A Python Implementation of a Shooting Algorithm With Numerical Continuation and Automatic Differentiation for Autonomous and Non-Autonomous Systems in Structural Dynamics

Abstract

Abstract. In this work, the open-source implementation of a shooting algorithm with numerical continuation and automatic differentiation is presented. The published toolbox is able to handle both autonomous and nonautonomous systems with a wide range of nonlinearities, enabling the calculation of frequency-response curves and of nonlinear normal modes (NMM), to better study the behavior of structures. This work gives an introduction to the most important aspects of shooting, numerical continuation, and automatic differentiation, providing a step-by-step guide on the implementation of a shooting algorithm for structural dynamics. In addition to its research-oriented capabilities, the implementation is designed to serve educational purposes: the code is structured in a transparent and accessible manner, allowing readers to follow the underlying computations and readily modify or extend the provided functionalities for their own applications. Furthermore, some of the systems already analyzed with the developed shooting algorithm are presented, namely, the Duffing oscillator, an Euler–Bernoulli beam with dry-friction and a reduced order model (using the direct parametrization of invariant manifolds approach (Vizzaccaro et al., 2024, “Direct Parametrisation of Invariant Manifolds for Non-Autonomous Forced Systemes Including Superharmonic Resonances,” Nonlinear Dyn., 112(8), pp. 6255–6290) of an archbeam.