In the hover a helicopter is statically...
Refer to figure.
Directional stability is the stability of the helicopter in the yawing plane and occurs about the normal axis.
Consider a helicopter with a clockwise rotating rotor. A hovering helicopter is subjected to a wind from the left, the disk will flap and the helicopter will start to move to the right. As it does so, it will try to weathercock into the relative airflow now coming from the right. In other words, it will try to yaw to the right.
But another effect of the gust from the left is to increase the induced flow into the tail rotor. This reduces the angle of attack and so reduces the tail rotor thrust. The reduction in the tail rotor thrust means that the torque reaction to the main rotor is no longer fully opposed, so the aircraft will yaw to the left. These two opposing forces give the helicopter static stability.
But, when the gust from the left ceases, the momentum of the fuselage continues to produce a tendency to yaw left. The relative airflow now opposes the induced flow through the tail rotor causing the tail rotor angle of attack to increase.
This increases the tail rotor thrust, so that the helicopter yaws back to the right. The aircraft weathercocks, the relative airflow into the tail rotor again switches sides and so the cycle is repeated. The yawing oscillation will continue with increasing amplitude. The helicopter is thus, dynamically unstable.
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