Abstract

The Poincaré map (point mapping) analysis approach is used to investigate the stability of multivariable periodic dynamical systems. The approach presented here is based on a procedure to obtain an analytical expression for the period-to-period mapping description of system’s dynamics. It also allows to find analytical dependence of point mapping on system’s parameters. Analytical stability and bifurcation condition are then determined and expressed as functional relations between various system parameters. The approach is applied to investigate the parametric stability of flap motion of a rotor. It is shown that the proposed approach provides very accurate results when compared with direct numerical results which are assumed to be an “exact solution” for the purpose of this study. It is also demonstrated that the point mapping method yields more accurate results than the widely used classical perturbation analysis. To emphasize the applicability of the method for studying higher order systems, bifurcation and stability analysis of a two degree-of-freedom cantilever beam subjected to harmonic base motion is performed. The results of our analysis exhibit an excellent agreement with the “exact” (numerical integration) solution. The ability to perform analytical stability studies of systems with multiple degrees-of-freedom is an important feature of the proposed approach since most existing analysis methods are applicable to single degree-of-freedom systems. Extension of these methods to higher dimensional systems is not straightforward, and is usually very cumbersome.

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