ENGM 670 & MECE 758

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ENGM 670 & ENGM 670 & MECE 758MECE 758Modeling and Simulation of Engineering Modeling and Simulation of Engineering SystemsSystems

(Advanced Topics)(Advanced Topics)Winter Winter 20112011

Lecture Lecture 5:5:Extra MaterialExtra Material

M.G. LipsettM.G. LipsettDepartment of Mechanical EngineeringDepartment of Mechanical Engineering

University of AlbertaUniversity of Albertahttp://www.ualberta.ca/~mlipsett/ENGM541/ENGM541.htm

© MG Lipsett, 2011 2

Department of Mechanical Engineering

ENGM 670 MECE 758 – Modeling & Simulation of Engineering Systems, Lecture 5: Advanced Topics Extra Material

ENGM670ENGM670--X5 Lecture 5 Extra MaterialX5 Lecture 5 Extra Material

• Engineering and management is all about resources• The design of a system is, at its essence, a decision about

using limited or valuable resources to achieve some objective

• The design is a choice for combining different types of resources and their interactions according to some rules (including physical laws)

• Three main factors :– Mechanics of the transformation process (input-process-

output in producing the design)– Values of resources– Values of products

• Values of resources and products are generally out of the designer’s control

http://www.ualberta.ca/~mlipsett/ENGM541/ENGM541.htm

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The Resource Allocation ProblemThe Resource Allocation Problem

Resourcevalue

Transformationsystem

Product value

Designer decision options




Production, Resource Cost, and Revenue Production, Resource Cost, and Revenue

• The resource allocation problem can be addressed by defining the production function, a mathematical representation of an efficient production function

• Three functions associated with the design factors:– Physical production process– Cost of resources– Value of products

• The production function is based on the mechanics of the process, without considering the external factors

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Production, Resource Cost, & Revenue Production, Resource Cost, & Revenue

Physicalresources

Production process or function

Physical goods &/or services

Resource valuationor cost function

Inputvalue

Product valuationor revenue function

Outputvalue

(Transformation system)




Production FunctionProduction Function

• Representation of how resources are transformed into products (goods &/or services)

• Mathematical description of the maximum output Z that can be obtained from a given set of resources (X1,…, XN) :

Z = g(X1,…, XN)• In a production function, money and value are NOT part of

this model• Inputs are tracked as resources themselves, but not with

any monetary value• Output is measured in units of production, not the price they

could command

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Production Function (2)Production Function (2)• Here is an example of a production function for a single

resource

• Production function is the boundary between regions

Prod

uct

(in u

nits

of p

rodu

ctio

n)

Resource (units of resource)

Region of feasible combinations

Region of infeasible combinations




Production Function (3)Production Function (3)

• Can relate any number of inputs to a product (or multiple products)

• Difficult to visualise for more than one input and output• The production function is a manifold of the boundary at the

edge of feasible solutions for the system• Linear and dynamic programming techniques can be used

to solve the function (when functions can be expressed algebraically)

• When an economic objective function is used, then the solution depends not only on the production function but also on resource and product functions

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Influences on Production FunctionInfluences on Production Function• A typical production operation has interdependent processes

• The production function has a number of contributing factors that will affect it in a given set of circumstances




Influences on Production Function (2)Influences on Production Function (2)

• The market will affect the production plan for the number and range of products

• The efficiency of the ordering process affects the short-term production plan (how much material to order for a run)

• The production function determines whether the production requirement (the customer orders) can be met, given the resources available, including raw material, facilities, and labour

• The long-term production plan affects labour requirements• The long-term plan also affects what facilities are required

to meet the plan• This is a very simple model, which ignores effects of delays

(of all sorts, internal and external), financing requirements, and many other factors

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Production RelationshipsProduction Relationships

• The representation of what processes take place during production is often defined as a set of sub-processes, from which the overall system mathematical formulation is constructed.

• This is analogous to the constitutive relationships of physical systems.

• By breaking the process into a number of functional sub-processes, a network model of the overall process can be built up

• A few examples:– The overall output of the process is the sum of the number of units

produced from individual trains– The throughput is constrained by the slowest rate that a sequential

set of operations can deliver




DiminishingDiminishing Marginal ReturnMarginal Return

• The behaviour of the production function (either the entire function or an observable sub-process) is usually described by the rate of change of output as resources are added or subtracted

• Marginal product MPi with respect to input Xj is defined in terms of partial derivatives for continuous processes:

• Or in terms of finite differences for discrete units of production

(which is more accurately referred to as incremental product)

ji X

ZMP

ji X

ZMP

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Sensitivity of Production Function to InputsSensitivity of Production Function to Inputs

• Production function is considered over a range of inputs• In general the range will have two parts:

– MPi increases as input Xj increases– MPi decreases as input Xj increases

• For low levels of input, it is sometimes possible to increase product per unit resource by distributing start-up uses of resources over a larger number of units of output.

• We are usually interested in getting the benefits of increasing marginal returns, and so the key question is when does the production function deliver diminishing marginal products.

• This is the point at which the extra production deliver less extra output than it would have at a lower production rate




The “Law” of Diminishing ReturnsThe “Law” of Diminishing Returns

• The marginal product of any resource eventually decreases as the amount of resource used increases and the quantity of other resources remains constant

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The “Law” of Diminishing Returns (2)The “Law” of Diminishing Returns (2)

• This empirical observation seems to hold true for use of large quantities of resources, which is why it is sometimes referred to as a law

• At increasingly negative marginal return, the cost associated with the extra resources eventually eats up all of the incremental profit

• At the extreme, extra resources overwhelm the capacity for any incremental production




Convex Feasible RegionsConvex Feasible Regions• If the feasible region (described by the production function)

is convex, then marginal analysis and linear programming techniques can be used to find the optimal production point:

• If the function is not convex, then dynamic programming can still find the optimal production point

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IsoquantsIsoquants

• Variable amounts of resources can combine to create different technically efficient designs

• An isoquant is a locus of all technically efficient combinations of resources for a given level of production

• This is like a contour line for the resource requirements associated with production Z*




Marginal AnalysisMarginal Analysis• Marginal analysis is an optimization process that looks for

the optimal solution by improving a trial solution in the direction of greatest improvement f. Gradient methods are used.

• Like walking up a hill, always taking the steepest path, to find the top.

• The optimality criterion is the relationship between the production function and cost of resources at the optimum

• The marginal product per unit cost if each resource must be the same for all resources at optimum (“the top”)

∆

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Resource Costs and BudgetResource Costs and Budget• There will be a feasible range of budget to be able to have

resources for production• This region exists because it is always possible to spend

more than necessary to get the resources




Resource Budget and Feasible Use of ResourcesResource Budget and Feasible Use of Resources

• At optimality the line of constant cost is tangent to the isoquant (which defines the ratio at which resources may be substituted for each other with no change in output Z*)

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Expansion of Resource Requirements & CostsExpansion of Resource Requirements & Costs

• Expanding production will have an increasing resource requirement path

• Budgets also have to change to meet the resource requirements




Optimal Production LevelOptimal Production Level

• The optimal production will be somewhere on the expansion path, and on a cost-effectiveness curve

• Optimal production maximises net benefits

Where V(Z) is the benefit from production Z,And h(X) is the cost of resources for production X. which can be alternatively expressed as a function of the output:

• This is the same as saying, the level of production that maximises net profits occurs when

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MECE 758MECE 758--X5 Lecture 5 Extra MaterialX5 Lecture 5 Extra Material

• Nonlinear dynamical systems can exhibit behaviour that is very counter-intuitive

• Often a system will behave fairly predictably within some region of the state space

• We see this in familiar ways, such as small variations in weather

• But sometimes the system will move into a very different part of the state space

• This apparently bizarre behaviour may be a completely natural consequence of the system, and not due to a particularly strong forcing input or random forcing

• Forcing inputs, however, can move the system into a part of the state space where big changes are more likely to occur




Chaotic SystemsChaotic Systems

• Nonlinear dynamical systems behave in unpredictable ways.

• In a non-chaotic system, the outset and inset regions are separate. Trajectories do not cross.

• When the behaviour of a system is characterized by patterns that almost – but don’t quite – repeat, then the system is chaotic.

• In a chaotic system, something about the dynamics allows the inset and outset regions to overlap.

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Limit Cycles in the PlaneLimit Cycles in the Plane

• Balthasar Van der Pol (a Dutch electrical engineer working for Philips) found that oscillations could occur in electrical circuits with vacuum tube elements. He developed a dynamical model for a radio transmitter using nonlinear amplifiers (triode vacuum tubes):

• This a homogeneous ODE, which has no forcing input, and so it has a critical point at the origin.

01 22

2

ydtdyy

dtyd




y

y

wikimedia.org

Van Van derder PolPol Oscillator Phase PortraitOscillator Phase Portrait• This model contains one limit cycle, which is attractive. That

means that this entire trajectory is an attractor, and all other trajectories (arrows in green) will tend to fall into this limit cycle (in red), with the exception of the trivial solution represented by the critical point at the origin)

• The blue lines that connect to the limit cycle are other possible states at this point (phase) in the cycle.

• At other phases in the cycle, the shape looks different.

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A Plot of a Van A Plot of a Van derder PolPol System CycleSystem Cycle• The behaviour of the system changes as we look at

different parts of a cycle (a cycle goes from 0 to 2π)• Each diagram on the left represents the possible states for

the limit cycle at a different part of the cycle• One part of the cycle is shown in red




Back to LimitBack to Limit CyclesCycles

• If we follow one part of the system, starting at 1/2 π, the location on the limit cycle at A (5/4 π) develops a bulge that turns into a “beak” at B (3/2 π), which then develops further into a pleat, which persists for most of the cycle. The beak spawns from the region of the limit cycle and then moves away from the limit cycle to form a pleat, which is a separate but connected part of the limit cycle at that particular phase of the cycle.

• This is divergence.• The pleat then becomes squished back into the limit cycle

after π/4 (the pleat at C disappears into the limit cycle again around D at π. (This sequence takes one-and-a half cycles.)

• This is folding.

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DivergenceDivergence ((Information Gain)Information Gain)

• This creation of new features during a cycle expands the space of the system, and so it is diverging.

• In information theory, this is a gain of information.• A measurement of the system state after divergence is as

accurate as an earlier measurement, and tracing backward in time to the time of the earlier measurement means that the earlier measurement can be refined more accurately.




FoldingFolding• Squishing of a trajectory feature into the thick band of the

limit cycle means that, over the course of multiple cycles, any trajectory feature will tend to come back to the limit cycle region.

• This folding means that outsets (divergence) and insets (attraction) can overlap. This is how the unusual behaviourof a chaotic system is generated.

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UnpredictabilityUnpredictability

• Trajectories spawn that diverge and move away from the limit cycle region, and then fold back in due to the presence of one or more attractors.

• The combination of divergence and folding means that randomness appears in the system, despite the fact that the behaviour is completely described by the governing equation.

• Such systems exhibit deterministic chaos.




Deterministic ChaosDeterministic Chaos

• Due to the nonlinear nature of these systems, irregular variations in the output signal are observed to occur near the natural frequencies.

• This is not due to any random fluctuation in the inpts to the system. This behaviour is inherent in the system itself.

• This phenomenon was one of the first discovered examples of deterministic chaos: a system that has unpredictable behaviour despite being completely described physically.

• Another simple – but nonlinear - system that behaves chaotically is the double pendulum. It has an attractor that is a cycle if there is no friction. If there is friction, then the attractor is a centre (where the pendula are not moving and pointing straight down).

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Simulating ChaosSimulating Chaos

• To simulate this system, express the ODE in first-order form with new variables

• And write

• Set µ=3 and assign initial conditions for [z1(0) , z2(0)] and use an ODE solver (such as Euler or Runge-Kuttaintegration) to produce a set of states as a function of time.

• Plot the set of data in state space to represent how the state variables change with time, and observe how the system wanders close to the attractor trajectory, and never exactly repeating.

• This is chaos.

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1

zyzyz

122

12

21

1 zzzzzz




Example: Lorenz MaskExample: Lorenz Mask

• Lorenz Mask is a famous deterministic chaotic system• The system is a representation of the heat and momentum

transfer associated with a small weather system.• The Lorenz equations describe a reduced-order (simplified)

model of the atmosphere as a fluid system between two parallel plates, with the temperature of the bottom plate hotter than the top plate. If the temperature difference is sufficient to overcome viscosity, buoyancy forces cause the fluid to move and form convection rolls. With a greater temperature difference, the movement becomes more pronounced.

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Lorenz Mask (2)Lorenz Mask (2)

• The Lorenz equations are:

• where x1 is proportional to the intensity (speed of the convective rolls), x2 is proportional to the temperature difference between ascending and descending currents, & x3is proportional to the vertical temperature profile from linearity

• σ is the Prandtl number (ratio of kinematic viscosity to thermal conductivity), b is a geometric factor related to the aspect ratio (height to width) of the convection roll)

,,

),(

2133

21212

121

xxbxxxxxrxx

xxx




Lorenz Mask (3)Lorenz Mask (3)

• r is the ratio of the Rayleigh number Ra to the critical Rayleigh number Rac. (when r > 1, buoyancy forces overcome viscosity forces and motion begins). The Rayleigh number is

• where α is the coefficient of expansion, H is the distance between the plates, g is gravitational acceleration, ΔT is the temperature difference between the plates, ν is kinematic viscosity, and k is thermal conductivity.

kTHgRa

3

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Lorenz Mask (4)Lorenz Mask (4)• A common set of parameters for the governing equations

yields

• And if we plot the trajectory of (x,y,z) over an interval of time we would set a line that moves in the neighbourhood of two attractors, and which never crosses itself, like this:

• This is an attractor that is neither a point nor a cycle.

wikimedia.org

Here is the region where divergence and folding occurs




Example: SelfExample: Self--Sustaining OscillationSustaining Oscillation

• This is the electrical system we looked at in lecture 3, except with a nonlinearity

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Example: Forced OscillationExample: Forced Oscillation

• This is a system that has an ongoing forcing input




PredatorPredator--Prey Populations DynamicsPrey Populations Dynamics

• Note that this system would actually have only integer solutions (you can’t have 0.317 of a wolf…)

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The Magnetic PendulumThe Magnetic Pendulum

• The simple pendulum system has additional attractors when the mass is ferromagnetic and a pair of magnets is added into the system; the governing equations of motion can be expressed as




The Magnetic Pendulum (2)The Magnetic Pendulum (2)

• To find the values of F1 & F2, let’s assume that the equilibrium is for one magnet only (the strong one on the right), and force F2 points along the line of the pendulum string:

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• We could set F2 to be slightly smaller than this, but if we make it too small then the magnet will not be able to hold the mass. (See if you can figure out what the minimum would be.)

• We could make the force stronger, too; but if we make it too strong then it will dominate the dynamics and the mass will tend to zoom in toward the attractor.

• Since we want the magnet on the left side to be smaller, we let F1 = 2/3 F2 (or some other number that is less than 1).

• Our constraints on setting the geometric parameters are that r1 and r2 must each be greater than zero.

• Now we can set up this set of functions in MATLAB to simulate the system.





• Recall that the vectorfield of trajectories will look like this:\

attractorsaddle

Not only does the system have energy dissipation, but there are now two new attractors, with a new saddle point between them. The basin of the rest point (shaded in green)is smaller for the small magnet, which means the likelihood that the system will come to rest near the attractor of the small magnet is less than for the large one.

saddle

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BifurcationsBifurcations & Catastrophes& Catastrophes

• There are many other fascinating behaviours where the system changes its behaviour dramatically.

• Bifurcations are splits in the behaviour, going from a stable cycle to a more complex cycle with at least one new repellerthat interacts with an attractor when it reaches a critical point.

• Catastrophes go the other way: the behaviour of the system changes to another mode when an attractor is lost (along with its corresponding basin). A fold catastrophe involves pairwise annihilation of a repeller and attractor. This can happen when an attractor moves to the separatrix at the edge of the basin.

• This is like a moth getting too close to a flame & getting drawn in to its death by its own phototropic attraction.




SummarySummary

• This is an extremely brief introduction to the concepts of nonlinear dynamics that manifest as chaotic systems.

• Even simple nonlinear systems can exhibit complex behaviours.

• The idea of randomness that is inherent in some nonlinear systems is a powerful repudiation of the classical idea that sufficient information about a system could allow accurate prediction of its future behaviour for an arbitrary time.

• In those cases, we have to rely on statistical representations to describe the system behaviour over period of time under certain conditions, rather than the specific trajectory.

• Statistics allows us to bracket our uncertainty about a system.

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ENGM 670 & MECE 758