Page 57 - Mechatronic Systems Modelling and Simulation with HDLs

46 3 MODELLING AND SIMULATION OF MIXED SYSTEMS

we can use the analogies introduced in Section 3.2.2 to associate electronic com-
ponents with basic mechanical elements. The other option is to model not the
mechanics itself, but rather the differential equations that describe the mechanics.

Analogy approach

InordertoconsidertheanalogieswemustfirstrefertoSection 3.2.2.Theforce/current
analogy is normally used. In addition to the basic elements, other mechanical phe-
nomena such as Coulomb friction have to be considered. These require behavioural
modelling based upon sources that can be controlled by arbitrary mathematical func-
tions. Such voltage and current sources are available in PSpice, for example. This
represents a rudimentary form of modelling in a hardware description language.
Yli-Pietil¨ a et al. [431] use this method to investigate mechatronic systems such
as a linear drive. They model a DC motor with an electronic control system and a
mechanical load. The same approach is further elaborated by Scholliers and Yli-
Pietil¨ a in [369] and applied to other examples, such as a double pendulum. In
[368] Scholliers and Yli-Pietil¨ a introduce a whole library of such models, which
expand the field of application of a circuit simulator such as Spice in the direction
of mechatronics.
Examples for the use of equivalent circuit diagrams in micromechatronics are
supplied, for example, by Ant´ on et al. [13] (pressure sensor elements), Garverick
and Mehregany [111] (micromotors), or Lo et al. [236] (resonators).

Modelling of differential equations using equivalent circuits

As an alternative to the analogy approach described above we can also find an
equivalent circuit for the underlying system of equations. In principle, this pro-
cedure is similar to the construction of a rudimentary analogue computer from
electronic components. In this context we can differentiate between explicit and
implicit methods, see Bielefeld et al. [31]. In the explicit version the values of the
state variables are represented as voltages in the network. In this, the highest time
derivative of each state variable is set depending upon lower derivatives and other
state variables using a controlled voltage source. In addition, there are integrators,
see the left-hand side of Figure 3.3, which again provide the low derivatives in the
form of voltages, see Herbert [139]. As an alternative to this, the implicit method,
see Paap et al. [312], solves a set of n equations in the form:

f(x, ˙x, t) = 0 (3.9)

where x represents a vector of n unknowns. As in Herbert [139] the states are
represented as node voltages. Each equation is defined by a current from a voltage-
controlled current source. This sets the input current of the differentiator in the