Page 168 - The Mechatronics Handbook

0066-frame-C09 Page 42 Friday, January 18, 2002 11:01 AM

When this form is expanded for either case into x, y, z components, then

h = h x i + h y j + h z k = ∫ ( xi + yj + zk) × ( [ ω x i + ω y j + ω z k) × ( xi + yj + zk)]dm
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
m
which can be expanded to
2
2
h x i + h y j + h z k = ω x ∫ ( y + z )dm ω y ∫ xy dm ω z ∫ xz dm i ˆ
ˆ
ˆ
ˆ
–
–
m m m
2
= – ω x ∫ xy dm + ω y ∫ ( x + z )dm ω z ∫ yz dm j ˆ
2
–
m m m
2
2
= – ω x ∫ xy dm ω y ∫ zy dm ω z ∫ ( x + y )dm k ˆ
–
–
m m m
The expression for moments and products of inertia can be identified here, and then this expression
leads to the three angular momentum components, written in matrix form
– I
I xx I xy – xz ω x
h = – I yx I yy I – yz ω y = Iω (9.25)
– I zx – I zy I zz ω z
Note that the case where principal axes are defined leads to the much simplified expression
h = I xx ω x i + I yy ω y j + I zz ω z k ˆ
ˆ
ˆ
This shows that when the body rotates so that its axis of rotation is parallel to a principal axis, the angular
h,
momentum vector, is parallel to the angular velocity vector. In general, this is not true (this is related
to the discussion at the end of the section “Inertia Properties”).
The angular momentum about an arbitrary point, Case 3, is the resultant of the angular momentum
about the mass center (a free vector) and the moment of the translational momentum through the mass
center,
p = mV x i + mV y j + mV z k = mV
ˆ
ˆ
ˆ
or
h = h G + r × p

r
where is the position vector from the arbitary point of interest to the mass center, G. This form can
also be expanded into its component forms, as in Eq. (9.25).
Kinetic Energy of a Rigid Body
Several forms of the kinetic energy of a rigid body are presented in this section. From the standpoint of
a bond graph formulation, where kinetic energy storage is represented by an I element, Eq. (9.25)
demonstrates that the rigid body has at least three ports for rotational energy storage. Adding the three
translational degrees of freedom, a rigid body can have up to six independent energy storage “ports.”

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