Dimensional Analysis Notes | EduRev

: Dimensional Analysis Notes | EduRev

 Page 1


1
The Physical Basis of
DIMENSIONAL ANALYSIS
Ain A. Sonin
Second Edition
Page 2


1
The Physical Basis of
DIMENSIONAL ANALYSIS
Ain A. Sonin
Second Edition
2
Copyright © 2001 by Ain A. Sonin
Department of Mechanical Engineering
MIT
Cambridge, MA 02139
First Edition published 1997. Versions of this material have been distributed in 2.25
Advanced Fluid Mechanics and other courses at MIT since 1992.
Cover picture by Pat Keck (Untitled, 1992)
Page 3


1
The Physical Basis of
DIMENSIONAL ANALYSIS
Ain A. Sonin
Second Edition
2
Copyright © 2001 by Ain A. Sonin
Department of Mechanical Engineering
MIT
Cambridge, MA 02139
First Edition published 1997. Versions of this material have been distributed in 2.25
Advanced Fluid Mechanics and other courses at MIT since 1992.
Cover picture by Pat Keck (Untitled, 1992)
3
Contents
1. Introduction 1
2. Physical Quantities and Equations 4
2.1 Physical properties 4
2.2 Physical quantities and base quantities 5
2.3 Unit and numerical value 10
2.4 Derived quantities, dimension, and dimensionless quantities 12
2.5 Physical equations, dimensional homogeneity, and
physical constants 15
2.6 Derived quantities of the second kind 19
2.7 Systems of units 22
2.8 Recapitulation 27
3. Dimensional Analysis 29
3.1 The steps of dimensional analysis and Buckingham’s
Pi-Theorem 29
Step 1: The independent variables 29
Step 2: Dimensional considerations 30
Step 3: Dimensional variables 32
Step 4: The end game and Buckingham’s ?-theorem 32
3.2 Example: Deformation of an elastic sphere striking a wall 33
Step 1: The independent variables 33
Step 2: Dimensional considerations 35
Step 3: Dimensionless similarity parameters 36
Step 4: The end game 37
3.2 On the utility of dimensional analysis and some difficulties
and questions that arise in its application 37
Similarity 37
Out-of-scale modeling 38
Dimensional analysis reduces the number of variables
and minimizes work. 38
Page 4


1
The Physical Basis of
DIMENSIONAL ANALYSIS
Ain A. Sonin
Second Edition
2
Copyright © 2001 by Ain A. Sonin
Department of Mechanical Engineering
MIT
Cambridge, MA 02139
First Edition published 1997. Versions of this material have been distributed in 2.25
Advanced Fluid Mechanics and other courses at MIT since 1992.
Cover picture by Pat Keck (Untitled, 1992)
3
Contents
1. Introduction 1
2. Physical Quantities and Equations 4
2.1 Physical properties 4
2.2 Physical quantities and base quantities 5
2.3 Unit and numerical value 10
2.4 Derived quantities, dimension, and dimensionless quantities 12
2.5 Physical equations, dimensional homogeneity, and
physical constants 15
2.6 Derived quantities of the second kind 19
2.7 Systems of units 22
2.8 Recapitulation 27
3. Dimensional Analysis 29
3.1 The steps of dimensional analysis and Buckingham’s
Pi-Theorem 29
Step 1: The independent variables 29
Step 2: Dimensional considerations 30
Step 3: Dimensional variables 32
Step 4: The end game and Buckingham’s ?-theorem 32
3.2 Example: Deformation of an elastic sphere striking a wall 33
Step 1: The independent variables 33
Step 2: Dimensional considerations 35
Step 3: Dimensionless similarity parameters 36
Step 4: The end game 37
3.2 On the utility of dimensional analysis and some difficulties
and questions that arise in its application 37
Similarity 37
Out-of-scale modeling 38
Dimensional analysis reduces the number of variables
and minimizes work. 38
4
An incomplete set of independent quantities may
destroy the analysis 40
Superfluous independent quantities complicate the result
unnecessarily 40
On the importance of simplifying assumptions 41
On choosing a complete set of independent variables 42
The result is independent of how one chooses a dimensionally
independent subset 43
The result is independent of the type of system of units 43
4. Dimensional Analysis in Problems Where Some Independent
Quantities Have Fixed Values 45
Cited References 49
Other Selected References 51
Acknowledgements
My thanks to Mark Bathe, who volunteered to perform the
computation for the elastic ball.  This work was begun with
support from the Gordon Fund.
Page 5


1
The Physical Basis of
DIMENSIONAL ANALYSIS
Ain A. Sonin
Second Edition
2
Copyright © 2001 by Ain A. Sonin
Department of Mechanical Engineering
MIT
Cambridge, MA 02139
First Edition published 1997. Versions of this material have been distributed in 2.25
Advanced Fluid Mechanics and other courses at MIT since 1992.
Cover picture by Pat Keck (Untitled, 1992)
3
Contents
1. Introduction 1
2. Physical Quantities and Equations 4
2.1 Physical properties 4
2.2 Physical quantities and base quantities 5
2.3 Unit and numerical value 10
2.4 Derived quantities, dimension, and dimensionless quantities 12
2.5 Physical equations, dimensional homogeneity, and
physical constants 15
2.6 Derived quantities of the second kind 19
2.7 Systems of units 22
2.8 Recapitulation 27
3. Dimensional Analysis 29
3.1 The steps of dimensional analysis and Buckingham’s
Pi-Theorem 29
Step 1: The independent variables 29
Step 2: Dimensional considerations 30
Step 3: Dimensional variables 32
Step 4: The end game and Buckingham’s ?-theorem 32
3.2 Example: Deformation of an elastic sphere striking a wall 33
Step 1: The independent variables 33
Step 2: Dimensional considerations 35
Step 3: Dimensionless similarity parameters 36
Step 4: The end game 37
3.2 On the utility of dimensional analysis and some difficulties
and questions that arise in its application 37
Similarity 37
Out-of-scale modeling 38
Dimensional analysis reduces the number of variables
and minimizes work. 38
4
An incomplete set of independent quantities may
destroy the analysis 40
Superfluous independent quantities complicate the result
unnecessarily 40
On the importance of simplifying assumptions 41
On choosing a complete set of independent variables 42
The result is independent of how one chooses a dimensionally
independent subset 43
The result is independent of the type of system of units 43
4. Dimensional Analysis in Problems Where Some Independent
Quantities Have Fixed Values 45
Cited References 49
Other Selected References 51
Acknowledgements
My thanks to Mark Bathe, who volunteered to perform the
computation for the elastic ball.  This work was begun with
support from the Gordon Fund.
5
Francis Bacon (1561-1628)
1
:
“I found that I was fitted for nothing so well as the study of
Truth; as having a nimble mind and versatile enough to
catch the resemblance of things (which is the chief point),
and at the same time steady enough to fix and distinguish
their subtle differences...”
“Think things, not words.”
Albert Einstein (1879-1955)
2
:
“… all knowledge starts from experience and ends in it.
Propositions arrived at by purely logical means are
completely empty as regards reality."
Percy W. Bridgman (1882-1961)
3
:
“...what a man means by a term is to be found by observing
what he does with it, not by what he says about it.”
                                                
1
 Catherine Drinker Bowen, 1963
2
 Einstein, 1933
3
 Bridgman, 1950
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