New View of the Euler-Bernoulli Equation

A special attention is paid to the motion of flexible links in a robotic configuration. The elastic deformation is a dynamic value which depends on the total dynamics of the robot system movements. The Euler-Bernoulli equation should be expanded according to the requirements of the motion complexity of elastic robotic systems. The EulerBernoulli equation (based on the existing laws of dynamics) should be supplemented with all the forces (inertial forces, Coriolis, centrifugal forces, gravity forces, environment forces, disturbance forces as well as coupling forces between the present modes) that are participating in the formation of the elasticity 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 as well as a damping matrix. The mathematical model of the actuators also comprises coupling between elasticity forces. A particular integral defined by Daniel Bernoulli should be supplemented with the stationary character of elastic deformation of any point of the considered mode, caused by the present forces. The 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. The 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.