Unformatted text preview: Srikanth Devanathan Karthik Ramani School of Mechanical Engineering Purdue University Familiarize with terminology associated with mathematical models in design Fallacies of math models ! " ! A model is a representation, simplification or estimation of a product’s realization to aid making product decisions. # % && '
Models make observations cheap Cheaper than physical models Compress time to observe phenomena Measure without disturbing or disrupting existing systems Experiment with unknown situations Analysis of models can generate insights Identify critical issues Models can be used to make predictions Predictions can be tested Model can (in)validate hypotheses Identify gaps in knowledge ( Reduce development time.
$ * + & Simulation involves subjecting models to various inputs or environmental conditions to observe how they behave.
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Main steps
1. 2. 3. 4. 5. 6. – Related areas Problem formulation and setup Mathematical modeling Solution algorithms and methods Interpretation of results Modification of current design / Decision making Verification and Validation Dealing with uncertainty Optimization Robust design and Reliability based design . Modeling is an iterative task depending on
Fidelity needed Information at hand Purpose and Assumptions!!! Models at the end need to address customer needs!
These can be very vague like “should be comfortable” / + ' Customer needs lists Activity diagrams Functional models Engineering specifications Teardown of past/competing products Product architecture Various options to choose from Experimental data Modeling guidelines from handbooks … 0 What to model? What is the causality? What kind of model is suitable? Is it accurate enough? / Descriptive enough? How much resource should be dedicated?
Construction difficulty
Time for modeling Tools needed for modeling (Domain) knowledge needed How do we know that the model is correct? What is the uncertainty in the model? Should we do some experiments?
To validate assumptions To calibrate model parameters To gather more data… Solution determination time – How long will it take to execute model? Is it reusable? It is very likely that you will revisit the model again and again. % ( You used QFD 1 to translate customer needs into engineering specifications And QFD 2 to translate engineering specs to part specs Models should address engineering specs or part specs %
Vary greatly based on fidelity, basis and purpose.
Qualitative relationships Simple algebraic calculations Detailed analysis ( l δ & h δ (read as δ → → decreases monotonically with h )
+ − δ=
8 Eh Wl 2 «property» a
«property» «parametricRelation» m Car.mass Earth.gravity F = ma F «property» «parametricRelation» Car.tire.cFriction Cf Fnormal Fresistive F
«parametricRelation» «property» Cf = Fresistive / Fnormal m F = ma a dstop
«parametricRelation» Car.minStoppingDistance «property» Car.speed v dstop =  ½ v 2 / a a 1 f= L τ Wt 2
p xhb # 1 $ 1 ) 1 # , 1 $ . Design variables – variables that we have direct control over (e.g. size of cylinder) Performance variables – dependent on design variables (e.g. weight of cylinder) Noise variable – we do not have control over choice (e.g. actual density of cylinder) / 4 ui xj
" # " pj 1 # 1 ' pi xi ! *5 $ Design Space D: The set valid designs of the described using design variables Performance Space P: The dependent set of evaluated performances at each point in D. Noise Space N: Set of possible configurations described using noise variables
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Available information Assumptions, simplification and causality Accuracy
How well does it match the physical observations? Resolution
Can two different inputs give correct different outputs? Uncertainty 3 4 Order of magnitude reasoning Dimensional analysis Algebraic simplification
Quantifier elimination Surrogate models
Response surface methods Qualify the relative importance of different phenomena on which the whole behavior of a system depends Consider impact of two masses M and m with velocities Vi and vi. Remember + ' 2 Quantifier elimination is a glorified name for eliminating variables from equations symbolically. Performed automatically by Mathematica with the resolve or reduce operations. Example:
In: Resolve[ForAll[x,ax^2+bx+c>0]] Out: Notice: no x! – x has been eliminated # *
Simplify a very complicated model into simple (algebraic) relations. A more general form of regression.
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Electric wok design Metrics
Steady state temperature Weight Handle temperature Heating and cooling times ) 5
Causality: Model of the energy flow System Boundaries:
Heater Space environment Wok Bowl Food 7* & 2 Let q be the heat entering through the heater Modes of heat transfer from Heater to Bowl
Conduction (qk) – through bowl material Convection (qh) – heat loss to air Radiation (qe) – heat loss through radiation
, 5 & " First order model, so assume lumped parameters Approximate the bowl as a disk! From classical heat transfer
dT qk = kA dx qh = hA(T − T∞ ) qε = εσ A(T 4 − T∞4 ) . 5
Inner region (center) (outer region similar)
qrad = εσπ rL2 (Tc 4 − T∞ 4 ) qconv = hπ rL2 (Tc − T∞ ) qL = 2kπ rL t (TM − Tc ) % 78 2 Middle region (heater)
qrad = εσπ (rR 2 − rL2 )(TM 4 − T∞ 4 ) qconv = hπ (rR 2 − rL2 )(TM − T∞ ) q = 2kπ r t (T − T )
L L M c qL − qrad − qconv = 0 qR = 2kπ rR t (TM − Tc )
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/ qin − qR − qL − qrad − qconv = 0 • • ( " 56 2
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c Lp = x + + r 2 Loverall = La + Lp H overall = 2r + 2 B + d B = hb + 3Dh f= xhb 6 xf x 1− 2 Wtm Lp r = 2tm Lpτ Wt 2 σ =K
cycles f max ≥ f d ≥ hb + 1.1t L ≥ 5t
a m W ≥ 5 Dh Design variables = {tm , Dh , hb , x, d }
σ max ≥ σ
0 ; ; < 1 " ;
x4 x4 x3 x6 x1 L1′34 ′ L346 x3 Blind modeling without thinking is foolish. Uncertainty plays a very big part in reality Need to understand the limitations of math models Read Hazelrigg (1999). Aleatory Inherent in any system Measure and model (usually as a probability distribution) Epistemic Did we forget to account for something? Lack of knowledge / limited understanding of complex physics Little or no experimental data
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Unclear objective Model either too complex or too simple Erroneous assumptions Undocumented or tacit assumptions Wrong distributions Bugs in simulation programs Replacing a distribution (stochastic) by its mean (deterministic) Assuming independence in input data Invalid model Poor communication $ Otto, K., and Wood, K., (2001), Product Design: Techniques in Reverse Engineering and New Product Development, Prentice Hall, Upper Saddle River, NJ. Hazelrigg, G.A., (1999), On the role and use of mathematical models in engineering design, J. Mechanical Design, vol. 121. pp. 336341. Cardella, M. E., (2006) Engineering mathematics: an investigation of students’ mathematical thinking from a cognitive engineering perspective, Doctoral dissertation, University of Washington Stump, G.M., Yukish, M., Simpson, T.W., and O’Hara, J.J., (2007), Trade space exploration of satellite datasets using a design by shopping paradigm, Proc. 2004 IEEE Aerospace Conference Wei Chen, (2007) – personal communication Anu Maria, (1997), Introduction to modeling and simulation, Proc. 1997 Winter Simulation Conference Devanathan, S., and Ramani, K., (2009), Creating polytope representation of design spaces for visual exploration using consistency technique, ASME 2009 International Design Engineering Technical Conferences IDETC, San Diego, CA ) ...
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 Spring '09
 Ramahni
 Mechanical Engineering, mathematical model, design variables, design variables Performance, critical issues Models

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