The Effect of a Buoyancy Gradient on the Longitudinal Flight Dynamics of a Superpressure Airship

Abstract

Some stratospheric airships have been designed to operate routinely above their pressure height. These superpressure airships are intended to fly at altitudes of 60,000 ft MSL or even higher. One outcome of operating above pressure height is that an airship's weight remains approximately constant while its buoyancy varies with altitude. This effect has not been included in the linearized longitudinal equations of motion that have been applied to airships. In this paper, linearized equations governing the two-degree-offreedom longitudinal motion of a superpressure airship are derived. These equations contain additional terms that involve the rate of change of airship buoyancy with altitude. The characteristic equation of the airship’s motion is also provided. Due to the additional buoyancy terms, the degree of the characteristic equation increased from 3 to 4. This means that the airship’s motion changed from 1 nonoscillatory mode and 1 oscillatory mode to a more complicated motion. As an example, the roots of the characteristic equation are determined as a function of airspeed for the High Altitude Superpressure Powered Aerostat (HASPA) which was designed, constructed, and tested during the 1970s. Predicted time histories of the airship's vertical displacement, angle of attack, pitch angle, and pitch rate are also presented. At low airspeeds, the airship’s motion consisted of 2 oscillatory modes. At higher airspeeds, the motion changed to 2 nonoscillatory modes and 1 oscillatory mode. At all airspeeds, one of the oscillatory modes was nearly identical to the pendulum mode that occurs when the buoyancy gradient is not included. At 63% of the airship’s maximum airspeed, one of the nonoscillatory modes became unstable. A formula is derived that can be used to estimate the airspeed at which a superpressure airship will become longitudinally unstable. Also, a controls-fixed longitudinal stability criterion is developed for superpressure airships.