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A core Course on Modeling

This course introduces the fundamental concepts of dimensional analysis and synthesis in modeling physical systems. It covers the importance of consistent dimensions when formulating equations, including examples such as length, time, and weight. Participants will learn to derive formulas using dimensional synthesis, which relates variables through their dimensions without the use of dimensionless factors. By the end, students will appreciate the necessity for dimensional consistency and be able to apply dimensional analysis in various modeling scenarios.

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A core Course on Modeling

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  1. A core Course on Modeling Introductionto Modeling 0LAB0 0LBB0 0LCB0 0LDB0 c.w.a.m.v.overveld@tue.nl v.a.j.borghuis@tue.nl S.10

  2. u1, u2 constant ratio p12  x1 , x2canbeconverted set convertible units  dimension http://commons.wikimedia.org/wiki/File:Big_and_small_red_phonebox.jpg {m, inch, light year, yard, m, …} defines‘length’ [L]; {sec, year, day, …} defines‘time’ [T]. Non-convertible units (e.g., ‘nr. sheep’) … … giverise to dimensionsontheirown [SHEEP]

  3. units ‘go with the math’  dimensions ‘go with the math’ differentdimensions: not + or -, hence not equated (a=b  a-b=0) http://commons.wikimedia.org/wiki/File:Zipper_animated.gif like a zipper, everydimension must match leftand right from ‘=‘ dimension analysis: check formula dimensionsynthesis: construct formula

  4. Dimensionalsynthesis: Oscillation time T [T] of pendulumcoulddepend on g [LT-2], m[M], l[L] : T  mlg [T]=[M][L][LT-2] =[M][L]+[T]-2 http://commons.wikimedia.org/wiki/File:Foucault_pendulum_animated.gif So: M: =0; L: +=0; T:1= -2 or: T2/(l/g) is dimensionless: T2 is proportionalto l/g.

  5. Dimensionalsynthesis: Wind force FW coulddepend on  [M/L3], v [L/T], A [L2]: http://www.morguefile.com/archive/display/213367

  6. Dimensionalsynthesis: Wind force FWcoulddepend on  [M/L3], v [L/T], A [L2]: QUIZ Derive a formulafor wind force usingdimensionalsynthesis http://www.morguefile.com/archive/display/213367

  7. Dimensionalsynthesis: Wind force FWcoulddepend on  [M/L3], v [L/T], A [L2]: http://www.morguefile.com/archive/display/213367 Fw  v A  [ML/T2]=[M/L3] [L/T] [L2] [M][L][T]-2=[M][L]-3++2 [T]- So: M: 1 = ; L: 1 = -3++2; T: -2= -  or: Fw/(v2A)is dimensionless: Fwis proportionaltov2A.

  8. Summary: • units with constant ratio  convertible • set of convertible units: dimension • examples: length, time, weight, ... • in anyformula, dimensionsleftand right must beidentical • dimensional analysis: check • dimensionalsynthesis: construct • dimensionalsynthesiswill not yieldformulas with + , - anddimensionless factors

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