# Computing the cable xz vertical angle

19oct20
plots:
Over plot platform xz cable angles for 3 computation methods (.ps) (.pdf)
Over plot backstay  xz cable angles for 3 computation methods (.ps) (.pdf)

Other pages:
Cable tensions from the p50 sag survey
Tower4 auxNorth failure

# Intro (top)

Let x,y be in the horizontal plane,and z  is positive up. The sag survey computation rotates the cable datapoints into the xz plane and the fits a catenary.
To compute the cable tension from the sag survey, you need to know the vertical (xz) angle of the cable.
Assume the catenary fit is:

• z=A + B*(cosh(x-C)/B - 1)
• Then the cable tension is:
• Tension=B*cableLinearWeight*cos*(xzVerticalAngle)

Prior to fitting for the catenary i did a 2nd order fit to remove outliers:

• z=c0 +c1*x +c2*x^2

Up until 19oct20 i was using the c1 coef from the fit to compute the xz angle.
When i distributed the results  of the cable tensions , pierre ghisbain pointed out that the vertical angles i reported were not consistent

• Different measurements of the same cables (say the mains) were getting vertical angles that were differing by up to a degree or more
• 1 degree difference at 575 feet gives a vertical distance difference of  10feet..

# Computing the vertical angles from the fits.

When fitting a catenary to each cable data set:

• we get the coef of the catenary fit to the data
• we also have the min,max x value used in the fitting
• Plugging the xmin,xmax values into the catenary fit will give you the vertical angle (atan(deltaz/deltax))
• In almost all measurements, xmax-xmin is less that the expected length (using drawings cable length and attachment points).
• This is mainly caused by the measured points close to the platform connection point being obscured by parts of the platform.
• using Xmax from the data set and xmin=(xmax-ExpectedXvalueFromDrawings) as the minimum value you can also compute the slope
• You could argue that the expected xmin is coming from the drawings angle.. except that i'm still using the y values from the fit...

Plots were made showing the (measured xzangle - drawingsXzangle) for the platform and backstay cables.

• The differences were computed  for:
• using the 2nd order fit linear term to get the xzangle
• using the catenary fit with the xmax,xmin data values.
• using the catenary fit  with the xmax from the values and a computed xmin from the drawings cable length and angle.
• colors were used to separate out the results.
• There are 3 frames for each page:
• Tower 12, tower4 and tower 8.
• Measurements sets are separated by dashed vertical lines.
• T12: 3 platform sets, 3 backstay sets
• T4 : 4 platform sets, 2 backstay sets
• T8 : 2 platform sets, 2 backstay sets.
• the xz angles from the drawings are: 12.77 deg mains, and 10.09 aux, so a .1 deg difference is about 1%.
• T12: main cables:  the catenary computations are .2 degrees larger than the drawings value
• T4:  main cables: the catenary with xmin from drawings is within .1 degrees of the drawings values
• T8:  main cables: the catenary values are within .1 degrees
• The  aux cables for the catenary values with xmin from drawings have differences of .1 deg or less
• the xz angles from the drawings are:
• main: 35.64,35.66,26.15 for T12,T4,T8
• aux:   35:45, 35.49,26.57 for T12,T4,T8
• a .1 degree difference is about .3%
• the xzangles are within .1 degrees for the catenary computation with xmin from the drawings.

# Summary:

• Using the linear term from the  parabolic fit to the cables gave xz angles that differed from the drawings angles by up to 3 degrees.
• Using the xmax,xmin from the data set and the catenary fit gave values that agreed with the drawings angle to better than .2 degrees
• Using the catenary fit, xmax from the dataset , and computing xmin from the drawings cable length and angle
• matched the drawings angle to better than  .1 degree .. except for tower 12 main cables that were consistently .2 degrees larger than the drawings angle.
• You might argue that using the drawings cable length and angle to compute xmin  might be forcing the angle to match the drawing angle.. BUT:
• Tower 12 had a .2 degree offset. At a 575 foot radius this is a vertical offset of  575*sin(.2) = 2 feet.
• The laser ranging data shows that platform corner 12 is 2 feet low (matching what the cable tensions show).

processing: x101/p50/cables/xzangle/chkxzangle.pro, chkxzangleplot_plat/bstay.pro

home_~phil