In this paper, a high-order curved mesh generation method for Discontinuous Galerkin methods is introduced. First, a regular mesh is generated. Second, the solid surface is re-constructed using cubic polynomial. Third, the elastic governing equations are solved using high-order finite element method to provide a fully or partly curved grid. Numerical tests indicate that the intersection between element boundaries can be avoided by carefully defining the elasticity modulus.

The motivation for this work is the state of modern structural mechanisms that are characterized by growing complexity and ever-increasing demands for rapid and accurate motion. These contradictory requirements are often achieved according to easier and easier structures characterized by flexibility segments. In most of cases, the elasticity of structures appears as an obstacle for a precise and rapid control of motion. The aim of this paper is to explore ways of implementation of structural properties of elasticity with the application of high fidelity models during synthesis and analysis of complex mechanisms. Precisely, the aim is to explore the possibility of using Euler–Bernoulli equation, if not in its original form, then to the same extent with the use of modern knowledge in robotics (based on the knowledge of classical mechanics), and to examine the affordability and confirmation of the method through simulation results for a typical robotic configuration.

The paper is concerned with the relationship between the equation of elastic line motion, the “Euler-Bernoulli approach” (EBA), and equation of motion at the point of elastic line tip, the “Lumped-mass approach” (LMA). The Euler–Bernoulli equations (which have for a long time been used in the literature) should be expanded according to the requirements of the motion complexity of elastic robotic systems. The Euler–Bernoulli equation (based on the known laws of dynamics) should be supplemented with all the forces that are participating in the formation of the bending moment of the considered mode. This yields the difference in the structure of Euler–Bernoulli equations for each mode. The stiffness matrix is a full matrix. Mathematical model of the actuators also comprises coupling between elasticity forces. Particular integral of Daniel Bernoulli should be supplemented with the stationary character of elastic deformation of any point of the considered mode, caused by the present forces. General form of the elastic line is a direct outcome of the system motion dynamics, and can not be described by one scalar equation but by three equations for position and three equations for orientation of every point on that elastic line. Simulation results are shown for a selected robotic example involving the simultaneous presence of elasticity of the joint and of the link (two modes), as well the environment force dynamics.

L'augmentation des performances des machines de positionnement passe par l'augmentation des accélérations et donc des sollicitations transmises à la structure de la machine. Ces contraintes sont susceptibles d'engendrer des déformations et des vibrations dégradant le suivi de profil ainsi que le positionnement final. Les commandes numériques avancées disposent de différentes formes de loi de mouvement (à jerk limité, polynomiale...) qui ont un effet notable sur le compromis entre la durée du mouvement effectif et la précision attendue. L'objectif de cet article vise à démystifier cet effet en proposant une analyse comparative de l'influence de différents types de lois de mouvement sur les vibrations, ainsi que sur la durée du mouvement d'un axe soumis à un mode de déformation prépondérant.