# 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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