On the Use of the Generating Series for the Impulse Response of Duffing’s Equation

Traditionally, solving the Volterra series for nonlinear differential equations subject to an impulse excitation warrants the use of contour integration or a method of exponential inputs. Both of these solutions may be problematic when implemented on a computer and involve error-prone calculations when performed by hand. This chapter aims to address these issues by presenting a computer-based approach for the impulse response of a nonlinear oscillator with quadratic and cubic stiffness terms.

The generating-series method offers a technique to transform nonlinear differential equations with polynomial nonlinearities into a domain where the calculus problem is converted into an algebraic and combinatorial problem. Within this transformed domain, an iterative scheme determines higher-order Volterra kernels, and the nonlinear terms are expanded using the shuffle product. The generating series is then converted back into the time domain. This chapter showcases the computation of the first six terms in the Volterra series expansion and provides an error analysis by comparing the results to numerical approximations. The findings presented in this chapter contribute to the broader field of nonlinear dynamics and modal analysis; namely, the proposed method not only enhances the understanding of Duffing’s equation but also presents a practical tool for analysing nonlinear differential equations.