Feedback Picard Iteration-Collocation Method And The Applications On Astronautical Engineering

Nonlinear dynamical systems exist extensively in many scientific studies and engineering applications.For the existence of complex nonlinear terms,the law of superposition of the solutions does not apply,and thus it is usually very difficult to generally analyzing nonlinear systems.In real-life engineering problems,there could be various kinds of nonlinear terms affecting the dynamical system,which complicate the analysis,while the missions often require accurate real-time solutions.Considering that,the high performance computational methods for solving nonlinear dynamical systems become the key to both engineering and scientific studies of nonlinear problems.To investigate the steady or the transient responses of nonlinear dynamical systems,the researchers normally use the asymptotic methods or the weighted residual methods.Based on these two large classes of methods,we proposed some computational schemes for analyzing the structural vibration and orbital motion in astronautical engineering.First,an efficient method is proposed to numerically investigate structural vibration involving freeplay nonlinearity.Then we developed a precise searching scheme for periodic relative motion between spacecraft.Further,a new kind of Feedback-Accelerated Picard Iteration method(FAPI)is proposed to generally solve strongly nonlinear dynamical systems.The FAPI method successfully remedies two major drawbacks of asymptotic method and weighted residual method,i.e.(i)complex symbolic calculation,and(ii)numerical difficulties in solving nonlinear algebraic equations.The specific researches conducted in this paper are listed as follows.· An efficient computational method for investigating structural vibration involving freeplay nonlinearity is proposed.The simplicity and universality of differential transform method allows us to apply it to dynamical systems involving complex nonlinear integrals.The Henon method is combined with the differential transform method to precisely determine the switch point of piecewise function.The resulted method can accurately predict the complex dynamical responses of freeplay nonlinear dynamical systems and its bifurcation diagram,while the Runge-Kutta method fails to achieve that.· We developed an algorithm for precisely searching for the periodic relative motions between spacecrafts on near-earth orbit.Based on time domain collocation method,the model of relative motion is reduced to a system of nonlinear algebraic systems,which can be easily solved to provide the periodic solution.This searching scheme can be used to replace regular shooting methods and it helps to improve the searching efficiency and accuracy.In addition,a fuel-saving relative-orbit-keeping strategy is proposed using this searching scheme.· Three asymptotic methods,i.e.,the Picard method,the Adomian decomposition method and the variational iteration method,are unified in this paper.The underlying relationship between these three methods are discovered by proving that Picard iteration method and Adomian decomposition method can be derived from variational iteration method,while the variational iteration method is inherently the general use of Lagrange multipliers.Based on this analysis,we further developed a local variational iteration method,which can be used to accurately predict long-term responses of nonlinear dynamical systems and simplify the symbolic calculations.· We proposed a new kind of Feedback-Accelerated Picard Iteration method for solving strongly nonlinear dynamical systems.This method is free from complex symbolic calculations and solving nonlinear algebraic systems.Compared with some commonly used methods in literature and practical tasks,such as the finite difference method,the collocation method,etc.,the Feedback-Accelerated Picard Iteration method is much superior on computational efficiency and accuracy.The simulation results show that the proposed method is much better than regular finite difference methods,in that the computational step size can be thousand times larger and the accuracy can be several magnitudes higher.· A high performance computational method is proposed for orbital integration and orbital transfer.The simplicity,high accuracy,fast convergence speed,and robustness of Feedback-Accelerated Picard Iteration method allow us to apply it to orbital mechanics involving various perturbations.It is shown in this paper that the proposed method achieves machine precision in orbital propagation.Compared with Runge-Kutta 12(10)method,which is commonly used in practice,the efficiency of Feedback-Accelerated Picard Iteration method is almost hundred times higher.The proposed method can also be used to accurately solve orbital transfer problems,of which the computational cost is much lower than the regular shooting methods.