Page 170 - The Mechatronics Handbook

0066-frame-C09 Page 44 Friday, January 18, 2002 11:01 AM

In order to use these relations effectively, the motion of the axes x a , y a , z a , must be chosen to fit the
problem at hand. This choice usually comes down to three cases described by how Ω relates to the body
angular velocity ω .
1. Ω = 0. If the body has general motion and the axes are chosen to translate with the center of
mass, then this case will lead to a simple set of equations with Ω = 0, although it will be necessary
to describe the inertia properties of the body as functions of time.
2. Ω ≠ 0 ≠ ω . In this case, axes have an angular velocity different from that of the body, a form
convenient for bodies that are symmetrical about their spinning axes. The moments and products
of inertia will be constant relative to the rotating axes. The equations become

˙
F x = mV x – mV y Ω z + mV z Ω y
F y = mV y – mV z Ω x + mV x Ω z
˙
F z = mV z – mV x Ω y + mV y Ω x
˙
(9.26)
T x = I x ω ˙ x – I y ω y Ω z + I z Ω y ω z
T y = I y ω ˙ y – I z ω z Ω x + I x Ω z ω x
T z = I z ω ˙ z – I x ω x Ω y + I y Ω x ω y

3. Ω = ω . Here the axes are fixed and moving with the body. The moments and products of intertia
relative to the moving axes will be constant. A particularly convenient case arises if the axes are
chosen to be the principal axes of inertia (see the section titled “Inertia Properties”), which leads
to the Euler equations, 4

˙
F x = mV x – mV y ω z + mV z ω y
F y = mV y – mV z ω x + mV x ω z
˙
F z = mV z – mV x ω y + mV y ω x
˙
(9.27)
T x = I x ω ˙ – ( I y – )ω y ω z
x
I z
T y = I y ω ˙ – ( I z – )ω z ω x
y
I x
T z = I z ω ˙ – ( I x – )ω x ω y
z
I y
These equations of motion can be used to determine the forces and torques, given motion of the body.
Textbooks on dynamics [12,23] provide extensive examples on this type of analysis. Alternatively, these
can be seen as six nonlinear, coupled ordinary differential equations (ODEs). Case 3 (the Euler equations)
could be solved in such a case, since these can be rewritten as six first-order ODEs. A numerical solution
may need to be implemented. Modern computational software packages will readily handle these equa-
tions, and some will feature a form of these equations in a form suitable for immediate use. Case 2
requires knowledge of the axes’ angular velocity, Ω .
If the rotational motion is coupled to the translational motion such that the forces and torques, say,
are related, then a dynamic model is required. In some, it may be desirable to formulate the problem in
a bond graph form, especially if there are actuators and sensors and other multienergetic systems to be
incorporated.

4
First developed by the Swiss mathematician L. Euler.

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