Planar oscillations of a boat in a tank

Abstract

Abstract Oscillations of solid bodies like boats interacting with liquids like water are commonly studied in the marine science community but less so in the broader mechanical sciences. This paper presents a clear and simple exposition of a basic problem in this field, for a general nonspecialist audience. A two dimensional analysis is developed from first principles of small free oscillations of a boat in inviscid water in a finite tank. The water motions obey the Laplace equation. The boat motions are not imposed, but rather found as a part of the overall solution using rigid body dynamics under time-varying pressures on the hull. The importance is clarified of the difference between nominal and displaced configurations of the boat. The equations for water, boat, and boundary conditions are assembled into an eigenvalue problem that determines the oscillation frequencies and mode shapes. The equations are discretized using a boundary element formulation, and the corresponding discrete version of the eigenvalue problem is constructed and solved. The mode shapes thus found include (i) a well defined roll-dominated oscillation, (ii) easily interpretible nearby equilibrium solutions (“rigid body” modes), and (iii) tank-scale sloshing-dominated modes. The roll frequency is found to vary negligibly with tank size, and goes to zero as the boat center of mass is moved to the metacenter. The net added inertia implied by the roll frequency is discussed. Finally, experiments conducted with two boat models, one round bottomed and one square, match roll frequencies predicted by the above calculations.