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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.
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