Page 115 - Mechatronic Systems Modelling and Simulation with HDLs

104 6 MECHANICS IN HARDWARE DESCRIPTION LANGUAGES

If we now move on from particles to rigid bodies we now have to consider the
moments of inertia in addition to the translational inertia. These are described by
the underlying Euler equations for the rigid body K j :

e
I j1 ˙ω j1 − (I j2 − I j3 )ω j2 ω j3 = M + M r
j1 j1
e
I j2 ˙ω j2 − (I j3 − I j1 )ω j3 ω j1 = M + M r j2 (6.10)
j2
e
I j3 ˙ω j3 − (I j1 − I j2 )ω j1 ω j2 = M + M r
j3 j3
In matrix form these equations look like this:
e
I j ˙ω j + ω j × (I j ω j ) = M + M r (6.11)
j j
where M j represents the applied and reactive torque vectors, I j represents the
tensors of the moment of inertia and ω i represents the angular velocities with
respect to the three principal axes of the rigid body K j .

6.2.2 System-oriented modelling

In system-oriented modelling two classical approaches can be distinguished, the
synthetic and the analytical, see for example, Kreuzer [207]. In the synthetic
methods we first draw up the Newton and Euler equations for each body. The
connections between bodies, e.g. joints, give rise to constraining forces, and the
elimination of these converts the Newton/Euler equations into equations of motion.
The analytical approach, on the other hand, is associated with the name Lagrange
and starts from an energy formulation. This is rearranged directly into equations
of motion without the constraining forces being considered.
Both approaches will be described in the following in a formulation using gen-
eralised coordinates. In addition to the above-mentioned approaches there is also
a range of further options, which are briefly described and compared by Kane and
Levinson in [177]. It should not go unmentioned that the equations that result from
the various approaches are ultimately the same. However, they are obtained at a
different level of complexity. The formulation is also of varying suitability for the
subsequent numerical simulation.

Newton–Euler approach
The Newton–Euler approach, see also Kreuzer and Schiehlen [208], should — just
like the Lagrange approach described subsequently — be represented in a
formulation using the generalised coordinates q 1 ,...,q n . From these the velocities
should be determined for each body K j in the x, y and z coordinates: