A tractor Gyro with vertical takeoff and landing.

[down under]

[This should not be a problem. The database form 'FORM: Exhaust - Tuned', shown on this web page was used and the frequency was changed from the engine at 6,000 cycles per minute to a rotor at 500 cycles per minute. The tuned length worked out to just over 50 feet.]

I think it may end up very complex to work efficiently. The gate would need to be opening just as a pressure wave is leaving the gate. If it takes 50ft for the pressure wave to travel up then back down a tube between gate opening and gate closing, a number that divides evenly into 50 would need to be used as the rotor length. If the rotor were 10ft long the gate opening would catch the 5th pressure wave.
If the rotor rpm needs to be in sync with the pressure waves or harmonics, the rotor would need to be kept within a very close rpm tolerance possibly changing slightly with air density and humidity. Athough there would be various combinations the rotor length and rpm would be govened by the pressure waves.

There is the possibility that the tip jets may dampen the pressure waves and that if the gate opens at about the 4th or 5th harmonic the pressure waves may not be strong enough to cause a problem. It would need looking into by somebody with more knowlage on the subject.

The other thing to consider is that the rotor with thrust would be levering the other rotor around the hub, making the hub a fulcrum. This may mean more strength and therefore more weight is required in the hub and rotor centre section.

A few of my thoughts.
My interest is for a low speed gyro with vertical takeoff and landing and partialy powered rotor for mustering work. Using a compressor to power tipjets would make a very reliable and versatile machine.

The Djin helicopter was incredibly simple with very few moving parts to fail or replace. I believe it was the only production tip jet helicopter. Its only downfall I think was excessive fuel consumption, due perhaps to an early gas turbine engine and possibly thermal losses through the rotors.

For the application you are looking at, a turbo prop engine with a centrifigal compressor that could engage directly to the turbine shaft without requiring a gearbox would be a very simple system. The compressor could be engaged for vertical takeoff, hover and low speed flight at a cost of heavier fuel consumption. In forward flight the compressor could be disengaged or possibly slowed, with a large percentage or all of the power going to the prop. This would make a more versatile machine, possibly lighter and simpler than a jump takeoff gyro that uses inertia from a heavy rotor.