A Python Implementation of a Robust Multi-harmonic Balance with Numerical Continuation and Automatic Differentiation for Structural Dynamics

Abstract

Simulations are used in vibration analysis to appraise the structure's functionality and to determine the loading effects, enabling partial optimization before actual prototyping. Oscillations are fundamental in nature, appearing in practical engineering applications. General nonlinear problems hardly have analytical solutions, requiring sophisticated techniques to reach approximate solutions. This works presents a robust Python implementation of multi-harmonic balance with predictor-corrector numerical continuation, Newton-Raphson root-solver, and forward automatic differentiation with dual numbers. This toolbox shows promising converging robustness when dealing with polynomial as well as sharp nonlinearities, especially in the construction of frequency response curves. The tool with its functionalities will be uploaded and made available to interested researchers upon request.