Nonlinear modal interaction in rotating composite Timoshenko beams

Nonlinear free vibration analysis of rotating composite Timoshenko beams featuring internal resonance is studied in this paper. Three nonlinear coupled equations of motion for flapping, shear and axial motions, are based on the assumptions of Timoshenko theory and the nonlinear von Karman strain–displacement relationships. Due to the small magnitude of low-order flapping/shear frequencies ratio, usually a big gap exists between the aforementioned frequencies especially in isotropic beams, while this gap reduces for composite beams which exhibit more characteristics of Timoshenko beams. Different material, geometrical and operational parameters effects on the frequencies of rotating beams are investigated. For the first time the possibility of internal resonance occurrence between low-order flapping and shear modes is proved. The direct multiple scales method is implemented for construction of the flapping nonlinear normal modes. Results for the flapping nonlinear normal modes are validated via comparison with the results of the fourth-order Runge–Kutta method. Depend on the amplitude ratios besides the nearness of the frequencies of the interacting modes, four bifurcation regions are defined through the nonlinear normal mode stability analysis; (a) one stable coupled mode, (b) two stable and one unstable coupled modes, (c) three stable coupled modes and (d) one stable coupled mode.