NOTE: No effort of any kind is necessary with this other than standard swinging motion.
A playground swing hanging from stiff bars instead of a chain. The seat has rollers so that seat can move up and down on those bars. No arm movement is required to move up and down those bars, the motion described
below occurs naturally.
The swing reacts like any other swing until it reaches beyond the horizontal level. At that point the swing slides TOWARDS the pivot point (because the angle is tilting towards that pivot point) reducing the resistance as the mass of the swinger moves inwards. As the swing goes back down, the swinger moves down the bars again, to the outer part of the swing path and adds momentum as the leverage changes.
At some point, the swinger at the top of the arc slides (upside down) almost to the pivot bar and as the swing reverses, they're hurled down to the bottom of the support bars very quickly.
Might need to have springs to absorb the impact when the seat hits the bottom of the bars as it would be moving pretty quickly.
Once a complete revolution was achieved, the swinger would move in an egg shaped course, narrower at the top and wider at the bottom and slower at the top and much faster at the bottom.
The basic idea is that when you're at the top of the arc, you move in and provide less resistance due to the change in leverage, then tap the additional energy of moving back to the outer end of the cycle.
It's a bit hard to describe so I'll use a clock face to describe what happens at each position of the swing:
6:00 position - Swinger is flying forward with full force, seat at the bottom of the slider bars.
9:00 position - Swinger is still at the end of the bars due to centrifugal force however at
10:00 position - The mechanical centripetal force is nullified and replaced by gravity because the swinger is slowing and thus falls back towards the pivot point away from the bar's extent that was holding them in the orbit.
12:00 position - The swinger falls completely into the pivot bar (stops would keep them from hitting it)
1:00 position - Swinger's weight due to leverage pushes the bars very quickly downwards to be met by the swinger decending the bars at:
4:00 position - where the cycle repeats.
Boy is that wordy. This really is something that a 5 minute to create animation could clarify.
ADDENDUM: A swing that uses springs (within a tube where the rider holds it) that moves up and down as well as swinging.