14
Ribbon Dynamics
The dynamics of the elevator, in general, are fairly straightforward but to ensure proper operation we need to examine the details of the elevator dynamics.
In 1975, Jerome Pearson published a technical article that included the a discussion on the natural frequency of the space elevator. Pearson found that the natural frequency depended on the taper ratio of the cable and in some cases would be near the critical 12 and 24 hour periods that could be problematic. Pearson also stated an ugly equation for calculating the shape of the cable as a function of the material strength, planetary mass, and planetary rotation speed.
We have taken Pearson’s original equation and attempted to simplify it into a more usable and intuitive form. However, this equation does not simplify well and like Pearson we have resorted to an analytical solution. In our case, however, we have ready access to spreadsheets that easily handle these types of calculations. We have composed a set of spreadsheets that produce ribbon profiles, tension levels, linear velocities, counterweight mass and total system mass. This spreadsheet is designed to handle different planetary bodies, rotation rates and applications.
Another spreadsheet we have composed is similar but for elevators with their anchors located off the equator. In this case the ribbon is found to sag toward the equatorial plane but remain entirely on the side as the anchor. This sag in the ribbon is due to the non-axial pull of gravity on the ribbon. The magnitude of the sag depends on the planetary rotation, planetary gravity and mass to tension ratio of the ribbon. In the case of a Martian cable, where anchoring the cable off the equator would allow it to avoid the moons this calculation is critically important. In the Martian case the cable extends parallel to the equatorial plane with only a 3 km sag back toward the equatorial plane when the cable is moved 900 km from the equator. This simple reanchoring of the cable would allow us to avoid any difficulties with the Martian moons.
What these and the dynamics work discussed below imply is that from a system stability and operations it is possible to move the anchor tens of degrees off of the equator if other factors (weather) permit.
In addition to the spreadsheets that we have assembled, David Lang has conducted computer simulations on the dynamics of the system. The code Lang is using was originally designed for modeling the ProSEDs experiment. Lang has modified it to examine the elevator scenario. The results from these simulations show that the elevator is dynamically stable for a large range of perturbations. The natural frequencies were found to be 7 hours for in-plane (orbital plane) oscillations and 24 hours for out-of-plan oscillations. The out-of-plane number is misleadinghowever. For any elevator or geosynchronous satellite a 24 hour period is found for the out-of plane because that simply implies an inclined orbit. For determining the stability, Lang gave the system various angular deviations, initial velocities and also quickly reeled in some length of the ribbon at the anchor. At some limit in each of these cases the elevator becomes unstable. What was found was that angles of tens of degrees were required to create a catastrophic failure. (The energy required to move the counterweight this far is equivalent to that required to lift 3000 loaded semi trailers kilometers into the air.) It was also found that reeling in 3000 km of ribbon in 6 hours will create a catastrophic failure. Each of these perturbations is well beyond any we expect to encounter. The events leading up to any of these are easily avoidable.
Lang also suggested that we consider a pulse type of movement for avoidance of orbital objects rather than a translational as we have been proposing. The difference is that in the pulse situation the anchor station is moved one kilometer and moved back to its starting position. This will send a wave up the ribbon to avoid an orbital object. The pulse will reflect off the counterweight and return to the anchor where an inverse pulse maneuver is conducted to eliminate the pulse. The result is a quiet system. In our proposal the anchor would be moved and remain there. This would send a long pulse that could oscillate up and down the ribbon for some time. Simultaneous pulses and a complex movement of the ribbon would result. This is a simplified explanation of a complex operation and response but the point is that there are operations that still need optimization.
Along with the computer simulations we have conducted some hardware tests of various ribbon designs and damage scenarios. The tests included several sets of ribbons with parallel and diagonal fibers composed of plastic fibers and epoxies or tape sandwiches. The ribbons ranged from two to four feet in length and were placed under high tension loads.
In the ribbon tests we found much of what was expected and predicted by our models. In situations where there is continuous rigid connection between adjacent axial fibers, aligned or diagonal, high stress points are created at the edges of the damaged area. These high stress points tend to be the starting point for zipper type tears and greatly reduce the optimal strength of the ribbon. On the contrary, ribbons with non-rigid interconnects between fibers had minimal stress points and yielded at high tensions and larger damage. A full description of the optimal ribbon design is found in our book. Similar tests are now being arranged at Rutgers to explore the degradation that might occur. We have also started to set up ribbons close to what will most likely be the final design.
:shy: