The vertex system provides a
voltage
from each motor amplifier that is proportional to the absolute value of
the torque. On 19jun02 the dome was moved from 2 degrees za to 19.5
degrees
za and back down to 2 degrees za at .02 degrees/second (half slew
rate).
This motion was then repeated with a velocity of .01 degrees/second.
The
torques were measured once per second and recorded.
The plots
show the torques by motor during the motion (.ps) (.pdf)
The motors are
numbered
11,12,21,21,31,32,41,42 going uphill to downhill.
The adjacent pairs
(11,12),
(21,22) etc. have backlash compensation at low zenith angle.
The torque
units are foot lbs and are computed as:ftlbs=13.3*(dToACnts*10./2047)^.73
This is a fit to the torque versus voltage done by jon hagen
05jul02 using his torque wrench.
Note: this measurement was done before the hydraulic brake was
installed
on the dome.
Fig 1 torques versus time
The top plot shows the torques for each of the eight
motors for za 2->19 (left half) and za 19->2 (right half) going
at .02
deg/sec. The data was sampled once a second. The plot has been smoothed
and decimated by 7. The purple line is za/2. There is a large
step
in torque when it changes directions at 19.5 from going up to going
down.
When you go up, the motors must push the weight
of the dome plus friction. When coming downhill, the motors must hold
the
weight of the dome - the frictional force (the frictional force helps
the
motors rather than opposing them). The step at 20 degrees za
should
be twice the frictional force. Since we move at constant velocity, the
non frictional force should be the same for moving uphill or downhill
(sin(za)*Weight
of the dome).
The middle picture is a blowup of the torques around
19.5 deg za where it changed direction. The data is plotted with 1
second
resolution. The purple line is a blowup of the za. The dome sat
at
19.5 degrees for about 20 seconds before it started down.
The bottom plot is the sum of the torques for
the
8 motors during the turnaround. At 16.350 you can see the torques
wandering
up while the dome remains stationary. This can happen if there is a
small
dc offset in the voltage sent to the amplifiers. The integrator will
then
accumulate this value. A simplified model of the torques needed are:
ForceUp= Fgrav + Ffriction
ForceDown=Fgrav-Ffriction
Half the difference should be the frictional force:
friction=(166-121)/2=22 or 22/166=13% of torque moving
upward at
20 degrees.
Fig 2 torques versus za.
The upper plot has has the torques versus za moving
uphill while the middle plot has the torques moving downhill. The
uphill
motors (11,21,31,41) have a steeper slope than the downhill motors.
When
moving downhill they go to zero torque around a za of 7 degrees and
then
the torque increases. Below 7 degrees the torque is changing direction
and the motors are pushing in the direction of motion (while moving
downhill).
The bottom plot is the sum of the torques for the
8 motors. Black is moving uphill and red is moving downhill. The
torques
below 7 za for the uphill motors have had their sign flipped. When
moving
downhill below za=4.5, the motors must push the dome to keep it moving
downhill. The frictional force becomes larger than the gravitational
force.
The sine of 4.5 is 8%. If the dome weighs 200000 lbs then the
frictional
force at 4.5 degrees is 16000 lbs.
Fig 3 summing uphill and downhill.. weighing the dome.
The top plot is the average of the uphill and
downhill
motion for each motor. If the frictional force reverses going uphill,
downhill,
then the average should be the weight of the dome times the sine of the
zenith angle. The .02 deg/sec and .01 deg/sec runs are over plotted and
they lay on top of each other. The average has done a good job of
removing
any velocity dependent frictional forces.
The force needed to push the dome uphill = DomeWeight*sin(za)
where the za in the center of mass za.
The encoders report the za of the optic axis.
the dome centerline za is 1.1113 degrees downhill
from the optic axis za.
The center of mass of the dome is .7827 degrees downhill from
the dome centerline za.
So the center of mass za is 1.904 degrees downhill from the
optic axis za (see
reference)
The center plot divides the mean by sin(zaCenterOfMass).
The motors break into two groups: the uphill motor of
each
pair, and the downhill motor of each pair.
The bottom plot sums the average of the 8 motors,
divides by 1/sin(zaCenterOfMass) and then multplies by the gear ratio
and pinion
radius
to get lbs. This should be a constant line that is the weight of
the dome. The .02 deg/sec and .01 deg/sec runs are over plotted and
agree. This means that we have corrected the velocity dependant
frictional forced correctly. There
is still a za dependence below 13 degrees za. This may be from the dome
Center of mass correction.
Fig 4 differencing the uphill and downhill motions.. looking at the
friction.
The top plot is a blowup showing the torques versus
za. You can see were the torques cross zero. The discontinuities are
from
dc offsets in the readouts (below za=7 moving down, i negated the
torques
for the uphill motor pairs).
The center plot is the difference
(torquesUphill-torquesdownhill)/2.
The uphill motors go to zero near 7 degrees and then increase.
The bottom plot is the sum of the 8 motors. This
should be the frictional force. The black plot is .02 deg/sec and the
red
plot is .01 deg/sec. There is a small difference between the two
velocities
so there may be a small velocity dependent frictional force. Above 10
degrees
there is a linear or sin(za) dependence in the friction. There is a
transition
from 10 to 5 degrees below which it levels out.
Summary:
At high za it take 27% less torque to move downhill than uphill.
This
implies
that friction is 13% of the force needed to move up at high za.
When the dome is sitting still, it looks like the PI integrator
is not
turned off and the torques build up from dc offsets.
The pairs of motors (uphill, downhill) have different torque
profiles.
It is probably the bias compensation to prevent gear backlash that
causes
this.
There is no velocity dependence after summing the uphill and
downhill
motions.
Dividing the mean of uphill,downhill by sin(za) does not give a
flat
line
for the dome weight. Whatever frictional force has not been removed, it
is not velocity dependent since the .01,.02 deg/sec runs overlay each
other.
The weight of the dome at high za is about 208000 lbs.
The difference of the uphill,downhill motions give a frictional
force
that
has a linear dependence with za above 10 degrees, a transition, and
then
a flat level below za=5.
The bias and gravity compensation needs to be looked into some
more.
There
is also probably a torque dependent frictional term for the gear box
that
would differ for the uphill, downhill motions. I've also assumed that
the
wind load is not a function of za nor direction.