Euler-Bernoulli equation today

Abstract

Special attention is paid to the motion of the flexible links in the robotic configuration. The elastic deformation is a dynamic value which depends on the total dynamics of the robot system movements. 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 elasticity moment of the considered mode according to the requirements of the motion complexity of elastic robotic systems. This yields the difference in the structure of Euler-Bernoulli equations for each mode. The stiffness matrix is a full matrix as well as damping matrix. Mathematical model of the actuators also comprises coupling between elasticity forces. Particular integral which defined 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 mechanism elastic line is a direct outcome of the system motion dynamics, and cannot 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 gear and of the link (two modes), as well as the environment force dynamics.