An Approach to the Mechanics of Fluids

by Frank C. Fung - 1st published in June, 2008.

Top Of Page

Section IV - 18 { Degrees-Of-Freedom } associated with a { General-Random-Type Six-Face Object }

Summary for the Section :

• In this Section , we shall look at the { Degrees-Of-Freedom } , or [ DOF's ] , associated with the { General-Random-Type Six-Face Object } , i.e. the { GRT-SFO } .

go to Top Of Page

Construction of the { GRT-SFO } :

• We shall now use the following construction method for our illustration and analysis of the { GRT-SFO } :
  • Let us take 3 sticks of any given lengths and construct a { Structural Frame } :

    

    • with the [ ends ] of the 3 sticks coming together at a [ single point ] ,
      • i.e. : [ Point A ] as shown in the diagram above .

  • We then have 6 { Degrees-Of-Freedom } associated with the [ size and shape ] of this { Structural Frame } :

    

    • 3 [ DOF's ] associated with the lengths of the sticks , i.e. [ L1 ] , [ L2 ] , and [ L3 ] respectively ;

    • 3 [ DOF's ] arising from the 3 intersection angles , i.e. :
      • the angles [ angle C-A-B ] , [ angle B-A-E ] , and [ angle E-A-C ] respectively ; as shown above .

  • Now that we have the { Structural Frame } in-place , let us now put in [ Point D ] , [ Point F ] and [ Point G ] :

    

    And we note here that :

    • [ Point D ] must reside on [ Plane C-A-B ] , which is already in-place , so that :
      • Only exactly 2 [ DOF's ] may be associated with the placement of [ Point D ] ;

    • [ Point F ] must reside on [ Plane B-A-E ] , which is already in-place , so that :
      • Only exactly 2 [ DOF's ] may be associated with the placement of [ Point F ] ;

    • [ Point G ] must reside on [ Plane E-A-C ] , which is already in-place , so that :
      • Only exactly 2 [ DOF's ] may be associated with the placement of [ Point G ] .

      Thus , a total of 6 [ DOF's ] will be involved in the placement of [ Point D ] , [ Point F ] , and [ Point G ] here .

  • Let us look at , next , the placement of [ Point H ] , the final { nodal-point } of the { 8 nodal-points } of the { Six-Face Object } .

    And we note here that , at this juncture , [ Plane F-E-G ] , [ Plane G-C-D ] and [ Plane D-B-F ] are already in-place .

    

    • Since [ Point H ] are always on all these 3 [ planes ] and therefore lies at the intersection of the said-3-planes ,
      • the number of [ DOF's ] associated with the placement of [ Point H ] is necessarily [ zero ] ,
        • its position being already fixed by the positionings of the said-3-planes above .

  • Thus , the number of [ DOF's ] associated with the [ Size and Shape ] of a { Six-Face Object } is [ 12 ] ,
    • i.e. [ 6 ] + [ 6 ] = [ 12 ] .

go to Top Of Page

Positioning and Orientation of a { Six-Face Object } :

• Let us now identify a point , [ Point P ] , inside the { Six-Face Object } so that :
  • this point would have been the { Center-Of-Mass } if the { Six-Face Object } were a { soild-object made-up-of a homogeneous material } .

  

• Let us also affix a { 3-othogonal-unit-vectors Reference Frame } to the said { Six-Face Object } at [ Point P ] ,
  • as per the diagram on-the-right , above .

• We see here that the position of { Six-Face Object } can be defined in terms of the position of [ Point P ] , and
  • as we rotate / re-orient the { 3-orthogonal-unit-vectors Reference Frame } , the { Six-Facce Object } will also rotate / re-orient accordingly .

  Thus , [ 6 DOF's ] are involved in the { positioning and orientation } of the { Six Faced Object } :

  • 3 [ DOF's ] arising from the positioning of [ Point P ] , i.e. the [ X-Y-Z co-ordinates ] , and

  • 3 [ DOF's ] arising from the rotation / re-orientation of the { 3-othogonal-unit-vectors Reference Frame } .

    ( please refer to the author's previous papers for further details here . )

go to Top Of Page

The 18 [ Degrees-Of-Freedom ] for the [ Six-Face Object ] :

• The 18 [ Degrees-of Freedom ] ( DOF's ) for the [ Six-Face Object ] are then :
  • 3 [ DOF's ] for { positioning } ,
  • 3 [ DOF's ] for { orientation } , and
  • 12 [ DOF's ] for the { Size and Shape } .

go to Top Of Page

go to the next section : Section V - General Stress-Strain Relations

go to the last section : Section III - The { Cube } and other { Six-Face Objects }

return to the Mechanics of Fluids HomePage

original dated 2008-6-08