Terminology:

• Wgr :  gregrorian weight. This includes everything that moves with the dome (motors, pigs, etc..)
• zaE,zaCm  :  Zenith angle. (more info)
• zaE: encoder zenith angle. This is the zenith angle reported by the encoders.
  
• It is 1.1113 degrees uphill from the dome centerline position
• zaCM: the zenith angle of the dome center of mass. This is .80 degrees downhill from the dome centerline.
• So zaCM= zaE - 1.913 degrees. This is  zenith angle  that is needed when fitting for the sin(za) term of the force (gravity).
  
• Sign of motor force. Let a positive force be uphill.
  
  

Forces that motors must overcome:

• Wgr * sin(zaCm) . This is the gravitational force.
• zaCm is the zenith angle of  the dome center of mass.
• C1  .. i have called this C1 in the fit.
  
• This force always points downhill (negative).
• C2*vel:  This is caused by  friction.
• vel will be measured in deg/sec
  
• C3*accel: This is the force needed to accelerate the dome
• It is measured in deg/sec^2
• C4*vel^2: The resistance provided by hydraulic brake when moving down hill.
• This term is not used since we limit the motion to uphill (see below).
  

• The velocity and acceleration are computed by differencing the position and velocity.
• The hydraulic brake is designed so that the motors must push downhill above a certain velocity.
  
• This velocity differs for the 8 motors (because of  torque biasing of pairs of motors).
• The zenith where this occurs differs for different motors.
  
• The torque measurements return the absolute value of the motor torque.
• I've limited the fit data set to za > 5deg, vel > .001 deg/sec because:
• we only have the absolute value of the torques
• moving downhill, different motors switch from holding back the dome, to pushing downhill.
• staying above .001 degs/sec will hopefully stay away from stiction (static friction).
  

Fitting the torques:

    The fits to the torque were limited to uphill motion (vel  > .001 deg/sec) and za > 5 degrees za.

The fit used was:

• motorTorqueFitFtLbs= C0 + C1*sin(zaCm) + C2*vel + C3*acceleration
• C0:  offset for fit
• C1: The motor torque that  is needed for the dome weight
• C2: The motor torque to overcome friction
• C3: the motor torque to handle the acceleration.
  

Potential problems with the fit:

• positions and torques are only measured once a second
  
• The maximum velocity is .04 deg/sec
  
• The maximum acceleration is .025 deg/sec^2
  
• With only 1 sec position measurements:
• the velocity and accel are computed by differencing the positions and then interpolating back to the time of the torque measurement.
  
• This will cause errors in fitting the friction and acceleration portions of  the fit.
  
• The motor torques are returned by the motor amplifier  as a voltage proportional to the torques.
• Jon hagen calibrated a number of these devices and found a non-linear response:
• ftLbs=13.3*(A/Dcnts/2047 * 10) ^ .73
• I've used this for the measured ft lbs.
• This was measured back in 200? . It may have changed when amplifiers were replaced.
• I've tried to avoid sticktion (static friction) grabbing and then letting loose at low velocities by forcing the vel < .001 deg/sec
  
• This may not have been enough given the errors in the velocity computation.
  

Monthly fits of the torques

    The data for each month was fit. 8 separate fits were done (one for each motor).

The fit used was :
     motorTorqueFitFtLbs= C0 + C1*sin(zaCm) + C2*vel + C3*acceleration

Each monthly fit plots contain:

• Page 1:
  
• The x  axis has the 8 motors 0.. 7 are motors:m11,m12,m21,m22,m31,m32,m41,m42
• The red * are the Uphill motors (of each pair)
• The blue * are the downhill motors  (of each pair)
• TopFrame: C0: constant term in the fit
• 2ndFrame: C1: The dome weight in foot lbs
• See below for converting torque to weight.
  
• 3rdFrame: C2 : The ft lbs needed to overcome friction (moving uphill).
• The velocity units used are deg/sec (max vel = .041 deg/sec).
• bottomFrame;C3;The ft lbs need for the dome acceleration.
  
• The acceleration units are deg/sec^2. Max Accel= .025 deg/sec^2
• Page 2:
• Top: The fit errors:
• This is the (data -fit) rms for each motor.
• 2ndFrame: The fractional error for each coef by motor.
• The x axis is the 4 coef used for the fitting: c0,c1,c2,c3
• The fit values for each motor have been over plotted (in  color).
  
• This is returned by the fitting routine. Since i didn't enter any measurement errors, it may not be too meaningful.
• I also divided by the value of the coef. to give a fractional error.
• This caused coefs close to 0 to blow up the fractional error.
• Bottom:The 4 fit coef for the 8 motors are printed.
  

• 2013:
• jun13: trouble with dome jun13. motor 22 replaced. (more info)
  
• 2003:
  
• july03 was after the hydraulic brake and before alfa was installed.

Yearly summary of fits to dome motor torques

• Page 1: The fit coefs for each month over plotted in color,.
  
• The x axis is the  coef: c0,c1,c2,c3
• Top Frame: C0 constant term of fit
• 2ndFrame: C1 sin(za) coef. dome weight.
• 3rdFrame: C2: frictional coef.
• bottomFrame: C3  acceleration coef.
• Page 2:
• TopFrame: Fit rms for each motor by month:
• rms of (yfit - ydata) for each motor and month.
  
• The x axis 0..8 are the 8 motors m11,m12,m21,m22,m31,m32,m41,m42
• Each color is a different month.
• 2ndFrame: fractional error in each fit coef.
• the x axis 0..3 are the 4 fit coef.
• Each color is a different motor.
• Each month is over plotted.
• These errors are returned by the fit. 
• BottomFrame: Dome weight from the fits:
• For each month, the C1 coef for each motor is added and then foot lbs is converted to lbs (see below).

Yearly notes:

• 2013
• 2012

Summary:

• What the fits can be used for:
• The measured dome weight
  
• 2012-2013 had dome weights of for 2012, 2013 of 228-232 kips
• the fitted value is stable to within a few percent so we can probably use it for relative changes.
• comparing the jul03 motor weight to the feb03 weight from  maldonado's measurement, the motor weight could differ by 5%.
  
• Changes in the fit coef's may show problems in the system.
  
• But probably only if the problems are persistent for the month.
• Comparing a daily fit to the monthly average may help troubleshoot problems.
• Some of the variations in the fit values may be a function of the telescope motion that was done that  month.
• Eg.a month with lots of galfacts slewing 2-19 degs at 1/2 slew rate may give different results than a month of on/off position switching.
  
• friction:
• The C2 coef for friction is between 1000 -1500 ft lbs.
  
• at .01 deg/sec this is about 1200*.01 * 8=96 footlbs.
  
• The coef for dome weight is 4*80 + 4*55=540 ftlbs
• So friction is about 96/540= 18% at .01 deg/sec.
• Acceleration :C3
• the sum of C3 for the 8 motors ranges from 550 to 900 ft-lbs
• as and example, used .0125 deg/sec^2 (1/2 the max acceleration)
  
• .0125*725=9 ftlbs. This is 10% of what is needed to hold the dome against gravity at za=10 deg.
  
• Comparing the dome weights
• 
  

• date	weight (kips) 1kip=1000 lbs	Notes
  jun02	208	motor torques before hydraulic brake
  jun02		hydraulic brake installed
  feb03	215.5	maldonado using load cells,  and jacks.
  july2003	225	motor torques
  apr04		alfa installed in dome
  jan12-dec12	228-233	motor torques
  jan-jun13	230-232	motor torques

  
  

Converting foot lbs to lbs:

• gear ratio of transmission: 190.07
  
• radius of pinion: .4375 feet
• Lbs=FtLbs*190.07/.4375 
• To get the total weight you need to convert and then sum the C1 coef from all 8 motors.

 home_~phil