New form of the Euler-Bernoulli rod equation applied to robotic systems

This paper presents a theoretical background and an example of extending the Euler-Bernoulli equation from several aspects. 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. The stiffness matrix is a full matrix. Damping is an omnipresent elasticity characteristic of real systems, so that it is naturally included in the Euler-Bernoulli equation. It is shown that Daniel Bernoulli's particular integral is just one component of the total elastic deformation of the tip of any mode to which we have to add a component of the elastic deformation of a stationary regime in accordance with the complexity requirements of motion of an elastic robot system. The elastic line equation mode of link of a complex elastic robot system is defined based on the so-called 'Euler-Bernoulli Approach' (EBA). It is shown that the equation of equilibrium of all forces present at mode tip point ('Lumped-mass approach' (LMA)) follows directly from the elastic line equation for specified boundary conditions. This, in turn, proves the essential relationship between LMA and EBA approaches. In the defined mathematical model of a robotic system with multiple DOF (degree of freedom) in the presence of the second mode, the phenomenon of elasticity of both links and joints are considered simultaneously with the presence of the environment dynamics - all based on the previously presented theoretical premises. Simulation results are presented.