Systems with mass explicitly dependent on position

Abstract

This chapter addresses an interesting type of variable mass systems. Those in which mass may be explicitly written as function of position. Two perspectives can be followed: systems with a material type of source, attached to particles continuously gaining or loosing mass and systems for which the variation of mass is of a “control volume type”, mass trespassing a control surface. This is the case if, for some theoretical or practical reason, partitions into sub-systems are considered. Whenever mass depends explicitly on position, the Lagrange equation has to be carefully re-interpreted. As a matter of fact, an extra non-conservative generalized force term, linearly proportional to the mass gradient and quadratic on velocities, emerges from first variational principles. Ignoring this term has been the cause of misleading derivations of equations of motions and even of many misinterpretations, not rarely provoking claims of false paradoxes. The present chapter derives such an extended form of the Lagrange equation, through Lagrangean and Hamiltonian approaches. Illustrative and practical examples are taken from two engineering fields, offshore engineering and civil engineering. In the first category are included: (i) the reel laying operation of marine cables; (ii) the dynamics of a water column inside a free surface piercing open pipe (and the analogous moon pool problem) and (iii) the hydrodynamic impact of a solid body against a free surface of water. In the second category, the governing equation of motion of vertically collapsing towers is properly derived.