On the application of the generating series for nonlinear systems with polynomial stiffness

TL;DR

This paper presents a geometric and algebraic method using generating series and shuffle products to analytically approximate responses of nonlinear systems with polynomial stiffness, enabling automation and new solutions.

Contribution

It revisits and extends a series-based method for nonlinear differential equations, facilitating automated derivation of system responses with polynomial stiffness.

Findings

Derived recursive formulas for polynomial stiffness systems

Applied inverse Laplace-Borel transform to obtain time responses

Presented new solutions for deterministic and stochastic excitations

Abstract

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