Multibody Systems – approaches and challenges

Abstract

The multibody approach for the mechanical problems is a new one and it is based on a few aspects: determination of all constraints applied to the system; the written of the matrix of constraints for each body; determination of the kinetic and potential energy for the mechanical system; the obtaining of the matrix differential equation of motion; the solving of the non-linear differential system. The written of the equations of motion implies the introduction of the column matrices of the Lagrange multipliers; these multipliers can be hardly eliminated from the equations of motion. The methods for this elimination presented in the literature, base on the assumption that the so called matrix of inertia is an invertible one, which is not always true. Scientists created different algorithms which select the complete rank sub-matrix from the matrix of inertia. Another problem is that of the complexity of the equations of motion, equations obtained by applying the Lagrange second order equations. There exist some main directions of study: the obtaining of the equations of motion starting from other expressions given by Kane’s equations, Maggi’s equations, Appel’s equations, fractional calculus etc.; the introduction of some pseudo-coordinates (fake coordinates) which simplify the form of the equations of motion, but having the disadvantage that the equations of motion are no longer independent; the selection of other parameters as generalized coordinates, method which is particularly applied in the case of mechanisms etc. The advantage of such method is a relative one, that is this advantage exists only for some particular mechanical systems (presented as examples in the literature); these methods may be very toilsome for other mechanical systems. The selection of the method is up to every scientist.