Page 129 - Mechatronic Systems Modelling and Simulation with HDLs

118 6 MECHANICS IN HARDWARE DESCRIPTION LANGUAGES

the structure of the mechanics. This must either be completed during the modelling
or in the circuit simulator.
A sensible starting point in the drawing up of the element matrices is the
principle of virtual displacement. A virtual displacement is a small displacement
superimposed upon the actual displacement, which fulfils the geometric boundary
conditions and otherwise brings about no gaps or overlapping of the continuum.
The principle of virtual displacements demands that the virtual displacement
energy is equal to the virtual work of the external forces for each permitted virtual
displacement. This yields the basic equation that is drawn up for the whole contin-
uum. Now the components of the individual elements in the basic equation should
be taken into account. This would require knowledge of the continuous displace-
ments over the entire element. However, because we want to operate using only
the displacements of the nodes of the finite elements, it is necessary to approximate
the continuous displacements from the node displacements. This is done with the
aid of interpolation functions that are often created in the form of polynomials.
Thus the continuous displacements are approximated from the node displacements,
and using the displacement/strain relationship these are transformed into the strains
of the element. Using the underlying law of matter we find the stresses from the
strains. Using the quantities determined in this manner, the strain energy can be
integrated over the element range and summed over all elements. The integration
is significantly simplified by the use of interpolation functions, which — as noted
before — typically are polynomial.
By contrast, the virtual work of the external forces is based upon the excita-
tion forces, stresses at the edge of the body, and body forces such as weight. The
associated proportions of (virtual) work are again calculated from the node dis-
placements by integration over the range in question and summed for all elements.
Finally, the total virtual strain energy is equated to the total virtual work of the
external forces. In the static case this yields the following equation system:

Ku = p (6.33)

where K represents the system stiffness matrix, u the node displacements, and p
the converted body and contact forces at the nodes. The system stiffness matrix is
found from the suitable addition of the element stiffness matrices. In the kinetic
case there are also inertia forces and the equation is formulated as follows:

M¨u + Ku = p (6.34)

where M represents the system mass matrix, which, in a similar way to the system
stiffness matrix, is found by a suitable summing of the element mass matrices. The
system mass matrix is linked with the accelerations of the displacements. In this
discussion both equation systems correspond with the equilibrium principle.
If we want to represent finite elements in hardware description languages, then
it initially appears logical to first draw up the differential equation system resulting