Coupled Pitch and Heave Porpoising Instability in Hydrodynamic Planing

Abstract

Porpoising is a form of longitudinal instability characterized by an unstable coupling between heave and pitch degrees of freedom. In connection with flying boat stability, it was first analyzed by Perring and Glauert.2 In this paper, a simple theoretical description is first presented, based on quasistatic forces and moments, to show that stability is a function of center of gravity (c.g) location with respect to the center of pressure; and to a lesser extent, on the longitudinal radius of gyration. It is found that there are two stable zones; one where the c.g. is well forward of the center of pressure (analogous to a longitudinally stable aeroplane), and a second with the c.g. close to the trailing edge, which has no parallel in aircraft stability theory. Buoyancy terms are found to have a favorable effect for all c.g. positions, so that porpoising is essentially a high speed phenomenon, when most of the weight is supported by dynamic forces. Expressions are then developed for the transient hydrodynamic forces due to heave and pitch rates and accelerations, and the complete pitch and heave equations of motion are studied in some detail. It is concluded that there are several different ways of achieving stability, only one of which (moving the c.g. forward) is intuitively obvious. Other solutions include moving the c.g. right aft, the use of a very large radius of gyration in relation to the length of the planing surface, and the use of a high aspect ratio planing surface. All of these solutions can be identified in successful high-speed boats and hydroplanes. The calculations in the main body of the report assume that the fluid is inviscid. It can be shown (the calculations are omitted for brevity) that skin friction on the planing bottom results in additional terms in the stability equations, but that these are an order of magnitude less than the inviscid terms for a typical planing hull. But at very high speeds, coupled with a high c.g. position, the skin friction term in the pitching moment equation can become important, and may then have a dominant effect on stability. Generally, skin friction terms increase static stability and degrade dynamic stability. Despite the many porpoising tests made in the past with specific flying boat designs, there is very little experimental data available for simple planing surfaces alone; in fact, only the work of Day and Haag12 is known to the writer. A comparison between the theory and their experimental data gives reasonable agreement under the circumstances, but also points to the need for more experimental work.