Constructing and Using a Stocker Final Glide Calculator



Constructing and Using a Stocker Final Glide Calculator

By Moshe Braner, last updated January 19, 1999

Overview

In the book "Cross Country Soaring", Helmut Reichmann describes a graphical final glide calculator that is designed to be laid over a map. The design is due to Stocker. (There are markings over the 'o' of "Stocker" in the German.) The explanation in the book is rather terse. Hopefully this write-up, along with the accompanying pictures, can clarify it.

The calculator is transparent and round in shape, and overlays a round cut-out from a map centered on the home airport. A pin is inserted in the center so that the calculator can be rotated relative to the map. In addition to the round disc, a transparent strip with a distance scale is also attached to the center. Thus, the line with distance ticks can be made to overlay the final glide course on the map. While the map and the strip are held fixed relative to each other, the main, circular part of the calculator can be rotated. When it is rotated to the correct position, the calculator shows the altitudes needed at each point along the final glide course. The correct position is determined using a curvi-linear grid of climb rates and wind speeds on the other half of the calculator. See the example pictures.

The glide courseline is depicted on the map, obviating the need for a separate map during the final glide. This is one nice feature of this calculator, as one needs a map anyway.

Using the calculator is really simple: Rotate the transparent strip relative to the transparent disc until it crosses the climb/wind grid at the chosen combination of climb rate and wind (headwind or tailwind). Then rotate both the disc and the strip, held together, relative to the base map, until the extension of the straight line overlays the final glide course line on the map, opposite from the climb/wind grid.

Glide curves on the calculator disc then cross the course line at the points where specific altitudes are needed to complete the final glide, assuming that the correct "speed to fly" will be used. These altitudes are graphically "computed" so that the wind, and the glider sink rate for the flying speed, are taken into account.

For example, in the picture accompanying this write-up, the line on the strip crosses the grid at the point corresponding to a climb rate of 2 knots and a headwind of 10 knots. (The positions for no wind and a climb rate of close to 4 knots, or a 10 knot tailwind and a climb rate of a bit over 5 knots, fall on the same line, i.e., they correspond to the same achieved glide ratio, about 27:1.) If one's position is over the mountain peak that is just over 20 nautical miles out, the altitude curves on the calculator show that one needs about 4500 feet of altitude. Since the home airport in the middle of the map (Post Mills, Vermont) is at about 700 MSL, and adding another 800 feet for pattern altitude, the MSL altitude needed to start the final glide from that mountain peak is about 6000 feet. (If you don't want to do the math while flying, mark the altitude curves with the total MSL altitude needed.)

The climb rate is used in the sense of a speed ring setting. On a day with a better climb rate one should start the final glide closer and higher, and fly a faster and steeper glide, to achieve the best overall task speed. If one just wants to see if a final glide can be made, without optimizing for overall XC speed, then a climb rate of zero can be chosen on the calculator. The wind can still be factored in. The calculator can also be used enroute, if one ignores the map, and just compares the distance scale with the altitude curves, assuming that the distance to the next turnpoint, or an alternate goal, is known.

It is assumed that the final glide is made in "dead" air. No glide computer, not even the ones that cost thousands of dollars, can truly predict the actual glide in advance, when lift or sink are encountered during the glide. If one finds oneself getting lower than the needed altitude, one then needs to slow down. Setting the Stocker calculator to zero-climb (best-glide) will show if the glide can still be completed. Conversely, if, as the glide progresses, you find yourself too high, you can speed up. Rotate the calculator so as to set the needed altitude equal to the available altitude. That will show the speed ring setting that will result in the fastest speed that will still allow completion of the glide.

The zero climb-rate curve in the climb/wind grid is for a glide done at the best-glide speed, not adjusted for the wind. (The speed is not optimized for the wind, but the glide ratio is certainly computed taking the wind into account!) If successful completion of the final glide is in doubt, one can further improve the actual glide slope by adjusting the flying speed for the wind, e.g., by using the rule of thumb of adding half the headwind. Except in the case of a strong headwind, the improvement is slight, and the glide slope computed by the Stocker calculator is only slightly conservative. So go ahead and add half the headwind to your flying speed, and your glide slope should be no worse, and possibly a bit better, than the computed slope.

Designing the Calculator

The size is arbitrary. I chose a diameter of 18 cm as a compromise between having a large map area to look at, vs. keeping the size of the calculator down. Assuming that the map will be an aviation "sectional" (as used in the USA), with a scale of 1:500000, the map cutout will extend to a radius of 45 km, or about 28 statute miles. Sometimes one may make a final glide that is longer than that, but it does not seem worthwhile to make the calculator larger. One can extrapolate the glide from the portion visible on the calculator.

One section of the calculator has the glide spirals. I chose to draw one for each 1000 feet of required altitude. For any given glide ratio, the points that require a certain altitude are a certain distance from the center and thus would fall on a circle. As one moves radially away from the center, the distance and the altitude required grow larger. The other, tangential, dimension, is used to represent different glide slopes. Therefore the curves are spirals, not circles. They are drawn for an arbitrary range of glide ratios. I first chose the glide-slope range of 15 to 50, but in a recent improvement to the software I made it automatically choose a range that fits the glider well.

Also arbitrary is the manner in which the glide ratios are arranged tangentially. The angle around the center can be proportional to the glide ratio, or to any other mathematical function of the glide ratio. To make the curves reasonably nice to look at and work with, I chose to make the angle proportional to the reciprocal of the glide slope, and to spread the range of glide slopes over 120 degrees of arc in order to use a fair portion of the available map area. These decisions result in a definite mathematical relationship between glide slope and angle around the calculator. This relationship is used later.

The glide-spirals half of the calculator is independent of the glider polar, it only depends on the map scale. I.e., one can re-use the same glide curves to construct a calculator for a different glider. Only the climb/wind grid, on the other half of the calculator, must differ for different gliders. Note, though, that because my software automatically chooses a range of glide ratios to suit the glider polar, the glide-spirals half of the calculator does come out looking different for each polar.

The range of the climb/wind grid is also somewhat arbitrary. I chose to represent wind speeds from 20 knots headwind to 20 knots tailwind, and climb rates up to 8 knots.

The climb/wind grid is a graphical computer, i.e., a "nomogram". It determines the glide slope as a function of speed ring setting and wind. The radial scale is arbitrary, to fit nicely within the radius of the calculator. The radial direction is the wind axis. I chose to put headwinds further away from the center and tailwinds closer in.

The tangential axis of the grid is the climb-rate axis, and it is implicitly determined by the choice of glide-spiral scaling.

Here is how the location of a point on the grid is determined, given a climb rate and a wind speed, and the glider polar. This explanation describes the mathematical basis, in practice I use a computer program to do all these computations. For more theoretical discussion, contact the author.

Step 1: Compute the speed to fly. The STF depends on the climb rate and the polar but does not vary with the wind. This computation can be done graphically if a graph of the polar is available. That is how Reichmann explains it in his book. Alternatively, the polar can be approximated by a mathematical formula. E.g., the polar can be defined as a sink rate Z dependent on the airspeed V (both in knots) via the formula:

Z = S0 + k*(V-V0)^2

where S0, V0, and k are constants that define the polar. In that case, the formula for the STF is:

V = sqrt{ V0^2 + (S0+C)/k }

where C is the climb rate (also in knots).

Step 2: compute the glider sink rate Z at that speed. This is simply read off the polar, or computed via the formula approximating the polar.

Step 3: compute the actual glide slope at this speed, including the effect of the wind. The formula is:

G = (V+W) / Z

where W is the wind speed, tailwind being positive and headwind negative.

A position on the round calculator is defined in polar coordinates, i.e., as an angle (from an arbitrary starting angle) and a radius.

Step 4: Compute the angle that corresponds to the glide slope computed in step 3. This angle depends on the arbitrary formula connecting glide ratios to angle around the calculator, as chosen when the glide spirals were drawn.

Step 5: Compute the radius corresponding to the wind, using the arbitrary relationship chosen in the design of the grid.

Using software to do the plotting

So, all one has to do is to repeat steps 1 through 5 for a whole slew of climb rates and wind speeds. Alas this is a rather tedious process, and has to be done all over again to create a calculator for a different glider, or even the same glider at a different flying weight. Fortunately, computers can help here. I wrote a computer program to plot the curves (both the grid and the glide spirals). The program is written in the C language and, for my use, compiled into an MS-DOS executable (.EXE) file.

The input to the program is 4 numbers that define the polar: the best-glide ratio and speed, and the sink rate at a specified other, higher, speed. The program uses these 4 numbers to approximate the relevant portion of the polar using the formula mentioned above. These numbers are provided on the command line, run the program with any ONE command line parameter, e.g.,

stocker /?

to get help on the command line syntax.

The output of the program is a text file in a special format called "XYZ" that I have invented. This format holds plotting instructions that can be interpreted by various software tools to actually plot the curves. In order to actually print the curves on paper, a given fixed text file, holding a "prologue", is prepended to this output file. The combined file is then a valid Postscript file and can be printed by a Postscript printer. One then adds labels, and photocopies the page onto a transparency from which the calculator is constructed.

As a bonus, at end of text file output by the program, the speed ring numbers for the given polar are reported (as Postscript comments). These can be helpful if one wants to construct a speed ring for the vario, or even just a written table of speeds to fly to carry in the glider. The glide calculation done by the Stocker calculator assumes that one flies at the speed-to-fly, at least approximately, so one must have a method of determining what this speed is. The STF numbers can be written on the Stocker calculator itself. To aid in that, my software optionally produces ticks that will be printed along with the curves, ticks that cross the zero-wind climb rate curve. These ticks mark the climb rates at which the STF (in knots) is 50, 60, 70, etc. The software also outputs ticks on the perimeter of the calculator, marking the angular location of the glide ratios 10, 15, 20, 25, 30, 40 and 60. (Numbers in this list that fall outside the glide-slope range are not depicted with ticks.)

Constructing the Calculator

Once the software output is printed, one needs to do the following to complete the construction:

Cut the straight line that is printed about 1/2 inch from the bottom edge of the page. This will become the center of the courseline strip of the calculator. It has distance ticks on it every 5 nautical miles.

Add labels, as the software only draws the curves, without labels. The needed labels include:

* Altitudes, labeling the glide spirals. The spiral closest to the center represents an altitude of 1000 feet, so I label it "1". The spiral next closest to the center is labeled "2", and so on. These are the altitudes used up in the glide. Altitude needed for the landing pattern needs to be added, along with the field elevation. You may want to label the spirals with the total MSL altitude needed at that point in the final glide.

* Wind speeds. In the climb/wind grid, the curves that are actually parts of circles around the center of the calculator are constant-wind curves. The one closest to the center is for a tailwind of 20 knots. The next-closest is for a tailwind of 10 knots. The rest are for zero wind, and 10 and 20 knots of headwind.

* Climb rates. In the climb/wind grid, the curves that cross the constant-wind curves are constant-climb curves. The curve at the clockwise end of the grid is for zero climb, i.e., for a glide done at the best-glide speed. The next curve to the counter-clockwise side if for a climb rate of 2 knots, and the other curves are for 4, 6, and 8 knots.

* Glide ratios. If desired, the ticks along the perimeter of the calculator, near the climb/wind grid, can be labeled. They mark the glide ratios 15 (at the counter-clockwise end), 20, 25, 30, 40 and 50 (at the clockwise end). These glide ratio numbers are not needed in order to use the calculator, but one may be curious as to what the ratio is.

* Speeds to fly. If one does not have a speed ring or other means to determine the STF, one can label the ticks that appear along the zero-wind curve in the climb/wind grid. They are for multiples of 10 knots. The one closest to the zero-climb curve (most clockwise) is usually for 50 knots, but for some gliders it may be for 40 or 60 knots, so check the speed-ring numbers at the end of the output file from the software (or any other source for the speeds). If one wants to use MPH instead of knots, one needs to move the ticks. The output from the software (comments in the output file, not drawn) includes speed ring values in MPH as well as knots. I label the STF numbers in red, to distinguish them from the other labels in the grid. The STF depends on the climb rate and not the wind, thus, the STF label next to which the extended course line crosses the zero-wind curve does NOT show the STF, unless there is no wind. Instead, one must look for the STF number that is next to the assumed climb rate, on the zero-wind curve, independently from the rotation of the courseline strip. If one were to draw constant-STF curves on the grid, they would parallel the constant-climb-rate curves. I drew one such curve, as a dotted line, in red, to remind myself of that.

* Sink rates at the speeds to fly, if desired. The sink rates in dead air for each of the speeds-to-fly labeled. If you see a different sink rate during the glide, you are not in dead air and should adjust the airspeed accordingly.

* On the courseline strip: label the distance ticks. These are only used to satisfy one's curiosity as to how far the remaining glide is. If you use a GPS, it will tell you the distance, and then you can find yourself on the map by locating the appropriate tickmarks on the courseline. Then the calculator will tell you how much altitude you should have. If you prefer to use units other than nautical miles, make your own strip and ticks.

* On the margins of the map: azimuths. All around the perimeter of the map, actually on a white disc of paper backing the map, I marked azimuth ticks every 10 degrees, and added labels every 30 degrees: "36", "03", "06" and so on. These labels let one set the courseline for the glide using a landmark, and then read the compass bearing from the azimuth labels. On a hazy day one will need to maintain a compass heading in order to keep the glide on course. Remember to correct the heading for crosswind. Anyway these ticks are optional.

Besides the example picture of a completed calculator, see the diagram that marks the meaning of each curve section.

Once all the labels are penned in, photocopy the page onto a transparency sheet. Be careful to use a copy machine that leaves the size exactly as-is. Measure the original and the copy. Make sure the circle on the transparency is truly round and has a radius of 9 cm.

Cut the transparency into a circle (near the outermost circle drawn by the software) and a strip.

Make a hole in the middle of both.

Cut a circular base out of cardboard or other stiff material. I made mine 19 cm (7.5 inches) in diameter.

Cut a disk out of a sectional map, centered on your home field. Make it slightly smaller than the base disk (18 cm).

Put it all together: I tape the map onto the base, and use a small bolt, washers, and wingnut to hold the pieces together. With the wingnut, I can loosen things to rotate as necessary, then tighten them gently to avoid slippage during the glide.

I also taped a piece of white paper, appropriately shaped, under the transparent disk, lining the climb/wind grid area. That makes it easier to read the grid markings, as the background map does not show through. See the example picture.

If you fly out of several airports, you can build more than one calculator, or, convert the same calculator by inserting an appropriate map into it as necessary. Several maps can be stacked in one calculator, and the appropriate one swapped to the top, possibly in-flight.

To contact the author, email mbraner@ or write:

Moshe Braner, 47 McGee Rd., Essex, VT 05452

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