Scaling Laws and Constraints • What governs the structure or architecture of systems? • Are we free to choose; are natural systems free to choose? • It seems that all systems evolve, grow, or are designed in the presence of contexts, constraints, laws of nature or economics, scarce resources, threats, failure modes • Can we trace system structure to these externalities? Internal Structure 2/16/2011 © Daniel E Whitney 1/45What Happens as Systems Get Bigger? • Systems do seem to get bigger over time • Can they retain the same structure as they do? • Typical barriers to growth: – Natural or biological systems • Resistance to mechanical and gravitational loads • Energy use and conservation • Internal distribution of resources or removal of wastes • Fight or flight tradeoffs – Man-made systems • Engineered products and product families: complexity, cognitive limits • Infrastructures: capacity, budgets, complexity, reliability Internal Structure 2/16/2011 © Daniel E Whitney 2/45An Early Explanation: Simon’s Fable • Tempus and Hora make different kinds of watches – One kind consists of separable subassemblies while the other consists of highly interconnected parts • If the makers are interrupted (analogous to environmental stress) the subassemblies survive but the integral units fall apart • Thus modular systems have survival value and gain prominence • Magically, they also develop hierarchical structure (presumably analogous to nested subassemblies) Internal Structure 2/16/2011 © Daniel E Whitney 3/45A Less Ambitious Goal • Instead of trying to explain where entire body plans (read: system structures) came from… • Let’s just try to explain some repeating patterns in these body plans that support scalability as a factor in survival • Examples include scaling laws for – Size and shape – Energy consumption – Heating and cooling – Distributive systems • Many of these models rely on network, hierarchy or fractal arguments or metaphors • A few engineering systems have been analyzed in a similar spirit if not necessarily using the same approaches Internal Structure 2/16/2011 © Daniel E Whitney 4/45Constraints as Drivers of Structure • There is almost always a constraint, a limiting resource, a failure mode • Systems do not waste resources and can’t violate limits on basic processes – Bandwidth, pressure drop, congestion, other flow limits – Energy balance, heat rejection to avoid temperature rise – Energy transfer rates across barriers, diffusion, radiation – Information processing, CPU speed, bounded rationality – Strength of materials • Kuhn-Tucker conditions state that constrained optimum the cost of violating the constraint ΔJ=-λΔCmin J= J +λC Jismaxhere a balances cost of missing the unconstrained optimum and ΔJ>-λΔC Internal Structure 2/16/2011 © Daniel E Whitney 5/45Multiple Constraints • When there are several constraints, a “balanced” design seeks to operate near several boundaries at once • Aircraft max passenger load + max fuel load + empty fuselage wt > max takeoff weight Max Pass MaxT.O.Wt. Max Fuel Internal Structure 2/16/2011 © Daniel E Whitney 6/45Scaling Laws • Geometric scaling (starting with Galileo) – Proportions are preserved as size increases • Allometric scaling (Buckingham and others) – Proportions are not preserved (baby to adult, shoes, etc.) – Instead, different elements of the system scale at different rates – Discovering what these rates are, and why they apply, is a research industry of its own in engineering, biology, sociology and economics • Scaling laws reveal the invariants of a system Internal Structure 2/16/2011 © Daniel E Whitney 7/45Proportions Change During Growth http://www.carseat.se/images/growthchartcarseat.jpg Internal Structure 2/16/2011 © Daniel E Whitney 8/45 Relative Proportions, Birth to AdulthoodAt birth 2 years 6 years 12 years 25 yearsImage by MIT OpenCourseWare.Two Uses for Scaling Laws • Predict how something big will behave using a small model – example of wave resistance to ships • Predict how system behavior changes as size (or some other variable) changes – Examples include power needed to run a ship as well as size or other limits on physical systems, including biological Internal Structure 2/16/2011 © Daniel E Whitney 9/4510/45© Daniel E Whitney2/16/2011Scaling and Dimensional Analysis • If physical variables are related by an equation of the form v2 = kv1 where k is a number, then this is equivalent to v2/ v1 = k which is a dimensionless scaling relationship • For example, when a ship goes through water it makes waves. The wave energy is related to the size of the wave by mV 2 = mgL or V 2/gL = k • This is called Froude’s Law of ships and it shows how energy loss rises as the ship gets bigger • For low wave energy losses, the Froude number F Internal Structure is kept below 0.4, which describes the wave speed F = V / gL = 1/ 2π= 0.3989Vwave = gL /2πWave energy Froude Law V = FgL F< 0.4 dominates Skin friction dominates F = 0.4 Short ships cannot go fast Max energy loss to waves when ship speed = wave speed and ship length = wave lengthhttp://www.globalsecurity.org/military/systems/ship/beginner.htm Internal Structure 2/16/2011 © Daniel E Whitney 11/45 Courtesy of Global Security. Used with permission.Adapted from “On Growth and Form” by D’Arcy Thompson Boilers, Coal, and Ship Speed Resistance to flow = R, a force R = Rskin friction loss + Rwave energy loss Froude found empirically that Rwave energy loss = k * Displacement (D) if the Froude number F = V / gL was kept constant L ∝V 2 (keep F constant) D ∝ L3 ∝V 6 Power P = R *V ∝V 7 ∝ D7/6 Fuel needed = P * time = P * dist /V ∝V 6 ∝ D So as your ship gets bigger, you have enough space for fuel but not for one boiler, so you must have several small boilers, each (2) paired with an engine and a propeller Internal Structure 2/16/2011 © Daniel E Whitney 12/45Scaling Laws in Biological Systems • Tree height vs diameter (Chave and Levin; Niklas and Spatz; McMahon) elephant legs and daddy long legs – Failure mode analysis: buckling Height = a * Diameter2/3 – Nutrient distribution • Metabolic rate vs body mass (Schimdt-Nielsen; Chave and Levin; West, Brown, and Enquist; McMahon; Bejan) – Small animals have so much surface area/mass that
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