Research Topic: Extension of the Euler Bernoulli equation from many aspects

The Euler-Bernoulli equation and its solution were defined under the assumption that the elasticity force is opposed only by the inertial force proper. Besides, it is supposed by definition that the motion in Euler-Bernoulli equation is caused by an external force, suddenly added and then removed. The solution of D. Bernoulli satisfies these assumptions. The Euler-Bernoulli equation and its solution were defined under the assumption that the elasticity force is opposed only by the inertial force proper.

Idealized motion of elastic body according to D. Bernoulli.

Fig. 1. Idealized motion of elastic body according to D. Bernoulli.

Besides, it is supposed by definition that the motion in Euler-Bernoulli equation is caused by an external force, suddenly added and then removed. The solution of D. Bernoulli satisfies these assumptions. Bernoulli presumed the horizontal position of the observed body as its stationary state (in this case it matches the position - axis). At such presumption, the oscillations happen just around the - axis. If Bernoulli, at any case, had included the gravity force in its equation, the situation would have been more real.

Possible positions of the tip of elastic line with modes.

Fig. 2. Possible positions of the tip of elastic line with modes.

Then the stationery body position would not have matched the - axis position, but the body position would have been little lower and the oscillations would have happened around the new stationery position.The Euler-Bernoulli equation and its solution need a short explanation that, we think, should be assumed, but which is missing from the original literature. Euler and Bernoulli wrote your equation based on ‘vision’. They did not define the mathematical model of a link with an infinite number of modes, which has a general form, but they did define the motion solution (shape of elastic line) of such a link. They left the task of link modeling with an infinite number of modes to their successors.

Dynamics of each mode is described by one equation. The equations in the whole model are not of equal structure as our contemporaries, authors of numerous works, presently interpret it.We think that the coupling between the modes involved leads to structural diversity among the equations in the whole model. This explanation is of key importance and is necessary for understanding our further discussion.

Robot mechanism.

Fig. 3. Robot mechanism.

The tip coordinates and the position deviation from the reference level.

Fig. 4. The tip coordinates and the position deviation from the reference level.

The Bernoulli solution describes only partially the nature of motion of real elastic beams. More precisely, it is only one component of motion. The Euler-Bernoulli equation and its solution should be expanded from several aspects in order to be applicable in a broader analysis of elasticity of robot mechanisms. By supplementing these equations with the expressions that come out directly from the motion dynamics of elastic bodies, they become more complex.

The environment force dynamics.

Fig. 5. The environment force dynamics.

The motion of the considered robotic system mode is far more complex than the motion of the body presented with Euler-Bernoulli equation. This means that the equations that describe the robotic system (its elements) must also be more complex than the equations formulated by Euler and Bernoulli. This fact is overlooked, and the original equations are widely used in the literature to describe the robotic system motion. This is very inadequate because valuable pieces of information about the complexity of the elastic robotic system motion are thus lost. Hence, it should be especially emphasized the necessity of expanding the source equations for the purpose of modeling robotic systems, and this should be done in the following way:

The elastic deformations.

Fig. 6. The elastic deformations.