Included as a source in recent answers was this paper which while focusing on a tethered ring design best described as ambitious included maths for a mostly mechanical space gun system starting on page 11 of 20. The concept featured a pair of counter rotating threaded drive shafts with thread pitch changing to achieve constant acceleration down the launch tube with the coupling of vehicle to guide rails and drives being magnetic.
(image from page 12 of paper)
This seems initially plausible, and attractive in that the energy storage issue of most space gun systems is bypassed by directly 'storing' the launch energy as mechanical inertia that is then directly used 'at source' to propel the vehicle down the launcher.
The magnetic field strength achieve low friction/high force coupling is a problem, as discussed on page 13 with needed magnet being questionable on current technology. A more substantial issue is at page 14 examining the forces during launch where along with the useful force propelling the vehicle down the launcher there is also a component at right angles across the launch tube of several hundred tonnes. This is a problem for the drive shafts, since while it might be possible with regularly spaced bearings to engineer the shafts to handle hundred tonne static loads these shafts will be spinning at high speed where deflection will tend to amplify and result in self destruction. The proposed solution is flywheels within the shafts that are used to generate a counter force during the launch.
This takes the design from one where the 'long' part of the launcher is a pair of cheap(ish) passive shafts with some bearings and drive motors to one with very not passive gyroscopes in massive numbers that need to operate with perfect reliability in a very not cheap manner.
Is there any alternative to the geometry in picture above (payload above two drive screws) where adding either more screws or more screw contact points on the payload sled would allow the forces to be balanced within the payload sled system and leaving just the 'push down the tube' forces on the drive screws?
It is to be noted that there are several other obstacles to viability of this system, those shared with all gun launch systems and some unique to it (a magnetic coupling to changing surface geometry). This question is just if the only solution to the lateral forces is flywheels every couple of meters of a 100+ kilometer launcher.