Generalized methods for deriving the harmonic balance equations are illustrated using a ship roll model with angle dependent cubic damping, and cubic stiffness terms. The balance equations are then solved subject to a phase constraint which identifies the resonant solution only. The amplitude of the response at resonance, and the frequency at which resonance occurs, can therefore be obtained without needing to compute the response over a wide frequency range. This provides an efficient tool for investigating the dependence of the resonant response on the level of input excitation, with results which agree well with more time-consuming simulation methods.

A simple linearized theory for the problem of a two-dimensional thin supercavitating hydrofoil in nonuniform motion is developed. General expressions are presented for frequency-response function, lift and moment coefficients. It is found that, in general, the lift and moment vary appreciably with the depth of submersion. At a depth-chord ratio of unity (typical value), the nonsteady characteristics are not well approximated by either the deeply submerged or surfaced hydrofoil results. Detailed numerical results for the cases of a rigid foil performing heaving oscillations or entering the sinusoidal gust, based on digital-machine calculation, are given.