Abstract
This paper presents the main analytical methods, in the context of current developments in the study of complex multibody systems, to obtain evolution equations for a multibody system with deformable elements. The method used for analysis is the finite element method. To write the equations of motion, the most used methods are presented, namely the Lagrange equations method, the Gibbs–Appell equations, Maggi’s formalism and Hamilton’s equations. While the method of Lagrange’s equations is well documented, other methods have only begun to show their potential in recent times, when complex technical applications have revealed some of their advantages. This paper aims to present, in parallel, all these methods, which are more often used together with some of their engineering applications. The main advantages and disadvantages are comparatively presented. For a mechanical system that has certain peculiarities, it is possible that the alternative methods offered by analytical mechanics such as Lagrange’s equations have some advantages. These advantages can lead to computer time savings for concrete engineering applications. All these methods are alternative ways to obtain the equations of motion and response time of the studied systems. The difference between them consists only in the way of describing the systems and the application of the fundamental theorems of mechanics. However, this difference can be used to save time in modeling and analyzing systems, which is important in designing current engineering complex systems. The specifics of the analyzed mechanical system can guide us to use one of the methods presented in order to benefit from the advantages offered.
Highlights
In recent years, research, led by practical applications in engineering, and which uses increasingly complex equipment, operates at higher speeds in difficult environmental conditions, and which is subjected to intense loads, has driven the development of methods for analyzing large deformable mechanical systems
In order to perform this analysis, the researchers developed and reinvented the existing methods to apply them to the new situations that has arisen. The development of this field is based on the numerical techniques related to the finite element method (FEM) as well as on the application of classical methods used in analytical mechanics
Such an analysis avoids the study of each element and the assembly of the obtained systems of equations, and it allows for the obtainment of the evolution equations, starting from the fundamental notions written for the whole system
Summary
Research, led by practical applications in engineering, and which uses increasingly complex equipment, operates at higher speeds in difficult environmental conditions, and which is subjected to intense loads, has driven the development of methods for analyzing large deformable mechanical systems. The methods of analytical mechanics fit well with such an analysis because of the problems of complexity and the possibility to study a mechanical system with a large number of degrees of freedom (DOF) Such an analysis avoids the study of each element and the assembly of the obtained systems of equations, and it allows for the obtainment of the evolution equations, starting from the fundamental notions written for the whole system (kinetic energy, potential energy, work, acceleration energy, momentum, Hamiltonian, etc.). This is possible due to the tools used by analytical mechanics, which generalize the way of presentation of motion. Analytical mechanics has proven to be a powerful tool for analyzing complex systems, providing a set of equivalent formulations from which the most appropriate formulation can be selected
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