Motion of a rigid body in an unsteady non-uniform heavy fluid

The Lagrangian formulation of the equations of motion of a body immersed in a steady but heavy perfect fluid is outline in Lamb [1] and a resulting set of equations for the unsteady fluid case, is given in Lewis et al [2] in a form suitable for the flight dynamics of underwater vehicles. These equations have also been used to model the motion of airships in a steady uniform atmosphere, Cook et al [4]. Recently however the author has had some difficulty in applying these equations to the motion of other vehicles. In principle they should be applicable to not only underwater vehicles but also to airships, parafoils and aircraft. Two major problems of the equations in [2] is that they do not reduce to the small perturbation equations that are used for aircraft in gusts [3],[13], plus as will be seen later, the fluids inertial velocity causes difficulty. This paper identifies the source of the problems as being the conventional approach of lumping together the inertial and added masses. It provides an alternative formulation that keeps them separate and so avoids the difficulties.