New Interpretation of the Euler-Bernoulli Equation

A new notion of joint, defined in terms of the state of motor (active or locked) and type of the elastic or rigid element, gear and/or link that follows after the motor, is introduced. Special attention is paid to the motion of the flexible links in the robotic configuration. The article is concerned with the relationship between the Euler-Bernoulli equation, the "Euler-Bernoulli approach" (EBA), and equation of equilibrium at the point of elastic tine 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 tine. 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.