THE POINCARE METHOD IN TEACHING THE THEORY OF NONLINEAR VIBRATIONS

Аннотация

Annotations. In the realm of continuum mechanics, Poincaré's theory provides a framework for understanding the nonlinear propagation of waves in diverse media, such as nonlinear acoustic waves, solitary waves (solitons), and shock waves. By characterizing the stability, bifurcations, and interactions of nonlinear waves, researchers can elucidate the underlying mechanisms governing wave phenomena and predict complex wave behavior in heterogeneous materials. Poincaré's theory finds application in the analysis of fluid-structure interaction (FSI) problems, where the dynamic coupling between fluid flow and structural deformation leads to nonlinear oscillatory behavior. By considering the mutual influence of fluid forces and structural response, Poincaré's methods enable the prediction of resonant phenomena, vortex-induced vibrations, and instability mechanisms in FSI systems, with implications for engineering design and optimization.