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Kinematics and dynamics of rigid bodies 59
∫
2 2 )
Izz ( x y d m
In addition we can introduce the products of inertia I xy , I yz and I xz :
(2.172)
Ixy Iyx ∫ xy dm
Iyz Izy ∫ yz d m
Ixz Izx ∫ xz d m
(2.173)
(2.174)
(2.175)
This allows (2.167) to (2.169) to be arranged in matrix form as follows:
⎡Hx
⎤ ⎡Ixx Ixy Ixz
⎤ ⎡
x ⎤
⎢
H
⎥ ⎢
⎥
(2.176)
⎢ y ⎥
I I I
⎢
⎢ xy yy yz ⎥ ⎢
⎥
y ⎥
⎣⎢
Hz
⎦⎥
⎢I I I ⎥
⎣ xz yz zz ⎦ ⎣⎢
z ⎦⎥
If we return to our earlier consideration of Body 2 shown in Figure 2.27 the
matrix equation in (2.176) would lead to
{H 2 } 1/1 [I 2 ] 2/1 { 2 } 1/1 (2.177)
In writing the vectors {H 2 } 1/1 { 2 } 1/1 we revert to the full definition of a
vector used here where the upper suffix indicates that the vector is measured
relative to the axes of reference frame O 1 and the lower suffix indicates
that the components of the vector are resolved parallel to the axes of
frame O 1 . The matrix [I 2 ] 2/1 is the moment of inertia matrix for Body 2
about its mass centre G 2 located at frame O 2 . The use of the upper and
lower suffix here indicates that the moments of inertia have been measured
relative to frame O 2 but transformed to frame O 1 . This is necessary so that
the vector operation in (2.177) is consistent. This is only possible if the
vectors and matrix are referred to the same frame, which in this case is O 1 .
Note that in this form (2.177) is not practical since the orientation of frame
O 2 relative to frame O 1 will change as the body rotates requiring the recomputation
of [I 2 ] 2/1 at each time step. The matrix [I 2 ] 2/2, or in simpler form
[I 2 ] 2 , is constant since it is measured relative to and referred to a frame that
is fixed in Body 2 and hence only needs to be determined once for an undeformable
body. When considering the equation for the angular momentum
of a body it is preferable therefore to consider all quantities to be referred
to a frame fixed in the body, in this case frame O 2 :
{H 2 } 1/2 [I 2 ] 2/2 { 2 } 1/2 (2.178)
Before progressing to develop the equations used to describe the dynamics of
rigid bodies translating and rotating in three-dimensional space the definition
of the moments of inertia introduced here requires further consideration.
2.10 Moments of inertia
From our previous consideration of the angular momentum of a rigid body
we see that there are three moments of inertia and three products of inertia