Editing Dimensional analysis (section)

=== A third example: demand versus capacity for a rotating disc ===
[[File:Dimensional analysis 01.jpg|thumb|upright=1.5|Dimensional analysis and numerical experiments for a rotating disc]]
Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness {{math|''t''}} (L) and radius {{math|''R''}} (L).  The disc has a density {{math|''ρ''}} (M/L<sup>3</sup>), rotates at an angular velocity {{math|''ω''}} (T<sup>−1</sup>) and this leads to a stress {{math|''S''}} (T<sup>−2</sup>L<sup>−1</sup>M) in the material.  There is a theoretical linear elastic solution, given by Lame, to this problem when the disc is thin relative to its radius, the faces of the disc are free to move axially, and the plane stress constitutive relations can be assumed to be valid.  As the disc becomes thicker relative to the radius then the plane stress solution breaks down.  If the disc is restrained axially on its free faces then a state of plane strain will occur.  However, if this is not the case then the state of stress may only be determined though consideration of three-dimensional elasticity and there is no known theoretical solution for this case.  An engineer might, therefore, be interested in establishing a relationship between the five variables.  Dimensional analysis for this case leads to the following ({{nowrap|1=5 − 3 = 2}}) non-dimensional groups:

: demand/capacity = {{math|''ρR''{{i sup|2}}''ω''{{i sup|2}}/''S''}}
: thickness/radius or aspect ratio = {{math|''t''/''R''}}

Through the use of numerical experiments using, for example, the [[finite element method]], the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure.  As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs.<ref>{{cite web|last1=Ramsay|first1=Angus|title=Dimensional Analysis and Numerical Experiments for a Rotating Disc|url=http://www.ramsay-maunder.co.uk/knowledge-base/technical-notes/dimensional-analysis--numerical-experiments-for-a-rotating-disc/|website=Ramsay Maunder Associates|access-date=15 April 2017}}</ref>