Leonardo Casetta

The Lagrange equations for systems with mass varying explicitly with position: some applications to offshore engineering

The usual Lagrange equations of motion cannot be directly applied to systems with mass varying explicitly with position. In this particular context, a naive application, without any special consideration on non-conservative generalized forces, leads to equations of motions which lack (or exceed) terms of the form 1/2(¶m/¶q.2), where q is a generalized coordinate. This paper intends to discuss the issue a little further, by treating some applications in offshore engineering under the analytic mechanics point of view.

Equation of Motion Governing the Dynamics of Vertically Collapsing Buildings

AbstractThe present paper aims at contributing to a discussion, opened by several authors, on the proper equation of motion that governs the vertical collapse of buildings. The most striking and tragic example is that of the World Trade Center Twin Towers, in New York City, about 10 years ago. This is a very complex problem and, besides dynamics, the analysis involves several areas of knowledge in mechanics, such as structural engineering, materials sciences, and thermodynamics, among others. Therefore, the goal of this work is far from claiming to deal with the problem in its completeness, leaving aside discussions about the modeling of the resistive load to collapse, for example. However, the following analysis, restricted to the study of motion, shows that the problem in question holds great similarity to the classic falling-chain problem, very much addressed in a number of different versions as the pioneering one, by von Buquoy or the one by Cayley. Following previous works, a simple single-degree-of-...

Systems with mass explicitly dependent on position

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.

A theorem on energy integrals for linear second-order ordinary differential equations with variable coefficients

Abstract We aim at demonstrating a novel theorem on the derivation of energy integrals for linear second-order ordinary differential equations with variable coefficients. Namely, in this context, we will present a possible and consistent method to overcome the traditional difficulty of deriving energy integrals for Lagrangian functions that explicitly exhibit the independent variable. Our theorem is such that it appropriately governs the arbitrariness of the variable coefficients in order to have energy integrals ensured. In view of the theoretical framework in which the theorem will be embedded, we will also demonstrate that it can be applied as a mathematical method to solve linear second-order ordinary differential equations with variable coefficients. These results are expected to have a generalized fundamental character.