An approximate analytical solution of the gear meshing problem based on MSM using DynPy Python library - Twardoch , Krzysztof; Kaczmarzyk , Krzysztof; Długopolski , Grzegorz; Chiliński , Bogumił (Bulletin of the Polish Academy of Sciences Technical Sciences, 2026)
The paper presents the mathematical and numerical analysis of a 1-DOF (one-degree-of-freedom) dynamic model of the helical gear with time-varying mesh stiffness (TVMS). The article aims to determine an analytical solution for the presented model using a proprietary computational environment and to verify the results with numerical simulations and other solutions available in the literature. The paper presents the determination of a 2-DOF (two-degree-of-freedom) dynamic model and its reduction to a 1-DOF model. The concept of the created environment, the applied libraries, and the application basics are discussed. Based on the work effects, an analytical solution using the multiple scales method (MSM) was found and positively verified. The article presents the convergence of the obtained results and the added value as an analytical solution. This confirms the effectiveness of the novelty approach, which provides a framework that bridges the gap between directly determining a solution and manual calculations. It should be noted that time complexity is especially important for performance computing. Observations suggest significant advantages to using an analytical solution due to its precision and relatively low computational cost. Although obtaining an analytical solution is more time-consuming, it reduces the possibility of errors with numerical methods.