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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

Vol. 74, nr 1

art. no. e155895

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.

Development of a dynamic vibration absorber in the form of a variable inertia pendulum

Loch Piotr, Bugajewski Karol, Przybyłowicz Piotr, Chiliński Bogumił

Journal of Applied Mathematics and Computational Mechanics

2025

Vol. 24, nr 4

37--50

Nowadays, to maximize space utilization, multi-story buildings are designed to occupy as little surface area as possible. However, the high slenderness of these struc tures makes them susceptible to forces caused by factors such as wind and tectonic plate movements. Prolonged exposure to vibrations reduces the building’s lifespan and leads to its degradation. To minimize the impact of vibrations while maintaining slenderness and relatively low structural rigidity, a solution in the form of a dynamic vibration absorber can be applied. Structures already use absorbers in the form of pendulums, as seen in the Taipei 101 skyscraper, for example. The subject of study is an absorber in the form of a pendulum with a variable moment of inertia, achieved by extending the cable, which allows the structure to adapt to current forces. This significantly reduces the time of exposure to harmful influences. A skyscraper model with a variable-length pendulum was developed and analyzed through simulations, confirming the hypothesis of a significant reduction in exposure.