[The following is an excerpt from a letter sent to Allen Meece.]

[Updated 4 September 2002]

Here's a rough summary of what you'd need to know about tether behavior to make an accurate drawing.

In a stable configuration, the dominant forces on the tether are: wind drag on the balloon, balloon lift, and the tether's own weight & tension. Balloon drag is mainly horizontal, and the lift & weight are vertical. Since the tether tension force is always parallel to the tether, a tether under the influence of only these simple forces will tend to behave as a catenary, moored at the balloon and at the ground station, when in equilibrium. There is another external force -- wind drag exerted on the tether. However, the tether's cross section is so slim that for many purposes (but not all) tether drag is negligible for the static case. The cross sectional area of the entire tether extended up to 20km altitude is not equal to that of the lifting balloons partially inflated on the ground. A graph of the expected tether shape when in equilibrium -- estimated for a 2-D approximation using a single day's soundings of upper level winds to approximate the static case -- shows that variation in the mid-level winds has little effect on the stable tether shape.

I'm including a graph of the tether shape and the spreadsheet I used to create it. The relevant data is an x-y plot of the last two columns (U and V), which roughly describe the integral of the displacement. In short, an x-y plot of these points gives the shape of the tether.

The effect of increasing excess balloon lift is to shorten the horizontal radius of the tether. For example, the following chart represents the tether profile after an increase of baseline tension by decreasing payload by one ton (about the same response as lowering the elevator), generated using this spreadsheet, which is a more versatile version of the one mentioned above.

This chart represents a top view of the tether under the same conditions.

Note the gentle curve of the tether near its base. This curve covers several kilometers, and is caused by relatively strong lower level winds blowing at a right angle to the upper level winds. It is most prominent near the ground station.

Changes in the displacement can be examined by changing selected entries in the SKNT or windspeed column (E), or the parameters of the aerostat (B1:B8).

The tether does vary slightly from an ideal catenary due to midlevel wind drag on the tether, but the variation is almost beneath the resolution of this computer generated graph. The effect is quite small, and even the sharpest variation is a smooth curve over nearly a kilometer.

Midlevel winds are capable of setting up oscillations in the tether, but due to the low density of the material of which it is made, the tether's motion is damped by what drag forces are exerted on it. This means that oscillations in the tether are heavily damped over the course of several kilometers. Here again, the wind forces exerted on the lifting balloons dominate wind forces on the tether. The main type of oscillation we can expect in the tether is pendular motion, and this occurs because the tether gets pulled around with the balloon as it bobs up and down.

Because there is some vertical tension at the ground station (about 10000N excess lift), the tether does not touch or extend horizontal to the ground. (The ground station is not the mathematical origin of the tether's idealized catenary curve.) During the ascent, higher wind drag is found at intermediate altitudes due to thicker air and faster wind speeds. This raises the horizontal tension and causes the tether to tend toward a flatter curve with a lower angle of altitude. However, the tether is not fully extended at the lower altitudes of the ascent, which means that the extended portion does not weigh as much as it will at its final float altitude. The lift force exerted by the balloons is relatively constant throughout the ascent, and when unencumbered by the weight of an added length of tether, the balloons provide enough added lift to keep the tether off the ground even when passing through fairly strong winds. In short, the tether weighs so much less when it first starts unreeling that it has the same effect as lightening the platform ballast.

The lowest tether altitude angle at the ground station encountered during ascent on this sample day is 19 degrees, experienced near the end of the ascent when the tether was nearly fully extended. Even in the highest wind forces the beanstalk is capable of withstanding during ascent, this value is not expected to drop below 10 degrees as long as at least 1000kg of excess lift is available.

Regarding forces on the tether:

There is no indication, from this spreadsheet approximation or any other calculations, that there is any advantage in tapering the tether until well into the stratosphere. At every point in the the troposphere, wind force is more important than tether weight.

Over the course of a year, the normal operating tensions range between 65000N and 150000N, with surges up to 200000N to be expected during the ascent. In this example, the average operating tension at float altitude is about 100000N at the platform and 75000N at the ground station.

I hope some of that helps. I'll work at expanding the graph later to try to illustrate expected motions.

CME