PECDimensionalAnalysis

# PECDimensionalAnalysis - Dimensional Analysis in...

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Dimensional Analysis in Engineering Mechanics J. W. Eischen Precision Engineering Center Lunch Seminar August 6, 2004

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2 Cantilever Beam L P b h h L b L Eh P Ebh PL EI PL 2 3 3 3 4 L or 4 3 = = = δ Note the leading constant is of “order 1,” this is typical for relationships expressed in terms of dimensionless parameters
3 Buckingham’s pi Theorem The number of dimensionless products (pi’s) equals the original number of quantities minus the rank of their dimensional matrix In applying this theorem we seek to reduce the number of both independent variables and parameters- this can provide simplification and insight into important combinations of parameters

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4 Cantilever Beam 1 ] [ ] [ ] [ ] [ ] [ ] [ ] [ 2 = = = = = = = - ν δ L b FL E L h F P L L L L P b h Dimensions: Buckingham’s Theorem: 0 1 1 1 2 0 1 0 0 0 0 1 1 0 - L F b h L E P 7-2=5 pi’s
5 Cantilever Beam L P b h Buckingham’s Theorem: ) , , , ( 2 EL P b L h L v f L = δ

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6 Drag on a Sphere ) , , , ( μ ρ a U D D = Propose: U ρ,μ a D 1 0 0 1 2 1 3 1 1 1 1 1 0 0 1 - - - - - T L M a U D 5-3=2 pi’s
7 Drag on a Sphere = = ν μ ρ Ua f Ua f a U D 2 2 Result: U ρ,μ a D or: C D = f (Reynolds number)

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8 Drag on a Sphere n m l k U a U a D D μ ρ = ) , , , ( Rayleigh’s Procedure:
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PECDimensionalAnalysis - Dimensional Analysis in...

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