This article proposes a new formalism for the dynamic modelling of cables that can even be applied when they are submitted to cross flow of water or air. An important application is the case of umbilical cables used in remotely operated vehicles. The primary basis for the formulation is to assume that the continuous flexibility is represented by a discrete approach, consisting of rigid links connected by elastic joints, allowing movement in three dimensions. Each elastic joint allows three independent movements, called elevation, azimuth and torsion (twist). A significant contribution of the proposed formalism is the development of a compact equation that allows obtaining the Lagrangian of the system directly and automatically, regardless of the number of links chosen to form a chain of rigid bodies connected by flexible joints to represent the continuous flexibility of the cable. This formulation allows the construction of an algorithm for obtaining the equations of the dynamic model of flexible cables. IntroductionCurrently, maritime cables are present in a wide variety of applications, such as in systems for anchoring and towing. There are also many applications of umbilical cables; for example, they may be used for communication between remotely operated vehicles (ROVs) with a surface vessel.To determine dynamic models for cables is a complex problem because of nonlinearities, their structural flexibility in three dimensions and their complex interactions with the environment in which the cable is submerged. Therefore, this article proposes a new formalism for cables modelling, the foundation of which is in assuming that the cable consists of several rigid links connected by elastic joints (fictitious joints) as if they were composed of multiple spherical elastic pendulums, and each junction between two consecutive pendulums was conducted using an elastic articulation allowing three distinct movements: azimuth, elevation and twist (linear extensions of links were not considered). Initially, a convention for the placement of the reference systems with origins coinciding with the fictitious joints is proposed. Kinematic relations between the various reference systems are derived from homogeneous transformation matrices and thus the positions and velocities of the centre of mass of each rigid part are obtained and represented in the inertial frame fixed at the top of the cable (e.g. sea surface). These variables allow the obtaining

Controlar de forma ativa estruturas flexíveis permanece um problema ainda em aberto, tendo sido objeto de muitas pesquisas, principalmente a partir dos anos oitenta. Apesar de existirem muitos trabalhos dedicados à modelagem dinâmica de tais estruturas, a obtenção de um modelo dinâmico realista e preditivo é de fundamental importância para o desenvolvimento de controladores. No presente artigo propõe-se um novo algoritmo para determinar de forma bastante simples, as equações diferenciais da dinâmica de uma estrutura do tipo manipulador com um único elo flexível. Desenvolve-se ainda funções de transferência analíticas para este problema específico, as quais são importantes para a validação do modelo discreto proposto na forma de um algoritmo.
Active control law development of flexible structures is still an open problem. It has been the object of many researches, mainly starting from the eighties. In spite of many works dedicated to the dynamic modeling of such structures, the obtaining of a realistic dynamic model is of fundamental importance for the control law development. In the present work it is introduced a new algorithm to determinate in a quite simple way, the dynamic differential equations of a one flexible link manipulator. It is still developed analytic transfer functions for this specific problem, which are important for the validation of the lumped mass approach model proposed in the form of an algorithm

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