Summary
This paper explores stability and instability in periodic responses within nonlinear dynamic systems under forced vibrations using the Efficient Path Following Method (EPFM). The method, known for its cost efficiency with a 60% reduction in calculations through an updating formula, is analyzed for its accuracy in computing Floquet multipliers. By investigating two 4-degree-of-freedom systems and a nonlinear beam, the analysis involves calculating periodic solutions and determining Floquet multipliers to comprehensively discuss stability and instability. The study includes examining a discrete 4-degree-of-freedom system, identifying stable branch locations based on frequency, and analyzing changes in phase space due to energy variations. Additionally, the research extends to a simply supported nonlinear beam with in-plane stress, accounting for substantial out-of-plane deflections. The paper introduces the Efficient Path Following Method (EPFM) as a solution to the computational limitations of the Pseudo-arclength continuation method in computing periodic solutions of nonlinear systems. While EPFM addresses the challenge of generating the monodromy matrix with an innovative updating formula, its accuracy in predicting solution stability is yet to be fully explored. The study investigates EPFM's capability in forecasting stability by analyzing Floquet multipliers in various system setups, demonstrating satisfactory accuracy despite utilizing auxiliary formulas. Additionally, EPFM significantly improves computation speed, marking a notable advancement in stability analysis methodologies.
