Page 135 - Mechatronic Systems Modelling and Simulation with HDLs

124 6 MECHANICS IN HARDWARE DESCRIPTION LANGUAGES

upon introducing two current sources between the nodes i and j for an LC branch,
which satisfy the following equations:

where m ij is from M
i ij,C = m ij ˙u ij
(6.47)

i ij,L = k ij u ij dt where k ij is from K

In addition there are two further current sources for each degree of freedom, which
represent the connection to the ground and — as demonstrated above — the external
excitations p i0 and p i1 at each beam element.

Composition of the system matrix
In the previous section an element matrix was put together for the beam element,
the four degrees of freedom of which are represented by the potentials at the
four terminals of the element. The currents at the nodes in question describe the
integral of the associated forces and moments, depending upon whether the degree
of freedom is a translational or rotational deflection. In particular, the components
of the exciting forces and moments that are assigned to the elements adjoining the
nodes are also added to the currents at a node. Thus it is not necessary to explicitly
draw up the system matrix. Its solution is found implicitly from the interconnection
of the finite elements.

Example: beam with various boundary conditions
Two examples will be considered to illustrate the element model described above,
a cantilever beam with and without an additional support point, see Figure 6.10.
The second case, in particular, cannot simply be mastered by either analytical
equations or finite differences. The beam of length l itself is modelled by 40 finite
beam elements. The excitation consists of the pulsed force F y , which is applied to
the beam for eight seconds and then removed again. The outputs are the deflections
in the y-direction at x = 0.25 l, 0.5 l, 0.75 l and 1.0 l, see Figure 6.11. In the first
case there is an oscillation, the amplitude of which is more strongly marked towards
the end of the beam, and which is in phase at each point. The additional support
in the second case fundamentally alters the behaviour of the beam. Firstly, the
natural frequency of the system increases, secondly the node moves downwards at
x = 0.25 l due to the lever effect of the free end of the beam, although the force is
acting upwards. We note that the deflection at x = 0.5 l becomes zero.
The same simulation was performed using the ANSYS finite element simulator
to verify the results. The differences amount to less than one percent and are
in principle attributable to differences in the numerical solution procedure. The
simulations were run on a SUN Sparc 20 workstation. The simulation time for the
first case amounted to 91 CPU seconds for Saber and 94 CPU seconds for ANSYS.