On the stability of a dirigible body

Abstract

When the Magnus effects are completely absent the oscillatory motion of a dirigible body is said to be plane-yawing. Such bodies are stabilised by attaching fins or control vanes at their rear ends. Initial choice of the fin-size is made, depending on a static stability condition: JM < O, where JM is the normalised overturning aerodynamic moment coefficient. A proper fin-size requirement should normally be found from an appropriate dynamic stability condition. Under the very severe aerodynamic restriction stipulated above, one would expect, if the static stability condition is liberally satisfied by attaching over calibre fins, perhaps dynamic stability requirements could be met. This, however, may considerably reduce the ballistic range of the body. One would, therefore, need some sort of an upper majorant for JM consistent with the dynamic stability of the body and this should be a function of the other associated aerodynamic forces. In the present note a suitable majorant function for JM has been worked out. For this purpose, the aerodynamic coefficients are assumed to be slowly ranging functions of the path length as is usually stipulated. The stability problem has been solved using certain known results in the oscillation theory of differential equations.