PRODUCTS OF INERTIA

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					  PRODUCTS OF INERTIA
Product of inertia:   Ixy   xy dA

                         y
                                     A
Ixy may not be               x   y
positive!
                                          x
                 The Rectangle
   Compute Ixy.              dA = dxdy
         y                    dIxy = xydxdy
                                   h b      
                   dy
     h                      Ixy      xdx  ydy
                                    0 0
                                            
                                             
                                   1 2 2
                        x   Ixy    bh
             b                     4
            Geometrical Meaning
   Product of inertia Ixy   xy dA
        y                              y
                      y

                 x             x                  x


       Ixy = 0       Ixy = 0       Ixy = h2b2/2
      Parallel Axis Theorem:
             Ixy  Ixy  Axy
   Virtually the same as for Ix, Iy.
              y   x   y’       dA
                           x’ y’
                  x        C           x’
                                   y
                      y
                                       x
       Principal Axes; Principal
          Moments of Inertia
   Consider:
     Ix   y dA Iy   x dA Ixy   xydA
           2           2

                                 y
            Ix Ixy 
     I  
           Ixy Iy 
    referred to coordinate                  x
    system x,y
                    Motivation
   The Rectangle
           x’ y
                                  Ix = bh3/12
                                  Iy = hb3/12
                        h
    y’              x              Ix’ = hb3/12
             b                     Iy’ = bh3/12
     Ix, Iy, Ixy values change when axes are rotated!
                    Problem 1
   How does [I] change when we refer it
    to a rotated set of axes x’,y’, i.e., how
    are Ix’ , Iy’ , Ix’y’ related to Ix , Iy , Ixy?
                  y’    y

                                    x’
                               q
                                   x
                Coordinate
              Transformation
   The coordinates of a point P in the
    x,y system and the x’,y’ system are
    related by:               y
                          y’
     x’ = xcosq + ysinq         P
                                         x
     y’ = ycosq - xsinq             q    ’
                                        x
              Transformation of
              Moments of Inertia
   Recall:    Ix   y2dA y’ = ycosq - xsinq

               Ix   ycosθ  xsinθ 2dA

        Ix  cos θ  y dA  2cosθcosθ xydA
                2    2


               sin 2θ  x 2dA

       Ix  cos2θ Ix  2cosθcosθ Ixy  sin 2θ Iy
      … more
   Final form:

     Ix  2 Ix  Iy   2 Ix  Iy cos2θ - Ixy sin2θ
           1              1



     Iy  2 Ix  Iy   2 Ix  Iy cos2θ  Ixy sin2θ
           1              1



     Ixy  2 Ix  Iy sin2θ  Ixycos2θ
             1



     Note: Ix’ + Iy’ = Ix + Iy
                                          The Rectangle
   Note: I  I  I   I  I cos2θ - I
                                 x
                                          1
                                          2       x       y
                                                               1
                                                               2   x   y         xy   sin2θ
     I  I  I   I  I cos2θ  I sin2θ
      y
             1
             2       x       y
                                      1
                                      2       x       y                xy

     I  I  I sin2θ  I cos2θ
      xy
                 1
                 2       x       y                        xy
                                                                                       y      x’
    Ix =     bh3/12                   Iy =            hb3/12                y’
    Ixy = 0                                                                             45o        h
    Ix  h  b 
             bh  2                    2                                                       x
    Iy  h  b 
             24
             bh  2
                 2


    Ixy  h  b 
             24
              bh   2
                   2                                                              b
              24
                 Problem 2
   What is the orientation of axes x’,y’
    which give extremal values of 2nd
    moment? What are those values?
                y’   y

                               x’
                           q
                               x
        Principal Values, Axes
    Extrema :    dIx                2Ixy
                      0  tan2θ 
                 dθ                        (Ix  Iy )
                              2 values of
                              2q 180O apart.
                 p              2 values of
           p/2
                     2q         q 90O apart.
                             2 values of q are
                             principal axes
       … more
   Recall:          Ixy  2 Ix  Iy sin2θ  Ixycos2θ
                             1


                2Ixy
       tan2θ            Ix’y’ = 0 products of
               Ix  I y   inertia vanish !

     [Ix’ - (Ix+Iy) /2]2 + Ix’y’2 = [(Ix-Iy) /2]2 + Ixy2

              Ix  Iy    Ix  Iy 
                                  2
                                             Principal
     I1,2                        Ixy
                                       2
                                             values !
                 2       2 
                        Summary
   Values referred to rotated axes:
      Ix  2 Ix  Iy   2 Ix  Iy cos2θ - Ixy sin2θ
            1              1

      Iy  2 Ix  Iy   2 Ix  Iy cos2θ  Ixy sin2θ
            1              1


      Ixy  2 Ix  Iy sin2θ  Ixycos2θ
              1



   Principal axes, values:
                                   Ix  Iy    Ix  Iy 
                                                       2
              2Ixy
     tan2θ               I1,2                        I2
             Ix  I y
                                                            xy
                                      2       2 

				
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