论文标题
关于具有多项式刚度的非线性系统的生成系列的应用
On the application of the generating series for nonlinear systems with polynomial stiffness
论文作者
论文摘要
非线性微分方程的分析解决方案（根本存在）通常很难找到。例如，具有立方刚度的系统的Duffing方程需要在精确解决方案中使用椭圆函数。如果某种解决方案甚至存在，则具有一般多项式刚度的系统将更加难以分析。扰动和串联解决方案是可能的，但是随着解决方案的增加，越来越多的要求变得越来越苛刻。本文旨在重新访问，介绍和讨论确定系统响应的几何/代数方法，该方法适合自动化。该方法最初是由于Fliess和同事，它利用了生成系列和洗牌产品，以差异几何形状和抽象代数建立的数学思想。考虑了具有多项式刚度的非线性微分方程家族。操纵串联扩展到生成系列的过程随后是随后的，并显示出具有递归示意图，这与计算机代数相当。然后，将逆拉动式 - 孔变换应用以得出时间域响应。针对具有多项式刚度的系统提出了新的解决方案
Analytical solutions to nonlinear differential equations -- where they exist at all -- can often be very difficult to find. For example, Duffing's equation for a system with cubic stiffness requires the use of elliptic functions in the exact solution. A system with general polynomial stiffness would be even more difficult to solve analytically, if such a solution was even to exist. Perturbation and series solutions are possible, but become increasingly demanding as the order of solution increases. This paper aims to revisit, present and discuss a geometric/algebraic method of determining system response which lends itself to automation. The method, originally due to Fliess and co-workers, makes use of the generating series and shuffle product, mathematical ideas founded in differential geometry and abstract algebra. A family of nonlinear differential equations with polynomial stiffness is considered; the process of manipulating a series expansion into the generating series follows and is shown to provide a recursive schematic, which is amenable to computer algebra. The inverse Laplace-Borel transform is then applied to derive a time-domain response. New solutions are presented for systems with general polynomial stiffness, both for deterministic and Gaussian white-noise excitation
