Conflicting objectives in design

Transcription

1 Conflicting objectives in design Common design objectives: Minimizing mass (sprint bike; satellite components) Objectives Minimizing volume (mobile phone; minidisk player) Minimizing environmental impact (packaging, cars) Maximizing performance (speed, acceleration of a car) Minimizing cost (everything) Each defines a performance metric. Example mass, m cost, C we wish to minimize both (all constraints being met) Conflict : the choice that optimizes one does not optimize the other. Best choice is a compromise. More info: Materials Selection in Mechanical Design, Chapters 9 and 0 ME Spring 008 Slides -

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 optimal by one metric (as B: optimal by one criterion but not necessarily by both Trade-off surface: the surface on which the non-dominated solutions lie (also called the Pareto Front) Cheap Metric : Cost C Expensive B Non-dominated solution Trade-off surface A Dominated solution Light Metric : Mass m Heavy There are several possible strategies for trading off or compromising among conflicting objectives ME Spring 008 Slides -

3 Finding a compromise: Strategy Make a plot of the conflicting objectives 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? Cheap Metric : Cost C Expensive Trade-off surface Light Metric : Mass m Heavy ME Spring 008 Slides -3

4 Cars: Cost-Performance Trade-off 4-quadrant Plot Reciprocal of performance (/Top speed) /Top speed 0.0 8e-3 6e-3 4e-3 Cheaper but slower Trade-off surface Smart Fortwo Fiat Punto. Toyota Yaris Renault Clio Citroën C Cheaper and faster Peugeot 307 Toyota Corolla Skoda Octavia (98-) Mercedes-Benz C30 SE Cost-performance trade-off: cars Land Rover Defender Your car Isuzu Trooper Slower and more expensive Toyota Land-cruiser Land Rover Range Rover Faster but more Porsche Boxster Jaguar 4. V8 SE Mercedes-Benz expensivecl Pence per mile Cost of ownership (cents/mile) ME Spring 008 Slides -4

5 Finding a compromise Strategy Reformulate all but one of the objectives 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 objective. Cheap Metric : Cost C Expensive Best choice Upper limit on C Optimum solution minimising m Trade-off surface Light Metric : Mass m Heavy ME Spring 008 Slides -5

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 δ = m M FL AE = ρla = ρ = E FL ρ ρl = L S Eδ E The material parameters to be minimized are different Ff σ f = A Ff A = σ m M f = ρla = = ρ σ f LF f ρ σ f ME Spring 008 Slides -6

7 000 Metals Density / Young's modulus Thallium, Commercial Purity Calcium Lead with 0.3% tin, cast Chemical Lead Corroding Lead Indium, Commercial Purity, min 99.97% 400 Lead, Arsenical, F-3 alloy, extruded and air cooled 00 0 Bismuth Metal, Commercial Purity Antimony metal, Commercial Purity, "Regulus" Density / Yield strength (elastic limit) ME Spring 008 Slides -7

8 Density / Young's modulus Molybdenum, 360 grade, wrought, 50 micron wire AlBeMet 6 Beryllium Aluminum Casting Alloy, Beralcast 9 Beryllium, grade 0-50, hot isostatically pressed Density / Yield strength (elastic limit) ME Spring 008 Slides -8

9 Density / Young's modulus Titanium alpha-beta alloy, Ti-6Al-4V, Annealed (generic) Wrought aluminum alloy, 606, T65 Carbon steel, AISI 00 (annealed) Density ME / Yield strength Spring 008(elastic limit) Slides -9

10 Finding a compromise Strategy 3 Consider a tie rod with L= m S = 3 x 0 7 N/m and F f = 0 5 N Then the two criteria may give rise to different materials with different values of m and m For each material, the larger of m and m is the better choice m*, and if all materials are considered, then the one with the smallest of m* is the optimum choice m = ρ L S E m = LF f ρ σ f Material ρ (kg/m 3 ) E (GPa) σ y (MPa) m (kg) m (kg) m* (kg) 00 steel AA Ti ME Spring 008 Slides -0

11 Density / Young's modulus Titanium alpha-beta alloy, Ti-6Al-4V, Aged Wrought aluminum alloy, 606, T65 Carbon steel, AISI 00 (as-rolled) Density / Yield strength (elastic limit) ME Spring 008 Slides -

12 Finding a compromise Strategy 3 In general, if we can have several equations for minimizing mass m m etc = F G M = F G M 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 m = m F G M M M F G = F G = C = F G M c M M C c is called the coupling constant between the two material parameters ME Spring 008 Slides -

13 Finding a compromise Strategy 3 The coupling constant depends upon the particular problem M = C c M Heavier Larger Constraint dominant Large C c Lighter Mass m Lighter Best Choice Mass m Heavier Index M Smaller Smaller Small C c Constraint dominant Index M Larger ME Spring 008 Slides -3

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? Cheap Metric : Cost C Expensive Z Z Z 3 4 Contours of constant Z Z Decreasing values of Z Optimum solution, minimising Z α Light Metric : Mass m Heavy ME Spring 008 Slides -4

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. Z = αm + C α = Z m C C C = Z αm α = m Z 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 ME Spring 008 Slides -5

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 obtained if a lighter vehicle were built Mass becomes very important in space craft, but not for a car Transport System: mass saving Family car (based on fuel saving) Truck (based on payload) Civil aircraft (based on payload) Military aircraft (performance payload) Space vehicle (based on payload) α ($ per kg) to 5 to 0 00 to to to 9000 ME Spring 008 Slides -6

17 Graphical representation The equation Z = αm + C 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. Linear scales Log scales Cheap Cost, C Expensive Lighter Decreasing values of Z -α mass, m Heavier Cheap Cost, C Expensive Lighter Decreasing values of Z mass, m Heavier ME Spring 008 Slides -7

18 Trade off: mass vs. cost for given stiffness Material cost for given stiffness Density x Price /Sqrt Modulus e Epoxy/HS Carbon weave CFRP epoxy laminate GFRP MAGNESIUM alloys Ni-based superalloys ALUMINUM alloys Ti-alloys Exchange constant Cobased superalloys α = 500 $/kg Tungsten alloys Lead alloys Bronze Copper alloys Exchange constant Zinc alloys α = 0.5 $/kg 0 HSLA steels CAST IRONS Density/Sqrt Modulus Mass for given stiffness ME Spring 008 Slides -8

19 Case study: casing for a minidisk player Electronic equipment -- portable computers, players, mobile phones are miniaturized; many less than mm thick An ABS or Polycarbonate casing has to be > mm thick to be stiff enough to protect; casing takes 0% of the volume Function stiff, light, thin casing Constraints Objectives Free variables bending stiffness EI at least that of existing case minimize casing thickness minimize casing mass choice of material casing thickness, t The thinnest may not be the lightest need to explore trade-off ME Spring 008 Slides -9

20 Performance metrics for the casing Function Constraints Objective Stiff casing Stiffness, S S = 48 E I 3 L with / 3 I = Minimize thickness t w t Adequate toughness, G c > kj/m 3 S L Metric t = 4E w / 3 E 3 t w L F m = mass w = width L = length ρ = density t = thickness S = required stiffness I = second moment of area E = Young s Modulus Objective Minimize mass m Metric m ρ / E 3 ME Spring 008 Slides -0

21 Relative performance metrics We are interested here in substitution. Suppose the casing is currently made of a material M o say ABS. The thickness of a casing made from an alternative material M, differs (for the same stiffness) from one made of M o by the factor t t o = Eo E / 3 The mass differs by the factor m m o = E ρ / 3 E. ρ / 3 o o t Explore the trade-off between and t o m m o Need a relative penalty function, Z Z = m m + α o t t (α now dimensionless) o ME Spring 008 Slides -

22 Four-sector trade-off plot The four sectors of a trade-off plot for substitution Mass relative to ABS, m/m o Mass relative to ABS 0 B. Thinner Trade-off but heavier surface Cu-alloys Ni-alloys Steels T i-alloys Al-alloys Al-SiC Composite Mg-alloys CFRP Lead PC PMMA GFRP AB S Polyester PTFE PP Nylon Elastomers Ionomer PE. Polymer foams D. Worse by both metrics 0. A. Better by both metrics C. Lighter but thicker 0. 0 Thickness relative to ABS Thickness relative to ABS, t/t o Is material cost relevant? Not a lot -- the case only weighs a few grams. Volume and weight are much more valuable. ME Spring 008 Slides -

23 Penalty lines for casing Plotting the penalty function, Z Assume mass and thickness are equally important: α = m m o = α t t o + Z 0 Mass relative to ABS Mass relative to ABS Ni-alloys Low alloy steel T i-alloys Al-SiC Composites Al-alloys M g-alloys CFRP GFRP AB S Decreasing values of Z Z Z Z Thickness relative to A B S Thickness relative to ABS ME Spring 008 Slides -3

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 (-) and then switching back to logarithmic scales. ME Spring 008 Slides -4