Wednesday, March 18, 2009

Friction Modelling near Striebeck Velocities

Many models were developed to explain the friction phenomenon.
These models are based on experimental results rather than
analytical deductions and generallydescribe the friction force (Ff)
in function of velocity (v). The classical static + kinetic + viscous
friction model is the most commonly used in engineering. This
model has three components: the constant Coulomb friction
term ( ) (v sign FC ), which depends only on the sign of velocity,
the viscous component ( v FV ), which is proportional with the
velocity and the static term ( S F ), which represents the force
necessary to initiate motion from rest and in mostof the cases its
value is grater than the Coulomb friction: (see Figure 1.)

The servo-controlled machines are generally lubricated with oil
or grace (hydrodynamic lubrication). Tribological experiments
showed that in the case of lubricated contacts the simple
static +kinetic + viscous model cannot explain some
phenomena in low velocity regime, such as the Striebeck effect.
This friction phenomenon arises from the use of fluid lubrication
and gives rise to decreasing friction with increasing velocities.

To describe this low velocity friction phenomenon, four regimes
of lubrications can be distinguished (see Figure 2). Static Friction: (I.)
the junctions deform elastically and there is no excursion until the
control force does not reach the level of static friction force.
Boundary Lubrication: (II.) this is also solid to solid contact, the
lubrication film is not yet built. The velocity is not adequate to build
a solid film between the surfaces. A sliding of friction force occurs
in this domain of low velocities. The friction force decreases with
increasing velocity but generally is assumed that friction in boundary
lubrication is higher than for fluid lubrication (regimes three and four).
Partial Fluid Lubrication: (III.) the lubricant is drawn nto the contact
area through motion, either by sliding or rolling. The greater the
viscosity or motion velocity, the thicker the fluid film will be. Until the
fluid film is not thicker than the height of aspirates in the contact
regime, some solid-to-solid contacts will also influence the motion.
Full Fluid Lubrication: (IV.) When the lubricant film is sufficiently
thick, separation is complete and the load is fully supported by fluids.
The viscous term dominates the friction phenomenon, the
solid-to-solid contact is eliminated and the friction is 'well behaved'.

The value of the friction force can be considered as proportional with
the velocity. From these domains results a highly nonlinear behavior
of the friction force. Near zero velocities the friction force decreases
in function of velocity and at higher velocities the viscous term will
be dominant and the friction force increases with velocity. Moreover
it also depends on the sign of velocity with an abrupt change
when the velocity pass through zero.

For the moment no predictive model of the Striebeck effect is
available. Several empirical models were introduced to explain the
Striebeck phenomena, such as the Tustin model [2]:

The model introduced in this paper is based on Tutin friction model
and on its development, the following aspects were taken into

- allows different parameter sets for positive and negative velocity regime
- easily identifiable parameters
- the model clearly separates the high and low velocity regimes
- can easily be implemented and introduced in real time control algorithms

For the simplicity, only the positive velocity domain is considered,
but same study can be made for the negative velocities. Assume
that the mechanical system moves in 0 … vmax velocity domain.
Consider a linear approximation for the exponential curve represented
by two lines: d1+ which cross through the (0,Ff(0)) point and it is
tangent to curve and d2+ which passes through the (vmax, Ff(vmax)
point and tangential to curve. (see Figure 3.) These two lines meet
each other at the vsw velocity. In the domain 0 … vsw the
d1+ can be used for the linearization of the curve and d2+ is used
in the domain vsw… vmax. The maximum approximation error
occurs at the velocity vsw for both linearizations.
If the positive part of the friction model (2) is considered (v>0),
the obtained equations for the d1+ and d2+, using Taylor expansion,

Thus the linearization of the exponential friction model with bounded
error can be described by two lines in the 0 … vmax velocity domain:

Same study can be made for negative velocities. Based on
linearization, the friction can be modelled as follows:

It can be seen that the model is linearly parameterized and it
can be implemented with low computational cost.


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