Keith,
I agree with Neville in that you would have to make a frame similar to a lathe so it could be rotated slowly by hand, either by crank or a wheel attached the end. My first thought was to attach the frame so it could be hung as a pendulum over a tablesaw to rough it to shape by spinning and swinging, but that would take some serious stabilizing. If you started with a frame that it could rotate on, and made a curved pair of rails for a router to follow, then when it's roughed down you might be able to adapt the curved rails to hold a belt sander or disk sander to further refine the finish. Nice challenge!
Gary thanks for the comment, if the frame was stiff enough then the drum wold rotate truly, the amount of work you put into any Jigs construction depend on how much you will use them or how good a job you want to do so if the jig is made very well then both things will happen, the drum will turn smoothly and the cuts will be very clean, as long as you are patient and do light cuts, that's the U frame, the arch is harder. you need a rise and fall system at both ends, if the jig was not going to be used that much then this could be held in place with a clamp to the post, just loosen it a little and then tap it down then tighten the clamp, it you wanted to use it a lot then more work could be done so that the ends of the arch went down a threaded rod, the real trick here is the router sled and arch, that has to be made very well as the router would vibrate and the more that it did move under load then the less perfect the cut would be, I wold attach the router box to the sled with a tilting plate so that the angle of the cut could be varied, doing that would be very easy, I would make the arch as a pair of rails that were held apart by the same thickness as the end posts and that way there would be two flat plates that straddle the post at each end and the cutter has to just hang down far enough below the sled to do the cuts, once that arch was done then there are a few ways to make the sled in any regard then after the turns and cuts were done I would use a compass plane to clean it up, a razor sharp blade and the curved base set correctly should get a very good finish, this is just a thought exercise for me, I can see that this idea will work, with all the work that did get put into the drum then more work will be needed to get the outside cut, the good thing is that only the final cuts really matter as all the rest are just to get rid of the waste material so if the arch and the sled are not correct then there is plenty of time to change them and get that so it does cut smoothly, this can work, any free hand method will result in an imperfect drum, it depends on how perfect the drum needs to be, I am sure that I could make a Router Lathe that would turn a near perfect drum, I did think about this one so much that the plan was perfect before I started making it I can't do it as I have three Router Lathe of my own to build, one is nearly done, two more to go but a rotating drum router lathe could be made without that much effort. If the Lathe is stiff and the rotation slow then you would get perfectly circular cuts, if you make the router box so that the cutter can tilt from side to side then you could change the angle of the cut as you need to and improve the final cut, either way the more work that got put into the jig then the less clean up would be needed. N