange constant α

The table below shows the exchange constants for mass saving The exchange constant for a passenger car was obtained from a

full life analysis, i.e., how much fuel cost savings would be

• btained if a lighter vehicle were built

Mass becomes very important in space craft, but not for a car Transport System: mass saving α ($ per kg) Family car (based on fuel saving) Truck (based on payload) Civil aircraft (based on payload) Military aircraft (performance payload) Space vehicle (based on payload) 1 to 2 5 to 20 100 to 500 500 to 2000 1000 to 9000

ME 474-674 Spring 2008 Slides 11 -17

Graphical representation

Linear scales

Lighter

mass, m Heavier

Cheap

Cost, C

Expensive

Decreasing values of Z

• α

Log scales

Lighter

mass, m Heavier

Cheap

Cost, C

Expensive

Decreasing values of Z

C m Z + = α The equation when plotted on a linear plot is straight lines with a negative slope of –α for different values of Z. On the log scale, these lines become the curves shown.

ME 474-674 Spring 2008 Slides 11 -18

Trade off: mass vs. cost for given stiffness

Density/Sqrt Modulus

50 100 200 500 1000 2000 5000

Density x Price /Sqrt Modulus

10 100 1000 10000 100000 1e6

MAGNESIUM alloys GFRP Epoxy/HS Carbon weave ALUMINUM alloys HSLA steels CAST IRONS Zinc alloys Lead alloys Copper alloys Tungsten alloys Bronze CFRP epoxy laminate Ti-alloys Ni-based superalloys Cobased superalloys

Material

cost

for given stiffness

Mass

for given stiffness

Exchange constant α = 0.5 $/kg Exchange constant α = 500 $/kg

ME 474-674 Spring 2008 Slides 11 -19

Case study: casing for a minidisk player

Electronic equipment -- portable computers,

players, mobile phones – are miniaturized; many less than 12 mm thick

An ABS or Polycarbonate casing has to be

> 1mm thick to be stiff enough to protect; casing takes 20% of the volume

• stiff, light, thin casing

bending stiffness EI at least that of existing case minimize casing thickness minimize casing mass choice of material casing thickness, t Constraints Objectives Function Free variables

The thinnest may not

be the lightest … need to explore trade-off

ME 474-674 Spring 2008 Slides 11 -20

Performance metrics for the casing

Function Stiff casing t w L F Metric 1

3 / 1 3 / 1 3

E 1 w E 4 L S t ∝ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Objective 2 Minimize mass m Metric 2

3 / 1

E m ρ ∝

m = mass w = width L = length ρ = density t = thickness S = required stiffness I = second moment of area E = Young’s Modulus

Objective 1 Minimize thickness t

3

L I E 48 S = Constraints 12 t w I

3

=

Adequate toughness,

G1c > 1kJ/m2

Stiffness, S

with

ME 474-674 Spring 2008 Slides 11 -21

Relative performance metrics

The thickness of a casing made from an alternative material M, differs

(for the same stiffness) from one made of Mo by the factor

3 / 1

• E

E t t ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =

The mass differs

by the factor ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ρ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ρ =

• 3

/ 1

• 3

/ 1

• E

. E m m

• m

m

Explore the trade-off between

and

• t

t We are interested here in substitution. Suppose the casing is currently made of a material Mo – say ABS.

Need a relative

penalty function, Z

• t

t α m m Z + = (α now dimensionless)

ME 474-674 Spring 2008 Slides 11 -22

Thickness relative to ABS

0.1 1 10

Mass relative to ABS

0.1 1 10

PTFE PC ABS PM M A PP Nylon Polyester PE Ionomer Ni-alloys Cu-alloys Steels Al-alloys Al-SiC Composite T i-alloys M g-alloys CFRP GFRP Lead Polymer foams . Elastomers

Thickness relative to ABS, t/to Mass relative to ABS, m/mo

Trade-off surface

Four-sector trade-off plot

Is material cost relevant? Not a lot -- the case only weighs a few

• grams. Volume and weight are much more valuable.

The four sectors of a trade-off plot for substitution

• A. Better by

both metrics

• C. Lighter

but thicker

• D. Worse by

both metrics

• B. Thinner but

heavier

ME 474-674 Spring 2008 Slides 11 -23

Thickness relative to ABS

0 .1 1 1 0

Mass relative to ABS

1 1 0

Low alloy ste e l Al-alloys M g-alloys G FR P CFR P Al-SiC Composite s T i-alloys AB S Ni-alloys

Thickness relative to ABS Mass relative to ABS Z1 Z2 Z3

Plotting the penalty function, Z

Penalty lines for casing Assume mass and thickness are equally important: α = 1

Z t t α m m

• +

− =

Decreasing values of Z

ME 474-674 Spring 2008 Slides 11 -24

How to get “Z” lines on a log-log scale

Plots like these require the use of the Advanced facility in the Select mode of

the EduPack software.

• Functions of properties to be created and plotted.
• The curved penalty selection line is obtained by first switching to linear

scales

• Plotting a selection line with the slope you want (-1) and then switching

back to logarithmic scales.