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.

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]:

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

consideration:

- 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,

are:

and on its development, the following aspects were taken into

consideration:

- 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,

are: