Hardware Reference
In-Depth Information
An alternative approach is to get a model of friction off-line and then use the
model on-line to estimate the friction force. Classical models of friction include
preload model and Dahl model [40] as defined by the equations below.
Preload model: In this model, the friction force is expressed as,
F = F
c
sgn(v)+F
v
v
(2.30)
F
c
and F
v
are the Coulomb friction and viscous friction, respectively. Coulomb
friction is constant irrespective of the velocity (v) of motion, but the magnitude
of viscous friction increases with increasing velocity.
Dahl Model:Thismodeldefines the derivative of friction force with respect
to position as,
dF
dy
= σ|1−
F
F
c
sgn( y)|
i
(2.31)
where,
σ =restslope,
F
c
= rolling torque, and
i = exponential factor.
The preload model of equation 2.30 is independent of the position variable
(y) whereas the Dahl model (equation 2.31) defines the friction as a function of
position only. Experimental results suggest that the frictional behavior of the
actuator pivot is not determined solely by velocity or position, but by a com-
bination of the two [1], [201]. This conclusion is reached after examining the
experimentally obtained frequency response of VCM actuator. The frequency
response of the double integrator model is expected to show -40 db/decade
slope and −180
◦
phase at the lower end of the frequency axis. But the experi-
mentally obtained frequency response show 0 dB/decade slope and 0
◦
phase at
low frequency suggesting a transfer function with no integrator (Figure 2.23).
In other words, the friction force, that opposes the applied force, is a func-
tion of both position and velocity. The experiments also show variations in
the low frequency gain with varying amplitude of the input signal used while
measuring frequency response, which requires a nonlinear model to describe
the actuator.
The method for finding a nonlinear model of the pivot friction described
in [1] exploits the relationship between the frequency response measurement
and describing function. Several models have been suggested that take feed-
back of both velocity and position into consideration. Some of these models
are given below.
