First some history on the gyroplane design. Looking at old photos of the gyroplane, called the autogiro (autogyro) we can see that they looked exactly like an aeroplane with a rotor blade on top much like a helicopter rotor. Later the main wings were removed from the design as they were unnecessary for flight. (Reference book available: P.W. Brooks, Cierva Autogiros: The Development of Rotary-Wing Flight.)

The purpose of a horizontal stabilizer on a "pusher powered" gyroplane is to make it easier for inexperienced gyroplane pilots to fly the gyroplane in windy conditions. Many experienced gyroplane pilots have flown for years making trim adjustments automatically. New pilots having to think and react, find learning to fly without a horizontal stabilizer very difficult. On a "pusher" gyro the horizontal stabilizer is fixed, meaning it has no pilot input controls. An aeroplane does have control surfaces on the horizontal tail stabilizers. A horizontal stabilizer only works for positive effect during fast forward motion of the gyroplane. It has some effect on landing when positioned directly center on center with a rearward facing propeller. However, normally, gyroplane landings are accomplished in a parachute style setting down of the airframe and flare. Many gyroplane pilots execute aeroplane style landings which are dangerous on unknown grass surfaces. Such landings can, and do result in dangerous nose up or down situations. see video

A study of flying without a horizontal stabilizer is a must for this discussion. Lets look at the "Feral* Gyrocopter" flown by a man in Oz. "Birdy" uses an open frame gyrocopter to "muster" cattle on a large ranch in Australia. His machine uses only a vertical tail and no horizontal stabilzer. He has been flying many years. A study of his flying technique: see videos part 1. Note the tail area of the gyrocopter. Watch his foot in the turns in video part 2.

It's interesting to note that the American inventor of the affordable gyroplane, Igor Bensen, did put a small stabilizer on his designs that was below the propeller, not behind. It was also of a specific design and size. Mr. Bensen is considered to be the inventor of the 'pusher-prop' style of gyroplane design, he trademarked as a 'gyrocopter'.*1 Later designs by other manufacturers of gyroplanes have been experimenting with bigger and bigger horizontal stabilizers placed behind the propeller in positions ranging from the top of the vertical tail fin to the bottom of the vertical tail fin. Some designs put the horizontal stabilizer in place of the vertical tail fin and add two vertical tail fins on either side of the horizontal stabilizer. All of these more recent designs are counter to the Bensen design. The noticeable difference would be that the Bensen design would force air over the top of the horizontal stabilizer where as the other designs use prop wash above and below the stabilizer 'wing'.
Is there a 'best postion' to mount a horizontal stabilizer when positioning it behind the propeller? Yes, and not where you think. Placed between the tip of the propeller blades and the center of the propeller thrustline would give the best airflow over most of the horizontal stabilizer.*3

So, what is the one big danger of horizontal stabilizers on gyroplanes? That would be in a situation of low forward air speed, close to the ground (landing) and receiving an unexpected gust of wind to the side of the gyroplane. Why would that be a problem? It wouldn't be a problem if the gyroplane airframe is allowed to vane and face the gust of wind. However the automatic response would be to try and correct the direction of the gyroplane, which in turn would cause the airframe to roll over on it's side. This is due to three things happening. First is a large horizontal stabilizer receiving pressure from below. Second is that the prop wash coming off the propeller is not like a cooling fan at home, it's coming off in a cork screw fashion. In sideways gusty conditions it would add to the upward horizontal stabilizer force into a roll over situation. Done quickly enough it would be too fast to counter. Did you ever try to carry a sheet of plywood across the yard to the house on a windy day? It's not difficult to see the effect of a sudden gust of wind below a large horizontal surface.
This happened to a pilot recently in Perth, Australia. On take off a gust of wind to side of the gyro caused it to loose control and flip on it's side. The man was injured and the gyro was a "write-off". December 18, 2011.
The final danger is twisting the rotor blade 'plane'. You know what happens to a spinning vertical bicycle wheel (gyro effect) when you suddenly try to tilt it in a large movement, it flips dramatically 90 degrees horixontal and 45 degrees to the direction of spin (try it).
The smaller the tail area, the bigger wind gusts it can handle, the larger the tail area the less stable it will be in gusty wind conditions. - "Food for thought".

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The number one fact to remember with gyroplanes is that you're really flying the rotor disc above your head. Be knowledgable about it, and it will behave properly for you.

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