# Computing unbalanced moments in the az arm

#### 30mar10

Abbreviations used:
W   : weight
Cm : center of mass
E    : encoder reading of az,za, or dome
gr   : gregorian dome
ch   : carriage house
za   : zenith angle
tq    : torque
R     : radius from center of curvature.
Eoff : encoder offset to go from optic axis  to center off mass

# Computing the unbalance:

The moments in the azimuth arm come from:
• -chCouterWeight*chCounterWeightDistance
• ResidualTorques: coming from asymmetries in the azimuth arm: vertex shelter, stairway topchord to bottom chord,etc.
Values we know from measurement:
• Wdome/Wch = 6.02
• the Dome center of mass za (grCmza) is 1.913 degrees downhill from the dome encoder za: grEZa.
• see y08-y09 fit to tiedown tensions.
• With the ch at 0, the dome balances the other side at zaEGr0=6.6  degrees or zaCmGr0=4.56 degrees.
• from td tension fits y08-y09
• When the zaEch = 8.83, the az arm is balanced with zaEGr = 7.968 degrees.
• Dome weight: 215 kips.
Values we use that have not been measured.
• ch counter weight: 45 kips
• ch weight: 35 kips
• ch za encoder to za center of mass. assumed to be 0.
• Radius for ch and dome center of mass: 430 feet.
Values to be computed:
• Residual azimuth arm torque (from vertex, stairway, etc.)
• compute using the grCmZa that balances when zaCh =0
• tqRes=Wgr*Rgr*sin(zaCmGr0) - Wcw*Rcw
• tqRes=215*430.*sin(4.56) - 145*45.
• tqRes=825 kip-Feet
• Compute using grEza=8 balances when zaECh=8.834
• tqRes=Wgr*Rgr*sin(grEZa-zaEOffset) - (Wcw*Rcw + Wch*Rch*sin(zaECh)
• tqRes=215*430.*sin(7.968-1.913) - (45*145 + 35*430.*sin(8.834)
• tqRes= 904 kip-feet
• Residual tq ch side: (904+825)/2=864 kip-feet.
• Equation to compute imbalance:
• unbal=(Wgr*Rgr*sin(zaEgr-zaEoff)) - (Wcw*Rcw + tqResCh + Wch*Rch*sin(zaECh))
• unbal=(215*430.*sin(zaEgr-1.913) - ( 45*145 + 864. + 35*430*sin(zaEch)

# Plotting the unbalance:

The plots show the unbalanced moments in the azimuth arm (.ps) (.pdf) for various configurations.
• Top: dome moves 1.06 degrees to 19.69 degrees in za:
• ch at 8.83 degrees za (stow). This is the current setup we use.
• ch at 20 degrees za.
• ch  tracks  dome za when dome above za=10.
• Bottom: ch moves 0 to 20 degrees in za, dome remains fixed.
• dome at 8.83 degrees za (stow). This is the current setup we use.
• dome at 13.5 degrees za.
• dome at 15.0 degrees za.
• dome at 19.69 degrees za.
• The dashed horizontal lines show +/- 12000 kip-feet.

### Notes:

• tracking with the ch when dome above 10 degrees za only saves 3000 kip feet when dome is at 19.69 deg. The resulting unbalance is still larger than the allowed 12000 kipfeet unbalance.
processing:x101/100308/unbtorques.pro

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