Page 138 - The Mechatronics Handbook

F
3
dry
friction
F 1 = F 3 F 2 = F 3
V 1
1 0 1
F 1
V V
V 1 F V 2
3 3 3
F 2
R
F = µΝsgn(V ) V 2 N
3 3
F 3 = F 1 = F 2
(a) (b)
FIGURE 9.11 (a) Classic coulomb friction for sliding surfaces. (b) Bond graph showing effect of normal force as
a modulation of the R-element law.
however, the velocity comes into effect only to determine the sign of the force; i.e., F = µN sgn(V), where
sgn is the signum function (value of 1 if V > 0 and -1 if V < 0).
This model requires a special condition when V → 0. Dry friction can lead to a phenomenon referred to
as stick-slip, particularly common when relative velocities between contacting surfaces approach low values.
Stick-slip, or stiction, friction forces are distinguished by the way they vary as a result of other (modulating)
variables, such as the normal force or other applied loads. Stick-slip is a type of system response that
arises due to frictional effects. On a bond graph, a signal bond can be used to show that the normal
force is determined by an external factor (e.g., weight, applied load, etc.). This is illustrated in Fig. 9.11(b).
When the basic properties of a physical element are changed by signal bonds in this way, they are said
to be modulated. This is a modeling technique that is very useful, but care should be taken so it is not
applied in a way that violates basic energy principles.
Another difficulty with the standard dry friction model is that it has a preferred causality. In other
words, if the causal input is velocity, then the constitutive relation computes a force. However, if the
causal input is force then there is no unique velocity output. The function is not bi-unique. Difficulties
of this sort usually indicate that additional underlying physical effects are not modeled. While the effort-
flow constitutive relation is used, the form of the constitutive relation may need to be parameterized by
other critical variables (temperature, humidity, etc.). More detailed models are beyond the scope of this
chapter, but the reader is referred to Rabinowicz (1995) and Armstrong-Helouvry (1991) who present
thorough discussions on modeling friction and its effects. Friction is usually a dominant source of
uncertainty in many predictive modeling efforts (as is true in most energy domains).

Potential Energy Storage Elements

Part of the energy that goes into deforming any mechanical component can be associated with pure
(lossless) storage of potential energy. Often the decision to model a mechanical component this way is
identified through a basic constitutive relationship between an effort variable, e (force, torque), and a
displacement variable, q (translational displacement, angular displacement). Such a relationship may be
derived either from basic mechanics [29] or through direct measurement. An example is a translational
spring in which a displacement of the ends, x, is related to an applied force, F, as F = F(x).
In an energy-based lumped-parameter model, the generalized displacement variable, q, is used to
define a state-determined potential energy function,

E = E(q) = U q
This energy is related to the constitutive relationship, e = F(q), by
U q() U edq = ∫ Φ q)d
∫
=
(

=
q
It is helpful to generalize in this way, and to identify that practical devices of interest will have at least
one connection (or port) in which power can flow to store potential energy. At this port the displacement
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