Research On Solution Methods Of Nonlinear Dynamical Systems And The Application On Aerospace Engineering

There is a great deal of nonlinear problems in the preceding research areas. Actually, most of the real-life engineering systems are unavoidably affected by nonlinear terms such as the gaps in mechanical systems, unsmooth friction, deformation of structures, etc. Normally, linearized model can provide good approximation to real-life dynamical behavior. But sometimes the neglected nonlinear terms will cause unacceptable error, especially to the long-term dynamical problems. To investigate the nonlinear dynamical behavior of the system, numerical or semi-analytical techniques are mostly engaged, which are tightly related to the computational mechanics.Nonlinear dynamical systems are also rarely common in satellite relative motion, and are receiving increasing research interests. To better understand the characteristics of spacecrafts relative motion and manipulate the vehicles more precisely, we have to study the complex nonlinear dynamical behavior of the relative motion models. The main research work in this dissertation includes the study of nonlinear response of van der Pol oscillator, the analysis of flutter response of two dimensional airfoil section, and the application of time domain collocation method in searching for periodic relative orbits of satellites.1. The time domain collocation method(TDC) is applied to solve the van der Pol equation. The results obtained by the TDC are compared with the results by the traditional harmonic balance method(HB) and high dimensional harmonic balance method(HDHB). We strictly prove that the HDHB is equivalent to the TDC and both of them can be regarded as special cases of the extended TDC method, where least squares method is used to handle an over-determined system. For the extended TDC method, appropriately increasing the number of collocation points can significantly relive the nonphysical solution phenomenon and generate a better solution. Compared with the HDHB, the TDC method is derived more strictly, and more convenient to implement. The extended TDC method with more collocation points can not only reduce the computational cost of the HB method, but also improve the resulting accuracy of the HDHB method.2. The differential transform method as well as the time domain collocation method are applied to analyzing the aeroelastic dynamic response of an airfoil section with freeplay nonlinearities. The original differential transform method is further developed and the adaptive grid size mechanism is introduced to solve the aeroelastic system. By comparing with the Runge-Kutta method, the accuracy and efficiency of differential transform method and time domain collocation method is verified.3. A numerical approach for obtaining periodic orbits of satellite relative motion is proposed based on using the time domain collocation(TDC) method to search for the periodic solutions of an exact 2J nonlinear relative model. The initial conditions for periodic relative orbits of the Clohessy-Wiltshire(C-W) equations or Tschauner-Hemple(T-H) equations can be refined with this approach to generate nearly bounded orbits. With these orbits, a method based on the least-squares principle is then proposed to generate projected closed orbit(PCO), which is a reference for the relative motion control. Numerical simulations reveal that the presented TDC searching scheme is effective and simple, and the projected closed orbit is very fuel-saving.