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.


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