Study On Theory And Applications Of Multibody System Dynamics In Improved Variant Space

There exists dispute whether the classical first integrals can be treated as constraints or not. In order to solve this problem, this paper analyzes the inner relationship between constraints and classical first integrals at first, and proves that constraints and classical first integrals belong to first integrals essentially. And then, based on the theory about Lagrange's equation of the first kind, the paper proves that first integrals (including classical first integrals) except constraints can be treated as constraints and that Lagrange multipliers of the integrals except constraints are equal to zero. The order reduced equations are built by substituting all the first integrals into the expression of kinetic energy.In order to disclosure the nature of treating the classical first integrals as constraints, the paper improves the variational principle of differential form from constraint variant space to variant space of integral groups based on Newton's classic mechanics and classical analytical mechanics. Universality of Newton's Second Law is described as the independence of the variations. Then, on the basis of the independence of the variations and improved variational principle in variant space based on integral groups, improved theoretical system of Lagrangian mechanics is established, which can be generalized to the event space. The improved theory solves the dispute whether the classical first integrals can be treated as constraints or not, moreover, the theory is in good agreement with Newton's Second Law.The paper obtains the periodic analytical solutions of a class of two-link, three-link and four-link system moving freely on a planar based on the maximum integrable group. The relationship between the integrable group and kinematical trajectory of the system is utilized to analysis the trajectory of a free floating space manipulator whose base is zero-coupled. Then, based on the improved theory and classical reduction method, a hybrid reduction method is presented: first establish the reduced equations based on the maximum integrable group, and then reduce the equations according to Routhian equation or Wittaker equation if there are cyclic integrals or energy integrals in the reduced equations.On the base of the improved theory, the maximum integrable group of a three-body system with center configuration are equivalent to its solutions, and it can be conclude that the solutions of planar three-body system with center configuration are periodic.The paper present a class of symmetric three-body systems, in which one of the particles moves back and forth on the straight and the other two particles with the same mass are 180°rotationally symmetric. Considering that the symmetric three-body systems satisfy some certain initial conditions, there are only 14 independent integrals. Because it is certain that there are some integrable curves among the curve family of the systems, there must be periodic solutions to the systems. The paper proves that the trajectories of the two symmetric particles are not plane curves but space curves.