Nonlinear flapping-torsional free vibration analysis of rotating beams considering the Coriolis force

The nonlinear free flapping-torsional vibration of rotating beams is investigated in this paper. The presented equations are based on the exact geometrical formulation in conjunction with the Cosserat theory for rods. The equations of motion are reduced to the flapping and torsional equations of motion for symmetric rectangular beams by neglecting the shear deformation. The governing equations are coupled to each other with the non-homogenous boundary conditions. Then by introducing the dimensionless parameters the non dimensional equations of motion are derived. By employing the direct method of multiple scales the effective nonlinearity coefficients of nonlinear natural frequencies are extracted. After validation of the current results, the effects of the rotating speed on the type and the value of the effective nonlinearity coefficient of natural frequencies are examined. The sign of the effective nonlinearity coefficient demonstrates the softening or hardening treatment of the corresponding nonlinear natural frequencies. It is concluded that ignoring the flapping-torsional coupling due to the Coriolis force, for odd modes makes some errors in the magnitude of effective nonlinearity but the type of nonlinearity is predicted correctly. On the other hand, in the even modes for average to high rotation speed in addition to incorrect estimation of the magnitude of effective nonlinearity the different type of nonlinearity is also predicted.