# Dynamic Calculations on Tethers

Since I’ve written a nifty chain dynamic model, it was time to look at some dynamic problems associated with rotovators. Someone coined the term “rotovator” to describe a rotating tether designed to dock with an upcoming spacecraft, and then release the spacecraft after a half-turn (or less) with a higher velocity, illustrated below.

A classical solution attributed to Hans Moravec gives a cross-sectional area distribution that results in a constant material stress. For a very idealized situation, this is a minimum mass design. The dynamic question addressed in this study is what happens when the spacecraft (the tether payload) releases. At release, there is a sudden drop in stress which propagates as a stress wave. The stress wave must ultimately be reflected at the far tip, which results in a higher stress than the equilibrium value. There are fairly simple ways of looking at stress waves (at least for constant area rods), but I wanted to do the numerical simulation for a couple of reasons. One, it was an opportunity to run the model with a variable cross-section tether with variable tensile force (stress is constant, force is not). Second, I wanted to see if anything unexpected happened with the rotating problem. For example, did the longitudinal wave find a way to couple with transverse motion? (Answer: no). The analysis notebook is in the main website. I left in all the Mathematica input so anyone interested could see how I set up the model. The results are what you might expect. The reflection at a free end basically doubles the tensile stress. This means that the equilibrium stress needs to be at least a factor of two below the material strength, along with additional factors-of-safety. The figure below shows an animation of the stress for one of the cases examined.The time-span of the plot covers one-half of a revolution of the rotovator. The animations require the Wolfram CDF Player plugin to view.

Design Case:  Material Characteristic Velocity (Sqrt[Strength/Density]) = 1.3 km/sec, Rotor Mass/Tip Mass = 406, Tip Velocity = 4 km/sec, Operating Stress = 3 GPa, Length CM to tip = 50 km, Sqrt[modulus/density] = 15 km/sec

Another dynamic problem considered in the study was the situation of a spacecraft docking with the tether with a slight mismatch in speed. I assumed that 20 m/sec to be a reasonable error in speed. This sets up a transverse motion in the tether. The point of the model was to see of this motion grew in some unusual or unbounded manner. In order to see the additional motion, I created a plot of the deviation from the rigid-body motion. The following animation is a plot of the displacement perpendicular to a rigid-body rotation. The maximum time in the simulation is equivalent to two full rotations. The plot shows that the initial 20 m/sec docking error quickly grows into a 2 km swing in the tether motion. Some active mechanism would be required to eventually damp this motion. Note that for this case, we have a assumed a lower tip velocity, which allows for a smaller mass ratio.

Design Case: Material Characteristic Velocity (Sqrt[Strength/Density]) = 1.3 km/sec, Rotor Mass/Tip Mass = 5.4, Tip Velocity = 2 km/sec, Operating Stress = 3 GPa, Length CM to tip = 50 km, Sqrt[modulus/density] = 15 km/sec

The simulations in the study did not include any damping. Besides material damping, my intuition is that a simple spring/damper combination at the far-end mass connection could go a long way to mitigating the stress doubling effect at reflection. The simulation also did not include a gravity gradient. For the curious; I was running these simulations with 200 nodes. The docking-error problem ran to a simulation time of 314 seconds (two full revolutions). The simulation required about 10 minutes of computation time on my elderly PC. So even though Mathematica is basically an interpretive language, it seems reasonably efficient for this fairly massive computation.

I’m not sure where I’m going next. It’s summer in Maine, and maybe it’s time to step away from the computer for a while.

# Update to whip problem

I caught a bad input value in my example analyses of a whip from the last post. I used 0.5 GPa as a working strength for a carbon fiber tether. That might be equivalent to the operating strength for a multidirectional composite laminate using current technology, including a lot of reduction factors normally considered. But for the theoretical strength of a pure tension tether under idealized conditions, it’s way too conservative. For some quick numbers, type “tensile strength carbon fiber” into Wolfram Alpha. It generates the following table for fiber products ranked by tensile strength.

Having spent a lot of time trying to gather together composite data into a handbook (MIL-HDBK-17), I’m pretty impressed that Alpha can now pull together this information.

I’ll assume these are mean values for the fiber strengths. There is typically a wide statistical variation in carbon fiber strength which makes the allowable strength much lower. And there should be a factor-of-safety. For the study, I’ll be ambitious and use a working stress of 3000 MPa.

The whip study was updated using the new strength. Obviously, the new value makes the release velocity of the projectile significantly higher, and thus improves the viability of the idea.

I’ve only been away from composites for three years and already it’s possible for me to write down a bogus material property. I’ll blame the metric system. American structural engineers (at least of my age) think in English units. There’s always the danger of not spotting a bad number when converting the units.

# Whips and Chains

As a kid, I played around with launching paper clips or folded bits of paper by placing the projectile in the middle of a piece of string, then quickly pulling on the ends of the string. With some practice, you can get some respectable velocity. Inevitably, I started to fantasize about using the same principal at a large scale to launch spacecraft. The figure below shows one possible configuration. Imagine two subsonic transports each trailing a tether that is attached to a projectile vehicle. At the beginning of the sequence, the transports begin to turn in opposite directions. A wave propagates down the tether, resulting in a whipping action that accelerates the projectile to a speed that is much larger than the original transport speed. As the tether pulls taught, the projectile reaches a maximum velocity and releases. At the same time, the two tethers disconnect from each other so that the transports do not yank themselves out of the sky. Pretty wild, and probably impractical, but the whole point of this blog and website is to numerically examine the possibilities. If I can prove it doesn’t work, then I don’t have to waste any more brain time imagining whipping payloads into space.

I now have some tools for simulating the process. The chain model discussed in the last posting works beautifully for setting up the problem (and thus my overly clever post title).  I didn’t spend a lot of time trying to optimize all the parameters. There is a study on the main website that goes into the details. Briefly, I assumed the initial speed of the transports was 300 m/sec (high subsonic speed), and the transports make a 2g turn (4.6 km turning radius). The tethers are graphite fiber, 14 km long, and 3.5 cm in diameter. The mass of a tether is 25,000 kg, which should be manageable for a 747 class transport. For a 1000 kg projectile, the simulation gave a 2.2 km/sec launch velocity. Pretty respectable. Note that I have not included aerodynamic drag. Drag will go up dramatically at supersonic speeds and that could well kill the whole idea. An animation of the tether trajectory is shown at the bottom of the post (visible if you have the Wolfram CDF add-on installed on your browser). In the animation, the red dot represents the payload. A velocity function is applied to the opposite end of the chain to simulate the path of the transports. In the simulation, the projectile is not actually released. We run the model a bit past the point of maximum velocity.

The trouble with the 2.2 km/sec launch is that just before the peak velocity, the stress in the tether is well above the possible strength of current materials. One trick is to release everything a bit before the peak. Releasing 2 seconds before the peak reduces the launch velocity to 1.1 km/sec, while keeping the tensile stress down to the range of possible. Thus, the whipping action can be used to magnify the transport speed by about a factor of three. Just before release, the acceleration of the projectile is around 15g’s; too high for people and delicate payloads. For the same tether and stress level, you can accelerate a 10,000 kg projectile to 870 m/sec.

In the end I failed to disprove the idea would work and thus exorcise the fantasy. At this level of simulation, it sort-of works, but there are lots of problems. As I mentioned, aerodynamic drag would be significant. Also, the simulation is two-dimensional. Even with aerodynamic lift, the tethers will drop below the altitude of the transports. And in the end, it is not clear whether flying multiple transports and dealing with long cables could be justified to gain a 800 m/sec kick-start on the way to orbit.

I did find one reference to a vaguely similar concept posted by Dmytry Lavrovon the web. He mentions a student competition paper, but I was not able to find the original paper. His study involved propagating waves on a tether as an accelerator, but in a space vacuum. Another distant relative is called the Kinetics Interchange TEther (KITE) Launcher which has a patent application by Dana Johansen. As the Kite name implies, it uses a combination of aerodynamic forces and a tether connected to a large transport to build up a projectile velocity larger than the transport initial velocity.

# A New Chain Model and Earth-Circling Rings

Once again, instead of a logical progression on any of the topics that I’ve already started, I got interested in the dynamic analysis of flexible structures using a chain model. A “chain model”, is a discretization into a series of point masses connected by elastic bars. Using numerical integration, the trajectory of each of the masses can be tracked in time. The model is useful for a more detailed look at various tether problems, and for the whole class of dynamic structures. I wrote a Mathematica software package to perform this type of analysis in a fairly general way.  The package is called chainDynamicModel.m, and it is available for download on the website Resources page.

I’ve been meaning to address the whole world of dynamic structures; sometimes called kinetic structures. These concepts use changes in the direction of a momentum stream to generate forces that can be used to support a structure. For example. consider a stream of high-speed projectiles that move vertically through an evacuated tube, using some form of levitation to avoid touching the walls. At the top, we somehow reverse the direction of the stream and pass the projectiles downward through another tube. The reversal in direction must be balanced by a force that in turn keeps the tubes in tension. A neat way to make a tall tower. Besides towers, it is easy to envision giant loops that are partially on the ground and partially high in the atmosphere. The largest loop would encircle the entire earth. I give a bibliography of some papers and other sources on a new website page devoted to dynamic structures.

An earth-circling ring has been proposed by a several authors, starting with Tesla. Tesla (and Larry Niven in Ringworld) conceived of a rigid ring. More realistically, consider a cable spinning at, or just above, the orbital speed at the cable altitude. At the orbital speed, the cable would be tension-free. So far, so good. But now consider a small perturbation in velocity, applied to a small portion of the cable. The numerical simulations show that without a restoring force, the cable trajectory grows increasing distorted. A 10 m/sec radial velocity applied to one node of the model results in the cable crashing in less than a revolution. The new page includes a brief study of these stability issues using the new chain model. The animation below (visible if you have the Wolfram CDF player installed) shows the evolution of the cable distortion over time. In this particular simulation, the cable is assumed to have the properties of graphite fiber. It is spinning at 1% above orbital velocity, which corresponds to a tensile force near the working limit for graphite. A 10 m/sec radial velocity perturbation is applied to the red node at the beginning of the simulation. Because of strobe effects, it may look like the black nodes are not moving, but the whole ring has a uniform tangential velocity.

I’m having fun with the chain model at the model. Expect to see several studies addressing the dynamic response of tethers and other long, flexible structures.

# Atmospheric Harvester Example Design

In the previous post, I introduced the concept of an atmospheric harvester that skims the exosphere to gather propulsion fluid mass for use by space tugs. I felt that an electrodynamic tether was a promising way to propel the device and make up for the drag losses. I did a fair amount of work over the last couple of weeks to prove to myself that the atmospheric harvester was practical, at least from the viewpoint of propulsion and power. I had a model for the tether equilibrium, but it didn’t include tether aerodynamic drag because I didn’t have a handy atmosphere density model that went high enough. So first I had to create a better atmosphere density function in Mathematica. This was a bit of a cheat; I found a well established Fortran code, J77sri, which embodies a 1977 model by Jacchia. The code is pretty old-fashioned Fortran, so converting directly to Mathematica would have taken too much time. Instead, I just compiled and ran the Fortran, put the tabulated results into Mathematica, and interpolated the results. Good enough. Then it was a small matter to modify the tether model to include the drag terms.

The next step was to actually size some of the elements of a harvester system. The idea was to arbitrarily pick a mass collection rate and then calculate the total system mass. For the study, I used a collection goal of 1000 kg/day of oxygen plus nitrogen (I like to think big). Unfortunately, I have no information yet on the actual collection mechanism and it’s mass. So to proceed, I simply assigned 5000 kg to the collection device. Aluminum seems like a good material for the tether. We are looking for a low density material with good electrical conductivity. It turns out that strength is secondary to conductivity. The maximum working temperature of aluminum then sets the collection altitude. I found that 125km worked with 100% aluminum. I also tried adding 30% steel strands to the cable to increase the strength at elevated temperature. This allows one to operate down to 115km. In the mass trades, the higher altitude system came out much lighter. I also looked at a range of tether lengths. Seventy five kilometers seemed to work out well. That particular combination gave a system mass of about 12,000 kg. That seems like a pretty good deal for collecting 1000 kg/day of fluids, assuming I’m anywhere near the ballpark on the collector mechanism mass. Other information derived during the design cycle:

Cable Length    75 Kilo Meter
Cable Dia    0.4 Centi Meter
Cable Drag    20.814 Newton
Collector Drag    89.8506 Newton
Power to Overcome Drag    859100. Watt
Drive Voltage Drop    18897.9 Volt
Resistance Voltage Drop    7054.31 Volt
Current    45.46 Amp
Total Power    2.30069*10^6 Watt
Cable Mass    2544.69 Gram Kilo
Powerplant Mass    3403.39 Gram Kilo
Bottom Node Mass    5000 Gram Kilo
Top Node Mass    4506.28 Gram Kilo
Total System Mass    12051. Gram Kilo
Strength F.O.S    2.46726
Top Mass Check    True
Cable Max. Temp    145.454

The summary document gives the basic assumptions and a series of these tables for different starting conditions. The actual steps to computing all the parameters and the iteration process are contained as a big function in the full Mathematica notebook (don’t bother downloading the notebook unless you’ve installed the free viewer from Wolfram, or have a copy of Mathematica).  I get a little lazy and I don’t feel like explaining in words all the steps of the algorithm. Hopefully the code is self explanatory. I’ve tried to spell things out pretty carefully.

These types of system studies are the fun part of this little hobby. I enjoy trying to learn the physics of all the different pieces, not just my specialty (structural engineering). However, that means I also run the risk to getting something wrong. I took some big leaps on the electrical side. So please, if anyone sees a problem let me know. That’s part of why I throw this stuff out there.

By the way, you’ll notice I freely interchange “tether” and “cable”. The literature has settled on “tether”. But the definition of a tether is something that anchors something movable to something fixed. So I tend to call it a “cable”. Personal quirk.

# Atmosphere Harvesting

Much of the mass that needs to be lifted into orbit consists of propulsion fluids, and the the majority of the fluid is the oxidizer. The total cost of operating in space could be reduced if there was a ready supply of liquid oxygen already in orbit. One application would be to refuel space tugs. The idea is that instead of blasting oxygen into space using rockets, gradually gather oxygen by skimming the extreme upper atmosphere. This idea goes back to work by Serge T. Demetriades, and it was published in the Journal of the British Interplanetary Society (Serge T. Demetriades, “A Novel System for Space Flight Using a Propulsive Fluid Accumulator”, J. British Interplanetary Society, 17 (1959) pp. 114-119.). The original idea was to gather gas, liquefy, and separate the oxygen from nitrogen. The nitrogen could then be used as a propulsive reaction mass using an MHD device. There are some nice cartoons of the device here. The whole thing would be powered by a nuclear reactor. Solar power would be impractical because of the added drag of the panels. A space tug would periodically dock with the device to transfer the liquid oxygen.The concept has come to known by the name PROFAC.

The concept has been reworked more recently by a team at Worcester Polytechnic Institute. From what I can gather, Paul Klinkman began reconsider PROFAC in about 2005. In particular, he found it beneficial to raise the collection altitude to 150-200 km where there is a higher concentration of oxygen, and he has proposed methods for collection. The most complete technical reference I’ve been able to find is an internal document by a WPI undergraduate. There’s also some information in this report, but the report focuses mostly on team dynamics. There is also AIAA Paper 2009-6759, but I have been motivated to purchase a copy. What originally intrigued me about the WPI work is the idea of using an electrodynamic tether for propulsion. This a particularly good application for tether propulsion because the system stays near the earth where the earth’s magnetic field is strong. And a long cable could raise solar panels far enough above the atmosphere to reduce the total drag. But mostly I liked the idea because I have a bunch of notes from 1983 that explore this combination of tethers and PROFAC (no claim of precedence; only publishing counts, and I didn’t publish anything, plus I don’t care). It appears that Klinkman has now moved away from tethers as the preferred propulsion for PROFAC, but I’d like continue to do some studies. As an aside, everyone assumes that nitrogen is a byproduct, but I think that any mass at orbital velocity is valuable in an integrated space transportation system. Below is my cartoon of what the system might look like.

Mass accumulator with electrodynamic tether propulsion

The most recent website update is a page on atmosphere harvesting, and the one study included to date is a solution for the shape a tether takes when used to drag a collector through the atmosphere. I was curious about the relation between the tidal forces that tend to keep the tether vertical and the drag at one end that will tend to bend the tether horizontally. The equations come from my 1983 notes, but at the time I didn’t have a handy way to solve the resulting coupled differential equations. The current Mathematica NDSolve function has no problem at all. There’s some heavy calculus involved, so I don’t particularly recommend the study as casual reading unless you need to do a similar calculation. I plan to use the results in a more complete system study, which should make for more interesting reading.

# Rotovator release angle study

I finally added some words to an existing study on rotovators (a rotating tether system) and posted the study on the main web site (see “Circular orbits from a rotating tether using variable release angles”). I wanted to consider some of the operational aspects of a rotovator. In particular, could one take advantage of different release points to achieve a range of circular orbit heights. The answer is definitely yes. A single rotating tether can be used to send payloads anywhere from near-earth orbit to geosynchronous by picking the release point. I focused on a configuration in which the lifter rocket that docks with the tether has to reach 50% of the orbital velocity at the tether docking altitude. This balance between what the rocket provides and what the tether adds allows for the tether to be built with existing materials without an excessive mass ratio. After release, the lifter must apply an additional rocket burn to circularize the orbit. An efficiency metric would be the total lifter delta V; the sum of the burn needed to catch up with the tether, plus the burn needed to circularize the orbit. For the 50% tether tip velocity selected, the total deltaV to orbit is less then 60% of the pure rocket deltaV for a wide range of orbit heights.

Posting stuff is partially slowed down by me still discovering the details of using Mathematica to create CDF and html pages.  For html, I discovered that some sloppy style formatting prevents images from being converted to gif files. For CDF, I’ve been discovering all sorts of ways to break an interactive manipulate window when there are complicated packages and functions involved. Basically, the manipulate cannot reference any external packages. That causes a lot of grief in the final stages of preparing the files.

So the big question is what to write up next. I have some interesting notes on suborbital refueling that I’ve always wanted to get organized. But there may be a big break while I get ready for a summer trip.