Solution Branching and Dive Plane Reversal of Submarines at Low Speeds

Abstract

The problem of multiple steady-state solutions in the dive plane of submarines under depth control at low speeds is analyzed. This phenomenon occurs regardless of the particular means used for depth control, manual or automatic, and linear or nonlinear. It is shown that the primary bifurcation parameter is a Froude-like number based on the vehicle speed and metacentric height. Generic solution branching is shown to occur below a critical Froude number. Singularity theory techniques are employed to quantify the effects that various vehicle geometric properties and hydrodynamic characteristics have on steady-state motion. It is demonstrated that a comprehensive bifurcation study provides a systematic and effective way of predicting the phenomenon of dive plane reversal at low speeds. A complete characterization of the parameters in the problem, both in deep water and at periscope depth, is achieved through the organizing center of the pitchfork singularity.