Drawer glide.
You could mount a drawer glide. The kind with the ball bearings. 18" throw about $12 at Home Depot. Mount your motor half way up the trash can with as large a crank as you can fit. An arm attaches to the crank and to the top or extending portion of the guide. You will need some micro switches to limit or stop the prop. A notch cut in the crank could be your stop circuit. The trigger would close a solenoid to provide power to the motor to bump it past the notch. The switch would be wired to provide power to the motor on the NC side. The prop would cycle. Rise up and then back down until the micro dropped into the notch. This would stop the prop until another trigger bumped the motor past the notch. You could also have a raised portion of the crank like a cam, and wire the switch to provide power in the NO position. Either way it will work. Before electronics, that is how washing machine cycle's were controlled. You could stop or delay the cycle at the top by using a timing circuit to control a relay that would open the power circuit. Another microswitch could trigger that. I think a 555 chip would work. You need the help of the circuit guru's for that. I am purely a mechanical guy.
