Stability of a Blade with Varying Rotation Speed by Considering the Lagging Motion

Principal parametric resonance in rotating blades with varying rotating speed is investigated in this paper. In the presented model, the lagging-axial coupling motion due to Coriolis force is considered. The governing equations of motion are the available equations in the literature based on the exact geometrical formulation for unshearable blades. The rotating speed of the blades is considered as a mean value perturbed by a small harmonic variation. The variation frequency of the perturbed value is considered twice the one of the lagging frequencies and/or one of the axial frequencies which causes the principal parametric resonance. The direct method of multiple scales is implemented to study the dynamic instability produced by the principal parametric resonance. A closed form relation which defines the stability region boundary under the condition of the principal parametric resonance is derived using the method of multiple scales. The current results are validated by comparison with the available results in the literature. After validation of the results, a comprehensive study has been adjusted for illustration of the rotating speed effects and mode number influences on the parametric stability region.