Page 119 - Mechatronic Systems Modelling and Simulation with HDLs

108 6 MECHANICS IN HARDWARE DESCRIPTION LANGUAGES

6.2.3 Object-oriented modelling

Introduction
The use of generalised coordinates in object-oriented modelling raises two prob-
lems. Firstly, it poses the question of how we should determine the generalised
coordinates from the very limited perspective of an element. Secondly, the local
Jacobi matrices, which describe how the local coordinates arise from the totality
of the generalised coordinates, have to be set up. Both questions necessitate the
global perspective of mechanics, i.e. the local consideration that has brought so
many benefits in electronics is lost. In other words: When generalised coordinates
are used the consideration of a multibody system generally results in the completion
of the drawing up of the Newton–Euler equation system

M¨q + k = Q (6.28)
using hardware description languages, based upon models for rigid bodies, springs,
dampers etc. A more promising approach would seem to be to add the automated
creation of symbolic equations of motion by a suitable programme and thus select
system-oriented modelling.
Object-oriented modelling thus cannot be performed directly using generalised
coordinates. However, if we free ourselves from the generalised coordinates and
in particular permit a greater number of unknowns, then the question is reformu-
lated. The work of Suescun et al. [391] provides a first approach to the modelling
of multidimensional mechanics in hardware description languages (VHDL-AMS).
Here the position of the body is given in natural coordinates, which occur in two
forms: Firstly, they are given as cartesian coordinates for certain points on the
body. These marked points may be contact points of joints, springs and dampers.
Secondly, unit vectors are introduced as natural coordinates, in order to specify
axes of rotation. According to Suescun et al. the mass matrix M of a body is con-
stant if a sufficient quantity of natural coordinates are considered. This represents
the vector q of the natural coordinates on the inertial force Q I (with respect to the
natural coordinates):

Q I =−M¨q (6.29)

The natural coordinates are modelled in the hardware description languages as
potentials (across), the forces and moments as flows (through). In addition there
are algebraic constraint equations in quadratic (for planar mechanics) and cubic
(for 3D mechanics) form, which hold constant the constellation of points in relation
to each other and the length of the unit vectors. In addition there is a VHDL-AMS
module for the gravitation that is suspended on the rigid body model. Also present
are models for joints, springs and dampers. The models mentioned are put together
using a circuit editor. The corresponding system of DAE is then solved in a circuit
simulator. Finally, we should also mention that no simulation results are shown
in [391].