Page 54 - Mechatronic Systems Modelling and Simulation with HDLs

3.2 ELECTRONICS AND MECHANICS 43

Figure 3.2. The following applies:
1
˙
Li s + Ri s + i s dt = u s (3.7)
C
If we formulate the equation with the aid of charge q, it becomes:

1
L¨q + R˙q + q = u s (3.8)
C
This equation too corresponds with the structure of equation (3.1). Now, however,
the inductance is linked to the mass, the resistance to the damping, and the spring
constant to the inverse of capacitance. The voltage u s of the source is associated
with the activating force F here.
We can thus differentiate between two types of analogy, which differ from
one another primarily in the assignment of variables and basic elements. The
force–current analogy that we investigated first has the advantage that it retains
the structure of the mechanical system, see Crandall et al. [75]. Parallel circuits
remain parallel circuits, series circuits remain series circuits. Kirchhoff’s current
and voltage laws apply accordingly, i.e. forces/currents at a node and (relative)
velocities/voltages in a loop cancel each other out. The two Kirchhoff’s analo-
gies do not apply, if — as in the second case — forces and voltages are identified
as analogous. Table 3.1 shows the most important relationships for the force-
current analogy.

3.2.3 Limits of the analogies

The analogies described above are based upon linear relationships. However, this
circumstance often cannot be guaranteed. For example, the Stokes’ friction or
viscous friction has a linear relationship with velocity in a first approximation and
can thus be represented as a resistance. However, this is very definitely not the case
for the Coulomb friction. Here we can differentiate between two states of static
and sliding friction, for which different coefficients of friction apply. Furthermore,
the Coulomb friction is not dependent upon velocity but on another variable — the
perpendicular force. The Newton friction of bodies moved quickly through a fluid
finally depends upon a few parameters, such as the frontal area, the drag coefficient
and the density of the fluid, but above all on the square of the velocity. In order to
construct an analogy for the Coulomb friction we need a resistance controlled via
the normal force, i.e. via the corresponding current, which switches the coefficient
of friction in an event-oriented manner upon the transition from static to sliding
friction and vice versa. The Newton friction of bodies moved through a fluid, on the
other hand, can best be represented as a resistance with a quadratic characteristic.
We have thus already dealt with a good proportion of the components normally
considered in analogue electronics.
The transition from one-dimensional to three-dimensional mechanics represents
the limit of the consideration of analogies. The analogies can no longer be used