Structure-preserving Method For Several Dynamic Problems

The nonlinear phenomenon,which widely exists in nature,is one of the most important features of the complex dynamic systems.For a long time,mechanical and mathematical researchers devoted efforts to solving a variety of nonlinear dynamic problems by mathematical and mechanics principles.For the calculation of nonlinear dynamic systems,the traditional numerical methods are difficult to preserve the real physical characteristics of the system,thus the result will lose the fundamental essence of nonlinear problems.As a rational structure-preserving numerical method,the symplectic geometry method in Hamilton system could preserve some geometric properties of the systems well,such as the symplectic structure,the energy conservation and the first integral.For long time simulations,symplectic geometry method is more stable and precise than the traditional numerical methods.Based on the symplectic geometry method in Hamilton system,the orbit-attitude coupled dynamic behaviors of space solar power satellite and their components,and the almost structure-preserving analysis for weakly linear damping nonlinear Schr(?)dinger equation with periodic perturbation are investigated in detail by symplectic/generalized multi-symplectic theory in this paper.The main conclusions of the thesis are as follows:1.Taking dynamic problems of some large stiffness and small size adapting pieces in complex spatial structures before on-orbit assembly as studying background,a simplified and classical coupled dynamic model describes the coupling dynamic behaviours of orbit and attitude is established for the spatial rigid beam.The symplectic Runge-Kutta method is applied to simulate dynamic behaviors of the spatial rigid beam.From the results obtained about evolution processes of the beam's orbit radius,true anomaly and attitude angle,it can be concluded that,with increase in initial attitude angle speed,the coupled effects become more obvious.The relative error of the system total energy in each time step is recorded and all relative errors are compared with those in the numerical results using the classic Runge-Kutta method,the correctness of the numerical results using the symplectic Runge-Kutta method and the long-time numerical stability of the symplectic Runge-Kutta method are verified.2.As the spatial components do not work in isolation,for the strong coupling dynamic problems of the sail tower solar power satellite on orbit,a simplified and classical model combined with spatial rigid rods and spring is established which could describe the coupling dynamic behaviours of orbit-attitude.The dynamic behaviours of the simplified model are analyzed by the symplectic geometry method and the numerical results could be verified indirectly by the energy conservation of the system.The symplectic Runge-Kutta method proposed could reproduce the dynamic properties of the sail tower solar power satellite associated with the Earth non-shape perturbation rapidly and preserve the energy well with excellent long-time numerical stability,which will give a new approach to obtain the real-time dynamic response of the ultra-large spatial structure for the real-time feedback controller design.3.Based on the dynamic properties of single rigid component and combined structures in the previous chapters,for some dynamic properties of the deploying process of the solar power satellite via Arbitrarily Large Phased Array(SPS-ALPHA)solar receiver,the symplectic Runge-Kutta method is used to simulate the established system model.For the exact form of Rayleigh damping,the system contained the Rayleigh damping is separated and transformed into the equivalently undamping system formally by the relationships between the mass matrix,stiffness matrix and damping matrix in the classical damping system.So the symplectic Runge-Kutta method could be used to simulate the improved nonlinear dynamic equations describing the deploying process for the solar receivers.The numerical results show that the proposed simplified model is valid for the deploying process for the SPS-ALPHA solar receivers and the symplectic Runge-Kutta method could preserve the displacement constraints of the system well with excellent long-time numerical stability.4.For further research and application of the structure-preserving method to the infinite dimensional system describing the flexible vibration of the ultra-large spatial structure,the nonlinear Schr(?)dinger equation describing the nonlinear wave propagating in flexible spatial structure is taken as an example.The dynamic behaviors of the damping nonlinear Schr(?)dinger equation with periodic perturbation and the conservation properties of this non-conservative system in the numerical simulations are analyzed based on the generalized multi-symplectic idea.Then the local energy/momentum loss expressions as well as the approximate symmetric form of the linearly damping nonlinear Schr(?)dinger equation with periodic perturbation are deduced.By analyzing the special nonlinear phenomena of the nonlinear Schr(?)dinger equation,that is the multi-soliton splitting process,the bifurcation of the discrete eigenvalues for the related Zakharov-Shabat spectral problem is obtained.And the excellence of generalized multi-symplectic method is further revealed in reproducing the local geometrical characteristics of the dissipative system.The novel almost structure-preserving method developed in this chapter will give a new approach to investigate the propagation of the nonlinear wave in the flexible spatial structure.