Symplectic Solution Of Flexible Multi-body Dynamics

With the continuous progress of science and technology, especially the continuous development of such industries as aerospace, robotics, etc., the development of multibody system tends to be gentle and dexterous because of the intelligentized and lightweight requirements, which endows flexible multibody system with much application background of engineering. As the universal phenomenon of rigid-flexible couple exists in the flexible multibody system, it can be often seen that the dynamics equations of flexible multibody system show the characteristics of time dependence, rigid-flexible couple, and nonlinearity, which makes numerical solutions difficult. Traditional arithmetic guarantees calculation stability by introducing artificial dissipative mechanism. And this feature of traditional arithmetic will lead to wrong calculation results when solving long tracking problems.The symplectic scheme, whose format propulsion mapping is symplectic, is a numerical algorithm within the Hamiltonian system, no dissipative mechanism introduced. The symplectic scheme can keep phase space invariable and movement unchanged, which can lead to the correct solutions of long tracking problems. Simultaneously, symplectic Runge-Kutta scheme based on Gauss-Legendre polynomial is also A-stable, making it fit for solving stiff problems very well.This paper aims at the study on symplectic scheme and symplectic geometry, according to the difficulties of numerical solutions in dynamics equations of flexible multibody system. It builds dynamics equations of central rigid-flexible couple system by Kane method, and also imports the systematic dynamics equations to Hamiltonian system, resulting in canonical formulation of systematic dynamics equations, then solving the dynamic respond of the system by means of third-class symplectic Runge-Kutta scheme. The main tasks of this paper are as follows.Firstly, summarizing the production, development, and research status of symplectic scheme of Hamiltonian system and flexible multibody system dynamics.Secondly, introducing Newton-Euler method and rotation matrix method describing flexible body dynamics, stating Lagrange equations of assumed mode method and free flexible body, and studying the process of establishing flexible multibody system dynamics with Kane method.Thirdly, studying the basic theories of symplectic geometry and symplectic space, comparing the similarities and differences between symplectic space and Euclid space, expounding symplectic structure of Hamiltonian system and Hamiltonian canonical equation, discussing explicit symplectic scheme and the construction, accuracy and stability of symplectic Runge-Kutta method based on Gauss-Legendre polynomial, and also analyzing dissipative mechanism of traditional Euler method and Runge-Kutta method with the numerical results proving that symplectic scheme can maintain the square of phase space invariable and guarantee calculation results correct.Fourthly, studying the dynamics problems of center rigid body-flexible space beam coupling system, accounting the coupling impact of transverse deflection of flexible beam on axial deflection of that by adopting arc-length coordinates and Cartesian coordinates to describe flexible beams transformation, and then introducing system dynamics equations into Hamilton system which lead to dynamics equations of canonical formulation by adopting assumed mode method to discretize flexible beams and by establishing system dynamics equation using Kane-method; expounding Butcher matrix transformation method of Runge-Kutta scheme while solving nonlinear equations in the iteration process to reduce calculation cost. Finally, solving the problem of the mode's dynamics response by applying symplectic Runge-Kutta scheme with level 3-rank 6th-order precision, from which validity and superiority of symplectic scheme can be reflected.