Dynamic stability in principal parametric resonance of rotating beams: Method of multiple scales versus differential quadrature method

• The principal parametric resonance analysis of the rotating beams is investigated.
• The method of multiple scales is applied to the equations of motion in order to the dynamic stability analysis.
• The differential quadrature method is employed to validate the method of multiple scales results.
• A closed form relation is obtained for the description of the stability region boundary.

Parametric resonance is one of the key topics in studying the dynamics of structures. In this paper, dynamic analysis of rotating beams with varying rotational speed in the presence of principal parametric resonance is investigated. The equations of motion are based on the von Karman strain–displacement relationship. The beam is made of isotropic material with rectangular cross section. The flapping and axial motions are considered along the thickness and length of the beam, respectively. The Galerkin discretization approach is implemented to determine the natural frequencies. The method of multiple scales is applied directly to the ordered equations of motion for the dynamic stability analysis. The method of multiple scales delivers a closed form relation for the stability region boundary in terms of the adimensional rotational speed, axial mode frequency and damping ratio coefficient. The differential quadrature method is employed to validate the multiple scales results. A comprehensive study is accomplished to find the effects of damping ratio coefficient and mode number on the critical parametric excitation amplitude and the parametric excitation frequency. Damping ratio coefficient, mode number and parametric excitation amplitude influences on the stability region boundaries are also examined.