Synthesis on the Acceleration Energies in the Advanced Mechanics of the Multibody Systems

Abstract

The present paper’s objective is to highlight some new developments of the main author in the field of advanced dynamics of systems and higher order dynamic equations. These equations have been developed on the basis of the matrix exponentials which prove to have undeniable advantages in the matrix study of any complex mechanical system. The present paper proposes some new approaches, based on differential principles from analytical mechanics, by using some important dynamics notions, regarding the acceleration energies of the first, second and third order. This study extended the equations of the higher order, which provide the possibility of applying the initial motion conditions in the positions, velocities and accelerations of the first and second order. In order to determine the time variation laws for the generalized variables, the driving forces and acceleration energies of the higher order are applied by the time polynomial functions of the fifth order. According to inverse kinematics also named control kinematics of the robots, the applications of polynomial functions lead to the kinematic control functions of mechanical motions, especially the transitory motions. They influence the dynamic behavior of multibody systems, in which robot structures are included.

A New Approach in Analytical Dynamics of Mechanical Systems

“…As stated in a previous paper [4], to perform the dynamic analysis of mechanical systems, some basic mechanical models are used: the material point and rigid body respectively. They are both characterized by three and six degrees of freedom.…”

“…where χ (0) represents the unit vector of the driving axis and the symbols δ χ and ∆ χ included in the expression (3) are defined according to [4], as presented below:…”

“…According to [4], the generalized expression that characterizes the three simple rotation matrices, components of the 0 s [R] generalized rotation matrix included in (1), is written as:…”

“…Mechanical systems characterized by ultra-fast movements also include serial structures of robots. According to Eager, Pendrill and Visser [1,2] and to the authors' previous research [3,4], the fast movements occur when the linear acceleration is according to the following condition: a ≥ 2 · g, where g is the acceleration gravity. As a result, the fast movements of mechanical systems are characterized by a time variation law for the acceleration, which can be plotted in the form of a higher-order parabola.…”

“…The main purpose of this paper was not to conduct an exclusive kinematic study on higher-order accelerations. According to [3,4], the higher-order accelerations become central functions in the expressions of the so-called higher-order acceleration energies. Appell determined the expression for the first order acceleration energy in the case of material particles, and for discrete material particle systems [17,18].…”